New Project Title: Counting Principles and Thinking in Math in Kindergarten
Students begin Year 1 of their FDK career at the age of three or four. They arrive at school with differing degrees of “numeracy” knowledge and different real-life experiences related to math. As a result, there is a wide range of student comfort and confidence working with numbers. It is important for kindergarten educators to help all their students build a firm foundation in numbers to 10 by setting up a mathematics program that focuses on number sense and especially counting.
We want to learn more about how different counting principles (e.g., stable order, cardinality) influence student learning. We hope that some professional reading and research would give us better insight into this. We want to gain a better understanding of how kindergarten students move along the learning continuum related to counting principles and to see if one counting principle must be mastered before a student could become proficient at another one (i.e., Do students need to master one-to-one correspondence before they can attain an understanding of stable order?).
We want to create and use a baseline assessment to identify each student’s (Year 1 and Year 2) possible strengths and weaknesses in various counting principles (i.e., one-to-one correspondence, cardinality). From this assessment and correlation of results, we are hoping to identify areas of improvement in delivering our numeracy program. This knowledge would drive our instruction for the rest of the year and better prepare us for teaching numeracy in September. For example, a variety of activities related to number paths would need to be introduced if a majority of students were found to be generally weak in the stable order counting principle.
We want to see if providing a “numeracy rich” math environment where students could play, explore and discover would foster their curiosity in numbers and have an effect on their proficiency in counting. We planned on creating small and large group activities, intentional centres, provocations and a student area dedicated to math materials (e.g., dice, Rekenreks, number lines, counters, wooden numbers) that were directly connected to different counting principles. We feel that this would improve our students’ comfort in talking about numbers.
We want to place more emphasis on asking more “numeracy” questions throughout the school day and increase our discussion/sharing time so students had an opportunity to see how “numeracy” is seen by others.
Thames Valley District School Board
Thames Valley District School Board
Professional Learning Goals
- By reading various professional books, we gained better insight to what children ages three to six are developmentally capable of in numeracy (i.e., how young students learn about numbers) and more specifically, in relation to different counting principles
- By examining different research, we determined which of the eight counting principles are of utmost importance for this age group (i.e., one-to-one correspondence, stable order, cardinality, abstraction, order-irrelevance) and we gained a better understanding of how students’ proficiency in these counting principles relates to their success in math in later grades
- We discovered that students need to master a particular counting principle before they can master another one. For example, stable order must be mastered before the cardinal principle can be attained.
- We assessed student proficiency (i.e., strengths and weaknesses) in the five early counting principles that were identified in our readings. We also collated the results between the two classes and determined a plan of action.
- We began revamping the delivery of our math program so that numbers to 10 are a focus and we altered our teaching practices (i.e., explicit teaching, large/small group activities, centres and provocations) so that the five early counting principles are also the focus in our math program. This is especially important for the beginning of the next school year.
- We investigated and ordered math materials (i.e., foam number path mats to 10) that further support the activities we planned and will plan in relation to improving student knowledge in numeracy and counting
Activities and Resources
- Our early reading and research in regards to a child’s early capabilities in numeracy and the importance of certain counting principles focused on two particular books:
- Early Childhood Mathematics Education Research: Learning Trajectories for Young Children by Julie Sarama and Douglas H. Clements
- The Child’s Understanding of Number by Rochel Gelman and C.R. Gallistel
- We also reviewed some online research (i.e., Interactive Stem: Mathematics in the Early Grades: Counting and Cardinality) and a presentation/research by Dr. Daniel Ansari, a professor at Western University
- We administered a baseline assessment to our students (i.e., activities, questions, watch fors) introduced to us by our school board to identify strengths/weaknesses in relation to early counting principles. This assessment was based on Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3, a book by Jessica F. Shumway. This allowed us to understand exactly what our students knew in terms of numbers and counting strategies, which in turn helped us begin to plan developmentally appropriate activities and lessons.
- Reading parts of Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades Pre-K-2 by John A. Van de Walle helped us begin creating a more numeracy and counting rich program that was appropriate for young students
- Using the new math materials during teacher lessons and group activities and allowing students to explore these materials during centre time provided opportunities where students could be observed in how they were using the materials, how they were talking about numbers, and how their comfort and confidence was changing when working with numbers to 10. Some of the more popular materials were wooden numbers and family counters, a “counting on” tabletop game, a number path pocket chart, number lines, dice, a light table kit, and a number path play mat to 10.
- The release time we scheduled allowed us to share and discuss what each of us had been reading and researching early on in our TLC project. It also allowed us to spend a good amount of time with individual students during the baseline assessment. We had a chance to talk and observe and not feel we had to rush through the assessment. It also gave us time to share the results of our two classes. We also had the time to discuss and plan what we wanted to achieve before our next meeting date.
- There is so much information out there in terms of professional reading books and online material that we found it difficult determining where to start and which professional materials to use and focus on
- Some of the material/research we read had differing opinions surrounding the counting principles. Some research considered the ability to subitize of paramount importance, while other literature contradicted this.
