My motto for this blog, and for my website, is “There is no one way”. It is a topic I have returned to many times, for example, in these posts:
- Catchphrases, where I mostly discuss the assorted slogans I have spouted over the years.
- How To, where I argue that there is no single “best way” to teach any particular topic.
- Eclectic, and the subsequent posts, where I challenge fads, but also say we can learn from them.
Of course, this does not mean I believe all “ways” are legitimate at all times or all ways are equally valid. For example, I’m a big user and promoter of learning tools. Shira Helft introduced me to the concept of the knife: a learning tool that is powerful and versatile, like a kitchen knife. (As opposed to specialized tools such as peelers, corers, slicers, pizza cutters, and so on.) Shira is right: teachers should prioritize multi-use learning tools, especially at the start of their career. Later, they can diversify, as I did. (I discuss this here.)
Still, I insist that it is a mistake to put all of one’s eggs in any one instructional basket. The main reason is pedagogy: no one tool, no one strategy, no one model works for all students, all classes, all teachers, all communities, every day. (See my cheerfully titled article: Nothing Works.) One’s growth as a teacher is largely about learning many ways to teach important concepts, so as to reach more students, and to deepen all students’ understanding. How many ways you can think of something is an excellent measure of how well you understand it.
Frequently asked question: doesn’t using multiple approaches confuse the students? Well, yes, it may. It is especially confusing when connections are not made between the approaches, or when the teacher’s understanding of an approach is superficial. But we run greater risks if using a single approach. Multiple representations (and thus multiple approaches) are dictated by the math itself. This is what I want to discuss in this post.
Here are some examples with whole numbers.
A good way to think about whole number subtraction is with the help of counters. If I start with ten counters, and remove three, how many are left? Another good way is to think of whole numbers as sitting on a number line. If I start on the 3 and count up to the 10, how many steps did I take? Or, I can use Cuisenaire rods: what is the difference between the orange rod and the light green rod?
For place value, I can use counters again, possibly using the “exploding dots” version, or an abacus. Or I can use base 10 blocks, which help the students see successive powers of ten geometrically. Or I can have students make their own base 10 materials using beans and tongue depressors.
For multiplication, I can use counters again: 3 times 5 can be represented as 3 sets of 5 counters. Or, I can arrange the counters in a rectangular array, which allows me to see that 3 times 5 equals 5 times 3, and previews the area model for multiplication. Or I do 3 hops of length 5 on a number line. Or I can learn to count by 5’s— a condensed version of the number line model.
The power of math is precisely that the same structure (in this case whole number arithmetic) describes many different phenomena. Understanding these representations, and their relationships with each other is a lot more powerful than teaching this subject in a single way. It respects students’ intelligence, and it is more true to the math: numbers are indeed all those things, whether you like it or not.
Is this confusing? If it is, it is because numbers are challenging, not because we approach them in multiple ways. If a teacher is comfortable with all the models, uses them strategically, and helps students think about the connections, multiple representations are illuminating, because they reveal the many meanings of the underlying math.
The same is true about fractions: the widely used “pie” representation provides a great connection to angles and to time as seen on a clock. A number line representation emphasizes that a fraction is a number, and reveals its relative size. A rectangle representation facilitates computation and helps to introduce the idea of common denominator (as you will see below).
What prompted me to write this post is that I recently expanded a page on my website about a particular representation of fractions that I suspect is not sufficiently used in upper elementary school: rectangles on grid paper. One question I was asked was whether it would be confusing to students to see yet another representation of fractions. I say: quite the opposite! This is an accessible model that can be the foundation of important discussions, and can be used for many different fractions topic. (It’s a fractions knife!) Of course, it should complement (not replace) other approaches. Here is an example from that article: which is greater, 2/3 or 3/5?
Using 3 by 5 rectangles (as suggested by the denominators) makes it easy to see both fractions as parts of the same unit. Counting reveals that 2/3 is 10 squares, or 10/15, while 3/5 is 9 squares, or 9/15. Which is greater has been made clear. Those very same rectangles can help us think about 2/3 + 3/5 or 2/3 – 3/5. Read the whole article (and watch my homemade videos) here.
Or take the use of letters in algebra. Sometimes, x is an unknown, and to find its value we learn to manipulate symbols. Sometimes we use it to make a general statement about algebraic expressions, for example 2x = x + x. Algebra manipulatives can help with that. Sometimes it’s a variable, and we see what happens to an expression as x varies. Tables, graphs, and function diagrams can help us understand that. Sometimes a, b, and c are parameters. They constrain an expression, but they are not unknowns or variables.
A well-intentioned but pedagogically clueless mathematician once commented that variables are easy to teach: just patiently explain to students that they behave just like numbers! In fact, the reality is that it is a good idea to use many representations: tables of values, graphs, manipulatives, symbol manipulation, and function diagrams. Done well, this is not confusing: it’s illuminating. The historically catastrophic failure rate in the traditional Algebra 1 course is largely caused by the naive belief that patient explanations of how to move x’s and y’s on a piece of paper is the only way to go, as long as you follow it by mind-numbing and meaning-free practice.
Or take trigonometry: the sine can be seen in a right triangle, or in a general triangle, or in the unit circle, or in a Cartesian graph. This is not because a teacher decided that. It’s in the math itself — and it gives us multiple approaches to the concept, with opportunities to make connections between them. (Here’s one such connection.)
Conclusion: Multiple representations reveal the many-fold meanings of math concepts. To make that less confusing, the answer is not to limit students to a single representation, even if you’re proud of how good you are at that particular one. Instead, give some thought to a proper sequencing of the representations, encourage students to use the representations that makes sense to them, and especially be sure to connect the representations to each other.