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MATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 3, N2 March 2009

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Editorial

We present the Editors‘ issue whose aim is to demonstrate the type, the standard and the variety of submissions we would like to receive in Mathematics Teaching-Research Journal on line. Each of our editors was invited to submit the paper written „the way they would like to see submissions prepared.“ Together with the Editors issue we are bringing several new changes whose purpose is to streamline the reception and the review of the manuscripts. Please check (click Submissions above) the division of the MTRJ on line into described sections whose names were derived on the basis of the submissions to the previous six issues. The sections are: Mathematics Education, Reflections on Practice, Teaching Practice, Teaching-Research and Teaching-Research Investigations. Notice also the rubric with the help of which the Editors will asses the manuscripts. Note that we address ourselves both to school’s practitioners or teachers as well as to academic scholars hoping to create the platform of mutual understanding and support. The new Welcome Mission statement addreses itself deeper into those issues. In the section Reflection of Practice we have the submission from Broni Czarnocha, a physicist and mathematics educator from Hostos CC who infoms about the completion of the international Professional Development of Teacher-Researchers supported by the grant of the Socrates Program of the European Community. He discusses international comparison of different approaches taken by different teams of the project to one chosen question from the international exam PISA. In the section Teaching Practice we have three papers, two of them addressing the same issue of the development of number sense amongst our students. One of them is written by a physicist turned mathematics educator, Irina Lyublinskaya from CSI, who takes a technological angle for the preparation of activities to develop the number sense, and the second one is written by Vrunda Prabhu, a mathematician/topologist from Bronx CC who takes the geometrical approach to tackle the same problem. The third paper in this section is by Nkechi Agwu, mathematics instructor from BMCC and her colleagues, who had created the course with direct relevance for the communities where the students live. Her paper has a mark, of what has been called the Teaching-Action-Research, a new methoodology created for the purposes of education in India, which combines classroom Teaching-Research with the Action Research in the surrounding community. The section Teaching Research has a thought provoking submission from Chun Ip, mathematics educator from the Hong Kong Institute of Education who is informing about the teaching-research project Teaching for Mathematizing (TFM), which produces and studies the instructional designs. Chun Ip’s submission starts a very important discussion, What are the Standards the Teaching-Research profession should adopt as its guiding principles? In the section Teaching-Research Investigations we present the investigation of students perception of teaching styles conducted by Gerunda B. Hughes from Howard University. It compares the student perceptions of the interactive element of teaching in experimental classes taught with a strong emphasis on teacher/student interaction with the perception of students in the control class taught primarily by didactic lecturing.




List of Content Reflection From Practice Initial Appraisal of the Professional Development of Teacher- Researchers…………………………………………………..Bronislaw Czarnocha Teaching Practice Developing Number Sense with Technology-Based Science Experiments: Reflections of Classroom Practice in Primary Grades and Pre-service Education……………………………Irina Lyublinskaya Story of Number – A Modular Approach…………………….Vrunda Prabhu Teaching Statistics Through Student Engagement in a Study on Alternative Medication Practices………………….Nkechi Agwu, Piotr Bialas,

Brahmadeo Dewprashad, Barbara Tacinelli

Teaching Research Research on Teaching Mathematics: A Practitioner Perspective……………………………………Chun Ip Fung Teaching-Research Investigations Student Perceptions of Teaching Styles in Mathematics Learning Environments…………………………………………………Gerunda Hughes




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INITIAL APPRAISAL OF THE PROFESSIONAL DEVELOPMENT OF TEACHER-RESEARCHERS

Bronislaw Czarnocha

Hostos CC, CUNY

ABSTRACT

The power of an international six country (EU-US) collaboration is shown through the preliminary comparative qualitative analysis of different teams‘ approaches to the investigation of student responses of a PISA 2003 problem, called the Carpenter.The international collaboration entitled, Professional Development of Teacher-Researchers took place in Europe 2005 – 2008. The project was conceived of as a large scale teaching experiment addressing, among others, problems student encounter during the international test PISA – a common concern for the participating countries of Hungary, Italy, Poland, Portugal, Spain and US.

Introduction

The question of effective professional development for teachers, and especially for teachers of mathematics, is far from being solved, despite many different opportunities for it. In fact (Wilson and Bernie, 1999) suggest that the profession knows very little what „do teachers learn across those many opportunities.“ Amongst the beliefs concerning an effective PD we have an important Ball’s statement: „Teacher development is considered particularly effective, when teachers are in charge of the agenda and determine the focus and nature of the programming offered“ (Ball, 1996). (Franke et al, 1998) learned that „engaging teachers in current reforms requires more than showing them how to implement effective practices“. The remaining question is what is that „more“ that many of the professionals sense and how to reach it.

We present here a preliminary report from the international Professional Development of Teachers-Researchers (PDTR), which with the help of the Socrates grant of the European community had attempted just that, that is to give the teachers the possibility of engagement into the classroom investigations of their own practice leading to open didactic problems and determination of their own agenda. „The Teaching-Research methodology involves the teacher in the scientific examination of classroom practices, in the design and assessment of the innovative instruction, and in the formulation and verification of hypotheses. By virtue of this genuine, investigative involvement in the process of teaching, the teachers are able to inspire students to equally genuine intellectual involvement in their mathematics classrooms. Teachers become empowered by TR practice.“ (Project Description, Comenius 2.1, 2005).

Six teams of mathematics teachers, one from Hungary, Italy, Portugal and Spain and two from Poland (PL1,PL2), who participated in PDTR 2005-2008 were mentored in the craft of Teaching-Research by the international cadre of accomplished teacher-researchers and university faculty (www.pdtr.eu). Each team included 8 – 12 mathematics teachers; the project brought together 110+ teachers and university faculty committed to the development and utilization of teaching research methodology in mathematics classrooms. While the idea of teacher-researchers and their

http://www.pdtr.eu/




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preparation for the the craft of teaching research is not new, the original aspect of this PDTR has been its international scope and the benefits that scope offers to the profession.The project had been anchored in Rzeszow University, Poland while the international meetings of the project have been taking place in different countries of Europe such as Debrecen in Hungary, Lisbon in Portugal, Barcelone in Spain, Modena in Italy and Warsaw, Rzeszow-Krakow (PL1) and Siedlce (PL2) in Poland. The Project Cooordinator was Stefan Turnau, a senior education professional from Poland, the Teaching-Research aspect was coordinated by the author with the help of TR experts Nicolina Malara from Modena University in Italy and John Mason from Open University, London.

Methodology. The PD was conceived by the author as the large scale teaching experiment (Lesh and Kelly,

2000), whose components were conducted by each team and whose work proceeded through the familiar cycles of action research. Its design was modeled on the Teaching-Research/NYCity model (TR/NYC) developed in the Bronx to address the issues of learning mathematics (Czarnocha, 2002).The duration of the project contained two yearly cycles of the teaching experiments and one year of writing teaching - research reports from the conducted work – chapters to two books produced by the PDTR1 . The first cycle was devoted to the analysis of student difficulties displayed at the international mathematics test PISA administered every three years. PISA exam was chosen because of the novel and challenging nature of PISA tasks.The countries invited to participate in the project had exhibited similar results in first two editions of the test, PISA 2001 and PISA 2003, hence, the teams had similar issues to address in this part of the project. Several PISA problems (Carpenter, Apple Trees, Antarctida) were chosen to better focus attention of all teams on the same issues. Such an organization of work allowed for the reflection upon different, culturally determined approaches to solving similar problems in each of the participating countries. Below we analyze different methodologies used in the analysis of the Carpenter problem (Turnau,2008).During the second cycle of the project, teams had chosen their own themes for the investigation, some of them were derived from the consideration of difficulties of PISA exam, some of them related to different problems of interest of the participants of the project. The teaching-research teams met in regular intervals, in general different for each team. During the span of two years, the team meetings included Teaching-Research seminars led by national coordinators of the teams and teaching-research mentors and international experts, specially designed mathematics course internationally coordinated by Harrie Broekman from Freudenthal Institute, research study groups designing and analyzing teaching experiments, and a course of Mathematics-based English - a common language of the project. Different teams had a different background and experience in the teaching-research methodology; Polish and Hungarian teams were novices to the methodology and originally relied on the Teaching-Research/NYCity model introduced by the author (Czarnocha, 2002,Czarnocha and

1 Handbook of Mathematics Teaching Improvement:Professional Practices that address PISA. (2008), s. Turnau (ed), University of Rzeszów, Poland. Handbook of Mathematics Teaching-Research: Teaching Experiment, a tool forTeacher-Researchers (2008). Czarnocha, B. (ed) University of Rzeszów, Poland. Note: Both books are downloadable from http://www.pdtr.eu/index2.php

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Prabhu, 2007). Portugese team had the background of action research approach to the investigation of practice while the Catalonian team relied initially on the reflective practitioner approach.

The variety of cultural and methodological backgrounds possessed by the participating teams has been both the source of the richness of the project as well as its main challenges. The possibility of investigating the differences and similarities of the methodological approaches dealing with the same mathematical problem (e.g.) Carpenter problem of the next section provides an example of new, culturally responsive comparative dimension of the analysis of mathematical activity, the dimension impossible to reach through purely uni-national or uni-cultural approach to it (Section Approaches to the Carpenter Problem). On the other hand, the very same differences, when viewed purely from methodological angle had contributed to the variety of, often heated, discussions on the very issue of Who is a teacher – researcher ? and What is the route of attainment the teaching-research craft? (Section Who is a Teacher-Researcher).

Approaches to the Carpenter problem The following problem was chosen as one of the common PISA problems to be investigated by all the teams participating in the project. CARPENTER PROBLEM A carpenter has 32 metres of timber and wants to make a border around a garden bed. He is considering the following designs for the garden bed: 6m 10m 10m

6m

Design A Design B

6m 10m 10m Design C Design D Garden bed design Using this design, can the garden bed be made

with 32 m of timber? Design A Yes/No Design B Yes/No Design C Yes/No Design D Yes/No

6m




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The task for the TR teams was to investigate the challenges middle school and high school students experience in solving the problem, focusing the investigation on the competencies involved in the solution process. The investigations belonged to the first cycle of the teaching experiment of the project; the Handbook of Mathematics Teaching : Professional Practices that address PISA contains four reports about the Carpenter problem, from Hungary, Italy and Poland (PL1). There are additional reports from Spanish and Polish PL2 teams contained in the Final Report of the project, which will become the public domain and available for inspection by the time of the conference. Here we are investigating interesting variations observed in the approach to the task by four relevant reports in the Handbook: (Tunde Kantor et al, 2008) and (Kristina Barczi, 2008) from Debrecen, (Maria Legutko et al, 2008) from Krakow and (Roberta Raimondi, 2008) from Modena. An interesting axis of comparison in all three approaches is the attitude towards the role of communication in understanding and solving the problem. All the teams addressed this issue but with significant difference in emphasis. Whereas Poles (PL1) inform us that „…we agreed that Polish students aged 16-17 have enough mathematical knowledge to solve these tasks; however the formulation of the task might be untypical for them.“ Hungarians colleagues move further in this direction informing that they dropped the word „carpenter“ from the text since „In Hungary, a carpenter makes roofs of houses and uses timber for its work. The gardener makes flower beds; he makes their wooden bordering, too, but it is not of timber.“ Italian approach included the change of the text of the problem to: “A carpenter has got 32 meters of wooden planks and wants to build a fence. He considers different projects but doesn’t know whether they are realizable. Help him, showing for each project, if its realizable with the 32 meters of wooden planks available and explain your reasons.“ We see the steady increase of concern for proper communication starting with Poles (PL1) who merely noticed the possibility of the difficulty in the formulation of the problem, but did not think it’s important to change the text. Hungarians push this concern further focusing on the factual agreement with cultural meaning of the text while Italians reinforced the problem with the appeal to the personal action of the pupil. Let’s add here that all three teams asked pupils to explain their reasoning in words in order to be able to evaluate the thinking process of pupils.

The second interesting axis of comparison are the changes in the structure of the Carpenter problem. Since PL1 team was more interested in the relation of the results to the average of OECD („What factors contributed to such results? How mathematics instruction can be changed so that students attain high scores?“), it was more interesting for the team to leave the problem unchanged. On the other hand, the Hungarian team wanting to make the problem more challenging, while bringing it closer to students intuition eliminated the Design D (rectangle) and changed the order of presentation A,B,C,D to the order C,B,A because „the original design C is more symmetric“ than the design A and therefore easier to address. It is interesting to note that the Italian team also noticed this difference and its order of the Designs was D,B,C,A. The difference in the assessment of the difficulty between the Hungarian and Italian teams is in the relative difficulties of the Design B and C. Moreover since the Italian team retained the easiest shape of the rectangle (the Design D) and post it as the first to consider, one can venture the hypothesis that there was difference in the estimate of student possibilites vis-à-vis Carpenter problem by the two teams.

Finally we can observe a very significant and unique approach to the presentation of the problem to students. Whereas Polish and Hungarian teams had presented the Carpenter problem as the only problem to solve, the Italian team had imbedded the problem in the larger instructional




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sequence of 5 problems, where each new problem addressed certain conceptual aspects of the original set up. „In particular, the didactic path on geomtery intended to give space to the empirical-inductive study of space and objects, rather than focus on axiomatic and deductive aspects. Specifically, the didactic path…focused on the conflict between perceptible and cognitive aspects.“ Hence the Italian teacher-researcher had extended the conceptual framework of the Carpenter problem to fully investigate its different aspects. We see now from all three criteria of comparison that the Polish team introduced the least amount of changes into the problem, most probably, because the team had judged the problem fully adequate to the capabilites of Polish students, the Hungarian team, changed the problem to make it more challenging to the students while the Italian team had incorporated the Carpenter problem into an instructional sequence demonstrating how to imbed a classroom investigation into the course curriculum.

It is interesting now to compare the methods of assessment and analysis of the three teams and contrast it with the differences in setting up the investigation described above. All three teams had a qualitative analyses of student responses, and all three teams had observed known perimeter/area difficulties; (Kristina Barczi, 2008) from Hungary relates the difficulty to the similarity of words „terulet (area) and kerulet (perimeter)“.

Hungarian and Polish teams had considerably developed the quantitative schema, in addition to qualitative assessment, introducing new coding and assigning points on that basis. They used percentage comparison between different experimental schools, in particular both teams were interested in investigating the differences between pupils in middle and high school. In addition to numerical comparisons, (Kantor et al, 2008) used bar graphs. Italian investigator limited herself to pure qualitative analysis. Finally, to obtain the fuller picture let’s add to this comparison the assessment of what was the issue that gave students the strongest challenge. Whereas Polish pupils had the „largest difficulty with the estimation of the perimeter of the parallelogram“, Hungarian pupils „do not recognize the importance and applicability of what they learned beyond the school walls“, the Italian work points out to the „dramatic difficulties that student had in reporting their own thinking processes and facing the metacognitive aspects of the question.“ We see here an interesting pattern, which, due to the small sample of the analysis,can not have yet the generality one would wish to obtain but nonetheless is worthwhile to observe. On one hand we have a team (PL1) which while being concerned about the basic cognitive competencies is minimally interested in the communicative aspect; on the other hand, we have an Italian team deeply interested in the communicative aspect but without much concern about the cognitive. (Note the essential difference between the metacognitive aspect which has the role of self-control rather than the problem solving, cognitive role.) The presence of such extreme separation of concern about communication and cognition might cause problems in full understanding of the learning processes in a given teaching experiment, since we know from the work of Vygotsky about the inseparable (though distinct) nature of both competencies.

Conclusions The initial comparative analysis of different national teams‘ approaches to the same Carpenter

problem of the PISA International exam in the context of PDTR was presented.




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Distinct differences in the emphasis on the relative importance of two essential competencies, communication and cognitive aspects of problem solving were observed in these approaches. While the discussed sample of the teams is too small to derive the general conclusions one could wish for, nonetheless one can observe certain absence of equilibrium in the approaches.While the team which focused primarily on the solving problem competency (PL1), was not particularly interested with the communicative competency, the team which focused the most on the communicative competency (Italian team), was not particularly concerned about the cognitive aspect of student performance. The presence of such dramatic extremes between communicative and cognitive competencies in education should be of concern to anyone, especially to those amongst us who adopt socio-cultural framework. It is (Vygotsky, 1986,p.7) who asserts the dialectical inseparability of two aspects. On one hand „…communication presupposes generalization and the development of word meaning..“ and on the other hand „generalization thus become possible in the course communication.“ In order for each of them to grow and develop, communication and cognition must be in this mutually reinforcing relationship to each other thus both of them have to be taken equally into account, if the improvement of mathematics learning is our goal.

References Ball, D. L. (1996) Teacher learning and mathematics reforms: What do we think we know and

what do we need to learn? Phi Delta Kappan, 77, 500-508 Barczi, K. (2008) A study on how Hungarian students solve problems that are unusual for them.

Handbook of Mathematics Teaching Improvement:Professional Practices that address PISA. (2008), s. Turnau (ed), University of Rzeszów, Poland.

Czarnocha, B. (2002) Teacher Researcher for the 21st Century. Short Oral Report, Proceedings of the 24th Annual Meeting of NA-PME, Athens, Georgia, October 25-28,2002

Czarnocha,B., Prabhu, V. (2007) Teaching-Research NYCity model, Dydaktyka Matematyki, vol.29, Krakow, Poland.

Franke, M.L.,Carpenter, T. P., Levi, L.W., & Fennema, E. (1998) Teachers as learners:Developing understanding through children’s thinking. Paper presented at the annual meeting of the AERA, San Diego, Ca.

