3 Tried and True Tips for Developing Number Sense

Have you ever paid the bill at a store or restaurant, then realized at the last minute you have change? But when you try to be helpful and give the cashier the change, which would help to not get so many coins back, they either give you a blank look or just state, "But I already punched in $20." Then when you try to tell them how much to give back to you,  they become confused or anxious because it's not what the register says. . . Those scenarios are just as disturbing to a teacher, as when you ask your class to estimate how many objects are in a jar and they blurt out, "1,000!" or some other nonsense number (when there are clearly less than 50 in the jar). Number sense. How to help our students develop number sense?
Number Sense is the ability to appreciate the size and scale of numbers in the context of the question at hand. The three major elements that fall under number sense are counting, wholes and parts, and proportional thinking.
 Warm ups and number talks build number intuition and fluency, while giving you (the teacher) insight into how your students think. They support the idea that math makes sense, and you can explain what you see to help it make sense to other students. Engage students with expressions and equations as a means to:
  • develop relational thinking
  • look for patterns in number
  • develop place value understanding and/or
  • build number fluency
The difference is that the students aren’t just looking for the answer: they’re trying to find as many different ways to solve the problem as they can. The key elements to number talks are less emphasis on speed and right answers and and more of an emphasis on their thinking process and communication.
Choral counting is a warm up that teachers can use to help students identify patterns, strengthen number sense, develop concepts of place value, and increase number fluency. Ask: How are you deciding what number comes next? What patterns do you see? It's often quite surprising what students notice (that was unintentional on the teacher's part!)
Another warm-up is to encourage wonderings through real life photographs and situations. On Twitter, search for #unitchat and hundreds of examples will come up!
When I first heard of counting collections, I flashbacked to when I used to teach primary grades and we counted our 100th Day of School collections. However, counting large numbers is so much more! "Research shows that although counting is one of the best ways we know to help children develop number sense and other important mathematical ideas, we do not do nearly enough of it in elementary schools. Children need lots of experience with counting to learn which number comes next, how this number sequence is related to the objects in front of them, and how to keep track of which ones have been counted and which still need to be counted." (Fuson, 1988)  The National Research Council (2001) states that having experience with counting gives children a solid foundation for  future experiences  with addition, subtraction, multiplication, and division.
Counting collections are an opportunity for students to practice counting, make connections to number and quantity, and engage in problem solving. Click HERE or on the image below to download a copy of Counting Collection recording sheet.
Former high school math teacher Dan Meyer gave a TED Talk called Math Class Needs a Makeover. If you have not seen it yet, you must watch it HERE. He compares the difference between reading a word problem in a textbook and demonstrating the same problem in real life. Meyer argues that "students lack initiative, perseverance and retention because they spend most of their time plugging in formulas without understanding the mathematics." (p. 114) Our goal should be for students to learn and understand math by working on problems, NOT just to get the write answers to textbook problems. Amen!

Author of Becoming the Math Teacher You Wish You'd Had, Tracy Zager claims that problems in textbooks often have too many parts, often missing the main math objective. Or the problems spoon feed students and give them too much information so that no thinking is necessary. I've noticed this when students blindly multiply numbers in a word problem because the chapter is about multiplication or the other practice problems were all multiplication. They need to develop an understanding of what the problems are asking, building on what they already know, and then figure out how to solve problems themselves. (p. 118)

I love the following "math problem as room" analogy by Seyour Papert (1980). The best math problems have low threshold (entry points accessible by all), high ceilings (allow for differentiation and deeper investigation), yet open middles (multiple solution paths to the same answer).
Check out this fantastic website, Open Middle . It's a free resource organized by strand and grade level. There are hundreds of "Open Middle" tasks posted!

Looking for more ideas to build students’ number sense? Check out these posts to learn new strategies and get a few awesome freebies to use with students as you work to develop number sense in all students!
3 Tried & True Tips for Developing Number Sense  | Tried & True Teaching Tools
Steps to Build Number Sense | The Owl Teacher

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