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.

Back when I was pursuing my teaching degree, I learned that students learn by doing. As we studied Bruner and Piaget and other great thinkers who influenced mathematics teaching and learning, we learned about three stages in which students represent their mathematical thinking:

Since then, I’ve learned that “students learn by doing” is only a partial truth. I’ve come to discover that students learn by thinking about what they’re doing (Kamii, 2007). It’s in the reflection that meaningful learning takes place, not simply in the “doing” itself. Furthermore, the three stages of representation aren’t really stages at all. They are classifications, or categories, of representation. They do not necessarily occur sequentially, nor is one category of representation necessarily more sophisticated than another. Instead, the thinking behind the representation is what determines the level of sophistication.

To that end, I’ve adapted an existing framework to explain how students represent their thinking. The Lesh Model, pictured below, includes the three original representations — concrete (physical), pictorial (visual), and abstract (symbolic). And it also includes two additional categories: verbal and contextual. The addition of these two categories introduces more inclusive ways in which we use math to describe the world around us.

 

 

WHAT’S THE POINT HERE?

We see from the diagram that each representation is connected to every other representation. Since each category is connected to every other category with a two-way arrow, one can see that the categories interact with one another rather than occurring in a set sequence. It’s when students make these connections between and among representations that true learning occurs.

Let’s take a closer look at the five representation classifications:

• Physical representations use concrete objects to facilitate thinking about thinking about, representing, and manipulating math ideas. Concrete objects may include tiles, counters, paper strips, etc. Note that just because an object is physical does not mean that it concretely represents a mathematical idea. For example, coins are physical objects, but they do not concretely represent their mathematical values.
• Symbolic representations convey ideas through formal math representations such as numerals, equations, variables, and other symbols.
• Visual representations such as pictures, sketches, diagrams, charts, number lines, graphs, etc. use visual means to promote thinking about math ideas.
• Contextual representations situate math ideas in real-life, every day, imaginary, or math contexts.
• Verbal representations use oral or written language to describe, discuss, interpret math ideas.

WHY DOES THIS MATTER?

When looking at the Lesh Model, you probably focused your attention on the five representational displays: physical, symbolic, visual, contextual, verbal. The real power of the diagram, however, lies with the arrows that connect those representations. With the added arrows, the Lesh Model, originally called the Lesh Translation Model, emphasizes the ways in which the different representations are connected. The arrows call our attention to the notion that meaningful learning takes place during the thinking, processing, and reflection upon what one has done. By connecting representations, students develop and display their thinking in new ways.

TIPS FOR TEACHERS:

• Take inventory of the supplies available to students that encourage engagement in all five types of representations. What manipulatives are available? Have you provided students with a variety of writing materials? Are there different types of paper available to them? Is there opportunity to contextualize math — both in your discussions and on their paperwork? Is there a list of various symbols and their meanings posted somewhere in the room?
• Create anchor charts for each of the five categories of representation. Be sure to list or add pictures of many examples for each category.
• Frequently ask students to verbally connect multiple representations. Put a Lesh Translation Model up in the room and have students trace the arrow that shows the connection they are making (e.g., trace the arrow between physical and visual when explaining the connection between a fraction tile representation and a number line sketch).

TIPS FOR LEADERS:

• Lead a conversation with your teachers about the different categories of representation. Ask them to solve a math problem together and then lead a discussion about how to connect the different representations to one another.
• When observing a math class, notice if students are connecting representations in addition to simply following a process of using manipulatives the way the teacher demonstrated.
• Ask your teachers if their inventory of materials for each of the five categories of representation is lacking. Are there any tools that need to be purchased or located to support mathematical representation?

More on this topic of representation in the coming weeks…stay tuned!

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