A Brief Blog on Number Sense

Number sense is essential for student understanding and application of computation and other concepts related to numbers in math. Development of number sense begins from young ages when students first learn to count, to recognize numbers, and to connect what they are counting with the numerical representation. Number sense provides for flexible thinking and for learning and understanding computation as compared to simply memorizing procedures (Boaler, 2016).

Place value is an important component of number sense as it provides the foundation for understanding the value of a number and how it can be composed and de-composed. For example, a student who can show different ways to represent the number 15 understands that it is about the value represented and the value can be maintained through different representations. In teaching computation, we typically teach different strategies for solving the computation. This approach is different from when many of us were students, as we were taught one way to compute the given operation and the instruction was typically not grounded in conceptual understanding. We were simply taught the algorithm and expected to memorize and use it. Unfortunately, this approach does not build flexible thinking nor does it provide students with understanding of the algorithm and why it works (Boaler, 2016). The strategies we teach today are grounded in conceptual understanding and provide explicit connections to place value. It is important to ground computation strategies and language in place value. The strategies also relate to each other and we need to ensure our instruction explicitly supports those connections, including precision with place value language.

For example, when teaching multiplication, we often start with arrays, transition to area models, then to partial products, and conclude with the standard algorithm. With arrays and area models, students can "see" why and how the multiplication works. They can also count, if needed. In transitioning from area model to partial products, we can make explicit connections to place value and strengthen the flexible thinking and understanding of numbers. We conclude with the standard algorithm because it is a short cut! The standard algorithm is the short cut version of partial products. In general, most standard algorithms (if not all) are short cuts of the steps grounded in the conceptual aspects of the computation. This is why we teach multiple methods - not because students need to know three or four ways to multiply, but because it builds flexible thinking, conceptual understanding of the computation, and number sense. It also supports retention and application of learning!

There are a plethora of books, games, strategies, online programs, etc. with recommendations and activities for building number sense. With any of these recommendations and activities, the emphasis should be on developing an understanding of number. Too often, we get sidetracked with following the steps or following the formula. Number sense isn't about steps or formulas. It's about the understanding of numbers, their representation, how they connect to each other, and other attributes.

Below are activities and strategies that I find helpful for building and strengthening student number sense.

References

Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.

Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.

Ways to Make a Number

One of my favorite activities to support students in developing their sense of number and value is finding different ways to make a number (or ways to represent a number). This activity can be completed individually or in groups. I recommend groups as it provides an opportunity for students to build on each other's thinking. The numbers used for this activity can be whole numbers, decimals, fractions, or integers. Below are steps for completing the activity individually and steps for completing the activity in small groups. Learning Objective: Students will explore and identify different representations of a number to build flexible thinking and understanding of values.

Individual

Give each student a number. Each student can be given a different number. Ask each student to use one sheet of paper and to draw a large circle that covers most of the paper. In the middle of the large circle, ask students to draw a smaller circle and to write their assigned number in the smaller circle. Then ask students to begin writing different ways to represent the number. For example, the number 25 can be represented as 20+5, 24+1, 23+2, 26-1, 30-5, 5x5, 5^2, two tens plus five ones, 1 ten plus fifteen ones, 25+0, 25-0, 25x1, 50/2, etc. This activity can also be completed with integers (for middle school students), decimals, and fractions. I recommend providing time for students to share their work and to build on each other's responses.

Small Groups

Use one piece of large chart paper for each group. The teacher can have the paper for each group set up ahead of time or each group can set up their own paper. As with the individual task above, there should be a large circle drawn on the large chart paper and a small circle drawn in the middle of the large circle. Each group will write their assigned number in the small circle. I recommend each group have a different number. Be sure to establish group norms for this activity to ensure that every member of the group is able to participate in the activity. I typically have one marker for each group: one student writes their answer, passes the marker to the next student, that student records their answer, passes the marker to the next group member, etc. For each turn with the marker, a student can record one response. This prevents one student from doing all the thinking and writing. Within the group, students are completing the same task as with the individual activity: students are finding as many ways to make the assigned number as possible. I recommend setting a timer for this activity - usually five to ten minutes. After each group records their answers in the time provided, if time allows, I have the groups rotate with their marker to another group's chart paper. A timer is set (usually less time than the original task), and the group adds to the original group's work - using their color marker. Each group's work is indicated based on the marker color. The group rotation can occur several times, depending on the time available. By having the groups rotate, they can see each other's thinking and build on each other's thinking. Be sure to provide time for group and class reflection to conclude this activity.

Pay Bills Activity

This activity could be included under the Problem Solving section on this website, but I included it here because of the emphasis on applying computation concepts and skills. The activity is designed for small groups. It provides real-life application of budgeting, addition, subtraction, and problem solving concepts. Each group will be "roommates" and they need to determine as a group how they are going to use their monthly income to pay the bills. Some expenses are required each month, and some expenses are optional. Students have to collaborate to prioritize their spending. At the conclusion of the activity, students are asked to reflect on the problem solving and decision making processes they used in their group. The reflection component is key for reinforcing learning and for supporting student learning of the problem solving and decision making skills used.

Learning Objective: Students will apply problem solving and computation concepts and skills in a real-life scenario to strengthen understanding, application, and connection of the concepts and skills.

Suggestions & Tips for Teaching Fractions

Understanding and application of fraction concepts are fundamental for higher level math concepts (Watts, Duncan, Siegler, & Davis-Kean, 2014). Ratio and proportion concepts taught in middle school are based on students' understanding of fraction concepts. Because of the importance of all students developing a strong foundation in fraction concepts, I am a believer in allowing students to use tactile tools and to draw models as often as possible to support them in strengthening their understanding of the concepts.

Integrating multiple representations of fractions is fundamental for developing student understanding of what fractions are and how to define a fraction (Boaler, 2016). Circle fractions, fractions tiles, pattern blocks, and different colored linking cubes are a few examples of representations that can be used. With the different representations, it is beneficial to connect the fraction concepts across representations to strengthen student understanding and flexibility with applying fraction concepts. It's important to ensure students understand that a fraction is about the whole - how the whole is defined, what makes the whole. It's also important for students to understand that when a whole is divided into pieces, the pieces must be equal sized - particularly for a common denominator. Rulers and measuring tapes can be helpful tools with teaching fractions, as they provide a number line with fractions (customary - halves, quarters, eighths, sixteenths; metric - tenths). I recommend checking out Dr. Juli Dixon's work on teaching fractions, including when to introduce the words numerator and denominator with teaching fraction concepts in her video on YouTube titled "Six UnProductive Practices in Mathematics Teaching" (2020).

Fractions can be challenging for students, particularly when the concept is first introduced. Prior to learning about fractions, students have only worked with whole numbers. Their understanding is that a larger number means more. So when they are introduced to the idea that 3/4 is larger than 10/20, there may be confusion because they focus on the 10 and 20 and not the value of what 10/20 represents, which is 1/2. This is why it is essential to integrate use of tactile learning with fractions as often as possible to help students understand that it is about how many parts into which the whole is broken up (or divided). I often use food as part of these fraction discussions. I hold up a candy bar and ask students if they want to share it with one other person, three other people, five other people, etc. It helps them to understand that the pieces will get smaller and smaller for each person to receive an equal sized piece of the candy bar.

References

Boaler, J. (2016). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.

Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). What’s past is prologue: Relations between early mathematics knowledge and high school achievement. Educational Researcher, 43(7), 352–360.