Integrating Instructional Approaches in Developing Number Sense
- Relationships exist between and within numbers
- Multiple instructional approaches are used to build student understanding of number relationships
- Comparing, composing and decomposing, estimating, visualizing, and representing are used in connection with each other and are vital for developing number sense
See chart below for elementary and high school task examples that promote multiple big ideas:
N2.1: Demonstrate understanding of whole numbers to 100 (concretely, pictorially, physically, orally, in writing, and symbolically) by:representing (including place value)describing skip counting differentiating between odd and even numbers estimating with referents comparing two numbers ordering three or more numbers.
Thinking mathematically means students are using the big ideas as they explore questions such as: How many ways can you show ‘21’?
How do your representations compare with each other?
Which representation shows that your number is odd?
How do you know?
P20.7 Demonstrate understanding of quadratic functions of the form y = ax² + bx + c and of their graphs, including:
domain and range
direction of opening
axis of symmetry
x- and y-intercepts.
FM20.9 Demonstrate an understanding of the characteristics of quadratic functions of the form y=a(x−p)2+q , including:
domain and range
axis of symmetry
In this activity, students predict whether various basketball shots will go through the hoop, and then model these shots with parabolas to check their predictions.
Specific Examples (within the big ideas):
Objects, shapes, equations and events are compared to define their attributes. Numbers and measures are compared in many ways to get a sense of order and relative magnitude. Sometimes they are compared to each other, benchmarks, properties or classifications, events, and data sets (equivalence, inequivalence).
Composing & Decomposing
The purpose of composing and decomposing is to simplify the mathematics. It is an essential skill for making calculating easier because it is all about making and then using friendlier numbers. Learners will encounter scenarios where they may need to join or separate, to identify the parts that make up a whole, and to compare the parts. Equivalence must be maintained within each manipulation. Thinking flexibly about putting numbers together as well as taking them apart leads to a greater understanding of relationships not only between numbers and operations but in all strands of mathematics.
There are times when an exact answer is needed and times when it is not. Approximation further develops an understanding of what is reasonable within a given context. Estimation is a mental process and can be an indicator of a student’s understanding of conceptual understanding.
“Mental math enables students to determine answers and propose strategies without paper and pencil. It improves computational fluency and problem solving by developing efficiency, accuracy, and flexibility.” 1
“Number visualization occurs when students create mental representations of numbers.”2 Students use concrete, pictorial, and symbolic representations to explore strategies and communicate their mathematical thinking. When students use a concrete or pictorial model, that model is a tool that students use to make their thinking visible.
“In order to develop every student’s mathematical proficiency, leaders and teachers must systematically integrate the use of concrete and virtual manipulatives into classroom instruction at all grade levels.” 3 NCSM, 2013
Visuals that are familiar to students can also connect a new concept to prior learning. For example, number lines are a visual of number order that students use when working with whole numbers. When fractions or decimals are introduced that same number line, which is now familiar, can be used to build an understanding of how decimals and fractions can fall between whole numbers.
“To develop a deep and meaningful understanding of mathematical concepts, students need to represent their ideas and strategies using a variety of models (concrete, physical, pictorial, oral, and symbolic).”4
When students make connections between mathematical representations, number sense is developed. These connections lead to a deeper understanding of the relationships that exist within and between numbers as well as broader mathematics concepts and problem-solving.
“Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas.”5
Here are some instructional approaches that encompass multiple big ideas and may contribute to the development of students’ number sense:
“Number sense goes well beyond being able to carry out calculations. In fact, in order for students to become flexible and confident in their calculation abilities, and to transfer those abilities to more abstract contexts, students must first develop a strong understanding of numbers in general. A deep understanding of the meaning, roles, comparison, and relationship between numbers is critical to the development of students’ number sense and their computational fluency.” 6
- Thinking Classroom7
- Number Talks8
- Fraction Talks9
- 3 Act Math Tasks10
- Which One Doesn’t Belong?11
- Math Lessons That Build Number Sense12
- Same or Different13
- What is Going On in This Graph14
- Open Middle Problems15
- Visual Patterns16
- Desmos Activities17
- Citizen Math20
- Yummy Math21
- 3 Act Math22
- Would You Rather Math?23
1Ministry of Education, S. (2009). Math 9. p. 11. Retrieved 20 June 2020, from https://www.edonline.sk.ca/bbcswebdav/xid-181799_1.
2Ministry of Education, S. (2007). Retrieved 20 June 2020, from https://curriculum.gov.sk.ca/bbcswebdav/library/curricula/English/Mathematics/Mathematics_7_2007.pdf p. 13.
3Improving Student Achievement in Mathematics by Using Manipulatives with Classroom Instruction. mathedleadership.org, 2013. NCSM Position Paper. [online] Movingwithmath.com. Available at: https://www.movingwithmath.com/ncsm-position-paper/#zoom=z [Accessed 19 August 2021].
4Ministry of Education, S. (2009). Retrieved 20 June 2020, from at https://www.edonline.sk.ca/bbcswebdav/xid-181799_1 p. 20.
5National Research Council. (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press. p. 94 https://www.nap.edu/read/9822/chapter/5#94.
6Ministry of Education, S. (2008). Retrieved 20 June 2020, from https://www.edonline.sk.ca/bbcswebdav/xid-181792_1.
7Liljedahl, P. (2020). Building Thinking Classrooms in Mathematics, Grades K-12: 14 Practices for Enhancing Learning. Thousand Oaks,California: Corwin Press, Inc.
8Parrish, S. (2010). Number talks. Sausalito, CA: Houghton Mifflin Harcourt.
9Fraction Talks. Retrieved 9 December 2020, from http://fractiontalks.com.
10Meyers, D. (2020). Dan Meyer’s Three-Act Math Tasks. Retrieved 9 December 2020, from https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/edit#gid=0.
11Danielson, C. Which One Doesn’t Belong?. Retrieved 9 December 2020, from https://wodb.ca/.
12Math Lessons That Build Number Sense. Retrieved 9 December 2020, from http://www.estimation180.com/.
13Bushart, B. Same or Different. Retrieved 9 December 2020, from https://samedifferentimages.wordpress.com/
14What’s Going on in this Graph?. Retrieved 9 December 2020, from https://teacher.desmos.com/curriculum/view/5c38d8b9b2e2c80d2230827c.
15Kaplinsky, R. Open Middle: CHALLENGING MATH PROBLEMS WORTH SOLVING. Retrieved 9 December 2020, from https://www.openmiddle.com/.
16Visual Patterns. Retrieved 9 December 2020, from http://www.visualpatterns.org/.
17Desmos Classroom Activities. Retrieved 9 December 2020, from https://teacher.desmos.com.
18Wyborney, S. Splat! – Steve Wyborney’s Blog: I’m on a Learning Mission. Retrieved 9 December 2020, from https://stevewyborney.com/2017/02/splat/.
19Wyborney, S. 51 Esti-Mysteries – Steve Wyborney’s Blog: I’m on a Learning Mission. Retrieved 9 December 2020, from https://stevewyborney.com/2019/09/51-esti-mysteries/.
20Citizen Math: MATH CLASS, REIMAGINED. Retrieved August 10, 2021, from https://www.citizenmath.com/.
21Yummy Math – real world math. Retrieved 9 December 2020, from https://www.yummymath.com/.
22Pearce, K., & Meyer, D. 3 Act Math Tasks By Kyle Pearce, Dan Meyer and Others. Retrieved 9 December 2020, from https://tapintoteenminds.com/3act-math/.
23Would You Rather Math. Retrieved 16 December 2020, from https://www.wouldyourathermath.com/.