kindergarten students exploring big ideas: an evolution for teachers

Proceedings of the 40th Annual Meeting of the Research Council on Mathematics Learning 2013 101

KINDERGARTEN STUDENTS EXPLORING BIG IDEAS:

AN EVOLUTION FOR TEACHERS

Melfried Olson

University of Hawai‘i

[email protected]

Fay Zenigami

University of Hawai‘i

[email protected]

This paper describes the experiences of a team of kindergarten teachers as they came to grips

with what ideas students could meaningfully explore, represent, and communicate. We share

how teachers collaboratively planned a problem-solving lesson for their students and in the

process explored the mathematics for themselves as well. We discuss students’ work

demonstrating their engagement in the mathematics of the lesson and teachers’ reflections on the

lesson, thoughts about the significance of students’ thinking, and efforts to orchestrate

communication.

Often kindergarten teachers think they have little influence on the mathematics learning of

their children because they do not see the significance of ideas with which children can work.

This paper describes a three-year effort made by all kindergarten teachers in one school to

understand and influence their students’ thinking about mathematical ideas. We discuss a

professional development program that had a major influence on how teachers felt they impacted

the mathematical development of their students. We explain how a Lesson Study process helped

facilitate growth in teacher knowledge as they sought appropriate tasks in which to engage

students, expand teachers’ efforts to orchestrate opportunities for students to model and represent

mathematical ideas, and make visible decisions teachers made to foster discourse to benefit the

learning of all students by choosing the order in which students shared their work.

Theoretical Framework

The goal of the three-year project (KARES) funded through a Mathematics and Science

Partnerships grant from the Hawai‘i Department of Education was to promote deeper student

understanding of mathematics through professional development designed to improve teachers’

mathematics content understanding and instructional strategies. The authors were the KARES

Project Director and Project Manager, respectively, and were involved in conducting all aspects

of the professional development. Carpenter, Blanton, Cobb, Franke, Kaput & McCain (2004)

state that the two most critical things teachers need to learn are content knowledge and student

learning trajectories specific to that knowledge. Ball (2003) and her colleagues found that when

professional development for teachers focused on the mathematical content knowledge they need

for teaching, the effects are maximized. Such professional development situated in context is a


dynamic approach to learning content that incorporates the idea that content cannot be presented

in a way that is divorced from what goes on in the classroom. The objectives of the project

included developing deep mathematics content knowledge for teachers; increasing teachers’ use

of instructional strategies that include problem solving, reasoning, and communication; create

and evaluate Educative Curriculum Materials (Davis and Krajcik, 2005) that strengthen teacher

and student mathematics content knowledge; and develop a professional teaching community

through the use of Lesson Study (Lewis, Perry, & Murata, 2006).

Methodology

KARES was conceptualized in three phases: Phase 1 involved an in-depth look at content of

elementary school mathematics. Phase 2 involved researching and investigating selected content

at each grade level, with a focus on developing related curriculum materials. Phase 3 involved

planning and conducting research lessons to investigate the interaction between teaching and

learning at each grade level. The phases were not necessarily disjoint. Because they more

directly relate to this paper, Phases 2 and 3 are described relative to the effort of the kindergarten

teachers.

Phase 2 - Researching the content at the kindergarten level

The kindergarten teachers began Phase 2 by investigating the Hawai‘i Content and

Performance Standards, National Council of Teachers of Mathematics (NCTM) focal points for

kindergarten (NCTM, 2008), and initial drafts of the Common Core State Standards for

Mathematics (CCSSM) (National Governors Association Center for Best Practices, Council of

Chief State School Officers [CCSSI], 2010). They discussed what it meant for kindergarten

students to understand the NCTM focal points and how the ideas compared and contrasted with

the kindergarten standards in the CCSSM. As they were researching the CCSSM kindergarten

standards the teachers engaged in lengthy discussions as to the meaning of some standards and

developmentally appropriate instructional sequences needed to implement the intent of the

standards. They eventually focused on operations and numbers for their research, materials

development, and lesson development. These discussions helped them in their planning for Phase

3. When documenting their planning, Higa-Funada (2011) wrote, “When planning for the first

lesson in 2011 – 2012 we examined the CCSD (her acronym for CCSSM). As it was early in the

year, we wanted to focus on multiple ways to know a number. Even though the CCSD standard

for this was more limiting than we thought, we decided to delve into...ways of knowing 10.”


