fifth-grade students use and preferences for mathematically and practically based explanations...

Fifth-grade students’ use and preferences for mathematically and practically based explanations

Esther Levenson

Educational Studies in Mathematics, Vol. 73, No. 2, pp.121-142

“there is a continuum of explanations beginning with practically based explanations that use everyday concrete objects, proceeding to semi-structured manipulatives, continuing to mathematically based explanations, and ending with formal explanations” (p. 124)

Key Issue:Find the right balance between using these two types of explanations and support students in the transition from explanations via arguments and justification to proof and formal algebra.

1. Why should you read this article?

2. Overview of the article

3. Guide to the article

6. Reflective Questions

4. Tasks used in this research

5. Overview of Results

7. Suggested Task 8. Further Activity

Click on the arrow to come back to this slide

9. Relevant Literature

1. Why should I read this article?

First reflect on your own teaching experiences:What type of explanations do you often use?

Think of a recent lesson where you have explained a mathematical idea to your students. How did you do this? Write down the explanation you used. (You will need this later)

This article • presents current research on students’ use and preference of

two types of explanations.• informs teachers of issues to consider when planning their

lessons based on students’ ways of thinking and preferences in relation to odd and even numbers and equivalent fractions in particular that could apply to other topics in mathematics

• debates about whether using real-life contexts in tasks is useful and/or makes a difference to students’ ways of thinking

• gives advice to introduce students to formal mathematics as early as elementary school

2. Overview of the article

SAMPLE – fifth-grade students

RESEARCH TOPIC – Mathematically based (MB) explanations vs. Practically based (PB) explanations

CONTEXT – Even or odd numbers and equivalent fractions

WHICH PARTS SHOULD YOU READ FIRST?• Introduction: to get an idea of what the difference between MB

and PB explanations is and what the problem is and therefore reflect on why this research is important for your teaching practice

• Conclusion: to see their research outcomes, their key outcome and their advice to teachers and reflect on your own teaching

3. Guide to the paper

Introduction

Setting the scene by citing the work of others on the use of explanations used in mathematics classrooms; arguing for the idea of introducing formal concepts early; defining MB and PB explanations; stating the rational of this paper in making teachers aware of their students’ preferences and use of such explanations

Background

Based on the existing literature classify explanations; define and discuss MB and PB explanations; mention students’ evaluations of explanations based on the existing literature; other researcher’s recommendations regarding explanations

Methodology

Describe their methodology when carrying out this research, the plan, the sample, the tools and questionnaires used, the data analysis process, examples of tasks used (TASKS)

Results

Summative result regarding student-generated explanations; students’ chosen preferred explanations on the parity questionnaire and the fraction questionnaire (also compared to their level of mathematical ability). Summative result regarding what students used as the basis for their choices; when choosing PB explanations and when choosing MB explanations (also compared to their level of mathematical ability)

Summary and Discussion

Summarizes the results from their research and deriving some general advice to teachers, but also suggests issues needing research

4. Tasks

The following slides introduce the differences between mathematically based (MB) and practically based (PB) explanations for two ideas:

- 14 being even and 9 being odd

- why

Work through the explanations. Summarise the key features of a mathematically based (MB) explanation and a practically based (PB) explanation. How does this compare to your explanation that you identified in Section 1?

4. Tasks

• MB explanations for 14 being even and 9 being odd

• PB explanations for 14 being even and 9 being odd

• MB explanations for why

• PB explanations for why

4.Tasks





Tasks – MB explanations for 14 being even and 9 being odd

MB1: Divisibility by 2

14 is divisible by 2 without a remainder.

14 ÷ 2 = 70 R0

Therefore 14 is an even number.

When dividing by 9 by 2 there is a remainder of 1.

9 ÷ 2 = 4 R1

Therefore 9 is an odd number.

MB2: Sum of 2 equal whole numbers

14 may be written as the sum of 2 equal whole numbers.

7 + 7 = 14

Therefore, 14 is an even number.

9 cannot be written as the sum of two whole numbers because 4½ + 4½ = 9

4 ½ is not a whole number. Therefore 9 is an odd number.

MB3: Whole number quotient

14 divided by 2 is equal to 7. 14 ÷ 2 = 7

Because the result when dividing by 2 is a whole number, 14 is an even number.