- The way in which educators asked the questions or demonstrated a task during the baseline assessment may have varied (between educators) which might have influenced the outcome of the assessment
- Educators noted that although students had acquired a specific skill (i.e., subitizing or cardinality), they rarely utilized that skill into a task (e.g., where subitizing could be used to complete the task) unless explicitly asked. Their thinking seemed “rigid” and seemed more comfortable, perhaps conditioned, to only compete the task in one way.
- Timing was a factor since we did not have our first meeting until mid-December. By this time, our attention was turning to the tedious task of completing kindergarten report cards. It was difficult to focus and provide attention to the project until these were completed.
Enhancing Student Learning and Development
- It was interesting to learn what children of certain ages are capable of in terms of numeracy and in what order the counting principles should be addressed. We now have a better grasp of which counting principles need to be focused on when teaching younger students. We can plan instructional tasks in sequence and that are developmentally appropriate for students, which in turn will improve student mastery and success (i.e., activities/lessons should first focus on one-to-one correspondence before moving on to activities that involve stable order).
- Research supports that students who can quickly identify that the second set of numbers are in correct order (4,7,2 versus 2,5,9) have more “math success” in later grades. Our lessons and activities will focus on number paths to 10, representing numbers to 10 in different forms (e.g., 3 – the word “three” – three tallies – three dots), and composing/decomposing numbers to 10. For example, tasks might include having students identify which numbers have been switched on a number path or which number has been covered up on a number line. We should see that the variety and repetition of tasks will have an impact on student ability, understanding and confidence.
- Providing students with a variety of learning tasks (i.e., centres, large group activities, teacher-directed lessons) and hands-on math materials allowed them to investigate, explore and communicate/share with their peers and educators. Student interest level increased and with that an improvement in their comfort and confidence working with numbers.
- In February, our school dedicated a large portion of a staff meeting to “Math in our Classrooms.” We met in our divisions and this gave Natalie and I the chance to share with other kindergarten educators what we had learned so far in relation to a child’s developmental understanding of numeracy and the importance of teaching the five early counting principles in sequence. We noted that one-to-one correspondence and stable order were of particular importance.
- We referred to Dr. Daniel Ansari’s work and conveyed that a student’s future math success depended greatly on their early understanding of numbers. For example, if a student could quickly recognize that the second set of numbers were in correct order (4,1 versus 2,7), then they had more success in math in later grades. A stronger indicator was if they could do the same with three numbers (4,1,8 versus 2,6,9). Kindergarten teachers were provided a Smartboard Number Path Counting On activity called “Elevator Ride” as well as, a tabletop version with labelled dice (e.g., one more, two more). We answered questions as the other educators used the materials and tried out the tasks.
- Kindergarten teams were also given time to meet on a PA Day during the morning of April 27. During the morning, we shared some more of our research and the results of the baseline assessment that we had administered to our students, noting some interesting facts and areas of improvement.
- During the year-end meeting, we plan on presenting an instructional task in detail (i.e., the materials used, step-by-step instructions on how to carry out the task, student look-fors, and how children’s learning was consolidated)
- We also plan on sharing our work (i.e., online articles, summary of readings and research, summary of purchased materials and suggestions for tasks) and created activities (e.g., SMART board “Counting On” games) by placing it in the “U drive” on our school’s computer network, so all kindergarten educators can have quick and easy access
We felt that the project was a success and we enjoyed taking part in it. We learned a great deal about students’ learning in numeracy.
- Inquiry-based. Educators were afforded sufficient time to research their questions surrounding counting principles in the early years.
- The availability of funds allowed educators to purchase current literature and quality classroom materials that were relevant to the focus of the study. As a result, the resources and literature aided in favourable research and helped to reinforce the acquisition of best practices.
- Time to collaborate with a colleague and discuss similarities and differences found between students in two classes – invaluable. It also allowed educators time to discuss pedagogy and determine best practices based on the outcomes of the student assessment.
- Inquiring into and reading current literature, as well as developing assessment tasks that were specific to the focus of the study, afforded educators an opportunity for deeper learning pertaining to the counting principles in the early years. It also gave insight into scaffolding the principles and realistic expectations (for students) with regards to learning these skills (i.e., the concept of 1:1 correspondence needs to precede the concept of stable order).
- A bank of resources and manipulatives were created by educators that can be used within the classroom and shared with peers
- The opportunity to meet individually and assess students in a quiet setting was also extremely beneficial as this type of engagement is not always possible due to everyday classroom occurrences and class size that hinder the possibility of a focused conference for student and educator
Interactive Stem Research Brief (2015)
Mathematics in the Early Grades: Counting & Cardinality
How the 5 Counting Principles Lay the Foundation for Flexible Thinking in Later Grades by S. Edgar
Website referring to the research/early screening tools by Dr. Daniel Ansari and colleagues at Western University
The Child’s Understanding of Number (2009) by Rochel Gelman and C.R. Gallistel
Link is for the Google book version (some chapters).
Early Childhood Mathematics Education Research: Learning Trajectories for Young Children (2009) by Julie Sarama and Douglas H. Clements
Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades Pre-K-2 (2017) by John A. Van de Walle
Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 (2011) by Jessica F. Shumway
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Counting Principle Strategies to Build Number Sense
Various tasks/activities that address different counting principles.
Paper Number Path and What's My Number Game
Particularly useful when reinforcing stable order. Use the number path to 10 and simplify the riddles/clues for kindergarten students.