While thinking about this problem, the teachers engaged in problem solving that expanded

their own mathematical knowledge. Even though the CCSSM kindergarten standards state that

students “fluently add and subtract within 5” (CCSSI, 2010, p.11), the teachers felt that it would

be more robust to explore 10. During their discussion, the question ‘How many ways can we

make 10?’ was raised and as a group they engaged in figuring this out. Although several

remembered studying ‘something like this’, no one readily knew the solution, but all wanted to

know. With some guidance, they explored the patterns found with addition by exploring ‘How

many ways can 1 be made?’ (One way: 1), ‘How many ways can 2 be made?’ (Two ways: 1+1,

2), ‘How many ways can 3 be made?’ (Four ways: 1+1+1, 1+2, 2+1, 3), etc., and were pleased

with their problem solving abilities to arrive at a solution. They also realized that the large

number of possible ways might make the problem more accessible for their students.

Phase 3 - Developing and planning a research lesson

Phase 3 was designed so teachers would prepare lessons, research how the lessons worked

with children, and reflect on the experience. Teachers engaged in the typical Lesson Study

process involving (1) defining an instructional problem, (2) researching the problem and

brainstorming possible lessons that could speak to the issues, (3) anticipating possible student

misconceptions that might occur, and (4) then, planning the lesson. While step (2) is often

viewed as the most crucial, all steps should occur before the lesson is taught. This was important

because the research and deep thinking the teachers did before the teaching gave them ownership

of a lesson they planned and prepared to teach. In planning the research lesson, teachers used the

research evidence they collected to prepare the lesson. When preparing the lesson they used a

format with three columns (Figure 1). In the first column they listed the planned sequence of

events that would occur as the lesson unfolded, including key questions to ask and the rationale

for asking them. In the second column, they listed reactions and responses they anticipated

students would make. These responses were based on evidence they found in research as well as

from their prior teaching experiences. Column 3 had suggestions for how the teacher could

respond to the anticipated responses. The third column was also used to record student responses

and to think about how to use these responses when conducting whole class discussion.


Kindergarten Lesson Format

How May Ways Can you Make 10?

Steps of the lesson: Learning

activities and key questions

with rationale (time

allocation)

Anticipated student reactions

or responses

Teacher responses to student

reactions. Things to

remember.

Figure 1. Format used for preparing lesson

Studying the Lesson

To maximize the opportunity to study and reflect on the lesson while at the same time

causing the least disruption for the teachers and students, the following process was used: one

teacher taught the lesson while all others observed; this teacher led a debriefing session that

occurred immediately after the lesson was taught; if deemed necessary, minor suggestions for

adjusting the lesson were made; a second teacher taught the same lesson; and that second teacher

led the next debriefing session. While not a part of the study sequence involving observers and

debriefing, the other three kindergarten teachers later taught the same research lesson.

The goal of the research lesson, How many ways can you make 10?, was to develop a child’s

sense for “ten-ness” which the teachers knew would support later work with numbers between 10

and 20. As they brainstormed how to provide a context for their students, they realized they had

a significant amount of scrip remaining from the school’s family fair. They created a story for

the students about how a collection of 10 scrip could be exchanged for an “ice pop” treat left

over from the fair, and prepared curriculum materials to deliver their lesson.

For the lesson, students were provided containers of scrip with multiple lengths of 1, 2, 3, or

4 scrip per container and asked to make a collection of 10 using any

combination of scrip they wanted. When students were sure they

had 10 scrip, they were to glue the scrip on a strip of construction

paper. Students were given enough time to make one set of 10 and

encouraged to make more collections if they could. This task was

accessible to all students, and the variety of responses was

interesting, but more interesting was the discussion in which the

teachers engaged the children. Samples of student work are in

Figures 2 – 4.