9 divided by 2 equals 4½. 9 ÷ 2 = 4½

Because the result when dividing by 2 is not a whole number, 9 is an odd number.

4.Tasks





Tasks – PB explanations for 14 being even and 9 being odd

PB1: Pairs of students

Fourteen students from grade six went on a class trip. The teacher asked the students to form pairs. Seven pairs were formed and no student was left without a partner.

Because 14 students can be paired up, 14 is an even number.

Nine students from grade six went on a class trip. The teacher asked the students to form pairs. Four pairs were formed and one students was left without a partner.

Because 9 students cannot all be paired up, 9 is an odd number.

4.Tasks





Tasks – MB explanations for why

MB1: Twice equivalent fractions 1 3

One quarter (―) is the same as ― 4 12

2 3 6Therefore, two quarters (―) is two times ―, which is ―

4 12 12 2 6

Therefore, ― = ― 4 12

MB2: Equivalent to a halfWe will reduce both fractions: 6 6:6 1― + ―― = ―12 12:6 2

2 2:2 1― + ―― = ― 4 4:2 2

2 6 1Both fractions, ― = ― are equal to ― 4 12 2

2 6Therefore, ― = ―

4 12

4.Tasks





PB1: Chocolate explanation without illustrationMom gave David and Miri identical chocolate bars.David’s bar is divided into 4 equal pieces.David ate 2 pieces.Miri’s bar of chocolate is divided into 12 equal pieces.Miri ate 6 pieces.Each of the children ate the same amount of chocolate.

2 6Therefore ― = ― .

4 12

Tasks – PB explanations for why


PB2: Chocolate explanation with illustration

Mom gave David and Miri identical chocolate bars.

David’s bar is divided into 4 equal pieces.

David ate two pieces.

Miri’s bar of chocolate is divided into 12 equal pieces.

Miri ate 6 pieces. Each of the children ate the same amount of chocolate.

2 6

Therefore, ― = ― .

4 12

PB3: Pizza explanation without illustrationRon’s pizzeria sells different kinds of pizzas.

Each kind of pizza is the exact same size, except that they are divided up differently.

Danny likes pizza with olives.

He bought 6 slices out of a pizza pie with olives divided into 12 equal size slices.

Elie likes pizza with tomatoes.

He bought 2 slices out of a pizza pie with tomatoes divided into 4 equal slices.

Danny and Elie ate the same amount of pizza.

2 6


4 12


PB4: Pizza explanation with illustrationRon’s pizzeria sells different kinds of pizzas.

Each kind of pizza is the exact the same size except that they are divided up differently.

Danny likes pizza with olives.

He bought 6 slices out of the pizza pie

with olives divided into 12 equal size slices.

Elie likes pizza with tomatoes.

He bought 2 slices out of the pizza pie with tomatoes divided into 4 equal slices.

Danny and Elie ate the same amount of pizza.

2 6


4 12


Read pages 134 – 138

Make a list of the advantages for choosing PB explanations and the advantages for choosing MB explanations.

5. Overview of the results

For the corresponding results section in the paper click here

• All students, indifferent to their level of attainment, were capable of using valid MB explanations in both questionnaires

• PB explanations were preferred in most situations on the odd or even number questionnaire

• No preference was found for either type on the fractions questionnaire

– Reason: the differences in the context might have resulted from the different contexts. Or it maybe be the age at which a student is introduced to a concept which is reflected later on in the types of explanations preferred.

• Mathematical attainment was only related to preference on the fractions questionnaire

– Reason: perhaps as a concept becomes more familiar to students, attainment becomes less of a factor when choosing the type of explanation to use.

5. Overview of the resultspage 2

For the corresponding results section in the paper click here

When and how should MB explanations be used in the classroom? • How could teachers use familiar concepts to encourage students of medium and low

attainment level to use more MB explanations in the classroom?• Should teachers introduce and use more MB explanations when teaching students as

young as 10-11 years old?

With which type of explanation, MB, PB, or an intertwining of both, should we introduce a topic? What is the proper balance?• Should teachers use PB or MB explanations?• Should teachers choose based on the specific topic?• How might PB and MB explanations complement each other?