Kantor, T., Balla, E., Kozarne –Fazecas (2008) PISA and PISA like problems. Handbook of Mathematics Teaching Improvement:Professional Practices that address PISA. (2008), s. Turnau (ed), University of Rzeszów, Poland.

Legutko, M., Migoń,J. Białek (Przęzak) (2008) Student mathematical literacy in PISA assessment: samples of PISA tasks in teacher-researchers work. Handbook of Mathematics Teaching Improvement:Professional Practices that address PISA. (2008), s. Turnau (ed), University of Rzeszów, Poland

Malara, N.A. & Tortora, R. (2008) A European Project for Professional Development of Teachers Through a Research Based Methodology: The Questions Arisen at the International Level, The Italian Contribution, The Knot of the Teacher-Researcher identity. Submitted to Cerme 6.

Malara, N.A. & Zan, R. (2002). The problematic relationship between theory and practice, in English, L. (ed.), Handbook of International Research in Mathematics Education. LEA, NJ, 553-580.




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Raimondi, R. (2008) The use of OECD PISA model questions in didactic: an innovative class experience. Handbook of Mathematics Teaching Improvement:Professional Practices that address PISA. (2008), s. Turnau (ed), University of Rzeszów, Poland

Wilson, S.M. and Bernie, J. (1999) Chapter 6: Teacher Learning and the Acquisition of Professional Knowledge: An Examination of Research on Contemporary Professlonal Development. Review of Research in Education 24: 173-209

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Developing Number Sense with Technology-Based Science Experiments: Reflections on Classroom Practice in Primary Grades and Pre-service Education.

Irina Lyublinskaya

Associate Professor

Department of Education

College of Staten Island, CUNY

Staten Island, NY, USA

Biography

Professor Irina Lyublinskaya received Master's degree in Physics in 1986, Ph.D. in Theoretical and Mathematical Physics in 1991 from the Leningrad State University and has published substantially in that field. She has taught at the university as well as the high school level for over 20 years. In recent years she has directed her professional endeavors to the curriculum development and research in the area of integrating technology into mathematics and science education and to the professional development of mathematics and science teachers, conducting grant-funded workshops to help teachers learn to use educational technology. She has received grants for these projects from such agencies as the Geraldine R. Dodge Foundation, the Bell Telephone Company, the Federal Eisenhower Professional Development program, The William and Mary Greve Foundation, The Best Practice in Education, Inc., The Clay Mathematics Institute, New York State Department of Education, and US Department of Education. Lyublinskaya is a recipient of Radioshack/Tandy Prize for Teaching Excellence Mathematics, Science, and Computer Science, NSTA Distinguished Science Teaching Award and citation, Education’s Unsung Heroes Award for innovation in the classroom, and NSTA Vernier Technology Award. She has published multiple articles and 10 books about teaching of mathematics and science. Her research interests are in the area of integrating instructional technology into mathematics and science education, pre-service and in-service professional development of teachers.

For the project described in this paper, Dr. Irina Lyublinskaya was a recent recipient of the 2008 National Science Teachers Association / Vernier Technology Award1 at the college level.

1 This award recognizes and rewards the innovative use of data collection technology using a computer, graphing calculator, or other handheld in the science classroom. A total of seven awards


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Abstract

Number sense refers to the ability to use numbers and quantitative methods as a means of processing, interpreting and communicating information. It requires an understanding of the number system, a repertoire of computational skills and an inclination and ability to solve number problems in a variety of contexts. The primary school years are crucial in providing the kind of positive start to students’ number sense learning that is needed to develop confident and capable lifelong learners.

This paper provides reflection on using inquiry based science lab activities designed for the purpose of development of the number sense with the students in the grades 3 – 5 and with the pre-service elementary teachers. These original inquiry based lab activities utilize Vernier probeware and various data collection interfaces, such as Go!Link interface that is used with computers, or EasyLink interface that is used with TI-84 graphing calculators. These lab activities address NCTM standards for mathematics and focus on such topics as comparing positive numbers, comparing decimals, placing numbers on a number line, addition of positive and negative numbers, commutative and associate properties of addition, etc. The unique feature of the lab activities is the natural connection that is made between a mathematics concept and a science topic, which allows the teacher to use these experiments to integrate mathematics and science learning in the elementary classroom and to make real-life connections for the children’s development of the number sense. These activities were implemented in 3rd and 4th grades of a public elementary school on Staten Island and in four sections of undergraduate methods course for pre-service elementary teachers at College of Staten Island. Qualitative data collected from the pre-service teachers suggests that both, teacher candidates and elementary school students, did develop greater number sense as a result of their participation in these classroom activities. Particular pedagogical innovations, such as those involving the use of science technology experiments, seem to have supported students’ development of number sense.

Introduction

In the past decade, elementary schools have been strongly pressed to put more and more emphasis on improving mathematics scores. Many studies have shown that students’ experiences related to the learning of number concepts at primary school level are of central importance in instilling their beliefs and values they associate with mathematics. If these experiences are meaningful they will further lead to attainment of positive attitudes, values and beliefs about

are presented: one at the elementary level (grades K–5); two at the middle level (grades 6–8); three at the high school level (grades 9–12); and one at the college level.


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number concepts. On the contrary, experiences that are not mathematically meaningful will lead students to believe that mathematics learning consists of memorizing activities devoid of meaning (NCTM, 1989). The main concern for most mathematics educators are that many students demonstrate little understanding of numerical situations in instances where they have to solve number problems (Leutzinger & Bertheau,1989; Burns,1989). The development of number sense is a widely accepted goal of mathematics instruction (NCTM, 2000; Reys, Reys, McIntosh, Emanuelsson, Johansson, & Chang, 1999).

Emerging technologies afford teachers and students new opportunities to grasp mathematical concepts. One of the examples of emerging technology is real-life data collection probeware. While this type of technology has been used at secondary and middle school levels since early and mid-90s, it started to find its way to the elementary schools only in the late 90s.

The purpose of this project was to develop and implement a set of inquiry-based lab activities that integrate science and data collection technology with mathematics in order to develop number sense in elementary school children. Another goal of the project was to develop pre-service elementary teacher’s understanding of numbers and their confidence in teaching elementary mathematics, as well as to impact their perspectives and attitudes towards using data collection technology. Most of the pre-service teachers who were enrolled in the elementary education program had mathematics and science phobia. Based on the initial survey, they also had very negative attitude towards the use of technology in the elementary classroom. Integration of data collection technology into mathematics methods courses had a significant impact on the pre-service teachers’ confidence level with mathematics, specifically number sense, and on their attitudes towards using technology. The data included pre-service teachers’ reflections, written lesson plans, and instructor’s observations of their teaching in the elementary school. The data also suggest that pre-service teachers did develop greater number sense as a result of their participation in the developed activities. Particular pedagogical innovations, such as those connecting mathematics and science and use of real-life data collection, seem to have supported students’ development of number sense.

Sample Activities and Classroom Practice Number sense "describes a cluster of ideas, such as the meaning of a number, ways of

representing numbers, relationships among numbers, the relative magnitude of numbers, and skill in working with them." (Trafton and Treisen, 1999). Number sense is a part of children’s daily mathematical lives and slowly grows and develops over time. In a problem-centered mathematics curriculum, number sense is closely tied to problem solving.

We know that real-life applications, especially, visual and hands-on demonstrations enhance students’ learning of the material, meet needs of kids with different learning styles, and create additional motivation for learning a discipline. The use of experiments, demonstrations, computer simulations, and inquiry based activities allows students to create visual image and practical


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understanding of science and mathematics concepts and relationships. Use of hands-on activities within science and mathematics content provides additional opportunities for teachers and their students to make connections and to master standards based concepts and skills. Another important part of this approach is use of various technology and different measuring equipment, necessary to meet the expectations of preparing our students for the challenges of 21st century.

The examples of developed activities (Lyublinskaya, 2009) that helped pre-service teachers and their students become comfortable with numbers are analyzed below. Each activity has been completed by the pre-service teachers in a classroom settings modeling collaborative group work they would be expected to facilitate as an inquiry based technology activity with their own students. The experiment was followed by discussions of the major concepts and common students’ misconceptions, the place of the activity in the mathematics curriculum, lesson planning, alternative assessment option, and other pedagogical topics. Pre-service teachers then used these activities with the 3rd and 4th grade students in local elementary school.

Example 1. Freeze! The Number Line Game. In this activity students use Go!Motion detector, LoggerLite software, and computer to

construct a number line on the floor and to explore locations of the positive decimal numbers on it by standing at different locations on the number line and observing the distance readings of the motion detector. The activity meets the NCTM standard: Compare and order decimals and find their approximate locations on a number line. (NCTM, 2000). The main objective for the students is to predict location of positive decimal numbers on the number line and verify that experimentally with a motion detector.

In the first part of the activity students construct their own number line on the floor. In order to do that, the teacher first puts the 6-m long masking tape on the floor to represent the number line. The zero mark of the “number line” is drawn right underneath the motion detector. The Go!Motion detector is connected to the computer that is projected on the screen for a whole class to see. The teacher calls a student to stand in front of the motion detector and move slowly back and forth until motion detector shows the distance of 1 m. (see Figure 1). Then, the position where student stands on the masking tape is marked as 1. Teacher then calls another student to locate position on the number line where motion detector shows distance of 2 m, and that position is marked on the masking tape as 2. This is continued for until students mark numbers up to 6. (Note: since it is very hard for children to stay still, the distance measurement will be slightly changing. Students were instructed to round measurements to the tenth place in order to find positions of integers on the masking tape).


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Figure 1. Experimental Setup for the Number Line Game.

Since the computer screen was projected onto the screen, all students could see the

measurements and participate in the construction of the number line. In the classroom students were giving their classmates suggestions on how to move (forward or backwards) in order to find locations of integers on the number line. The construction of the real number line on the floor helped children and pre-service students to understand the fact that positive numbers get larger as they move away from the motion detector (from zero mark). It also helped them to start making connections between location of the number on the number line and distance between the zero and the number. The major difficulty for both, pre-service teachers and elementary school students, was the fact that the distance measurements were changing even when student stood still. Although changes were in the thousandths of a meter, students had a hard time to realize that within the errors of measurements, the distance was constant and that they could just take that measurement and round it up. For example, when student was standing at 1 m away from the motion detector, the measurements were changing between 0.995 and 1.027. Rounding to the tenth would lead to the same number 1.0, however, students did not come up with that on their own. In the second part of the activity teacher gives students several decimals between 0 and 6. Before experimenting, they are asked to predict the locations of these decimals on the number line drawn in their worksheets and explain their predictions. Children are also asked to compare each given decimal to the nearest integers on both sides and use symbols < and > to record their comparisons. In order to verify their predictions, children were asked to find locations of each number that they were given on the number line on the floor. Teacher then called several volunteers to stand at the position that they predicted. With the pre-service students instructor asked one student at a time to stand in front of the motion detector and let class observe the readings and if needed help the student to adjust his/her position. In the elementary classroom, where pre-service teachers conducted this lesson in small groups, they decided to ask all 3 or 4 students to stand at the positions they predicted for different decimals given to them. The motion detector showed the distance to the student who stood closest to the motion detector (smallest number). After this number location was verified, the student sat on the floor so that the motion detector measured the


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distance to the 2nd student, and so on. This helped students to reinforce the concept that larger decimals are located at a larger distance from the zero mark. In addition, children were asked to check the integers to their left and to their right in order to verify their comparisons. For example, student standing at a location 1.6 noticed that his number is right between 1.0 and 2.0. This observation led to the class discussion on how to compare 1.6 to 1.0 and 1.6 to 2.0, to the discussion of place value, and the discussion of distance measurements on the number line. To stretch students’ thinking, children were given three decimals and were asked the following questions:

1. Order these numbers _______, ________, ________. Can you compare them? 2. How much bigger is 2.7 than 1.5?___ What is the distance between 1.5 and 2.7? ____ 3. How much smaller is 1.5 than 3.3? ___ What is the distance between 1.5 and 3.3? ___ 4. Did you notice any patterns as you answered these questions? Record your observations.

Majority of students independently made connection between distance from zero to a given number and the magnitude of a number, however, only about half of the students were able to answer questions 2 – 4 above that compared different decimals to each other. However, after teacher let students measure distance directly on the floor, many of these students were able to connect result of subtraction and distance between two decimals.

The experience provided by this activity helped the 3rd grade students to learn that moving towards the motion detector reduces the distance and thus the displayed number is smaller. At the same time, moving away from the detector increases the distance, and thus the number gets larger. When predicting position of the decimal on the number line, children were able to move in a right direction to correct their initial prediction and to find correct location for a given decimal. In addition, they were able to compare their positions and the numbers they were representing and make conclusion how their decimals related to the closest larger and smaller integers. Answering extension questions helped students to generalize their experimental observations and make connections between relative location of decimals on a number line and distance between these decimals. Example 2. Mix’Em Up! Comparing Decimals and Exploring the Average.

This particular activity is a follow up of another activity where students use containers with water at different temperature for ordering numbers. In the preceding activity the focus is placed on the understanding of place value and how to compare numbers by comparing digits in corresponding places.

The Mix’Em Up! activity provides students with the opportunity to compare decimals “experimentally” and to discover the meaning of the arithmetic average. It meets the following NCTM standard: Compare and order decimals. The activity is designed as a game, where each student works with a partner. Each group is given three containers, labeled A, B, and C. These containers are filled with same amounts of water at different temperatures. Students are also given an empty container labeled “Mixture”. One child in each group turns away, while another one takes


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equal amounts of water from each container and mixes water from only two containers in the “Mixture” container, and discards the water from the 3rd container. The objective of the activity for students is to measure temperature in each container (See Figure 2) and compare temperature values in order to find out from which two containers water came from. Students then switch the roles. To make it even more like a game, students use timer to find out who was able to determine from which containers water was used for the mixture in a shorter period of time.

Figure 2. Measuring Temperature with Temperature Probe.

Here are two most common approaches that pre-service teachers used in this activity: 1. Student measured temperature of water in all four containers and placed these values on the

number line. Students then measured distance from the point M (mixture) to each point representing temperatures for containers A, B, and C. Whichever distances were equal, these were the containers selected.

2. Student suggested that the temperature should be an average of two temperatures, so she found averages of all possible combinations and determined which average is the closest to the experimental value in the ‘Mixture” container. Elementary school children started with measuring the temperature of the water and

ordering the numbers representing the temperature values. They checked their ordering by touching the water with their fingers and determining which water felt colder and warmer. (Figure 3).


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Figure 3. Checking Order of Temperatures by Touch.

Based on “touch” some students had to correct the order of decimals before they proceeded to the next step. Children then chose different approaches to solve the problem. For example, some kids used number line approach very similar to one used by the pre-service teachers. They put all four temperatures on the number line and then eyeballed the point “right in between” two different temperatures, until they found which of these mid-points was the closest to the experimental temperature of the mixture. Another group was using subtraction to compare the differences between different temperatures. None of the children used the formula for the arithmetic average to solve the problem. This provided the teacher with the opportunity to facilitate students’ discovery of the fact that average of two numbers lies exactly between these two numbers on the number line. Example 3. Stack’Em Up, Switcheroo, and Groupis. Commutative and Associative Properties of Addition.

This activity uses AA batteries and differential voltage probe to measure the voltage across stacked batteries. It meets the following NCTM standards:

• Understand the meaning and effects of arithmetic operations with decimals • Discover the commutative property of addition • Discover the associative property of addition

The activity has three parts. In Part 1 students explore properties of batteries as they are stacked together and extend that to the idea of adding signed numbers. In Parts 2 and 3 students discover commutative and associative properties of addition respectively as they measure voltage of batteries stacked in different order or combined in different groups.


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Only positive decimals were used with elementary school children in this activity. Total of three AA batteries were given to each group of students. Each battery was marked with the letter, A, B, and C. Children were asked to place each battery in the groove of a rule, align the negative end of the battery with the zero mark on the rule, and measure voltage of each battery. (See Figure 4).

Figure 4. Using Voltage Sensor to Measure Voltage of A Battery

Due to the fact that some batteries were used and some were brand new, the voltage of each battery was slightly different. Each voltage value was considered to be a number A, B, or C. Students were then instructed to keep the orientation of the battery constant. Then children were given addition number sequences shown below and asked to use the numbers that represented voltage measurements in order to complete these number sequences:

A B+ = + = A C+ = + = B C+ = + = A B C+ + = + + =

Students then were asked about strategies that they use to complete the number sequences. In order to verify their computations, children used voltage probe connected to the computer

via Go!Link interface and Logger Lite software. Students stacked batteries according to the number sequence and checked their results. (see Figure 5). This exercise helped children to verify their skills of adding positive numbers and also provided them with the opportunity to practice measurements using voltage probe.


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Figure 5. Measuring Voltage of Stacked Batteries

In part 2 students explored voltage of two batteries that are stuck together in different order. They were asked to write number sequence for each case (see Table 1):

Part II Data Table 1

Stacking Order Measured Voltage Number Sequence

V

V

Based on their observations, students filled the table above and were expected to generalize their findings and to complete the following mathematical statement

A B B+ A+ with the correct symbol <, =, or >. Since voltage measurements for both orders of stacking batteries came up to be the same, students had no problems to come up with commutative property of addition A B B A+ = + .