Figure 2


In Figure 2, both students worked with the largest scrip piece (4), but one used two 3s while

the other used three 2s. This aspect later proved interesting as when asked to count to verify

totals, some students used a skip counting process rather than counting by 1s. In Figure 3, a

student used three 3s and a 1 to make 10. Several students used a similar configuration, but in a

different order. This provided an opportunity for the teacher to compare and contrast how

arrangements were similar or different. Figure 4 shows a student had completed two

combinations of 10 and was working on a third.

After allowing time for each child to make a collection of 10, the teacher called the class to

sit on the carpet at the front of the room with their work. Individual students were asked to show

their collection and tell the numbers involved. For example, the student who showed the

collection in Figure 3 would say, “3, 3, 1, 3.” As was planned in the lesson, the teacher at this

point, while holding two different collections, engaged students in exploring, “Is 3, 1, 3, 3 the

same or different from 3, 3, 1, 3?” Teachers were excited by the appropriate students’ responses.

On a day following the research lesson, one teacher

worked with a group of students to further talk about

similarities and differences. She had students share an

example they had made or could make and recorded their

suggestions on poster paper as shown in Figure 5. She made

the effort to both draw the arrangement and to use the

numerical values in the discussion and had students describe a

collection, such as Jaycha’s, by saying, “One scrip and two

scrip and two scrip and two scrip and three scrip make a

collection with 10 scrip.”

Teacher reflection on the Lesson Study Process

After all five teachers had taught the lesson, they responded to the following prompt, “How

successful do you think the problem your team selected for the lesson study addressed your

Figure 3 Figure 4

Figure 5


team’s goals? As much as possible, cite specific examples from your observations.” The

collective response of the kindergarten teachers is given in two parts below. One part pertains to

how the lesson matched their intended goal, and a second part pertains to what the teachers felt

they learned from the work of the children during the lesson.

Response related to how the lesson matched the intended goal:

The K team’s goal was to develop number sense and experience making groups or

combinations of 10. The problem we chose addressed a wide range of academic abilities

(we were able to differentiate between levels). Examples are:

1. Some children from various classes took the initiative to add the sets of 10 together.

2. We were also able to identify students who need more support and practice with one-

to-one correspondence.

3. Even the ones which were wrong (were) used as teachable moments in which the

students were taking the lead as problem solvers.

4. Students developed oral communication when sharing their combination of 10. Even

the less verbal students felt confident in sharing.

5. Problem was relevant and had a personal connection to their own experiences using

scrip at the Kapālama Family Fair/Chuck E Chesse/Fun Factory. (Sakumoto, 2011).

Responses related to what was learned from student responses:

The Kinderettes (Author Note: What they called themselves.) felt surprised that the

problem we chose exceeded our expectations in the following ways. Children were able

to: self correct/identify how to help others; collaborate with each other; show many

combinations of 10, compare and contrast their own combinations with their peers;

subitize 1-4 scrip (Author Note: This is something they learned about in their research.);

see the big idea that there are a variety of ways to make 10; verbalize their thinking; and

take the initiative to extend the lesson. (Author Note: One example of this initiative

occurred when students had more than one scrip of the same length, such as 2 + 2 + 2 + 2

+ 2, and saw they could skip count to validate this was a collection of 10. A second

extension occurred when one child suggested they count the total of all the collections

made. While the student did not count by 10 to determine his solution, his final total was

close to the correct value.) (Sakumoto, 2011).

Findings, Discussion, and Conclusions

The teachers and authors made observations related to several aspects about the research

lesson that was conducted. These comments demonstrate that teachers made advances in the way

they thought about the mathematics they were teaching, the lessons they were preparing, and in

their analysis of student learning. The comments, with supporting evidence, are listed below:

1. Teachers designed an appropriate lesson. Each time the lesson was taught, the children

were actively engaged in the problem and intent on learning. Students were on task and most


found multiple correct solutions. This is a tribute to the depth of thinking the teachers did in

deciding on the task to investigate, and in knowing the significance of the mathematics involved.

2. The effort teachers made to fine tune the task paid off. The task was challenging but

accessible. While the CCSSM only focuses on knowing numbers 1 – 5, each child was able to

work successfully on this task. Even those students who basically could only approach the

problem by putting tickets one at a time were able to engage meaningfully with the task.

