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
Fay Zenigami
University of Hawai‘i
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
Proceedings of the 40th Annual Meeting of the Research Council on Mathematics Learning 2013 102
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.”
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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.
Proceedings of the 40th Annual Meeting of the Research Council on Mathematics Learning 2013 104
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
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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
Proceedings of the 40th Annual Meeting of the Research Council on Mathematics Learning 2013 106
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
Proceedings of the 40th Annual Meeting of the Research Council on Mathematics Learning 2013 107
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.
Proceedings of the 40th Annual Meeting of the Research Council on Mathematics Learning 2013 108
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
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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.