How could teachers promote the smooth transition from explanation to justification to proof? • When and how should teachers move from PB to MB explanations?• How would this move help them introduce the notion of proof to students?• How could teachers use the presented tasks for a constructive classroom discussion to

evaluate their students’ understanding of formal reasoning or proof in mathematics?

6. Reflective questions

Work through the task on the following slides noting the differences between the practically based explanations, mathematically based explanations and formal algebra.

Could you use this task in your classroom?

7. Suggested task

Task

There is a square pond of width 5m, which needs to be surrounded using tiles. The teacher gave to each student the following tiles of the same height 1m, but different lengths:

4 tiles of length 5m



and asked his students to surround this pond.

Steve used the 4 tiles of length 5m to surround each side of the pond and for the corners he used the 4 tiles of length 1m.

Mary used the 2 tiles of length 7m to surround the top and bottom sides of the pond and used 2 tiles of length 5m to surround the remaining 2 sides.

Steve and Mary placed their tiles in a row to compare their solutions

They both agreed that their solutions are correct. Do you agree? Are both solutions correct and why?

What is the solution for a different width for the pond?

Steve’s solution

Mary’s solution

Task

There is a square pond of width 5m, which needs to be surrounded using tiles. The teacher gave to each student the following tiles of the same height 1m, but different lengths:





Steve knows there are 4 sides of the same width so he said “I need to repeat the width 4 times and then add the 4 squares for the 4 corners”. So, he wrote 4 × width + 4.

Mary noticed that she needs 2 tiles for the top and the bottom of the pond that are 2m longer than the sides of the pond. She said I need 2 tiles of (width + 2) length. She then decided to use 2 tiles of the same length as each side of the pond. So, she wrote (width + 2) × 2 + width + width

They both managed to surround the pond, so they both found a solution. Do you agree? Which solution is correct and why?

Steve’s solution

Mary’s solution

Task

There is a square pond of width (w), which needs to be surrounded using tiles. The teacher gave to each student the following tiles of the same height 1, but different lengths:

4 tiles of length (w)

4 tiles of length (w+2)

4 tiles of length 1


Steve knows there are 4 sides of the same width so he said “I need to repeat the width 4 times and then add the 4 squares for the 4 corners”. So, he wrote down 4 × w + 4 and my solution is correct for any width.

Mary said that she needs 2 tiles for the top and the bottom of the pond that are 2m longer than the sides of the pond. Also, she used 2 tiles of the same length as each side of the pond. So, she wrote: 2 × (w + 2) + w + w.

They both managed to surround the pond, so they both found a solution. Do you agree? Which solution is correct and why?

Is 4w + 4 = 2(w + 2) + w + w True or False?

Steve’s solution

Mary’s solution

Reflection:What are the implications of this task for your classroom practise? Pick one or more of these bullet points to work on in the coming

weeks:• Consider introducing algebraic concepts and formal mathematics earlier; perhaps

through linking these concepts to simple tasks you have already used in your classroom

• By using the tasks presented in this paper you could lead a classroom discussion to explore which explanations are valid, formal and mathematically correct.

• You could generalise from the presented tasks in this paper and devise similar explanations regarding other topics.

• You could lead classroom discussions that would shed light on your students’ ways of thinking based on their preferences and reasoning skills when faced with such tasks and introduce them gradually to formal mathematics.

• Read other research which focuses on introducing 10-11 year old students to algebra

8. Further activity

Research papers

Goldenberg, E. P., Mark, J. and Cuoco, A. (2010). The algebra of little kids; a Mathematical-habits-of-mind perspective on elementary school. Education Development Centre, Inc.

Stylianides, A. J. (2009). Breaking the Equation. Mathematics Teaching Incorporating Micromath, n213 p9-14.

Carraher, D.W., Martinez, M., & Schliemann, A.D. (2008). Early algebra and mathematical generalization. ZDM – The International Journal on Mathematics Education (formerly Zentralblatt fur Didaktik der Mathematik), 40(1), 3-22.http://ase.tufts.edu/education/earlyalgebra/publications/2008/mathGeneralization.pdf

8. Relevant literature

Abstract

Abstract

Abstract

8. Relevant literature

Books

Kaput, J., Carraher, D. & Blanton, M. (2008) (Eds.). Algebra in the Early Grades. Mahwah, NJ, Erlbaum

Greenes, C. E. & Rubenstein, R. (2008) (Eds.). Algebra and Algebraic Thinking in School Mathematics. National Council Teachers of Mathematics.