3. The teachers were able to use discourse and communication in a meaningful way. Because

of the specific structure incorporated into the lesson, with anticipated responses and suggestions

for handling the responses, communication options occurred they never considered before.

Students who struggled to find a solution still were comfortable sharing what they made and why

they thought they made 10. Similarly, when a student had an incorrect solution, he or she could

count and determine that it was not 10. When either of these things happened, teachers asked

other students, “How can we help _____ make 10?” Interestingly, students were able to provide

multiple answers to this question. This shows the benefits of effective planning for student

responses and actions the teacher should be prepared to take.

4. The task proved well suited for the English Language Learners students in the class. They

could do the mathematics expected of them while being coached through the words needed to

explain their work. The decision the teachers made to have students record the value of each

length of scrip used when making their collections provided a representation that helped prepare

the students for the explanations they were to give. As a result of their research, teachers became

more aware that a focus on communication in mathematics supports other parts of the

curriculum.

5. Interesting discussions were forthcoming when the teachers asked, “How are these the

same, and how are they different?” While children realized they were all showing 10, the

discussion of same and different led to excellent comparisons. Some children said a 2, 3, 3, 2 and

a 2, 2, 3, 3 were the same because they each used two 2s and two 3s, while others said they were

different because the order was different. Some saw that 1, 2, 3, 2, 2 and 2, 2, 3, 2, 1 were

‘reversed’. The language used in the discussions exceeded teachers’ expectations, suggesting that

the preliminary discussions the teachers had about these same questions filtered into their lessons

and instruction.


6. While preparing the lesson, teachers became focused on how to use a context to which

their children could relate. This led them to focus on the development of the scrip and the “ice

pop” story. We all observed that throughout the lesson the students were so engaged in the

thinking and reasoning involved that the story had become irrelevant.

When reflecting on how focused the student were on this lesson and they forgot the story the

teachers used to get started, one teacher summed up the positive aspects of their effort to create

this lesson when she said, “It was not about the ice pops, it was about the learning.” This

statement captured the essence of the evolutionary journey to involve kindergarten students in

the exploration of big mathematical ideas.

References

Ball, D. L. (2003). What mathematical knowledge is needed for teaching mathematics?

Prepared for the Secretary’s Summit on Mathematics, US Department of Education, February 6,

2003; Washington, DC Available at http://www. ed. gov/inits/mathscience.

Carpenter, T. P., Blanton, M. L., Cobb, P., Franke, M. L., Kaput, J., & McCain, K. (2004).

Scaling up innovative practices in mathematics and science. Madison: University of Wisconsin-

Madison, NCISLA. Retrieved July 15, 2004, from

http://www.wcer.wisc.edu/ncisla/publications/reports/NCISLAReport1.pdf

Davis, E. A. and Krajcik, J. (2005) Designing educative curriculum materials to promote

teacher learning, Educational Researcher 34 (2005) (3), pp. 3–14.

Higa-Funada, H. (2011). Personal communication, November 10, 2011.

Lewis, C., Perry, R., & Murata, A. (2006). How should research contribute to instructional

improvement: The case of lesson study. Educational Researcher, 35(3), 3–14.

National Council of Teachers of Mathematics. (2000). Principles and standards for school

mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2006). Curriculum focal points for

prekindergarten through grade 8 mathematics. Reston, VA: Author.

National Governors Association Center for Best Practices, Council of Chief State School

Officers. (2010). Common core state standards for mathematics. Washington, DC: Author.

Sakumoto, A. (2011). Personal communication, November, 2011.

Acknowledgement

With their permission, we identify and give special thanks to the following whose work,

dedication, and collaboration made this paper possible: kindergarten teachers Mia Grant, Heather

Higa-Funada, Kannette Onaga, Stefanie Won, and Donna Wong; curriculum coordinators,

Atsuko Sakumoto and Cheryl Taitague; and Principal, Patricia Dang of Kapālama Elementary

School, Honolulu, Hawai‘i.

kindergarten students exploring big ideas: an evolution for teachers

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