In part 3 students completed similar activity to discover associative property of addition. In this part of the experiment students were asked to combine three batteries in as many different ways as they could think while keeping orientation of each battery the same. In order to help them, the lab worksheet asked students to measure voltage of each individual battery first, then to stack each two batteries and measure the voltage of each “double” battery. Students were then asked to write number sequence for each method that they used to create a triple battery and use parentheses to indicate how they combined batteries in each method. The voltage probe was used to determine voltage of the “total battery” in each method. The table 2 below shows cumulative result from the


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4th grade students’ groups completed this activity (the cases when students switched the order of the batteries are not included):

Part III Data Table 2

Method Number Sequence Measured Voltage

+ + A + B + C 4.25 V

+ (A + B) + C 4.25 V

+ A + (B + C) 4.25 V

Here, the individual voltage measurements were A = 1.45 V, B = 1.57 V, C = 1.23 V, AB = 3.02 V, BC = 2.8 V.

Both, 4th grade students and pre-service teachers were able to discover commutative and associative property of addition when using positive decimals. At both levels some students had difficulty in using voltage probe and understanding where to attach the leads on the batteries in order to measure voltage. Elementary school students had difficulty writing number sequence using parentheses to indicate how they combined the batteries. Taping combined batteries together, so kids could physically see the “double” or “triple” battery, helped students to make a connection between use of parentheses and combining batteries together.

For more advanced students the activity includes experiment that involves negative decimals. In order to create negative decimals, batteries are flipped, so that positive end of each battery is placed towards the zero mark of the rule, and the ground (black) lead of the voltage sensor is attached to the positive end of the battery. In this case the voltage is measured as a negative number. The following questions were used to extend students’ thinking:

1. Will the commutative property hold true if you add positive and negative numbers? 2. Can you think of other arithmetic operations that will have commutative property? Give

examples. 3. Does associative property of addition hold true for the negative numbers? Why? 4. Will the associative property hold true if you add positive and negative numbers? 5. Can you think of other arithmetic operations that will have associative property? Give

examples. Since negative decimals are not part of the elementary school curriculum, negative numbers

were not used with the children. Only pre-service students were asked to complete the activity for negative numbers, and it was particularly difficult for them. They had a hard time to add negative


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decimals and they had no understanding of operations with negatives. The experiment helped them to see what happens when you add negative numbers, but due to time constraints, it was not possible to develop true conceptual understanding of operations with negative decimals.

Discussion

In order to analyze the impact of these activities on the pre-service teachers’ learning,

qualitative and quantitative data were collected over three semesters. These data consisted of reflections about activities used in class and videos of short presentations that each teacher candidates made in class on specific topic related to the numbers and operations. In addition teacher candidates’ lesson plans and teaching were evaluated in terms of their understanding of content. The elementary school students’ mastery of the mathematics concepts was evaluated based on teacher candidates’ self-evaluations completed upon teaching of each lesson. The videos were analyzed for the content and conceptual understanding of numbers demonstrated by the teacher candidates. It revealed that pre-service teachers gained better number sense through the activities used in class. Analysis of lesson plans and in-class observations showed that pre-service teachers better understood the concepts they were teaching. This was inferred from the types of activities and questions that they were using while teaching children about numbers and operations with numbers. In self-evaluations pre-service teachers were asked to analyze how their students learned the content they were teaching. Based on these self-evaluations and examples of students’ answers to the questions that teacher candidates provided, it could be inferred that elementary school children also benefited from the use of the real-life science activities in learning about numbers and operations. Both groups commented that they enjoyed using hands-on experience. Kids really liked using technology, while use of technology was intimidating for some of the teacher candidates. Conclusion

These interdisciplinary activities can be used as a teacher’s demonstration or class exercise, as it was shown above for the Freeze! The Number Line Game activity. They can also be used as a hands-on exploration in small groups – this approach was illustrated with the activity Mix’Em Up!. Using an activity as a demonstration allows a teacher to talk about real-life applications without necessity to have multiple sets of equipment for the students groups. When activities are used as class exercise or small group explorations, students have an opportunity for teamwork and interaction with each other as well as learning skills of using measuring devices; however, any group work is more time consuming. Some activities may be divided up in parts and completed within two or three lessons. Teacher may use one part of the activity as a class demonstration and another part as an independent small group work.

In teaching a particular topic, teacher has an opportunity to introduce the activities in different place within the topic. Activity could be great exploration type introduction to a new topic


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that would be followed by the teacher’s instructions and explanations. The experiments could also be used as review exercises. In some cases experiments allow for more engaging way to exercise learned skills. Often activities are used at the end of studied topic when students are expected to use what they learned for applications and problem solving.

The connections between mathematics, science, and technology offered in these activities allow for interdisciplinary learning. Whatever place these activities are used within the context, they can enhance students’ learning of the mathematics and science to allow students to see real-life applications and allow the teacher to have performance-based assessment of students’ understanding of learned material.

This project leads to the conclusion that real-life problems presented as science experiments can help the pre-service teachers to strengthen their content knowledge, especially in the area of numbers and operations. Use of technology is not necessary, but does enhance their experience, if used appropriately. The perspectives and attitudes of pre-service teachers strongly depend on the intensity and length of experience they have with the use of data collection technology. More experience and practice provides students with better understanding of the benefits of this type of technology for their future students as well as for their own learning and reinforces teaching and learning of mathematics concepts References Burns, M. (1989). Teaching for understanding. In P.R. Trafton and A.P. Shulte (Eds.), New

directions for elementary school mathematics. Reston, VA: NCTM. Ghazali, M., Rahman, S.A., Ismail, Z., Idors, S.N., Saleh, F. (2003). Development of a framework

to assess primary students’ number sense in Malaysia. Proceedings of The International Conference on Mathematics Education into the 21st Century: The Decidable and the Undecidable in Mathematics Education. Brno, Czech Republic.

Leutzinger, L.P. & Bertheau, M. (1989). “Making sense of numbers”. In P.R. Trafton and A.P. Shulte (Eds.), New directions for elementary school mathematics. Reston, VA: NCTM

Lyublinskaya, I., (2009) Technology Activities for Elementary Mathematics and Science. (lab manual). Long Island, NY: Whittier Publications.

National Council of Teachers of Mathematics (1989). New directions for elementary School Mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

Reys, R., Reys, B., McIntosh, A., Emanuelsson, G., Johansson, B., & Yang, D. C. (1999). Assessing number sense of students in Australia, Sweden, Taiwan, and the United States. School Science and Mathematics, 99(2), 61-70.


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Trafton, P.R. & Thiessen, D. (1999). Learning through problems: Number sense and computational strategies. Portsmouth, N.H.: Heinemann.




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Story of Number : A Modular Approach Number and Number Line

Vrunda Prabhu, Bronx Community College, City University of New York

Abstract

Number, the very essence of mathematics is strangely alienated from many students and even teachers of mathematics. Reasoning based upon number, a natural thinking process which could spontaneously provide intellectual enjoyment such as that acquired through solving of puzzles is unfortunately not a commonplace occurrence. For the past few semesters, teacher-researchers of the City University of New York teaching at community colleges of the Bronx have been developing Story of Number, discovery-based instructional sequences for the arithmetic and algebra courses that are research-based, classroom-tested and draw on the perceptive base of the learner. Alienation from number must have an impact on absence of “number sense”. The present article provides an overview of the modular instructional sequences contained in Story of Number which have shown improvement in students’ understanding of the use of number.

Section 1. Introduction Gersten and Chard (1999) state, “the number sense concept acts as a lens to reveal reasons for relative successes and failures of past attempts at innovations”. Gersten and Chard articulate Cobb’s (1995) conceptualization of constructivism

mathematical learning occurs as students (a) learn the conventions, language, and logic of a discipline such as mathematics from adults with expertise: and (b) actively construct meaning out of mathematical problems (i.e., try a variety of strategies to solve a problem)

How does the research and teaching community view this concept called “number sense”? Quoting from Gersten and Chard’s article, we have the following as a point of view of the profession:

Number sense is an emerging construct (Berch, 1998) that refers to a child's fluidity and flexibility with numbers, the sense of what numbers mean and an ability to perform mental mathematics and to look at the world and make comparisons. Struggling to define number sense, Case (1998) stated the following:

Number sense is difficult to define but easy to recognize. Students with good number sense can move seamlessly between the real world of quantities and the mathematical world of numbers and numerical expressions. They can invent their own procedures for conducting numerical operations. They can represent the same number in multiple ways depending on the context and purpose of this representation. They can recognize benchmark numbers and number patterns: especially ones that derive from the deep structure of the number system. They have a good sense of numerical magnitude and can recognize gross numerical errors




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that is, errors that are off by an order of magnitude. Finally, they can think or talk in a sensible way about the general properties of a numerical problem or expression-- without doing any precise computation. (p. 1)

Most children acquire this conceptual structure informally through interactions with parents and siblings before they enter kindergarten. Other children who have not acquired it require formal instruction to do so (Bruer, 1997).

The approach described briefly below and carried out either in classrooms in the Bronx or, in a community development project in rural Tamil Nadu, India where the young women and men may have 10-12 years of schooling and are engaged in developing after-school child-knowledge-centers for the children of the community, is one of reasoning together. Learning is a natural outcome for both the student and the teacher-researcher in this investigative approach, where, as the student learns, the teacher-researcher learns the conceptions the student brings to the discussion. The obstacles to learning for one with some hesitation toward mathematics are often confusions caused by conventions of mathematics. The conventions could be simple such as the naming convention of numbers, i.e., a student quite familiar with the number pattern 1, 2, 3, etc, might be at a loss when the term “natural number” is mentioned and may state, he/she does not understand and has never been good at mathematics. The hindrance of “not knowing/not understanding” in the provided example above and the next more involved example is one of not knowing the terminologies and conventions adopted by the profession, and not of not knowing concepts. The conventions that hinder learning could be more complex such as not “seeing” how the following statement would make sense “ There is a one-one correspondence between the set of real numbers and points on the line”. Generally attributed to Dedekind for setting analysis on rigorous foundations or bringing “continuity to the number line”, one sees that the above statement can lead to doubts in student’s minds and obstacles to learning. Looking to Dedekind’s work, one encounters his words on the same topic:

(Dedekind Pg. 11., 1963) In the preceding section attention was called to the fact that every point p of the straight line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the following principle: “If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.” As already said I think I shall not err in assuming that every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed. To this I may say that I am glad if every one finds the above principle so obvious and so in harmony with his




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own ideas of a line; for I am utterly unable to adduce any proof of its correctness, nor has any one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line. If space has at all a real existence it is not necessary for it to be continuous; many of its properties would remain the same even were it discontinuous. And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point-individuals and would have to be effected in accordance with the above principle.

The assumption made by Dedekind, and one which is accepted by the mathematical profession as a convention, is not clear to the student who cannot reconcile her/his understanding with the convention that he assumes is mathematics he/she does not comprehend. Bringing deliberate attention to conventions used so that the learner distinguishes between that which he/she can perceive and “make sense of” and that which must be considered convention is one aspect of the instructional sequences being developed. While the students in the Bronx or rural Tamil Nadu may have had exposure to the topics in Basic Mathematics through College Algebra in their prior schooling, the failure or low passing in these prior courses have left numerous gaps in their understanding, so that familiarity with terms is not accompanied by the meaning of those terms, their connections to other concepts, and the careful interplay between concept and procedure, all of which are being attempted to be incorporated into the instructional sequence, Story of Number, developed via the iterative cyclic teaching-research methodology. The objective is to enable all learners to successfully navigate understanding of the concepts in question. In such a guided navigation is present

1. our use of Vygotsky’s Zone of Proximal Development (ZPD) 2. passage from the existing spontaneous concepts of the learner to the required scientific

concepts (Vygotsky, 1956) via 3. instructional material in which is embedded the concrete/enactive – iconic – symbolic

phases of concept development (Bruner, 1966).

Thus, the learner regardless of the current stage of her/his concept formation, has available the scaffolding needed to progress. Given the natural unpredictability of each classroom dynamic, the varied conceptions of students give rise to numerous investigations in the course of a semester occurring in the context of teaching with the aim of improvement of learning, and each is referred to as a Teaching Experiment (Prabhu & Czarnocha, 2007). In one semester, several concepts would be investigated, and new understandings arrived by the teaching-research team, each could result in refinements to the instructional material.




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Section 2: Outline of Story of Number

Fig. 1 The above concept map, Fig. 1, represents the modular structure of the instructional sequence, Story of Number1 that integrates arithmetic, geometry and elementary algebra. Each of the nodes or concepts in the concept map above form a self-standing module, i.e. a set of problems with the needed definitions or explanations, in the style of R.L. Moore or Mahavier’s (Mahavier, 1999) discovery-based problem sets2.

1 Story of Number was partially developed by funding from the CUNY Collaborative Incentive Award during the academic year 2007-2008. 2 The problem sets/instructional sequences are labeled Story of Number: Fraction, or Story of Number: Ratio and Rate, etc, and range in length, with the Fraction instructional sequence being the longest of 21 pages. They can be requested from the authors, Vrunda Prabhu or Bronislaw Czarnocha. The intent of the instructional sequences is that the classroom instructor uses them as part of her/his classroom teaching experiment. The instructional sequence is thus a tool that is tested in every classroom with the group of students in question and depending on the nature of questions or difficulties of the students, the instructor/teacher-researcher provides the guiding hints and/or supplements it with explanation to ensure the learning and mastery of concepts for the particular student audience.




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The exposition echoes Montessori’s approach of considering mathematics from three points of view (Montessori, 1946):

1. Arithmetic – the science of number 2. Algebra – the abstract of number 3. Geometry – the abstract of abstract

In the present article, we illustrate an annotated classroom discourse for the module, Number and Number line. The teaching-research classroom approach is illustrated (in the next Section) in the way of motivating student responses, and in its coordination with the educational research literature of Vygotsky and Bruner. First, we briefly remark on each of the modules of the instructional sequence Story of Number. Beginnings Module of Story of Number The iterative cyclicity of the teaching-research approach implies that one learns the language used by the students. This language is often imprecise, and communication proves to be difficult in the early part of the semester. The Beginnings Module (being designed) begins the integration of Mathematics and Language by

(i) learning the students’ vocabulary (ii) creating our database of words that will occur in the concepts of the semester (iii) creating an awareness of the meanings to be used in the classroom.

Starting from the first day of classes students will be responsible for collecting words they think involve: enumeration, order, proximity, inclusion, reasoning, and computing. These and other categories that emerge will create the basis for an evolving Math and Language module in English and Tamil. Quality analysis of word meaning in class begins from the first day of classes.

The Number and Number Line Module of Story of Number is laboriously sketched as an annotated classroom discourse in the next section. Before completing this module, the module on Prime Numbers and Divisibility would be taken up in the classroom and so is included in the description. Operations of number are reviewed as needed and not treated separately. The module on Number and Number Line generalizes to variable and variable types, and this is the beginning of the formal integration of arithmetic and algebra. This module serves the very important purpose of organizing students’ scattered notions about number from prior exposure to mathematics. The Number Line becomes the organizing principle via which part-whole relationships are understood and connected with the computational aspects via operations on number. The Number Line serves to distinguish those numbers that are easy to grasp and anchor on the number line and those that require the additional machinery of conventions or more abstraction than is available at the current level of mathematical sophistication. Prime Number and Divisibility Having constructed the Number Line, and created a connection between naming conventions and number patterns, even and odd numbers are explicitly mentioned. The reason is to realize that within the constructed patterns of counting or natural numbers and




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integers, also exist other patterns, which are useful in their own right. Naming even and odd numbers also paves the way to think of prime numbers. Students connect prime numbers and divisibility by the table included below as Fig. 4, and this is followed by the application of the Fundamental theorem of arithmetic that every composite number can be written as a product of its primes. Given students’ trouble with non-mastery of the multiplication facts, the activities of this part are enjoyed tremendously, with students wanting more numbers to factor. Ratio and Rate Module of Story of Number is a quantitative comparison of like entities, and hence the earliest means of developing discernment. The discernment, or critical thinking foundations developed in this module form the groundwork for the module on Proportional Reasoning. It is also the basis for thinking of operations on number, and the simplest entry into thinking about the part-whole relationship. The question of Part-Whole relationship is known to be difficult for learners (Inhelder & Piaget, 1964), and ratio is a very useful scaffolding (Vygotsky, 1956) mechanism. It connects well with the development of concepts via language and is extremely intuitive for students to be initiated to problem solving. The “word” problems start by being elementary, ones in which everyone can engage and the complexity of the problems rapidly progresses to more involved problems which challenge students and equipped with the skills developed from the previous problems lend a puzzle-solving environment to the discovery-based sequence. Operations on Number builds upon ratio where possible as illustrated in Fig.5 of Mathematics in Pictures. This module is being worked out at present. Fraction is known to be troublesome for learners and over the past 4 years, the FractionsGrid (Fig. 3) has been developed in several versions as the didactic tool for the visualization of relative sizes of fractions and later in operations on fractions. A 21-page instructional sequence Story of Number: Fraction3 is available that builds on the reasoning process developed in the (i) Number and Number Line Module, with the Number Line as the organizing principle (ii) the part-whole concept in the Ratio and Rate Module. This module has been tried out in three teaching-research cycles. The fraction concept is the first abstraction students encounter. The abstraction is in the fact that the fractions or rational numbers cannot be listed as the counting numbers or integers. Secondly, in order to test whether the number 4 (for example) or 0 (as another example), is a rational number, learners must coordinate several processes. They must consider what does it mean for a number to be a fraction, i.e., a quotient of two integers, and at the same time, they must think whether the given number 4 or 0, could be expressed as a quotient of two integers, where the denominator is necessarily a non-zero integer. Decimal Number System is developed in accordance with the concept map below:

3 Available from authors.




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Fig. 2 Space, Shape and Symmetry includes scale drawings (magnification, diminishing in the given ratio) on a prespecified grid in addition to problems determining areas, perimeter, etc. The Module on Proportional Reasoning based on the simple Rule of Three (Patwardhan, et.al. 2001), or Ratio rule has been useful for students in transfer, i.e., word problems that initially were considered non-doable are not so anymore. Given new situations, students not only get started but find the solution effectively. The NCTM considers the significance of proportional reasoning in problem-solving to be such as to devote extra time if need be to this important concept. This module creates a comparison of the whole or the unit in the context of fraction, decimal, percent, and circle graph. In the usual discovery approach, initial problems are within all students’ reach and the challenge is increased with the complexity. Navigating across any of these representations with the notion of ratio developed earlier, the proportion then becomes a tool for reasoning with any given representation in question.




