In backwards mapping a unit on integers with a new teacher, I went through the following thought process:

- Students need to be able to solve problems involving integers
- They need to remember integer “rules” by understanding them
- They need a context: Hot Rocks & Cold Rocks (teacher guide & slides)
- They need manipulatives to represent the Hot Rocks & Cold Rocks: the unit ones (constants) in algebra tiles
- They need pictorial representations of hot rocks & cold rocks (chip models)
- They need a construct to transition to the full symbolic representation so they can reference hot rocks and cold rocks when working; I use drawing circles around the terms to activate hot rock /cold rock thinking
- They need to be able to do the mathematics when presented in a full symbolic representation perhaps by thinking about the context provided, hot rocks & cold rocks

My process for teaching integers will be the focus of a future presentation; however, let me make a couple of points before getting to THE point, how I introduce integer operations. Notice the use of multiple representations & context. Context provides a foundational for conceptual understanding and aids in retention/recall by making connections and providing a single focus point for students’ memory – I remember the one context, not four integer rules. The other, teaching through multiple representations, is one of NCTM’s Effective Mathematics Teaching Practices.

When I first starting teaching, I made some mistakes on my quest to help students do & understand the mathematics. I would teach with multiple methods, not multiple representations. I found introducing a topic and showing various methods too quickly confused some students. Notice in the bulleted list above that each of the multiple representation is a *representation of the same thing*. It seems obvious; however, early in my teaching I found it easy to get off track saying, “Think of it this way… or this [other] way…” rather than building understanding of how to think of it one way through multiple representations before moving on to connect and explore others methods or ways of thinking. For example, I want my students to fully understand integers via a “chip method” before moving on to number lines. In fact, I transition them by drawing the hot rocks/cold rocks (chips) over the number line.

One more digression, some argue against using a chip model for the very reason I did not use the chip model to teach all of integers. It seems to break away from how people really think of it just to make the model work. I have found that people usually think of integers two or three ways when I ask them, they either remember the rules or think of zero pairs & the chip model, possibly with a couple of number lines thrown in. So, if those who understand it think of it as a chip model, then what is the problem?

The problem is exercises like 2 – (-3) and others where you are adding zero pairs just to take some chips away. The argument is that the chip model is too confusing at this point and fails. I agree, especially in the following case. Consider 2 – 3; I would never add one zero pair to solve this problem mentally. Instead, I “see it” as 2 and negative three. Back to 2 – (-3). I used to simply move to another level of abstraction before dealing with these types of problems by having students see it as negative meaning opposite, thinking of it as 2 and the opposite of negative three. Many adults I have asked seem to think of it this way (besides those clinging only to rules). We dealt with this issue in Hot Rocks & Cold Rocks using it as a way to realize that subtracting a negative is the same as adding a positive then we move away from it as quickly as possible once students come to that realization. In fact, we teach both ways of thinking, chip models (seeing it as subtraction in a context) & seeing the sign as a negative, concurrently. A final point about the chip model, it is strengthened by providing a context.

Enough digression! How do you introduce integer operations?

I would argue the “digression” above actually helps inform the answer. The hardest thing for students to understand in the unit on integers is usually subtracting a negative. Notice I said understand. If you are teaching just integer rules, the hardest thing for students is remembering them. By the way, two good resources for the issue of knowing the difference between the “rules” is moving right into teaching problems like 2(-3) – 4 where both “rules” must be applied in the same expression like those found in Russel F. Jacobs Designs in the Coordiante Plane and Symmetry in the Coordinate Plane. The X Marks the Spot activity in Bill Lumbard’s & Brad Fulton’s Simply Great Math Activities: Number Sense is another way to have students internalize both “rules” at almost the same time. Yet another digression; is the suspense killing you?

So, I was looking over what we had laid out before:

- Students need to be able to solve problems involving integers
- They need to remember integer “rules” by understanding them
- They need a context: Hot Rocks & Cold Rocks
- They need manipulatives to represent the Hot Rocks & Cold Rocks: the unit ones (constants) in algebra tiles
- They need pictorial representations of hot rocks & cold rocks (chip models)
- They need a construct to transition to the full symbolic representation so they can reference hot rocks and cold rocks when working; I use drawing circles around the terms to activate hot rock /cold rock thinking
- They need to be able to do the mathematics when presented in a full symbolic representation perhaps by thinking about the context provided, hot rocks & cold rocks

I thought to myself, “Where is the place for student thinking?” Yes, we had given the students tasks to complete surrounding the manipulatives and multiple representations; however, we had not given them any rich math tasks to examine their thinking on the subject. I would argue tasks like this fit anywhere especially as embedded formative assessments; however, it is great to explore how students are thinking at the beginning of a unit so you can connect your instruction to it, which is the true definition of scaffolding. If the hardest thing for students to understand in the unit on integers is subtracting a negative, then why not start there? Hit the issue dead on from the beginning! Therefore, I created the following task using the Badwater Ultramarathon and Mt. Whitney as a context to explore the issue. Directions are in the notes of the slides. THIS is how I would start a seventh grade unit on integer operations. See I finally got there!

Let me know what you think. I would love to hear about what cognitive dissonance it creates in your students, and how you are able to refer back to it when teaching “double negatives” or subtracting a negative.

Hoping this meaning creates memories,

Dave