## Inappropriate Trimming ☆

One of the things that makes mathematics as a discipline relatively unique is the constant progression of ideas – that is, mathematics from elementary school onward continues to build on previous developments. As a teacher, this means that one of the aspects that mathematics teachers in particular need to attend to relates to the future development of ideas: presenting current ideas in light of these future developments. Doing so often involves a balance of presenting ideas at an appropriate cognitive level for the students in your classroom, but also in a way that still maintains the integrity of the concept. This is similar to what McRory, et al. (2012) describe as trimming: removing complexity while maintaining integrity. With that said, one of the ways that teachers’ knowledge of the mathematical horizon impacts their instruction would be in their attention to and phrasing of mathematical ideas. Below I highlight a number of examples of common adages or descriptions that, unfortunately, may lead students to some misconceptions – what could be considered examples of inappropriate trimming.

**“What you do to the top, you do to the bottom.”** This mantra is often repeated to help students generate equivalent fractions (something they notoriously have difficulty with.) Unfortunately, the adage, while true for multiplication, does not work with addition – which can be a source of confusion.

**“Just add a zero when you multiply by 10.”** Indeed, multiplying by ten in our base ten system is frequently easy than multiplying by other numbers. However, while it is often true, the result of multiplying by 10 is not always simply adding a 0 to the end of a number – 3.2 x 10, for example, is 32.

**“Multiplying makes bigger.”** In elementary mathematics, with natural numbers, this is often the case (not with 0 or 1, though). However, this notion may make students’ future work with multiplication of fractions more difficult, since this idea does not necessarily hold. (Example from McCrory, et al. (2012).)

**“You can’t subtract a larger number from a smaller one.” **

**“Anything to the zero power is 1.”** When students first have to expand their understanding of exponents to broader number sets, in particular to those that doesn’t make intuitive sense as “repeated multiplication”, there are many ways teachers try to help students learn these ideas. Why we define 5^0 as 1 takes some genuine work. And while most numbers to the zero power are one, both 0^0 and ∞^0 are indeterminate, and have some real implications in terms of developing calculus ideas from limits.

**“Perimeter is just the sum of all the sides.”** This idea works really well with polygons. However, for circles it makes no sense. Perimeter is the distance around a two-dimensional object, which may or may not be composed of straight sides. In fact, this may be part of the difficulty transitioning students to understanding how we calculate the circumference of a circle – the relationship is a multiplicative one (a comparison), not an additive one, where one can find the total from summing smaller lengths.

**“There are half as many even numbers as whole numbers.” **While it is true that half of the whole numbers are even and half are odd numbers, comparing the relative size of infinite sets is less obvious. In fact, based on bijective mappings, what we find is the the set of even numbers has the same cardinality as the set of whole numbers. In fact, it has the same cardinality as the set of integers, and even the set of rational numbers.

**“If it fails the vertical line test, its not a function.”** This is certainly true with graphs of functions on a Cartesian coordinate system. However, move to polar coordinates and functions start having very interesting shapes (lemniscates, cardiods, limacons, rose curves), very few of which pass the ‘vertical line test’. Functions have to do with every input having a unique output – on a Cartesian coordinate system, your inputs are points on a number line (not, say, angles), which makes the vertical test useful.

These are just some examples – probably only the tip of the iceberg. The larger point is that part of the mathematical work of teaching revolves around knowing the ways that the ideas we talk about as teachers get complicated in further developments, so that we provide proper attention to the details of how we describe and conceptualize them for students, making the overarching idea explicit rather than over-reliance on mnemonic devices. If you have other examples to share, post a comment!

Reference: McCrory, Floden, Ferrini-Mundy, Reckase, & Senk (2012). Knowledge of algebra for teaching: A framework of knowledge and practices. Journal for Research in Mathematics Education, 43(5), pp. 584-615.

Nick,

I completely agree that if you teach these simple tricks to students without teaching them why it works, they will make mistakes when they apply it to scenarios that don’t work. If you teach students a simple trick and then give them pre-selected problems that will allow the trick to work, then students develop a false sense of security that they know how to do something. This allows for short-term gains but also results in long-term misunderstandings.

However, some of the shortcuts are actually quite helpful when students are just learning concepts such as zero as an exponent or functions vs. non-functions. Within the confines of Algebra 1, these “shortcuts” work. While I’m not a proponent of teaching tricks and I am all for teaching mathematics in a meaningful way, I think that if these conclusions are presented in a way such that students discover the pattern that works for say 90% of the cases, an “asterisk” extension presentation of when it doesn’t work can give students a glimpse of why such shortcuts may not always work in higher-level mathematics. Thus students can continue to use the shortcut with the understanding that it may not work all the time, and if they forget the shortcut, there is another way to approach a problem without the shortcut.

Realistically not all my students will enjoy the extra explanation of why such rules may not always work, but I feel it’s my responsibility as a teacher to give all of my students as much information as I think they can handle so that math makes sense.

A common example I see is when students cross-multiply or use the “butterfly method” whenever they see two fractions next to each other. When students do not recognize that cross-multiplication is only appropriate for proportions, they will use it even to add and multiple fractions! If students do not understand that cross-multiplication is simply composed of the multiplicative and division properties of equality–which apply to equations, not simply operations involving fractions–they are more likely to misuse it.

Thanks for your comment, Diane. You’re absolutely correct, in many situations, some of these phrases can be useful for introducing ideas and conveying information to students. Finding ways to explain concepts to students is a crucial part of being a teacher, and in this regard, many of these could be useful in some situations. But the instruction and learning shouldn’t necessarily end there if its not the end of the story – understanding concepts, not tricks (like the butterfly method example you gave), needs to be the emphasis of instruction. And it is important to make sure concepts are introduced in ways that allow them the proper room to grow as students progress in mathematics, without running into contradictions.