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Patterns, Sequences utilizes the FractionsGrid in developing numerical sequences through observation of patterns. The module Story of Number: Graphing follows the above development and creates excitement among learners at the meaning that is now understood in plotting points or seeing the logic of the emerging graph. Section 3: Outline of an actual classroom development for Number and Number

Line All mathematics classes start with the teacher-researcher leading a discussion with the students, in which the intention is to draw out via questioning, student’s existing conceptions on the topic being studied. In the present exposition, we try to stay as close as possible to an actual classroom presentation, building upon the experience of teaching this material at community colleges in the Bronx, and the classroom in question imagined here consisting of the teachers of community-based schools of rural Tamil Nadu participating in the ongoing teaching-action-research project. Each question is followed by the responses it motivates, and a brief coordination with the existing knowledge of the field. The remarks and comments indicated in italic font. Initial questions:

1. What is a number? This question is very interesting to ask in any group and usually brings out/clarifies how inherent/natural/intuitive is the conception of number.

2. Give some examples of number. Usual examples if of the type 2, 3, etc., are then followed by the question: Are there simpler examples? The objective is to get a response of 1, and is possible in every group. Once 1 is agreed upon by everyone in the group, we ask for other examples. Here our focus is not on the distinction of numeral and number, or a concrete or symbolic form, but rather to begin a discussion in which everyone participates, and to build all the notions from the conceptions that exist, are voiced, and agreed upon through discussion. Note that the teacher-researcher facilitates the discussion, guides it along the spontaneous – scientific track, but intervenes very little in providing answers. If the prompt for more numbers generates 2, 3, 4, the next question is:

3. In the numbers 1, 2, 3, etc, do you see a pattern? What would you say is the next number in this pattern?

Once there is a group consensus that the pattern can be discerned and that 4, 5, 6, etc, need not be written we introduce the first convention.

4. Numbers in the pattern 1, 2, 3, … are called counting numbers or natural numbers. Our objective in developing the concept of the number line begins at this stage, however, for that we first need to motivate the number 0.

5. Can you think of a number other than in the above pattern? If there is no answer leading to 0, then the questioning continues with prompts till there is a suggestion of 0.




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6. How can we begin to anchor the numbers we have so far on the number line? Consider the line as follows Note that the arrows on both sides indicate that the line extends indefinitely in both directions.

7. Which number from those we know shall we anchor first? Almost invariably the answer is 0. If not, the question is asked whether we could anchor 0 first. 0

8. The concept being motivated in 8 and 9 is that of a unit, and there is more “direction” from the teacher-researcher.

The development of the unit is an example of the attention paid by the teacher-researcher to the Zone of Proximal development (ZPD) (Vygotsky, 1956) of the learner and to ensure that the spontaneous concepts traverse the span to the scientific concepts. The terms spontaneous and scientific concepts are used in the sense of Vygotsky as follows: Spontaneous concepts are the existing conceptions of the learner Scientific concepts are the accepted meanings of the profession for the concept in question. Where should 1 be anchored on the number line? The usual response is “near 0”. This is followed by the question: Can you anchor 1 to the left of 0?

0 1 The distance between 0 and 1 is called the unit or the unit of measurement. Once we determine that the distance is as above, that is what we will use for all subsequent measurement.

9. Now, where on the number line should 2 be anchored?

The response to this is near or to the right of 1. In student work, there is generally not much attention paid to the unit, i.e., the distance between two consecutive natural numbers may not be the same as the distance between two other consecutive natural numbers, and this is emphasized in this step with the following picture and the question: Is it appropriate to place 2 at the tick mark shown in the next picture? 0 1 When the response is negative as it always is, there is a short enquiry why it is not appropriate and what would be appropriate and the discussion of the unit is continued from step 8. 2 is anchored in its appropriate place as are 3 and 4. 0 1 2 3 4

10. Do you see the next few numbers in the pattern 1, 2, 3, 4 etc? What would they be and is there any necessity of marking them on the number line, and could we mark them if needed?




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11. What is a number other than those in the patterns we already have on the number line? There is sometimes a response of 5, or even 8 or 25, and the inquiry begins: are the numbers 5, 8, 25 in the pattern we presently have? If we continued writing the numbers in the pattern of the counting numbers, would we encounter 5, 8, and 25?

12. The response for a number not in the patterns already existing would draw prompts of a negative number or a fraction, and we begin with the negative number, asking for the simplest which would draw –1. The question then would be where do we anchor –1 and what relationship does it have to 1? Is it possible to see –1 as a reflection of 1 about 0?

13. What are other numbers we could now mark? Responses of –2, -3, -4 would be drawn and marked appropriately. The second naming convention is introduced. Numbers of the type …-3, -2, -1, 0, 1, 2, 3, … are called integers. …-3, -2, -1 are called negative integers and 0, 1, 2, 3, … are non-negative integers. The positive integers are also known by the name of counting numbers.

14. Is every counting number an integer? The part-whole relationship is a regular feature to bring student’s attention to wherever the situation appropriately presents itself.

15. What is a number that does not appear in any of the patterns we presently have?

A response of the fraction 21 is not uncommon and the question is asked:

16. Where should 21 be anchored?

A response of “between 0 and 1” is not sufficient and as in #9 above, attention is drawn to the

anchoring of 21 exactly between 0 and 1. The FractionsGrid (Fig. 3) is extensively used in the

classes of Basic Mathematics or in the professional development workshops of teachers in TN. The FractionsGrid is a picture consisting of 20 line segments each of the same length arranged parallel to each other and with each line segment from left to right not divided at all, or divided into exactly two equal pieces, three equal pieces, …20 equal pieces respectively. Each line segment is labeled with 0 and 1 at the bottom and top respectively, the second line segment has additionally the label

21 , the third has the labels

32,

31 etc.




11

Fig. 3

Labelling the fraction 21 provides an opportunity to count by halves, so that 1 gets labeled as

22 ,

and we have the picture of the number line relabeled and counting by halves takes on significance.

17. Similarly counting and labeling by thirds and fourths is also carried out.




12

The characterization of rational numbers is now carried out and this is utilized in asking students which of the following numbers are rational and why. This is the first abstraction in the sense, that the list description can no longer be used, and secondly, to verify that a given number is rational involves coordinating several processes simultaneously.

18. Are there any other numbers other than those in the patterns already encountered? The intention is to draw out students’ existing conceptions of irrational numbers and usually there is no response in that direction. Rather than introduce it without student response, prime numbers are introduced, and irrationals can naturally be developed after that.

19. Prime numbers and divisibility An activity is lead as follows. Consider the following table in which in the left hand column we enter counting numbers and in the right hand column we enter the multiples of the number. Note that numbers such as 4 which appear in the right hand column are not used when they appear in the left hand column. NUMBER Multiple of NUMBER 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 3 - 9 - 15 - 21 4 not used 5 - - - 25 - 35 - 6not used 7 - - - - - 8not used

Fig. 4 Discoveries to be made from the above table and student’s own work making it.

(i) prime numbers (ii) the first number to appear in the right hand column

for a prime number is its square 20. Discussion of squares: The square of a number is the number multiplied by itself. Fill in the

following table: NUMBER Square of NUMBER

1 1 2 3 4

21. The square root of a number is discussed next with the corresponding table and is compared with the table of squares of the given number.

22. Square roots of prime numbers as the first irrational numbers we can “construct” by repeated bisection. The sketch for 2 is as follows:

1 < 2 < 4 1 < 2 < 4




13

1< 2 < 2 At this point, we can say that 2 lies between 1 and 2. Taking the number 1.5 as that exactly between 1 and 2 as a guess, we check whether 2 could be 1.5 or a number greater or smaller than it. We find that 1.5 multiplied by itself gives 2.25 which is greater than 2, hence 2 must be smaller than 1.5. At this point our estimation window for 2 has changed to

1 < 2 < 1.5 Continuing the bisection argument students computations provide opportunity to review multiplication while also creating the conceptual underpinning for irrational numbers. 23. Irrational numbers via continued fractions requires that students have familiarity with the

quadratic formula to determine the roots of a quadratic equation. An example of Mathematics in Pictures

. .

Ratio 1:1

. .

Ratio 1:2

Scaling down

extend indefinitely scaling up(dilation) Reflection about 0

on both sides division

Continued

fraction

Unit of

0 1

-1 0 1

0 1

Counting

0 1/2 1

0 1/3 2/3 1

integers

0 1

-2 -1 0 1 2

Reflection about 0




14

Rational numbers irrational

Fig. 5 Real numbers

Note, the reasoning process in Fig.5. The first node/rectangle enquires of the learner: Same or Different? The same question also applies to the node/rectangle to its right, still on the first row of Fig.5. In this enquiry, the learner would look either at the dot or the line segment, i.e., he/she would choose either the discrete or the continuous representation based on her/his predisposition. Either way, the learner learns the concept of comparison, and learns about the representation not chosen by her/him. In the second row, the line segment being compared is given a name: the unit or unit of measure, and in that same row, placement begins on the number line. Geometric processes of reflection, duplication, dilation, etc, are used to motivate the progression. Naming conventions are indicated by the labels. The conventions connected with the irrational numbers either as continued fractions or as square roots of the prime numbers are difficult to express in the concept map above.

Section 4: Conclusions In this article, we have

• Outlined an approach to an integrated arithmetic-algebra development that draws on language and geometric supports

• Laboriously sketched the development of a possible conceptual development of the number line in the form of a classroom discourse, in which our emphasis is upon 1. the spontaneous through scientific development for a group of learners whose prior mathematics background might contain significant gaps, 2. for strengthening the connections between concepts 3. for eliminating any formerly held misconceptions about number.

The traditional errors such as “32 is a number between 2 and 3”, estimation problems, and others

arising in the classroom or from the literature are being sought to be eliminated by this development. Further the concept map above is part of an attempt to create “Mathematics in Pictures”, and is designed for ready reference or review, or preparation for one’s own teaching to a roomful of multi-age children often in pretty stark circumstances. We have demonstrated that when the instruction is supported in a Just-in-Time manner with appropriate supports which incorporate the spectrum of possible phases of concept formation – i.e., concrete/enactive – iconic – symbolic, then the learner is able to find the appropriate foothold needed to continue with their learning and fix the gaps in their background with the appropriate assistance in the classroom discourse from the instructor. We have attempted to briefly provide snapshots of our emphasis on the geometric aspect




15

of a balanced geometric-computational treatment that also strengthens the conceptual-procedural connections. The geometric didactic tools of Fractions Grid has several variations to the one we have illustrated in Fig. 3, via which we expose our students to the scaffolding that seems necessary. In the rural schools in Tamil Nadu, the attention span of the teachers in the teacher professional development workshops was extremely short. A culturally responsive (Mukhopadhyaya and Greer, 2009) didactic tool Fig. 6 below, was also available, and its multiple uses for varying stages of the professional development process becomes a new teaching experiment. In this, we draw from Indian yantras, i.e. the geometrical patterns drawn with amazing beauty of form, which are the iconic representation and a meditation tool in the yogic teachings to dwell upon that which the icon represents.

Fig. 6

In this article, we have demonstrated the several opportunities for students to develop greater familiarity with the concepts of number via the numerous instructional interventions.

References Bruner, J. (1966) Toward a Theory of Instruction, Harvard University Press.




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Czarnocha, B. (2008) Remarks on the role of a theory in the work of teacher-researchers, in Bronislaw Czarnocha (ed) (2008) Handbook of Mathematcs Teaching-Research – A tool for Teachers-Researchers. University of Reszów, Poland, Part 2, p 123

Czarnocha B., & Prabhu V. Focus: Women of Rural Tamil Nadu. Teaching-Research Project.

Proceedings, Homi Bhabha Centre for Science Education, Subramanian, K. (ed) Dedekind, R (1963) Essays on the Theory of Numbers. Gersten and Chard (1999) Number Sense : Rethinking Arithmetic Instruction for Students with

Disabilities. Journal of Special Education, 3. p.18-29. Greer, B., Mukhopadhyay, S., Powell, A., Nelson-Barber, S (2009) Culturally Responsive

Mathematics Education, Routledge Publications Inhelder, B., & Piaget, J. (1964) The Early Growth of Logic in the Child Mahavier, W (1999) What is the Moore Method, Primus, Vol. 9. Montessori, M (1946) Education for a new World, Clio Press, Oxford, England. Patwardhan, K., Naimpally, S., Singh, S. (2001) Lilavati of Bhaskaracharya (A Treatise of

Mathematics of Vedic Tradition), Motilal Banarasidass Publishers Prabhu, V., Czarnocha B. (2007) Teaching-Research NYCity model, Dydaktyka Matematyki,

vol.29, Krakow, Poland. Vygotsky L. (1956) Thought and Language, MIT Press.




1

Teaching Statistics Through Student Engagement in a Study on Alternative Medication Practices1, 2

Nkechi Agwu, Piotr Bialas, Brahmadeo Dewprashad and Barbara Tacinelli

ABSTRACT. This paper presents an approach to teaching statistics through student engagement in a study on alternative medication practices in Caribbean immigrant communities in New York City (NYC), funded by a CUNY Community Collaborative Incentive Research Grant. The authors discuss the opportunities and processes that lead to this interdepartmental collaboration of faculty and students from the Mathematics, Science and Nursing Departments at the Borough of Manhattan Community College (BMCC), City University of New York (CUNY), faculty from the QueensBridge to Medicine, Sophie Davis School of Biomedical Education at City College of New York (CCNY), CUNY, and two high school teachers from the NYC and New Jersey State Departments of Education. They identify the challenges that arose from the timeframe required to submit the proposal and accomplish the project. They discuss the educational benefits of this project to BMCC, CUNY, through provision of research internships for students and development of curriculum materials for teaching introductory statistics. They also discuss the educational benefits to the students who engaged in this study. Lastly they highlight how this approach could be replicated in the future with high school students or adapted to a high school environment.

1. Introduction

In a number of immigrant communities, a combination of insufficient health insurance, shortage of (community based) trained medical personnel, frustrations with “western” therapies, an inclination to experimenting out of the realm of medications prescribed by a medical doctor (M.D.), culture/tradition and a paucity of health education/awareness leads to a myriad of health problems. A triad of these are related and might be significant but are not usually addressed. They are the practices of self-medication, the use of medications manufactured overseas (and not United States Food and Drug Administration (US FDA) approved), and the use of alternative medicines without due consideration to their efficacies and possible contraindications.

Numerous medications manufactured overseas are sold in many neighborhood urban groceries and are also readily obtained from friends and relatives visiting from abroad. They are used in preference to similar products manufactured in the United States of America (USA). However, they may be less efficacious. Some of these products are manufactured in countries where the composition and label claims are not subject to the same level of scrutiny as drugs produced specifically for markets in the USA. Also, prolonged shipping and unsuitable storage conditions may lead to loss of potency. Recent reports indicate that there is an increasing use of alternative medicines (e.g. saw palmetto and St. John’s Wort) and supplements (such as creatine, androstenedione, shark cartilage and glucosamine) [1], [2], [3]. It is estimated that Americans spend about $12 billion a year for these products [4]. Inappropriate use of these substances may lead to untoward risk. The literature indicates that toxic ingredients including pesticides, non-declared drugs, and added chemicals, are sometimes found in herbal preparations [5]. From January 1993 to October 1998, the Food and Drug Administration (FDA) received 2,621 reports of serious problems including 101 deaths linked to supplements [6]. In immigrant communities, there is (probably widespread) use of herbs and supplements as they are abundant in neighborhood groceries. The latter practice exists but has not attracted the attention of investigators [7].

A preliminary investigation of medications sold in immigrant neighborhoods in Queens and Brooklyn indicated that there were numerous products that did not have declared active ingredients that would support the label claims [8]. For example, there were products sold to treat Herpes simplex that did not contain any known anti-viral

1 2000 Mathematics Subject Classification. Primary 97D02; Secondary 92E02, 92C50. 2 This project was supported by the CUNY Community College Collaborative Incentive Research Grant Program and the following programs/offices at BMCC: the Nursing Program, the Collegiate Science and Technology Entry Program (CSTEP), the Computer Science Engineering and Mathematics Scholarship Program (CSEMS), the Louis Stokes Alliance for Minority Participation Program (LSAMP), the Minority Science Engineering Improvement Program (MSEIP), the Office of Instructional Technology, the Grants and Development Office, the Office of Academic Affairs and the Project Kaleidoscope Leadership Program.




2

agents and there were a host of products for peri-menopausal symptoms that did not contain active ingredients known to be effective. There was even a product “Study-Power” which claims to “increase intelligence”.

