I’ve been teaching some courses at Drexel University, and in those course we really focus on the language of mathematics and using students prior knowledge to help them make connections and build on their mathematical understanding.
In a current course, we are exploring integer addition and using manipulative’s to provide both a visual and concrete connection to the idea of creating zero pairs, and then progressing to the more abstract addition of integers without manipulatives and how do you support students understanding and language. What has come up frequently is the terms ‘cancelling out’ and ‘disappearing’ and ‘opposites’ to explain or help students understand that creating zero pairs allows you to use the additive identity property. But – what’s really happening, mathematically, is that we are using additive inverse (opposites) to create these zero pairs, which are NOT cancelling out or disappearing, but instead, are creating the quantity zero. And, once we have created this quantity zero, the remaining value can be added to those zeros using the additive identity property. Cancelling out or disappearing implies they don’t exist, which, when we expand the idea identity into multiplication where we use the multiplicative inverse to create a 1 (not a zero), cancelling out really seems confusing.
Seems like it shouldn’t make a difference, but think about it – if we use terms like opposites, cancel out, or disappear in one grade, and then the next grade or future courses, the same ideas are referred to as additive inverse, zero pairs, additive identity property, students will be confused and think they have never seen these concepts before. While it is very important to use language students understand to start the process, I think as mathematics teachers, no matter what grade level, we need to model proper mathematical language in conjunction with the ‘student-friendly’ terms we tend to rely on or fall back on. This way, students can relate, but then learn, build on, connect & utilize mathematical language so that we are all communicating on an equal playing field.
Something as simple as -3 being referred to as negative 3, minus 3, or even 3 negatives. That’s confusing. Let’s learn to be consistent with our mathematical language and use the correct vocabulary – both in our own teaching, but also in what we expect to hear from our students. So if a student says that the opposite of 3 is -3 (positive 3 and negative 3), let’s acknowledge that they are thinking correctly, and in mathematics we refer to that as an inverse. This will then help them make that connection when we talk about inverse operations. Consistent mathematical language supports students understanding as they progress into more abstract mathematics.
We want our students to communicate mathematically with the language of mathematics and become proficient mathematicians. Let’s then make a conscientious effort to use and model correct mathematical language instead of the ‘short-cuts’ or ‘simplified language’ we tend to use. Again – important to start out with this type of language to help students connect prior knowledge, but then more important to model using and support students use of, correct mathematical language. I think it would go a long way in preventing some confusion students experience as they move from grade to grade or course to course.