This is the first in a series of posts from myself and Michael Pershan about teaching integer operations. In this series, Michael and I are going to discuss the many contexts and strategies that teachers use to teach integers. We will then try to categorize integer problems into several problem types and analyze how these problem types are explained using these different contexts and strategies. Hopefully along the way we will actually learn how to teach integers, but no promises on that count!
Integers feel like a topic in middle school math that is awash in metaphors, visual strategies, and quick tricks. It can be hard for a teacher to figure out which contexts and strategies will provide students with the best foundation for a conceptual understanding of integers. Specifically, some of the metaphors for integer addition become quite confusing when they are used to explain integer subtraction.
Personally, I have always felt like I get my students about 80% of the way there with my various analogies and number line strategies, but I’ve never finished an integer unit with the feeling that I had nailed it. I am hoping that by deeply investigating integers, I will find something useful to use in my own classroom.
But first: The four major contexts I have found for teaching integers.
Elevation problems often place the student in the mind of a rock climber, scaling cliffs that stretch from deep canyons below sea level to high altitudes. Other elevation problems discuss submarines diving beneath the sea and helicopters rising up above the ocean. At other times, students are asked to compare the heights of various mountains with the depths of various ravines and trenches.
One particularly evocative elevation context involves a hot air balloon basket that is held up by balloons and weighed down by sandbags. If the balloons and sandbags equal each other, the basket remains at its baseline height (zero, for the purposes of the problem). Adding a balloon causes the basket to rise one foot, while adding a sandbag causes the basket to fall one foot. Removing a balloon causes the basket to fall, while removing a sandbag causes the basket to rise.
Temperature is a common context that teacher use to introduce students to integers, in part because negative temperatures are some of the few negative numbers that students encounter outside of the math classroom. This prior knowledge is less substantial in states like Alabama, where we are more used to temperatures with three digits than those with one. Regardless, most students have some prior knowledge about temperature and know that 23 degrees below 0 is colder than 8 degrees below 0.
More abstractly, some teachers use the invented context of “hot and cold cubes” which are fictional cubes that either cause or lower the temperature of water by one degree. This context lends itself to student investigations with manipulatives.
Money problems are common because their application to the real world is so strong and resonant. Every child has some prior knowledge about earning, spending, and owing money. Even though they may not have formal experience with debt, they are familiar enough with borrowing money from their parents to understand the idea.
Money is also useful because money can be earned and spent, and debts can be created and forgiven. These varied scenarios seem to provide a contextual support for many types of integer problems. Some teachers allow students to create a budget for themselves and keep track of the debits and credits to their savings over time.
Piles and Holes
James Tanton uses the metaphor of piles of sand and holes to teach integers. Piles represent positive numbers, while holes represent negative numbers. This image helps understand integer addition such as 5 + (-3), since every pile can be used to fill in a hole. Also, piles and holes can be removed, which can represent subtraction.
I have found a couple of examples of teachers using James’s analogy, but it definitely does not feel as widespread as the other three contexts. I include it because I think it would be valuable to compare the most common contexts with a context that is new to most people.
I have found a couple of other contexts, such as “good things” and “bad things” or golf scores above and under par, which were less common or relied on more specific prior knowledge. I can’t imagine a lot of my students knowing much about how golf is scored, for example.
So there you have it! If you think of any other contexts that might be useful, please add them in the comments.