In the 2004-2005 academic year the authors engaged in an interdisciplinary research study “An Investigation Into the Patterns of Uses and Effects of Self-Medication in Caribbean Immigrant Communities” at the Borough of Manhattan Community College (BMCC), City University of New York (CUNY), funded by a CUNY Community Collaborative Incentive Research Grant. The aim of this study was to determine the pattern and extent of self-medication in Caribbean immigrant communities in New York City (NYC). To accomplish this goal, a survey was developed and conducted on 339 self-identified non-minor persons of Caribbean heritage living or working in NYC about their self-medication practices. In addition, six community-based medical doctors were interviewed on their knowledge of their patients’ self-medication practices. A convenience sampling method was employed in all data collection activities. The results of the survey and the doctors’ interviews were used to compile a checklist of alternative medications. Medicines that were not prescribed by medical practitioners and those that are sold over the counter but not United States Food and Drug Administration (US FDA) approved were considered as alternative medications. Some of the medications on this checklist were purchased and chemical analyses were conducted on the medications purchased.

This project was a collaborative endeavor between faculty and students from the Mathematics, Science and Nursing Departments of BMCC/CUNY, faculty from the QueensBridge to Medicine, Sophie Davis School of Biomedical Education at City College of New York (CCNY), CUNY, and two high school teachers from the NYC and New Jersey State Departments of Education. The study involved teaching statistics to students through research internships. The pedagogical approach used here has implications for replication with high school students. It is adaptable for project-based teaching at the high school level. Additionally, it has potential for establishing or enhancing collaborative partnerships between colleges and high schools.

2. Opportunities and Processes Leading to the Interdepartmental Collaboration This project evolved as a consequence of the following existing CUNY-wide or local BMCC, CUNY, programs

and initiatives:

2.1. The CUNY Community College Collaborative Incentive Research Grant In Spring 2004, the CUNY Office of Academic Affairs announced a new program to support the research efforts of

faculty at the Community Colleges and to encourage collaborations with faculty within and across CUNY campuses. The PI’s responded to this opportunity by submitting the proposal for this project. Their project was selected for grant funding under this program. 2.2. The Project Kaleidoscope (PKAL) Leadership Program at BMCC, CUNY

This program served as the impetus for bringing together the mathematics and science PI’s for the grant that funded this study. The PKAL leadership group organizes faculty development seminars and monthly meetings for faculty from the science, mathematics and computer information systems departments to discuss science, technology, engineering and mathematics (STEM) initiatives at BMCC, CUNY, geared towards promoting “Science for All” students. 2.3. A previous partnership with the faculty from QueensBridge to Medicine, Sophie Davis School of Biomedical

Education at CCNY, CUNY The science PI had previously conducted a pilot study with the faculty from QueensBridge to Medicine, Sophie

Davis School of Biomedical Education. Thus, the initial idea for the writing of the proposal for this project was to involve high school student interns from the QueensBridge to Medicine, Sophie Davis School of Biomedical Education in conducting the interviews for this project. However, due to unexpected bureaucratic delays and the timeframe limitations for proposal submission the plan to involve high school students in this project was replaced by an alternative plan of working with students enrolled in the nursing 112 course (NUR 112), Nursing Process I: Fundamentals of Patient Care at BMCC, CUNY. See 2.4 below for a discussion of why this was considered a good option.




3

2.4. The Nursing 112 community-based internship requirement Nursing 112 provides nursing majors at BMCC, CUNY, with an introduction to the bio-psycho-social and cultural

factors that influence the nursing care of any patient or client who needs minimum assistance in the maintenance of health. The community-based internship requirement of this course presented an ideal opportunity to involve the Nursing Department in this project given that their faculty members were not participating in the PKAL Leadership Program, and that this project directly relates to socio-cultural factors that may have influence on nursing care. 2.5. Various STEM Undergraduate Research Programs at BMCC, CUNY

Various STEM undergraduate research programs at BMCC, CUNY, viz., the Collegiate Science and Technology Entry Program (CSTEP), the Computer Science Engineering and Mathematics Scholarship Program (CSEMS), the Louis Stokes Alliance for Minority Participation Program (LSAMP), the Minority Science Engineering Improvement Program (MSEIP) require participating students to engage in research internships under the mentorship of a STEM faculty member. These programs provided opportunities to recruit students to engage in this study under the mentorship of one or more of the collaborating faculty members and high school teachers.

3. Internships 3.1. The Nursing 112 internships

A survey questionnaire was developed in English for the purpose of collecting data on the pattern and extent of self-medication in Caribbean immigrant communities in NYC, together with a participant informed consent form and an interviewer informed consent form. Given that the language of business communication in some Caribbean countries is Spanish or French, it was perceived that there might be potential survey participants who could not understand English. Coupled with the fact that some of the NUR 112 students indicated fluency in either of these two languages, Spanish and French translations of the questionnaire and participant informed consent form were made with assistance from two other BMCC, CUNY, faculty members (one from the Modern Languages Department and another from the Mathematics Department) for potential survey participants that did not understand English but understood one of these two languages. However, the translated versions of the questionnaire and participant informed consent form were not used since all the surveyed participants understood English. The Modern Languages faculty member indicated the potential of using the questionnaire and the participant informed consent form as a translation assignment in teaching students Spanish.

This questionnaire was administered through face-to-face interviews with 339 non-minor persons of Caribbean heritage living or working in NYC. The interviews were conducted by 120 students enrolled in three sections of NUR 112 as their community-based internship requirement for this course, during the period of March 2005 through May 2005. Each NUR 112 student registered in the participating sections of this course was expected to interview three subjects. Student interviewers were supposed to follow the interview instructions and guidelines scrupulously. They were supposed to index their subjects’ forms according to previously established rules. The questionnaire and participant informed consent form for each subject was placed in an envelope that was marked with the name of the interviewer. The student interviewers returned the envelopes with the completed questionnaires for the interviews they conducted to the instructor of their NUR 112 section, who in turn returned them to the PI’s for further evaluation. It was envisioned that this type of involvement would develop the students’ patient-interview skills.

The three instructors for the three NUR 112 sections coordinated the logistics of the interview process. They selected sites highly populated with people of Caribbean heritage throughout the boroughs of NYC, for their students to conduct the interviews for this project. Some of the sites selected were churches and community centers. They obtained permission from the administrators at these sites for their students to conduct the interviews in rooms that guaranteed privacy for the participants during their interview. They disseminated information to their students about the interview locations, dates, times and directions to these locations, and also escorted them to these locations.

The subjects were selected based on voluntary participation and self-identification as belonging to the target population. This questionnaire sought answers to the frequency of usage of alternative medications, the names of such medications, the conditions for which they are used, the basis for their usage, and the subject’s perception of the effectiveness and side effects of the medication.

The student interviewers and all other personnel engaged in this project (PIs, research associates and research




4

assistants) completed CUNY’s approved training program for the protection of human subjects in research (see http://www.rfcuny.org/ResConduct/CBT) and received a certificate upon completion to indicate that they were employable for any CUNY research study involving human subjects. This was a requirement stipulated the BMCC, CUNY, Institutional Review Board (IRB). In addition, the student interviewers were trained and practiced in interview techniques within a half-day workshop conducted at BMCC, CUNY in February 2005. They were provided with instructions and guidelines for conducting interviews with participants. Lastly, the interviewers were required to sign an interviewer informed consent form. 3.2. Internships Within Various Undergraduate Research Programs

Statistical Package for Social Sciences (SPSS) software was used to create a database for participant responses to the survey questionnaire. The participants’ responses to the questionnaire were entered into the database and analyzed with the support of student research interns from the following BMCC, CUNY, undergraduate research programs, viz., CSTEP, MSEIP, LSAMP and CSEMS. These student interns were not required to have a pre-requisite background introductory statistics nor prior experience in undergraduate research.

Based on the participants’ responses to the questionnaire, a checklist of alternative medications was developed. The popularity of these medications were ranked, the reasons for use by participants were identified together with their knowledge about the efficacy of these medications.

A majority of the alternative medications that participants reported to have used in the past or are currently being used by them were critically evaluated. Medications selected were those that were available in multiple groceries in Caribbean immigrant communities and whose analysis would provide information that would lead to new knowledge about them. Labels (where available) of the most frequently used products were inspected and their label claims critically evaluated. The pharmacology of each of the active ingredients was researched from the literature and a critical evaluation was done in order to determine whether the label claims (and the purported usage of these medications) were justified. Students the identified undergraduate research program who were taking or who had taken CHE 230 - Organic Chemistry I, and CHE 240 - Organic Chemistry II at BMCC, CUNY, were involved in the analysis. This was done in order to provide them with training in undergraduate science research.

4. Curriculum Materials for Teaching Introductory Statistics

The following materials developed and/or used for this study served as curriculum materials for teaching students in one section of a 2005 Summer I MAT 150 – Introduction to Statistics course at BMCC, CUNY, about survey design and methodology, ethical practices for conducting statistical studies, data collection, organization and analysis, and reading graphs, tables and charts. Students in this class indicated at the end of the semester that these materials were valuable in helping them to develop an appreciation for the field of statistics, its real-life applications and its practice. 4.1. The CUNY Training Program for the protection of human subjects in research, the survey questionnaire,

the handout with instructions and guidelines for interviewing participants, the participant and interviewer informed consent forms Since the NUR 112 students who engaged in this project developed positive attitudes toward the research process

and ethical considerations associated with conducting a study involving human subjects through the training process and use of the above-mentioned materials, this was replicated with students in one section of a 2005 Summer I MAT 150 – Introduction to Statistics course as part of the coursework. The MAT 150 students underwent formal training within the CUNY training program for the protection of human subjects in research and they had to write a graded report on ethical considerations for the use of human subjects in research. By virtue of being formally trained through this program, each of the MAT 150 and NUR 112 students became certified and therefore employable on CUNY research studies involving human subjects.

The questionnaire, the handout with instructions and guidelines for interviewing participants, the participant and interviewer informed consent forms were used to teach students about survey questionnaire development, interviewing techniques and ethical issues related to conducting interviews. In addition, the questionnaire was used to teach students about various types of data, coding schemes for categorical data and methods of data analyses.

http://www.rfcuny.org/ResConduct/CBT




5

4.2. Two examples of graphical displays with relevant statistical summaries from the survey Example I – A qualitative variable (Medication Type) with seven levels

This example was selected to teach students how to code and analyze a qualitative variable using some statistical software. Students were provided with the raw data for this example. They had to use the raw data to create frequency, percentage frequency and cumulative frequency tables and an appropriate graphical presentation of the data. They had to write a brief report summarizing the statistics in their frequency, percentage frequency and cumulative frequency tables and in their graphical display. Question - Please indicate any medication (including herbs or alternative medicines) that you are taking or have taken in the past year that has not been prescribed by a medical doctor? Medication Type-Table 1

Medication Type

Frequency Percent Cumulative Percent

Tea

87 26.2 26.2

Prescription medication 6

1.8

28.0

Non-herbal OTC

36 10.8 38.9

Bitters

31 9.3 48.2

Laxative

8 2.4 50.6

Non-Caribbean alternative medication

26 7.8

58.4

Miscellaneous

138 41.6 100.0

Total

332 100.0




6

Figure I

Medication Type

Mis

sing

Mis

cella

neou

s

Non

-Car

ribea

n al

tern

Laxa

tive

Bitt

ers

Non

-her

balO

TC

Pres

crip

tion

med

icatTe

a

Perc

ent

50

40

30

20

10

0

Example I I– A numerical variable (Cost of Medication)

This example was selected to teach students how to analyze a quantitative variable. Students were provided with the raw data for this example. They had to use the raw data to obtain a summary statistics and create an appropriate graphical presentation. They had to write a brief report discussing the summary statistics for this example. Question - What is the cost of this medication? Cost of medication – Table II

Statistic

Statistic’s Value

Mean 11.2606

Median 5.0000

Variance

540.907

Std. Deviation 23.25741

Minimum 0.00

Maximum 210.00

Range 210.00

Interquartile Range 10.0000




7

Cost in Dollars

195.

0 - 2

05.0

175.

0 - 1

85.0

155.

0 - 1

65.0

135.

0 - 1

45.0

115.

0 - 1

25.0

95.0

- 10

5.0

75.0

- 85

.0

55.0

- 65

.0

35.0

- 45

.0

15.0

- 25

.0

-5.0

- 5.

0Cost of Medication

Freq

uenc

y

160

140

120

100

80

60

40

20

0

Std. Dev = 23.26

Mean = 11.3

N = 283.00

Figure II




8

283N =

Case number

Cos

t in

Dol

lars

300

200

100

0

-100

2031421311061048951369411629331955230162522326632327331326394

79

222

308

3024

Figure III

5. Educational Benefits to Students 5.1. Benefits to Nursing 112 students

One hundred twenty NUR 112 students were involved as student interviewers in this research project as part of their community-based learning experiences. The students developed positive attitudes toward the research process and ethical considerations associated with conducting a study involving human subjects. They increased their knowledge about self-medication with over-the-counter (OTC) agents (e.g. medicines and nutritional supplements) and home herbal remedies, and the potential for adverse drug interactions with prescribed medications. Integrating a research experience into a community-based learning experience proved to be an effective learning strategy to teach research and nursing communication skills, while also helping to promote the health and well being of individuals in a community.

This community research project allowed these nursing students to apply their readings and class discussions of communication, cultural diversity, ethics, and legal aspects of confidentiality to their lived experiences with individuals in Caribbean communities, thereby enriching and affirming the classroom theory through first-hand application. The experiences provided students with an opportunity to learn the basics of research, implement communication skills with increased confidence, and meet course learning outcomes related to community concepts and the role of the nurse in the community. These experiences translated to increased awareness related to the health of a community and the need for advocacy by nurses.

Student feedback was extremely positive. Students identified increased confidence in communicating.




9

Student comments reflected an appreciation of this community learning experience in allowing them to implement their knowledge and skill in the “real world” in a short time frame. Participation as interviewers in this community research project gave students a better understanding of the functions of the professional nurse in community settings, which was an expected outcome of this community learning experience. 5.2. Benefits to the Undergraduate Research Interns

The students who worked on the raw data preparation and analyses phases of the project were instructed on how to code different categorical variables, how to enter categorical variables’ values into a file, how to correct spelling errors and how to clean data (remove cases that remained incomplete, partially incomplete and/or illegible). They were instructed to label each variable with concise and meaningful information for further evaluative purposes. They were also instructed on selected SPSS procedures involving frequency analysis and graphical analysis of the data, viz., how to construct and format frequency distribution tables in the SPSS file and how to create/format various graphical displays of numerical and categorical data. In addition, they were instructed on how to transfer different SPSS displays into a Microsoft Word file while writing a report about the study. Lastly, they participated in drafting and proof reading the project report and the accompanying Power Point presentation. The fact that these student research interns were not required to have a pre-requisite background in introductory statistics and/or have prior undergraduate research experience provides an ideal opportunity for project-based adaptation to the high school learning environment.

The research interns who engaged in the chemical analysis of some of the alternative medications were provided with hands-on exposure to isolation and characterization of active constituents of medications. The techniques that were used built upon laboratory skills and theoretical concepts learned in CHE 230 - Organic Chemistry I and CHE 240 - Organic Chemistry II. It also helped to provide students with real-life applications of organic and general chemistry. The students involved indicated that this experience has helped to motivate them to pursue science research careers.

6. Replication With High School Students and Adaptation to the High School Environment

The authors believe that this project is replicable in the high school environment within a one semester or a year project-based teaching/learning involving students of mathematics, biology, chemistry, health sciences and social studies working together to determine the pattern and extent of self-medication of various ethnic groups. Some challenges that may arise in adaptation to the high school environment involve school/parent consent for students to conduct the interviews, staff to assist students during the duration of the project and recruitment of teachers to participate in this interdisciplinary collaboration. As indicated previously in section 2.3 in discussing the challenge related to involvement of the high school interns from the QueensBridge to Medicine, Sophie Davis School of Biomedical Education, this project has potential for high school and college STEM related partnerships. It also has potential for interschool, interstate and international collaborations and can be adapted for research projects in the area of sociology and cultural anthropology. In particular, the curriculum materials used for teaching MAT 150 can be replicated for use in teaching the same concepts in introductory statistics at the high school level. Lastly, this project is ideal for motivating high school students interested in health care and STEM related professions and professions in sociology and cultural anthropology.

7. Conclusion

The findings of this study revealed some alternative medications that were commonly used among those surveyed to treat many severe medical conditions for which they have not been proven to be effective. The pharmacology of some of the selected alternative medications showed toxicities and contra-indications associated with their use. As such, their use might directly lead to adverse health risks. The survey participants indicated that elders, family and friends propagated their use of alternative medications. Active community education may be needed to advice on relative merits of such medications. Education of community health care workers on such practices is also recommended. The participation of students in various aspects of this study and the positive feedback received about their participation, together with the use of project materials for teaching introductory level statistics emphasize the significant educational value of student engagement in scientific research and in teaching through real-life applications.




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On the whole, this study provides a framework for future research endeavors related to the patterns of uses and effects of self-medication in Caribbean immigrant communities and serves as one model for engaging students and teachers in interdisciplinary science research.

References

1. Hadley, S.K.; Petry, J.J. Hospital Practice, June 15, 1999, p 105-123 2. Consumer Reports, March 1999 3. Consumer Reports, October 1999 4. Eisenberg, D. M.; Kessler, R. C.; Foster, C. et. Al. “Unconventional medicine in the United States: prevalence, costs,

and patterns of use. New England J. of Medicine, 1993: 328(4): 246-52 5. Tyler, V.E. “Herbal medicine in America.” Planta. Med. 1987; 53(1):1-4 6. Chan, T.Y.; Chan, J.C.; Tomilson, B. et. Al. “Chinese herbal medicines revisited: A Hong Kong perspective.”

