In my own private tutoring I often work with students who are confused by their math lessons. These students are in grades 1 through 8 and come from both public and private schools. Some of these students have clear learning difficulties such as a language-processing problem like dyslexia, or attention-deficit disorder. But many have no such difficulties, and indeed, have mastered much other academic material easily and so should be relatively easy to teach.
This anecdotal and limited clue that there are some problems with math instruction generally is echoed by several broader measures, including a failure to improve 12th grade math scores over the last 20 or so years in spite of considerable national efforts to do so, persistent mediocre to poor performance of U.S. students on internationally administered math-tests, and the content of remedial math courses offered at colleges and the number of students who have to take these courses. I might add, as well, that of all the school subjects, it is when helping their children with math that there is the most difficulty and confusion and, let’s be honest, tears.
In order to help my struggling math students, I continually study the research literature regarding learning and cognition, and I also study and use a wide variety of specific math programs. I have distilled what I’ve learned from this study, as well as from the successes and failures that I’ve had with my students, into 8 guiding principles for evaluating and constructing efficient and engaging math lessons.
I’ve been thinking about these principles almost daily over the last couple of years as I have been writing the new ABeCeDarian Fractions Books. As part of the launch of these new materials, I will be writing a series of blog posts to share these principles. This blog is the first of this series. I hope you will find that these ideas give you a useful perspective for understanding any struggles your math students might encounter, as well as give you some tools for helping overcome their confusions or avoid unnecessary confusions in the first place.
Principle # 1: New concepts in arithmetic should be presented initially as manipulations of quantities and described in ordinary language
Mathematics consists of abstract patterns and relationships. The equation 2 + 3 = 5, for instance, doesn’t mean simply that if we combine 2 pencils with 3 other pencils we have 5 pencils, or if we have 2 dogs and 3 more come along, there are are dogs. It means that if we combine 2 things of any type with 3 other things, we invariably have a total of 5 things.
One of the amazing and powerful aspects of mathematics is that we can represent these abstract patterns by symbols that can be manipulated without reference to physical things.
For instance, when people use the procedure for “borrowing” when doing pencil-and paper shortcuts to do calculation, they are following a procedure, a set of steps, that allows them to manipulate the symbols productively without paying attention to the quantities the symbols represent. For instance, I will describe the steps of the standard procedure for calculating the difference, 63 - 24
We can’t take 4 from 3 so we add 10 to the 3 and calculate 13 - 4 and write the result, 9 in the ones place. Then we look at the tens place, where we no longer have 6 tens but 5 tens because we’ve put one 10 with the 3 ones, so we calculate 5 - 2 and put 3 in the tens place.
All of these steps, of course, are based on valid number relationships, but the number of steps and the abstractness with which they are expressed is, I hope you can appreciate, rather daunting, especially for a 7-year-old. We know that this is the case, moreover, when we look at the common errors students make when they are taught this procedure and the length of time and amount of practice they need to stop making these errors. An extremely common error, for example, is to come up with 41 as the difference between 63 and 24. The error here, of course is that the student subtracts the 3 ones from the 4 ones, even though she should do the opposite.
The relationship and sequence of these steps are much easier to understand for students if before confronting this procedure they have ample opportunity to perform this type of subtraction as actions on models such as snap cubes, base ten disks, and number lines. Throughout exploration of this kind, the quantities involved are always at the fore, and the operation is simplified to 2 steps: removing some cubes and then counting the remainder. Not only are these steps simpler to follow, they are much more familiar and hence comprehensible to the student because it is very much related to some of their ample, non-school experiences.
There is another crucial difference between focusing too soon on presenting calculation procedures on symbols versus allowing students ample time to explore a type of calculation by manipulating quantities, a crucial difference in the relationship of the student to the subject matter. In the former type of instruction, the teacher demonstrates a procedure, and the students mimic it, while in the latter, the students are given tasks (count out a certain number of cubes, remove some cubes, etc.) and then, to reflect on their experiences. In the one case, the emphasis is on remembering without much opportunity for sense-making, and in the other, the emphasis is precisely on doing things and then analyzing them, looking for (and uncovering!) patterns and relationships. I will be talking more about this crucial difference in the future blog posts as well.
The difficulties I have outlined with regard to teaching the “borrowing” procedure, exist as well for most of the K-8 curriculum, including multi-digit multiplication, long division, and all the calculations regarding fractions, decimals, and percents.
I suspect it is not too difficult for readers to see the underlying action of subtraction, namely, the removal of a quantity. Identifying the underlying manipulations of quantity at the heart of some more complex mathematical ideas in the K-8 math curriculum, however, may be less obvious or familiar. For example, what manipulations of quantity are represented by the concept of simplifying fractions or the concept of dividing a number by a fraction? I am not referring here to understanding a procedure to correctly calculate how to simplify a fraction or divide a number by a fraction, which most adults know how to do, but rather, understanding what specific manipulations of quantity these concepts involve, in the form of sorting, grouping, categorizing, and counting things. I suspect that for many readers this task will be difficult.
In my next blog post I will share some ways to represent the underlying actions of simplifying fractions and dividing fractions. This is extremely important because not only these concepts, but indeed all of the content of K-8 mathematics can be represented in similar ways, and, if I'm right, this sort of representation is critical to teaching the ideas efficiently, with little or no student confusion.