Schliemann, A.D., Carraher, D.W., & Brizuela, B. (2007). Bringing Out the Algebraic Character of Arithmetic: From Children's Ideas to Classroom Practice. Lawrence Erlbaum Associates.

Asking “When should algebra be taught?” is like asking “Is technology harmful or helpful?” There are lots of technologies and lots of uses of them. Some are harmful; some are helpful. Refining the question—asking about a particular use of a particular technology for a particular purpose in particular contexts and at particular stages in one’s learning— makes the question researchable and potentially answerable. Similarly, there are many “algebras”—algebra the course, algebra the discipline, algebraic ideas, algebraic language, early algebra, “patterns, functions, and algebra”—and many different takes on the learning and teaching of each of these. Treating algebra as an indivisible whole obscures the options. It’s more useful to ask what ideas, logic, techniques, and habits of mind algebra entails, and then, about each of these, ask when and to what extent that one item can be learned with intellectual integrity and how a coherent whole can be woven out of these learnings. The answers we get are that some of these ideas do have to wait for eighth or ninth grade, but that others—even including aspects of algebraic language—are already there, early in the primary grades. Moreover, children who get to apply, refine, and strengthen those ideas and skills as they emerge gain the advantage.

The algebra of little kidsGoldenberg et al 2010

Continued over

Any credible claim about habits of mind must surely accord with features of mind: children’s cognitive development. For a charmingly written scientific account of the ways that babies and young children think, read The Scientist in the Crib, by Gopnik, Meltzoff, and Kuhl (2000). The habits of mind approach to curriculum that we first described well over a decade ago (Cuoco, Goldenberg & Mark, 1996; Goldenberg, 1996) and have continued to refine (Goldenberg & Shteingold, 2003 and 2007; Cuoco, Goldenberg & Mark, 2009; Mark, et al., 2009) does accord well with children’s thinking and became a central design principle behind Think Math! (2008), the newest NSF‐supported elementary curriculum, developed at EDC.

Recognizing, enhancing, and building on developmentally natural habits of mind lets us dissect algebra and sort the resulting bits and pieces in a developmentally natural way, while preserving the content, concepts, and skills that schools (and states, parents, workplaces, and colleges) expect. The fact that it is possible to organize algebraic ideas, logic, and techniques around the development of mind makes clear that we are truly talking about thinking—habits of mind—rather than “features of mathematics” or “idiosyncrasies of mathematicians.” This article describes two of these natural habits of mind.

The algebra of little kidsGoldenberg et al 2010

Relevant literature

Breaking the equation Stylianides, A.J. (2009)

A proof's potential to promote understanding and conviction is one of the main reasons for which proof is so important for students' learning of mathematics. Unless students realise the limitations of empirical arguments as methods for validating mathematical generalisations, they are unlikely to appreciate the importance of proof in mathematics. In this article, the author describes working with students on a journey to appreciate the difference between being convinced and "proof". The author describes and discusses a mathematics lesson in a high-attaining Year 10 class that aims to help the students begin to realise the limitations of empirical arguments. The lesson is an adapted version of one developed by a research project in the context of a university course. The lesson is approximately 60 minutes long and is taught over two consecutive 45-minute periods. It involves three activities: (1) the Squares Problem; (2) the Circle and Spots Problem; and (3) the "Monstrous Counterexample" illustration. (Contains 5 figures.)

Relevant literature

Early algebra and mathematical generalisationCarraher et al (2007)

We examine issues that arise in students’ making of generalizations about geometrical figures as they are introduced to linear functions. We focus on the concepts of patterns, function, and generalization in mathematics education in examining how 15 third grade students (9 years old) come to produce and represent generalizations during the implementation of two lessons from a longitudinal study of early algebra. Many students scan output values of f(n) as n increases, conceptualizing the function as a recursive sequence. If this instructional route is pursued, educators need to recognize how students’ conceptualizations of functions depart from the closed form expressions ultimately aimed for. Even more fundamentally, it is important to nurture a transition from empirical generalizations, based on conjectures regarding cases at hand, to theoretical generalizations that follow from operations on explicit statements about mathematical relations.

Relevant literature

fifth-grade students use and preferences for mathematically and practically based explanations...

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