Lancet, 1993; 342 (8886-8887): 1532-4. 7. Borins, M. “The dangers of using herbs.” Postgraduate Medicine, July 1998, 104(1): 91-100. 8. Dewprashad, B.; Martin, C.; unpublished data

BOROUGH OF MANHATTAN COMMUNITY COLLEGE, CITY UNIVERSITY OF NEW YORK, 199 CHAMBERS STREET, NEW YORK, NY 10007 [email protected], [email protected], [email protected], [email protected]

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Research on the Teaching of Mathematics:

A practitioner’s perspective

Chun-Ip FUNG ([email protected])

Department of Mathematics, Science, Social Sciences and Technology

The Hong Kong Institute of Education

Researching the non-researchable

In the field of education in general, and mathematics education in particular, the issue of linking research and practice has a long history. From the practitioner’s point of view, the problems highlighted are almost always the same: (a) Research conducted by researchers often does not address the problems of the teachers (Heid, et al., 2006; Kennedy, 1997). This could happen in three ways. Firstly, in order to get research outputs published, researchers pay so much attention to meeting the rigorous standard of academic paper that they prefer adjusting the research questions, and sometimes the research context, to fit a rigorous methodology instead of adjusting the research method, or even the research context, to cope with practitioner-generated research questions. Secondly, lacking a repertoire of wisdom of classroom practice, many researchers fail to comprehend the genuine context and problems of classroom teaching. Thirdly, teachers need practical solutions, often in the form of a detailed instructional design, not just a long list of general guiding principles. While the latter can be found in many academic papers on teaching, the former is often missing. (b) Research outputs are criticized of not being accessible to teachers (ibid.). Apart from jargons, research papers build theories upon previous works. In order to comprehend what is written in one single paper, the teacher reader may need to read through some more papers. Very often, before getting any insightful idea, a huge amount of time might have been spent. The teacher must then decide whether it is worthwhile to invest the time and effort. In essence, are we suggesting that in order to be able to apply research results to improve teaching, a teacher must devote the effort to study the research literature, to an extent comparable to, or even beyond, that is needed for acquiring a higher degree? At the extreme, do we need a doctorate to teach well in schools? Experience shows that it is neither necessary nor sufficient.

One of the many questions that confront teacher educators is whether teaching is researchable. No matter how tempting it is to say yes, we understand that works to this end could be challenged in many different ways. The rigor of the research is often troubled by the formulation of research questions, the identification of hidden assumptions, the definition of the targeted population, the


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examination of relevant context parameters, the separation of the effect of the instructional design from the instructional skill of the teacher, … In a highly simplified model, teaching involves the teacher, the learners, and the instructional designs. Apparently, both the characteristics of the teacher and the group of learners could exhibit so great a degree of variation across classrooms, schools, or districts that the number of combinations arisen is practically infinite. At the end, as St. Clair (2005) put it, “the credibility of the discipline [education] is threatened by the inability of educational researchers to make generalizable statements that address a significant cluster of educational settings ⎯ for example, that one instructional method is better than another” (p.437). Built upon philosophical inquiry, he argued that there is “no logical basis for believing that it is possible to transfer empirically based knowledge between educational settings” (ibid., p.436). The existence of super-unknowns has prevented attempts to develop generalizable propositions from being successful. He stressed that “it is imperative to recognize the limit of our knowledge and to understand that no approach to research can take us beyond the need for informed human judgment” (ibid., p.436). He further brought forth the notion of ‘empirical heuristics’ as “models of relationships between factors based on empirical evidence, but without a claim to universality” (ibid., p.436). The implication of his arguments is that research results “should not be presented as answers to questions, but as contributions to the empirical heuristics supporting judgment” (ibid., p.437).

In other words, research on teaching should aim at providing insight and wisdom, not definitive answers to questions, or instructional recipe to follow. In this paper, I will give two examples to illustrate how the teaching of elementary mathematics is studied in Hong Kong, under the ongoing developmental research project ⎯ Teaching for Mathematising (abbreviated TFM hereafter). Before proceeding, it is helpful to give a brief introduction to the cultural context in which the research is done.

The cultural context of research

Hong Kong is a city inhabited mostly by Chinese. Although it has long been influenced by western culture, how general public perceives teaching is still predominantly grounded on Confucian heritage. The following beliefs distinguish the community from a western one: (A) Chinese teaching tradition emphasizes content as a vehicle on which process abilities could be developed. (B) Chinese aspire to the pleasure derived from significant achievement built upon hard work. (C) Chinese communities expect a mathematics teacher to have a good command of the subject and to demonstrate how it should be learned. (D) Chinese place equal emphasis on the student, the teacher, and the subject (Leung, 2001). The consequences of these characteristics include: (i) The community welcomes challenges given to students. (ii) The teacher has good authority to put high


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expectation on students and his/her autonomy to employ instructional design is generally respected, as long as students’ learning needs are carefully addressed. (iii) Teachers’ inadequate mathematical knowledge may be challenged by parents.

The object of research

If we could accept that research may not produce immediate answers to questions and problems about teaching, and refrain from looking for immediate measurable effects brought about by teaching, we will then not be restricted to study the relationship between the performance of the teacher and the immediate achievement of students (often through pre-test and post-test). Instead, we study the instructional design, and its impact on teaching. Wittmann (1995, 2001) proposed that the study of teaching units (called substantial learning environments) should form the core of mathematics education. “The design of substantial learning environments around long-term curricular strands should be placed at the very centre of mathematics education. Research, development and teacher education should be consciously related to them in a systematic way [bold-face original]” (Wittmann, 2001, p. 4).

TFM produces and studies instructional designs. Instructional designs are investigated at two levels, namely the framework of the instructional design which carries the major ideas underpinning the design, and the implementation plan of the instructional design which includes the fine details of the instructional design to the extent that other practitioners could paint a full picture of the teaching process and thus replicate. All TFM instructional designs are developed and scrutinized based on Freudenthal’s notion of ‘mathematising’. He maintained that to learn mathematics means to engage in the process of mathematising through which mathematics is re-invented (Freudenthal, 1973, 1991). His emphasis on mathematising carries a strong evolutionary stance and is further elaborated subsequently by Wittmann (1995, 2001) who proposed that mathematics education should be viewed and studied as a ‘systemic evolutionary design science’.

TFM studies the continuum of the evolution of mathematical understanding. On one end of it is the state that a piece of mathematical knowledge does not yet exist in the mind of the learner, called the ‘state of no mathematics’. On the other end of it is the state that the learner has acquired the specific piece of mathematical knowledge, very often in a mature and refined form meeting curriculum specification, called the ‘state of refined mathematics’. For discussion sake, the term ‘state of primitive mathematics’ is often used to refer to a state in between the two ends at which the learner does learn some mathematics, though not necessarily at an ultimate form commonly

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found in assessment or curriculum documents. The complete experience of learning mathematics portrayed by TFM can be summarized by Figure 1:

Figure 1

In between the state of no mathematics and the state of primitive mathematics, the learner progresses through the stage of evolving mathematical ideas. From the state of primitive mathematics to the state of refined mathematics, the learner progresses through the stage of refining mathematical ideas. After the state of refined mathematics is reached, the learner enters the stage of applying mathematical ideas.

While many teachers tend to put their major effort to the last stage of applying mathematical ideas, which typically requires students to complete tasks that frequently appear in territory-wide assessment, TFM teachers emphasize the first two stages, which collectively form the ‘teaching gap’, a term the TFM community borrows from Stigler & Hiebert (1999).

The research question

Research questions in education typically cover a certain not-too-small population. Here is where statistical techniques come in. Acknowledging the existence of super-unknowns, such paradigm is dismissed, at least at an early stage. TFM focuses on small and specific questions raised by practitioners. These questions have high impact on the day-to-day work of teachers, but generally do not attract the attention of researchers coming from universities. Without a slight concern of generalizability, teachers as practitioners simply want to tackle problems arising from the specific


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context they are working in. Whether certain solution works for the next classroom may not be of great concern. Teachers participating in TFM work hard to solve their own problems, not others’.

Developed on a practitioner’s perspective, research questions for TFM usually have the following structure:

Given the characteristics of the student group, and under the constraints dictated by various context parameters, how could the teaching of certain curriculum topic be designed and/or executed in order to bring about certain specified teaching and learning effects?

For communication within a community sharing most of the context parameters, such as a small place like Hong Kong, only a brief description of student characteristics is needed. Otherwise, research questions may read like the followings:

(Q1) How could the teaching of division at Grade 2 (7-8 years old) be designed so that students could make good sense of the symbolic computation they are required to perform?

(Q2) How could the teaching of basic fraction concept at Grade 3 (8-9 years old) be designed so that students can interpret m nth of a certain quantity, as an amount equals to that quantity first divided by n, then multiplied by m?

The phrasing of these questions depends not only on the context in which teaching occurs, but also the language through which learning happens. For instance, the phrasing of (Q2) in English is less natural than in Chinese because in naming a fraction, the denominator comes before the numerator in Chinese, and vice versa in English.

The method of research

The ‘how’ type research questions described above necessitate the generation of solution, usually in the form of instructional design. To come up with a solution as such, expertise in mathematics, curriculum, and classroom teaching skills is needed. Thus, there is a clear role for teachers, who should be most familiar with curriculum and classroom teaching skills, to input in the research team, though other academics or professionals may help as well. In other words, teachers are the main researchers, not subordinates of any academic outside the school. The mentality of having researchers develop theories and teachers apply them creates the gap between theory and practice. TFM avoids this consequence by letting teachers raise question, carry out investigation, and crystallize theory, all by themselves, with possible guidance and assistance from academics.


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Before arriving at any design framework, the research team begins by analyzing typical teaching processes, hoping to spot the weaknesses. Teachers argue, compare, negotiate, and hopefully come up with some conjectures about the inadequacy of teaching and learning. For example, (C1) and (C2) below are conjectures come under research questions (Q1) and (Q2) respectively.

(C1) Students fail to understand the meaning of the symbolic computation of division because during the teaching process, there is no step-by-step correspondence between the use of manipulatives and column division to find the result of a sharing or grouping activity.

(C2) Printed teaching materials have the intrinsic limitation of not capable of requiring students to divide something into arbitrary equal parts according to the denominator, causing students to just shade the number of shares defined by the dotted line (printed in order to bypass the trouble to divide into arbitrary equal parts) according to the numerator alone. In other words, by requiring students to shade, according to a specific fraction, on a figure with dotted lines dividing the figure into specified equal parts, student could do it right by just considering the numerator!

To test these conjectures, the TFM research team must exercise some creativity that stems from an amalgamated knowledge of mathematics, curriculum, and teaching skills. Coupled with necessary thought experiments, hopefully instructional frameworks could be developed to bring these conjectures to field test.

Before actual field test, there is a final step to convert the instructional framework to a detailed implementation plan. At this stage, the teacher doing the field test should fill in the missing details of the instructional framework and make necessary adjustment in order cope with all context parameters that need to be handled. Different teachers will inevitably come up with different implementation plan. In essence, different teachers are testing the same instructional design framework based on different implementation details. In addition, variation of personal qualities and classroom skills across teachers further multiply the diversity of the field test. These super-unknowns eliminate the possibility of making generalizable statement. Instead, the common observations, the interpretation of discrepancies, and teachers’ self evaluation of the whole experience all contribute to the development of empirical heuristics that could support teachers’ professional judgment.


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The research result

Research result includes a range of outputs generated throughout the whole research process. At the first place, it is the instructional design framework. In other cases, a teaching/learning aid could be the major product of research. I will explain the former with (Q1) and the latter with (Q2).

In Hong Kong, the first encounter of division concept appears at Grade 2 (7 or 8 years old). Students have already learned addition, subtraction of three-digit numbers, and are able to apply multiplication table to compute multiplication of single digit numbers. The following instructional design framework (D1) is created to tackle the weakness of correspondence between the use of manipulatives and symbolic computation:

Principles of design of (D1)

• Sharing and grouping are introduced as two separate activities, without the division sign appears before the two can be linked up conceptually.

• The design tries to convey the message that the symbolic column division is just an elegant recapitulation of the process of sharing or grouping

• The design also seeks to make the use of multiplication table evolve in a natural way.

• There is an important section of the teaching, which requires both the teacher and the students to demonstrate the step-by-step correspondence between the use of manipulatives or diagrams, and the application of multiplication table.

The flow of teaching of (D1)

• Students perform sharing and grouping activities using manipulatives, and are required to describe the process both in written form (by filling in the blanks in Appendices 1A and 1B) and orally (by reading aloud the statements printed on Appendices 1A and 1B).

• Students perform sharing and grouping activities on paper, and are required to describe the process both in written form (by filling in the blanks in Appendices 2A and 2B) and orally (by reading aloud the statements printed on Appendices 2A and 2B).

• By comparing the two processes using the same set of data in Appendix 3, students are guided to discover that in general the two processes, when applied to the same set of numerical data (dividend and divisor), yield the same two numbers (quotient and remainder), though corresponding units are not identical.


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• Teacher introduces the division sign “÷” and explain how these results could be expressed in terms of the division operation.

• Students work on Appendix 4 and are guided to discover how the multiplication table helps when doing division. This is done by having the teacher and the students voicing out the process step-by-step, and extract the three crucial numbers that come from the multiplication table: 3, 1, 3 from the statement “In teams of 3, forming 1 team requires 3 persons”; 3, 2, 6 from the statement “In teams of 3, forming 2 teams requires 6 persons”; 3, 3, 9 from the statement “In teams of 3, forming 3 teams requires 9 persons”; 3, 4, 12 from the statement “In teams of 3, forming 4 teams requires 12 persons”; and 3, 5, 15 from the statement “In teams of 3, forming 5 teams requires 15 persons”. By noting that 15 exceed 13, the number of persons given, students could tell that they need to stop when 4 teams are formed.

• Teacher summarizes the procedures to carry out division symbolically: With the help of multiplication table, we determine the largest multiple, kb, of b that does not exceed a. Then a ÷ b = k … r, where r = a – kb.

• Teacher explains the format of column division and guide students to finish Appendix 5.

Some evidence of implementation of (D1)

Figure 2 was taken when students were working on Appendices 1A and 1B.

Figure 2


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Figure 3 shows a PowerPoint animation used to illustrate the correspondence between using manipulatives and multiplication table.

Figure 3

Figure 4 was taken when the teacher summarized the evolution of the column division.




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Figure 4

Figure 5 was taken when student’s work on Appendix 5.




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Figure 5

For (Q2), TFM came up with the idea of using an elastic string (abbreviated (A1) hereafter) that could enable students to arbitrarily divide a line segment or rectangular strip of length from 4 cm to 9 cm inclusive, into 2 to 20 equal parts. The success of this approach depends on the assumption that any consecutive intervals with equal separation marked on the elastic string will retain the same length ratio upon any extension not beyond the elastic limit. Laboratory tests show that this assumption is generally valid. (A1) has four different scales printed on it. Table 1 below shows the technical specification:

Scale Number of intervals needed

Length of each interval when the string is just taut (cm)

Length of each interval when the string is at the elastic limit (cm)

1 20 0.198 0.44

2 20 0.45 1.00


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3 9 0.99 2.20

4 4 2.16 4.80

Table 1

(A1) could be applied to divide a line segment into equal parts only when it is extended, but not beyond elastic limit. Figure 6 was taken when students were working in pair, using the string to divide a rectangular strip into the number of equal parts specified by the denominator.

Figure 6


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Evaluation of research result

How successful is the research? There is no easy answer to this question. However, TFM does have some guidelines by which the quality of work could be evaluated. For the instructional design framework, the following are core questions to be carefully studied:

(G1) Does the development of the main content follow an evolutionary path?

(G2) Does the development of the lesson(s) facilitate pupils re-inventing mathematics based on their previous knowledge and appropriate scientific means?

(G3) Does the development of teaching illustrate how mathematical thoughts and methods interact with learning?

For (Q1), the instructional design framework (D1) given in this paper does follow an evolutionary path. TFM teachers all agree that the design passes (G1). Others could also scrutinize accordingly. As seen from video records, most students could discover the patterns, declare that multiplication table could help to do division, and explain to others. Many could invoke their previous knowledge, and argue for the validity of claims. Thus (D1) passes (G2). Requiring students to use real objects, then pictures, and finally symbols alone is an evolutionary approach, exemplifying the usual thought process of abstraction commonly seen in the discipline of mathematics. Labeling to share, circling to group are systematic methods that the design seeks to introduce to students. They are both means and ends of the intended teaching. In (D1), all these intertwine with the contents to be delivered. Thus the design also passes (G3).

It is pretty obvious that the effectiveness of any instructional design could not be scrutinized without considering also the detailed classroom implementation plan, which is exclusively the work of the teacher. Before the lesson, the teacher fills in all the details of teaching, including but not limited to the design of worksheets, the preparation of any possible demonstration or group work and materials required, the decision to insert suitable consolidation activities at suitable time point, the precise sequencing of all teaching activities and time allocation etc. TFM considers the following points worth extra attention:

(G4) Is the detailed instructional design feasible in the classroom context?

(G5) Are there measures to cope with individual differences?

(G6) Does the detailed instructional design take care of developing appropriate language?


