“You Can’t Take a Bigger Number from a Smaller Number”

Blog post #6 in the series, Lies We Tell Our Students

As a specialist in elementary mathematics teaching and learning, I learn so much by talking to my high-school-level colleagues. I learn about the things we say and do in the early years that must be “undone” in later years. Today I would like to introduce you to Donna, a colleague I have long worked with and whose work I admire.

The Story

Currently, Donna works with a group of pre-algebra students struggling with integer operations. Specifically, they struggle with negative integer operations. As we talked, it became evident that many of the difficulties her students face stem from misconceptions they developed in primary school. She shared a blog post with me that resonated. Here’s just a brief clip from that post:

“I’ve suddenly realized that negative numbers aren’t really the problem. Subtraction causes the disconnect, as a result of the tremendous bait and switch we pull when moving from basic math to the abstractions needed for advanced math.

“In elementary school, kids learn addition and subtraction. They are not told that they are learning addition and subtraction of positive integers. Nor are they told that they are only learning subtraction when the subtrahend is less than the minuend…

“At no point are kids told that everything they’ve been taught is temporary, and that much of it will become irrelevant if they move into advanced math.”

The (Inadvertent) Lie

Far too often, we inadvertently communicate ideas that are not easily unlearned in later years. In this case, when we say, “You can’t subtract a greater number from a smaller number,” we  know somewhere down deep that this will not be true later on. That said, during instructional planning and delivery, we tend to be short-sighted. We don’t stop to think about implications for the future because we’re too focused on the here and now.

Let’s take it just one step further. Our words may come back to haunt us within our grade-levels, as well. We’ve often followed “you can’t subtract a larger number from a smaller number” in a problem solving context and then had the children transpose the numbers. So then, when they face the standard algorithm for subtraction, why wouldn’t they simply subtract “bottom-up” when they get to a place where the digits appear to need transposition?

The Truth

What we really mean is that when working with whole numbers (positive integers plus zero), one cannot subtract a greater number from a lesser number and get a positive difference. What we really mean is that in K-5, we’re learning rules that apply to whole numbers, and these rules will not hold constant with all number sets.

So how do we “tell the truth” to young children and still help them build proficiency subtracting whole numbers? It’s simple. We can be forthright with them by letting them know that there are lots of different kinds of numbers and that the rules they are learning now may not apply to other sets of numbers.  Regularly show them negative numbers and fractions to let them know that there are numbers that are less than zero and some numbers that fall between integers on a number line.

Next Steps for Teachers

• When talking to students about subtraction, talk about the number sets used at your grade level. For example, “Today we are working with the positive numbers 0-100 for subtraction.”
• Introduce your students to the notion of numbers “less than zero” (negative integers) and numbers that “show parts” (fractions and mixed numbers). You don’t need to “teach” these numbers or operations to young children; simply expose them to the notion.
• Help students make sense of context problems before they write them symbolically. When students reverse numbers in a subtraction context problem, ask them, “Will this situation give you a solution that is more than zero or less than zero?” Example: Kris gave 3 pencils to her brother. She had 7 to begin with, so how many does she have now? Ask students to act out the situation until they realize that she started with 7 and gave away 3, which should be recorded 7 – 3. If they write 3 – 7, ask them, “Your expression shows that Kris started with 3 pencils and gave away 7. Is that what happened? Will she have less than zero pencils in the end?”
• Emphasize that the commutative property of addition works only for addition. Students can add in any order and the total remains the same. This is not true for subtraction – they cannot subtract in any order with the difference remaining the same. They will learn more about this when they start working with negative integers in later grades.

Next Steps for Leaders:

• When observing instruction, listen for teachers who inadvertently communicate rules that will fall apart later. Talk with them about how they might communicate these ideas more effectively.
• During a grade-level meeting, PLC, or staff meeting, post statements that communicate misconceptions. Then ask the teachers when these “rules” fall apart. This might include the following:
  • Adding makes bigger and subtracting makes smaller.
  • When subtracting, the larger number always goes first.
  • When multiplying, the product is always greater than the factors.
  • When dividing, the smaller number goes on the “outside” and the greater number goes on the “inside.”
• After discussing the above statements, facilitate the creation of school-wide agreements about how everyone will talk about these ideas at different levels.

In Summary

Attending to the language we use with young students is critical. When planning instruction, think beyond the rules and procedures you’re discussing with students. Ask yourself, “Is this rule always true or just sometimes true?” This could make a world of difference for your students and for their future teachers!

Kimberly Rimbey, Ph.D., works with teachers and leaders to develop system-wide change in mathematics teaching and learning.