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Before actual implementation, teachers exercise their judgment to examine how feasible the detailed instructional design is. In TFM, the teacher actually teaching the class is the one to fill in the details. He or she is the first one to ensure reasonable feasibility. Very often, the detailed design prepared is passed to another teacher experienced with TFM for second opinion. This is the second mechanism to ensure feasibility. We must not forget that with serious feasibility problem, nobody will ever be implementing the design! Thus, in a long run, the more teachers are willing to implement the design (or a close copy of it), the higher feasibility could be projected. From experience in Hong Kong, (D1) is gaining popularity. It was tested in many different classrooms, with promising effects. Nevertheless, it occasionally happened that certain ideas, whose feasibility was challenged by teachers, did not ever get down to implementation level. Guideline (G4) could be seen as a measure to eliminate any possible gap between theory and practice.

Individual differences can cause frustration. With some students getting fast ahead and bored, while others are still far behind the teaching, is a big headache to teachers. TFM is cautious on this, though there is no panacea. In (D1), the hypothetical learning trajectory has no steep slope. The whole process is gradual and evolving. It takes about eight teaching periods, 35 minutes each, to finish. If solving application problems is included, not more than 12 periods are needed. Field observation indicates that even the weakest students could learn effectively, though the speed and accuracy of computation may vary a bit over the class (of size around 30). Indeed most teachers working on (D1) did not encounter significant pressure arising from individual differences. They considered their detailed designs passed (G5).

Developing appropriate language has a high priority in TFM. Students are expected to be able to use a few simple but mathematical terms, phrases, or sentences after learning a topic. In the case of (D1), the statements in Appendices 1A, 1B, 2A, and 2B are typical ones. Descriptions of different parts of the column division in Appendix 5 are other examples. In sum, (G6) is handled with care, even at the framework level.

The instructional design alone does not define the whole teaching course. Teacher’s on spot interaction with students is crucial in bringing out any intended effect of the design. TFM consider the following questions require careful study:

(G7) Does the teacher manage to bring out main points in a fast, accurate, and clear manner?

(G8) Does the teacher manage to develop appropriate language via sufficient demonstration and feedback?


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(G9) Is the cultivation of good habit and attitude given high priority?

Most Hong Kong schools adopt a subject teacher system. Teaching periods normally last for 35 to 40 minutes. When the bell rings, the lesson must be suspended till the next meeting, which is often one day later. This is in contrast with what we see in many western countries, and has posted additional constraints on classroom teaching. Indeed, teachers must bring out main points quickly, and allow time for necessary consolidation before class dismisses. Here comes (G7), which could be seen as a consequence of local policy.

Development of mathematical language takes more than just its inclusion in worksheets. Demonstration by teachers is of utmost importance. TFM requires teachers to use precise and concise mathematical language, regard the development of mathematical language as part of the core of teaching, and be able to provide appropriate feedback to students who are learning to use it. (G8) focuses on how the teacher performs in terms of language development of students. The underlying belief is that students needs training on the use of mathematical language before they could use it to discuss mathematics with other people. Lacking such training, students may not know how to ask questions, respond to challenges and queries, or consult other people when they have problems with their own study.

Finally, (G9) examines the culture building aspect of the whole teaching. Does the teacher bring forth values that could promote good learning habits and attitudes? Are the lessons managed in ways that facilitate the formation of good habits and attitude?

(G1) to (G9) form a set of guidelines by which teaching of mathematics could be evaluated. On one hand, they carry the cultural beliefs held by the local community summarized in a previous section. On the other hand, they collectively reflect the core values carried by TFM. Unlike many approaches of teaching research, students’ achievement in assessments is not the only measure of the effectiveness of teaching under TFM. Admitting that there are many variables not under the control of the teacher, TFM emphasizes the building of a ‘right’ image of mathematical activity among students instead of looking for short-term rise of assessment score. Downplaying the importance of quantitative indicators results in the phenomenon that only a handful of numerical data are present in reports of TFM field test. Could we arrive at any meaningful conclusion in this way? The answer could be yes or no, depending on what constitutes a conclusion. With (D1) above, the conclusion of any individual implementation could be: “(D1) works for my class in the


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sense that students were more engaged as compared with previous classes, and were able to submit classwork and homework with significantly higher accuracy as compared with previous classes.”

Can TFM research be supported by more solid quantitative data? The answer is yes, if it is done with a huge amount of research grant, substantial enough to save every teacher the trouble of collecting data for the preparation of a rigorous academic paper. It would not surprise anybody if such a grant is not secured. “Experience shows that funding bodies and policy makers often set priorities in research which favor certain regions (e.g. educational management) and marginalize others” (Tsatsaroni, 2006, p. 190). Schoolteachers are very busy, and their job does not require them to publish. Since teachers work on TFM to convince, at the first place, themselves, data collection to convince other people is not necessary. When a teacher engages in the work of TFM, they will soon see for themselves how effective their teaching is or could be. If TFM does bring about promising improvement to teaching and learning, more and more teachers will participate. We call these scattered, unorganized individual attempts “local” proof (Lewis, Perry, & Murata, 2006). In essence, while (G1) to (G9) confirm that the instructional designs created meet the standard of sound mathematical activity for the relevant age group, the degree of adoption of these designs signifies, though indirectly, the effectiveness of these designs in helping teachers from different schools to uplift the academic achievement of their students. If TFM could be put under the category of research, the validation of research results could take years. When more and more qualitative reports surface, and concur to roughly the same findings and conclusions, research results are then considered validated. It is only at this point that quantitative methods have a role to play.

Qualitative research is gaining ground, but we are still far from the point where mathematical methods can add a finishing touch to qualitative knowledge, and many researchers are even farther from the insight that mathematics is not able to do more than just this. But it is hard to fight the prevailing superstition that ready made mathematics can solve all problems. (Freudenthal, 1991, p. 152)

In Hong Kong, conjecture (C1) and (C2) are generally considered confirmed. By requiring students to orally describe what they are doing at the same time when they put down symbolic record (which at the end turns into column division), (D1) confirms (C1) by making students learn the algorithm well, as evidenced by high computation accuracy and long retention time. (A1) overcomes the difficulty of not enabling students to take action to subdivide a rectangular strip into arbitrary equal parts (below 20 of course) according to the denominator. Evidences confirm (C2) because with (A1), students are observed to be cautious enough to first divide according to the denominator, and then multiply by the numerator.


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Concluding remarks

What is this paper for? If you stick to a high standard of academic rigor, it is definitely not a research paper. It only tells how a group of people tackles problems of teaching elementary mathematics in Hong Kong. I cannot even tell the size of this group because some work closely together, and some work in isolation without much interaction with other people. Some borrows ideas of TFM and generate their own ways of teaching. TFM does not attempt to solve problems of mathematics teaching in general. At best, it helps to shed light on how Hong Kong elementary mathematics teachers could improve the quality of their teaching. Teachers learn the designs and the underlying mathematical principles and methods. Then they proceed to tackle their own problems, not the grand problems of the mathematics education arena in broad terms.

Who is this paper for? For average teachers, they need more details. Even teaching materials are much needed. In this paper, I include just the instructional design framework of (D1), not the fine implementation details. Thus if the reader is not a teacher, he or she may have difficulty figuring out the whole idea. However, for an experienced teacher who is much concerned about (Q1) or (Q2), this paper is enough for him or her to fill in the details and tackle his or her own problem.

Now I come to my last point. If a paper aims at improving teaching, should it be evaluated by the general standard of academic rigor? Or should it be evaluated by professional relevance? It may not be easy to strike a good balance between the two. For instance, in academic work, one concerns very much about originality. However, for teaching, one can hardly identify the first one to exercise that particular idea of teaching. With that many years of history of school education in different parts of the world, it’s next to impossible to do a comprehensive literature search, given that a significant amount professional knowledge of teaching is not systematically documented. More importantly, professional teachers do not care much about who is the first one to do it! So long as an idea is applied to a new setting, there is certain element of the attempt that is not a mere replication of previous work. If we are serious about developing the professional competence of teachers through publication of a journal, we should carefully consider who should be qualified to be the ‘peers’ in a peer review system.


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References

Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, Netherlands: Reidel. Freudenthal, H. (1991). Revisiting mathematics education : China lectures. Dordrecht ; Boston:

Kluwer Academic Publishers. Heid, M. K., Middleton, J. A., Larson, M., Gutstein, E., Fey, J. T., King, K., et al. (2006). The

challenge of linking research and practice. Journal for Research in Mathematics Education, 37(2), 76-86.

Kennedy, M. M. (1997). The connection between research and practice. Educational Researcher, 26(7), 4-12.

Leung, F. K. S. (2001). In search of an East Asian identity in mathematics education. Educational Studies in Mathematics, 47, 35-51.

Lewis, C., Perry, R., & Murata, A. (2006). How should research contribute to insturctional improvement? The case of lesson study. Educational Researcher, 35(3), 3-14.

St. Clair, R. (2005). Similarity and superunknowns: An essay on the challenges of educational research. Harvard Educational Review, 75(4), 435-453.

Stigler, J. W., & Hiebert, J. (1999). The teaching gap : best ideas from the world's teachers for improving education in the classroom. New York: Free Press.

Tsatsaroni, A. (2006). Mathematics and science education research against the audit culture. International Journal of Science and Mathematics Education, 4(2), 187-193.

Wittmann, E. C. (1995). Mathematics education as a 'Design Science'. Educational Studies in Mathematics, 29(4), 355-374.

Wittmann, E. C. (2001). Developing mathematics education in a systemic process. Educational Studies in Mathematics, 48(1), 1-20.


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Appendix 1A

Divide the counting cubes in the bag equally among 4 persons, how many could each get?

Record the process using the following table:

Divide among ____ persons, when each gets ____ piece(s), ____ pieces are distributed. The left

behind (could/could not) allow each person to get at least 1 piece more. Thus we (need/don’t

need) to continue.



need) to continue.



need) to continue.



need) to continue.



need) to continue.



need) to continue.



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need) to continue.

Result: Each person gets ____ piece(s), ____ piece(s) remain.


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Appendix 1B

Group the counting cubes in the bag into groups of 3, how many groups could be made?

Record the process using the following table:

In groups of ____, when ____ group(s) is/are made, ____ pieces are used. The left behind (is/is

not) sufficient for making at least 1 group more. Thus we (need/don’t need) to continue.













Result: ____ group(s) is/are made, ____ piece(s) remain.


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Appendix 2A

11 pieces of sweet are divided equally among 4 persons. How many does each get and how many remains?

Distribute step-by-step the following sweets to the 4 persons A, B, C, and D by labeling the sweets accordingly. Then fill in the records on the right.

Divide among ____ persons, when each gets ____ piece(s), ____ pieces are distributed. The left behind (could/could not) allow each person to get at least 1 piece more. Thus we (need/don’t need) to continue.






Result: Each person gets ____ piece(s) of sweet, ____ piece(s) remain.


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Appendix 2B

t into groups of 5. How many groups can be made and how many rem

tep by circling into groups of 5. Then fill in the records on the right.

In groups of ____, when ____ group(s) is/are made, ____ pieces are used. The left behind (is/is not) sufficient for making at least 1 group more. Thus we (need/don’t need) to continue.

12 pieces of sweet are puains?

Group the following sweets step-by-s







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In groups of ____, when ____ group(s) is/are made, ____ pieces are used. The left behind (is/is not) sufficient for

continue. making at least 1 group more. Thus we (need/don’t need) to

Result: ____ group(s) of sweet is/are made, ____ piece(s) remain.




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Appendix 3




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VOL 3, N2

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Appendix 4




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VOL 3, N2

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Appendix 5




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Students’ Perceptions of Teaching Styles in Mathematics Learning Environments

Gerunda B. Hughes Howard University

Abstract: The use of interactive assessment strategies along with interactive instructional strategies in order to enhance student learning makes good educational sense. In fact, the two are inextricably linked to one another. The definition formative assessment, for example, contains many “actions” that students and teachers can take independently and collaboratively during the instructional process. The actions of the students and teachers produce feedback that is used to make adjustments either in teaching, in learning or in both and thereby, create successful interactive learning environments.




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INTRODUCTION

An effective mathematics learning environment is one in which students and teachers interact in ways that allow students to have an opportunity to maximize how much they learn. There are a variety of ways in which students and teachers interact in a learning environment. Some interactions result in student learning, however, others have very little effect on student learning. Classroom discussions, teacher and student initiated questions, cooperative group work, peer tutoring and a host of other feedback systems such as assignments, examinations and electronic response systems such as the personal response system (PRS) and the personal data assistant (PDA) are instructional strategies that provide a measure of two-way communication in which information about what is taught and what is learned is exchanged between two people. On the other hand, there are instructional strategies in which students sit passively in classrooms where there is one-way communication – from teacher to students. On many college and university campuses, for example, the professor operates as the proverbial “sage on the stage” and the didactic lecture is the modal way of teaching. And although the lecture is an efficient method for transmitting information from a teacher to a large group of students, telling information to someone does not mean that learning takes place. In order to determine whether learning is occurring – in fact, to ensure that learning is taking place, there must teacher-student assessment interactions along with the instructional interactions.

Assessment interactions between students and teachers occur when teachers gather information about student learning and use that information help students better understand concepts and principles and apply knowledge, not just learn facts. This type of assessment interaction referred to as formative assessment is defined as follows: Formative assessment is a process used by teachers and students during instruction that provides feedback to adjust ongoing teaching and learning to improve students’ achievement of intended instructional outcomes (Council of Chief School State Officers, 2008). It is clear from this definition that formative assessment is a process. It is a process that may employ tests or various other types of assessments, but it may also employ interactive instructional strategies such as classroom discussions, assignments, homework, quizzes, projects, investigations, electronic response systems or oral questions to gauge and improve student learning (Angelo & Cross, 1993; Fennell, 2006).

Creating an interactive learning environment inside the mathematics classroom in which students are engaged in mathematics learning can be challenging. Students may experience discomfort about their own level of mathematics content knowledge and may shy away from participating openly in class discussions or responding to teachers’ oral questions. Further, the complex negotiation of teacher talk, student talk, and classroom dynamics while remaining on-task requires certain skills and know-how. In some models of “best practices” in mathematics teaching and learning, these classroom dynamics are viewed as a social endeavor (Cobb & Bauersfield, 1995) in which the classroom functions as a learning community where thinking, critiquing, debating, disagreeing and agreeing are encouraged. When these dynamics work well, the result can be the creation of a learning environment in which critical thinking and quantitative reasoning develop, student learning thrives, and students take increasing responsibility for their own learning.

This paper discusses the various ways in which interactive mathematics classrooms can be created with the use of formative assessment strategies. In addition, it will report on students’




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responses to a questionnaire in which students enrolled in several sections of a pre-calculus course were asked to share their opinions about current and preferred teaching styles.

The Interactive Learning Environments and the Role of Feedback

According to Motani and Garg (2002), a successful learning environment is one in which students and teachers interact easily, continuously and without any inhibitions. In this type of learning environment, student learning is not left to chance; rather, teachers know whether their students understand intended concepts. The key to this success is the implementation and use of an instantaneous feedback system. Instantaneous feedback enables teachers to intervene immediately when students misunderstand a concept or principle which is important in meeting the learning objective. A teacher may have to adjust a teaching strategy, provide different examples or offer alternative explanations. In making these adjustments, teachers show that they recognize and appreciate that previous attempts at teaching the concept or principle were not effective. Furthermore, making adjustments in teaching instantaneously with the aim of reaching all students, and especially less successful students, leads to improved learning for all students (Guskey, 2003).

Mathematics teachers can use several strategies to get and give feedback about how well students are learning material that is being taught. Motani and Garg (2002) observe that there are electronic and non-electronic mechanisms for getting feedback. Non-electronic mechanisms may include class discussions, cooperative group work, board-work, seat-work or answering questions that are posed orally. While these interactive strategies are effective, a major shortcoming is that at any particular time, only a subset of the students in the class are actively providing information to the teacher about their learning and are receiving feedback from the teacher. Class discussions are a good example of how an interactive strategy can work for some, but not all students. Silverthorn (2006), a professor or physiology, noted, “There was always a group of students, usually sitting at the front of the room, who would answer questions and talk to me as if we were chatting in my office, while the remainder of the class sat passively at the back [of the room] and listened and took notes” (p. 136). The students who sat at the front of the room benefited from the verbal interactive exchange with the professor. The students in the back of the room did not benefit from the exchange of information about their learning with the professor. Therefore, in order to engage more students in the interactive activities in the class, Silverthorn (2006) suggests that professors employ more student-talk, less teacher-talk and more class time for problem-solving activities.

Even when teachers employ interactive assessment strategies such as assignments or examinations to determine what and how much students have learned, care must be taken so that these strategies are effective in improving student learning. One reason that care must be taken is because the feedback to students from teachers is often delayed – that is, the feedback to students does not occur during the instruction. When students respond to questions on an assignment or examination, they may not get feedback for several days or weeks. Thus, by the time they receive feedback, they may have moved on to “learning” new content. If understanding of the new content is dependent on understanding of the old content, and if there were misunderstandings of the “old” content that were not addressed immediately when it was presented, then the cumulative effect of misunderstandings coupled with no corrective feedback could put students at risk of underperformance or even failure. A second reason is that students generally focus on doing what is necessary to get the highest grade possible on an assignment. Strategies used by students in this




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context may result in very little learning. Should either of the above scenarios exist, the goal of improved student learning would be compromised.

Over the last 25 years, technological advances have provided opportunities for creating interactive mathematics learning environments using electronic mechanisms. Some examples of electronic mechanisms are personal response systems (PRS) and personal data assistants (PDA). These e-mechanisms allow interaction between fellow students and students and teachers and enable teachers to provide instantaneous or immediate feedback. For example, the PRS allows students to respond privately (and anonymously) to questions posed by the teacher during instruction. The student responses are collected, analyzed, summarized and displayed as a histogram. The teacher is able to use the results to make adjustments in teaching in real time and correct any misunderstanding of concepts or principles among students. “Low tech” strategies that accomplish similar interactive learning objectives as the “high tech” PRS and PDA are colored flash cards and quiz games. Notably though, PRS and PDA provide students who would otherwise be reluctant participants in an open interactive classroom with a safe and non-threatening way to participate in classroom activity and still provide the teacher with information about their learning that will ultimately help themselves and others maximize learning of intended instructional objectives.

The Interactive Lecture: An Oxymoron?

Research has shown consistently that traditional lecture methods in which professors talk and students listen dominate college and university classrooms. Nevertheless, for at least sixty years, and particularly during the current standards movement in education, there has been considerable effort in identifying more effective methods and procedures to enhance learning (Tyler, 1949; Hake, 1998; Robinson & Maceli, 2000; Webb, 2003; Morton, 2007).

In the 1940’s, Tyler (1949) noted that learning takes place through the active behavior of the student. According to Tyler, it is not what the teacher does, but through what the student does that learning takes place. That said, (Mazur, 2009) suggested that a modification of traditional lectures is one way to incorporate active learning in the classroom. For example, if a faculty member allows students to consolidate their notes by pausing three times for two minutes each during a 60-minute lecture, students will learn much more information (Silverthorn, 2006).

Several alternatives to the lecture format not only increase student achievement but also raise levels of student engagement in the learning activity. For example, the feedback lecture consists of two mini-lectures separated by a small-group study session built around a study guide. In the guided lecture, students listen to a 20- to 30-minute presentation without taking notes, followed by their writing for five minutes about what they remember, and then spending the remainder of the class period in small groups clarifying and elaborating the material. Morton (2007) calls the latter third of the guided lecture the “active review.” Students can also become involved during a lecture by completing short, un-graded exercises followed by class discussion.

Clearly, there are a variety of teaching methods and styles that can be used in mathematics classrooms. They range from mostly interactive to mostly lecture. Information gathered from students can provide incite into the degree to which teaching styles are interactive and the effect of those styles on their learning. The purpose of this study was three-fold: (1) to determine if there were differences in students’ perceptions of the amount of interaction that occurred between themselves and their teachers across sections of a pre-calculus course; (2) to report on the type of




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teaching style students experienced in their last mathematics course, and the type of teaching style students would prefer in the next mathematics course if they were to enroll in one in the future; and (3) to examine the relationship between students’ perceptions of the amount of interaction in the teaching style of their pre-calculus course and the extent to which that teaching style helped them understand concepts taught in the course.

METHODOLOGY Participants The sample consisted of 117 students who were enrolled in three different sections of a pre-calculus course during the spring semester. At the beginning of the semester, enrollment in the three sections was approximately the same. The different sections of pre-calculus were randomly assigned to experimental and control conditions. Two instructors were assigned to the experimental condition and one was assigned to the control condition. By the end of the semester, 101 students were enrolled in two experimental sections and 16 students were enrolled in the control section. There were 41 males and 74 females. Two students did not identify their gender. Experimental vs. Control Sections of Pre-calculus

Instructors who were assigned to the experimental sections of pre-calculus employed a teaching style that incorporated frequent use of questioning that was designed to engage students, provide them with greater opportunities to learn, and improve their understanding of concepts. This teaching style also involved more examples in real world contexts. Longer “wait times” were embraced and most questions sought to determine if students had developed conceptual understanding of concepts rather than just knowledge of simple facts. In contrast, the instructor who was assigned to the control section employed a traditional lecture approach to teaching pre-calculus. Teaching Style Questionnaire At the end of the semester, students completed a seven-item questionnaire in which they were asked to respond to questions about their perceptions about the amount of interaction characterized by the teaching style of various mathematics courses – past, current and future. In particular, the questions were grouped according to the teaching style of the instructor for the current course, the teaching style of the instructor for the mathematics course taken previously, the students’ preferences for teaching style in a future mathematics course, and their perceptions about the extent to which the teaching style employed in the mathematics course helped them to understand the concepts that were taught (See Appendix). At the time the questionnaire was administered, all instructors were teaching the same mathematical content--in fact, all were at the same point in the syllabus.




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RESULTS

Null Hypothesis 1: “There are no differences in students’ perceptions of the amount of interaction that occurred between themselves and their teachers across experimental and control sections of a pre-calculus course.”

Descriptive statistics were used to calculate the percent of students who responded on a continuum from “strongly disagree” = 1 to “strongly agree” = 5 for Item 2 which states: “The teaching style in my (current) pre-calculus course is mostly interactive.” The results indicate that among students enrolled in the control section of pre-calculus, 25% “strongly agreed” or “agreed” that the style of teaching was interactive; whereas, among students enrolled in the experimental sections, 74.3% “strongly agreed” or “agreed” that the style of teaching was interactive.

To test the null hypothesis of no differences in students’ perceptions about the amount of interaction that occurred between themselves and their teachers across the control and experimental sections, a one-way analysis of variance was performed on the sum of the items that dealt with lecture (Item 1) and interactive styles (Item 2), with corresponding contrasts to test the differences in means between the combined experimental sections and the control sections. The results revealed that (1) there were significant differences in the teaching styles among the three pre-calculus instructors (F= 9.36, df = 2, 114, p = .0002), and (2) when the combined means of the experimental sections were compared with the control section, the difference was significant (t = 4.28, df = 114, p < .0001). The results indicate that the teaching style in the experimental sections was significantly more interactive than in the control section. Thus, the null hypothesis of no differences in teaching styles among the (current) pre-calculus instructors was rejected. Null Hypothesis 2: “There are no significant differences in students’ perceptions of the teaching styles employed by instructors in students’ last (previous) mathematics course.” Sub-hypothesis 2.1: “There are no differences in the proportions of students across control and experimental groups who prefer ‘at least some interaction’ in the teaching style of their future (next) mathematics instructor.”

To get a sense of the style of teaching the students had experienced just prior to their enrollment in the pre-calculus course, the students were asked to respond on a continuum from “strongly disagree” = 1 to “strongly agree” = 5 to Item 6 which states: “The teaching style of the instructor for the last mathematics course was mostly interactive.” Descriptive statistics were used to calculate the percent of students who responded at each level on the continuum. Among students in the control section, 27.5% “strongly agreed” or “agreed”, 18.8% were “unsure”, and 43.8% “strongly disagreed” or “disagreed”. Responses from students in the experimental sections indicated that 38.8% “strongly agreed” or “agreed”, 12.9% were “unsure”, and 58.5% “strongly disagreed” or “disagreed”. A one-way analysis of variance was performed to test the hypothesis of no differences among the means of the three classes. The results indicate that there were no significant differences (F = 1.54, df = 2, 113, p>.05). Thus, regardless of whether the students were enrolled in the experimental or control sections, they perceived that the teaching style of the instructor in their previous mathematics course was “mostly lecture.” Thus, the null hypothesis of no differences in the teaching styles of instructors of the students’ last or previous mathematics course was supported.

To get a sense of their preferred teaching style, and address Sub-Hypothesis 2.1, students were asked Item 7 which states: “If you were to enroll in a mathematics course in the future, which




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style of teaching would you prefer?” Students selected among: “mostly lecture,” “some lecture, some interaction,” and “mostly interactive.” Of all students surveyed, 98.3% preferred a teaching style that has “some interaction” or is “mostly interactive.” Interestingly, among the students in the control section, 100% responded that they would prefer some type of interaction in a future mathematics class. Among the students in the experimental sections 98% responded similarly; hence, there was no difference in the proportion or percent of students across control and experimental sections who prefer some type of interaction in their next mathematics instructor’s teaching style. Null Hypothesis 3: There is no relationship between students’ perceptions of the amount of interaction in the teaching style of their current pre-calculus course and their perceptions of the extent to which that teaching style helps them understand the concepts taught in the course.

Students were asked to respond on a continuum from “strongly disagree” (1) to “strongly agree” (5) to Item 3 which states: “The teaching style of the instructor for this mathematics course helps me understand the concepts taught in the course.” Students enrolled in the control section of pre-calculus, 31.3% “strongly agreed” or “agreed”, 12.5% were “unsure”, and 56.3% “strongly disagreed” or “disagreed.” For students in the experimental sections, 81.1% “strongly agreed” or “agreed”, 8.9% were “unsure”, and 8.9% “strongly disagreed” or “disagreed.”

The results also reveal that there is a significant correlation (r = .37, p< .01) between the students’ perceptions of the extent of interaction in the teaching style employed in the current pre-calculus course and their perceptions of the extent to which the teaching style in the course facilitated understanding of concepts. This correlation coefficient indicates that the more interactive the teaching style in their pre-calculus course, the more that teaching style helped them understand the concepts taught in the course.

To test the hypothesis that there are no differences among the three pre-calculus sections in students’ perceptions about the extent to which the teaching style of the mathematics course helps the student understand the concepts taught in the course, a one-way analysis of variance was performed with a corresponding contrast to test the difference in means between the combined experimental sections and the control sections. The results reveal that (1) there were significant differences in the students’ perceptions of the effectiveness of the teaching style in facilitating understanding of concepts taught in the course (F = 13.66, df = 2, 113, p< .0001), and (2) when the combined means of the experimental sections were compared with the control section again the difference was significant (t =5.12, df = 113, p<..0001). The results indicate that students in pre-calculus felt that a teaching style that is” mostly interactive” facilitates understanding of concepts to a greater degree than a style of teaching that is “mostly lecture.” Finally, when asked if the teaching style in the last mathematics course facilitated understanding of concepts, there was a significant correlation (r = .58, p< .01) between the students’ perceptions of the extent of interaction in the teaching style employed in the last mathematics course taken by the students and their perceptions of the extent to which the teaching style facilitated understanding of concepts. The results also revealed that there were no differences in the mean responses of the three classes (F = 1.43, df = 2, 114, p>.05). Additionally, among the students in the control section 43.8% “strongly agreed” or “agreed”, 12.5% were “unsure,” and 43.8% “strongly disagreed” or “disagreed” that the teaching style in the last mathematics course helped them understand the concepts being taught. Among the students in the experimental sections 44.5% “strongly agreed” or “agreed,” 10.9% were “unsure,” and 44.6% “strongly disagreed” or




8

“disagreed.” When these results are considered along with the results of Null Hypothesis 2, we find that less than half of all students “strongly agreed” or “agreed” that a teaching style that is “mostly lecture” helps them understand concepts that are being taught in their mathematics course.

DISCUSSION

Interest in improving the quality of mathematics teaching and learning in American primary

and secondary schools and in colleges and universities has grown tremendously in recent years. Part of the growth is due to a realization that the world is getting “flatter” (Friedman, 2006) and that a key to remaining competitive in the midst of globalization is to have a mathematically literate citizenry. The making of a mathematically literate citizenry will not happen by chance or overnight. Without an instructional focus on teaching for understanding, students are at risk of viewing mathematics as a collection of rules and procedures to be memorized, regurgitated and eventually forgotten. Teaching mathematics for understanding, on the other hand, engages students more fully in the learning process by making use of interactive assessment and teaching strategies (Silver, et al, 2009).

Learning and teaching are iterative processes that ideally continue until a desired goal has been reached. In the interactive learning environment, there are many strategies that can be used to produce feedback that can be used to reach the goal. Before the information is used, however, a “gap analysis” is performed to determine the “next steps”. Feedback fills the gap iteratively until the goal is met. The feedback might come instantaneously with the aid of e-mechanisms such as the PRS or PDA or it may be delayed when more traditional methods of assessment are used such as assignments or examinations. In either case, the interactive mechanism might engage a few students at a time for example through class discussions or cooperative groups or it may allow form maximum participation through the e-mechanism.

The use of interactive assessment strategies along with interactive instructional strategies in order to enhance student learning makes good educational sense. In fact, the two are inextricably linked to one another. The definition formative assessment, for example, contains many “actions” that students and teachers can take independently and collaboratively during the instructional process. The actions of the students and teachers produce feedback that is used to make adjustments either in teaching, in learning or in both and thereby, create successful interactive learning environments. For students, that information can come from teachers, other students or from the student himself. Interacting with teachers or other students to get feedback about a task helps a student to remain on target toward the desired goal. Interacting with oneself in the assessment/instructional process builds self-monitoring, self-reflection, self-assessment, and self-regulation skills. When students develop these “self-” skills, they become independent, life long learners.

Comfort level with certain types of interactive assessment mechanisms may be generational-based. If the learning environment is made up of millennials (ages 27 and younger), there is a good chance that they have grown up with technology. Millennials make up the general population of undergraduates and graduates on most college and university campuses. For this group, education through technology is the norm (Eshleman, 2008). On the other hand, if the learning environment consists primarily of traditionalists (over age 60) or boomers (ages 48 to 62), they expect their educational experiences to resemble the didactic lecture format and may view the ubiquitous




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PowerPoint presentations as “high-tech.” They may also feel threatened by interactive Web-based or Internet technologies. Still, they are good at leading cooperative groups. Finally, there are the GenXers (ages 28 to 47). They, like the millennials, are generally comfortable with technology and are good at thinking outside the box (Eshleman, 2008). A successful interactive learning environment will meet the needs of all of these diverse components of the general population. As the twenty-first century unfolds, the United States is faced with an extraordinary challenge – that of providing a quality education for all of its diverse citizens. The diversity of its citizenry is what makes it unique; but failure to nurture, appreciate and educate that citizenry will mean the lost of valuable human capital – capital the United States cannot afford to lose. Education is the tool for developing that human capital. Furthermore, making sure that learners actually learn with understanding regardless of gender, socio-economic status, race/ethnicity, or even generational membership may be a challenge, but it is one that can be overcome successfully. By implementing effective interactive assessment and teaching strategies, educators can ensure that all citizens are able to make meaningful contributions to their local, national, and the global communities.




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REFERENCES Angelo, T. A., & Cross, K. P. (1993). Classroom assessment techniques: A handbook for college

teachers (2nd ed.). San Francisco: Jossey-Bass. Cobb, P., & Bauersfield, H (1995). Introduction: The coordination of psychological and

sociological perspectives in mathematics education. In P. Cobb & H. Bauersfield (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 1-16). Hillside, NJ: Lawrence Erlbaum Associates.

Council of Chief State School Officers (2008). Attributes of effective formative assessment. Washington, D.C.: Author

Eshleman, K. Y. (2008). Adapting teaching styles to accommodate learning preferences for effective hospital development. Progress in Transplantation, 18(4), 297-300.

Fennell, F. (2006). Go ahead, teach to the test! NCTM News Bulletin. Reston, VA: The National Council of Teachers of Mathematics.

Friedman, T. L. (2006). The world is flat: A brief history of the twenty-first century. New York, NY: Farrar, Straus and Giroux.

Guskey, T. (2003). How classroom assessments improve learning. Educational Leadership, 60(5), 6-11.

Hake, R. R. (1998). Interactive-engagement versus traditional methods: A six-thousand-student survey of mechanics test data for introductory physics courses. American Journal of Physics, 66(1), 64-74.

Mazur, E. (2009). Farewell, lecture? Science 2, 323(5910), 50-51.

Morton, J. P. (2007). The active review: One final task to end the lecture. Advances in Physiology Education, 31, 236-237.

Motani, M., & Garg, H. K. (2002, August). Instantaneous feedback in an interactive classroom. Paper presented at the meeting of the International Conference on Engineering Education, Manchester, UK.

Robinson, E., & Maceli, M. J. (2000). The impact of standards-based instructional materials in mathematics in the classroom. In M. Burke & E. Curio (Eds.), Learning mathematics for a new century (2000) Yearbook of the National Council of Teachers of Mathematics, pp. 112-126). Reston, VA: National Council of Teachers of Mathematics.

Silver, E. A., Mesa, V. M., Morris, K. A., Star, J. R., & Benken, B. M. (2009). Teaching mathematics for understanding: An analysis of lessons submitted by teachers seeking NBPTS certification. American Educational Research Journal, Retrieved from http://aerj.aera.net, DOI: 10.3102/0002831208326559.

Silvertorn, D. U. (2006). Teaching and learning in the interactive classroom. Advances in Physiology Education, 30, 135-140.

http://aerj.aera.net/




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Tyler, R. W. (1949). Basic principles of curriculum and instruction. Chicago, IL: University of Chicago Press.

Webb, N. (2003). The impact of the Interactive Mathematics Program on student learning. In S. Senk & D. R. Thompson (Eds.), Standards-based school mathematics curricula: What are they? What do students learn? (pp. 375-398). Mahwah, NJ: Lawrence Erlbaum Associates.




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Appendix

Student Questionnaire on Perceptions of Teaching Style and Understanding of Mathematics Concepts

Directions: For each statements in Items 1-6, indicate the extent to which you agree with the statement, where “strongly disagree” = 1; “disagree” = 2; “unsure” = 3; “agree” = 4; and “strongly agree” = 5. For Item 7, please indicate whether for a

ITEMS Strongly Disagree

Disagree Unsure Agree Strongly Agree

1. The teaching style in my (current) pre-calculus course is mostly lecture.

2. The teaching style in my (current) pre-calculus course is mostly interactive.

3. The teaching style of the instructor for this mathematics course helps me understand the concepts taught in this course.

4. The teaching style in my last mathematics course was mostly lecture.

5. The teaching style in my last mathematics course was mostly interactive.

6. The teaching style of the instructor in my last mathematics course helped me understand the concepts taught in that course.

Mostly

Lecture Some Lecture

Some Interaction

Mostly Interactive

7. If you were to enroll in a mathematics class in the future, which teaching style would you prefer?

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