# “Table” method for expanding brackets vs “each term in the first bracket gets multiplied by each term in the second bracket”

Hi I've just discovered mathseducators stackexchange.

As a maths tutor in the UK, I am irritated that some of my students - particularly GCSE and sometimes below - use the table method for expanding brackets, rather than "each term in the first bracket gets multiplied by each term in the second bracket", the latter method being what I'm familiar with.

I also prefer my method because:

a) expanding brackets using my method is quicker. Drawing a table is unnecessary and tedious.

b) It is easier to see that my method can be derived from repeated applications of the Distributive Law (although students usually don't care about stuff like this).

c) It is easier to simplify algebraic expressions, for example showing where the "2b" comes from when completing the square.

Maybe it's the way they are teaching in schools nowadays, but the table method seems to me to be unnatural and unnecessary. My students insist on using it, however, and any attempts to get them to use my method are shut down because they are so used to their method being the "right one". They know theirs works, so they don't need my method. But then, for example, teaching some of them how/why "completing the square" works is more difficult if they only use their table method.

So I guess my question is, should I try to get students to use my method and/or abandon their method, or should I accept their method and get over it?

I think what makes this question different to usual "should I accept their method (assuming it works) or get them to use my method?" questions is that, they have probably used their table method for many years and for them, it is probably a fundamental part of multiplication.

• You said you are a tutor. What method is their teacher using? (Or am I confused about the use of the term tutor?) You would definitely want to work with what they are learning in class, if I understand your situation correctly. I give a fuller answer below about the value of using area models. – Sue VanHattum Oct 6 '19 at 17:47
• @SueVanHattum They are using the “Table method” as in the link in my question. – Adam Rubinson Oct 6 '19 at 17:56
• It is connected to the area model. Making that connection clearer may help your students more than trying to push them (before they're ready) to a purer algebraic method. (See my answer below.) – Sue VanHattum Oct 6 '19 at 17:58
• I wonder why the rule I use, namely “each term in the first bracket is multiplied by each term in the second bracket” isn’t sufficient for everyone. It was sufficient for me, and seems succinct and simplest. Is this rule ambiguous or unclear? After all, algebra is somewhat abstract. You can’t graphically draw: x^2 -3x + 6 = 9x^2 + 3x - 7. You have to manipulate this equation using algebraic rules/laws. So why do we have to go all graphical when expanding brackets? Surely in order to expand brackets, which is just something else in algebra, you just need to learn another rule. – Adam Rubinson Oct 6 '19 at 18:22
• However, It is probably a good idea to make a graphical connection before making it abstract. I think I see your point now – Adam Rubinson Oct 6 '19 at 18:32

## 3 Answers

I teach in the U.S. at a community college. Although I prefer distributing without drawing an area box when I'm doing math myself, I often show the box in class to help students see how things work. When we use an area model to help students visualize the workings of the distributive property, it makes much more sense for many students.

In fact, the box seems to help them understand completing the square. You may want to watch a few of James Tanton's videos on this. (Completing the Square - Part I is here.) The 2b term is the diagonal of the area box.

I have found that his method of deriving the quadratic formula, which avoids fractions, is much easier for students to understand than the method as usually presented in textbooks. He completes the square with his area boxes in this video also. I think it is lovely.

There is much evidence that having a visual understanding of algebraic procedures deepens understanding for many students. (Possibly for everyone?) I understand that Jo Boaler has done some work on this.

It may help you to shift if you discover how many of your students don't really get the distributive property, especially when it is used in more complicated situations

• In James Tanton’s example for completing the square in the video, how would you explain what x=-6 means graphically? Wouldn’t that confuse students? – Adam Rubinson Oct 6 '19 at 18:16
• @AdamRubinson Negative lengths do test the "reasonability'' of the area model. I have never seen anyone else do this, but one can make sense of signed length and area'' by thinking of rectangles as being oriented by the righthand rule. Each length gets marked with an arrow which is either up/down (+/-) or right/left (+/-). Then the sign of the area is determined according to the right hand rule. This convention agrees with how we do line integrals. An example of higher level math providing a useful model for elementary math! – Steven Gubkin Oct 6 '19 at 19:35
• It's true that it doesn't make complete visual sense with negatives. But I find that hasn't bothered my college students, and that it still helps them see how/why the distributive property operates. – Sue VanHattum Oct 6 '19 at 22:35
• I always liked this guy: youtube.com/watch?v=7oRgA0ChiIY – Rusty Core Oct 8 '19 at 16:31

I run into this issue frequently. As a high school in-house math tutor, students visit to show me their quiz/test scores and ask about their work.

The FOIL method is fine, if it works for the student. For those who are prone to making mistakes, I show them the Box method (call it what you will, that just my name for it). The benefit, if any, to this method is that it feels less prone to error, as whatever number of terms there are, each cell must be filled in.

For me, the ends justify the means. When a student returns, and offers evidence of success, or, at the time I offer this method, blurts out “that makes sense, why doesn’t my teacher teach it this way?” I can only trust my gut, that students are not a homogeneous bunch, and that we don’t all learn the same way. Again, call FOIL (distributing) the preferred method, but keep an open mind, there’s more than one way to present a concept.

So I guess my question is, should I try to get students to use my method and/or abandon their method, or should I accept their method and get over it?

To clearly answer this - If students are successfully using a particular method, pushing them away from it can be harmful. There are teachers for whom "the method" is very important. For example, when teaching the various ways to solve two equations in two unknowns. I've seen tests where the instructions rigidly request each method to be demonstrated. If the teacher feels their instructions are being ignored, you may have created an issue that was avoidable.

On reflection, offering a student what you feel is a superior (in this case, faster) method, also depends on the student. Those functioning at a higher level may always be seeking tips and tricks to improve their skills. At the lower level, I'd highly suggest mirroring the teacher's method.

To demonstrate the distributive law, we often use an area model. Then $$(2+4)\times(5+7+9)$$ can be visually decomposed into the sum of 6 subproducts.

Rather than actually drawing a grid, you might eventually start just labeling the edges and the products, without really caring about the relative sizes of the pieces.

Finally, this table method'' could appear as a final abstraction.

So students may prefer this table method because it is the end result of a logical progression. Your more algebraic approach may feel more disconnected from the way they are actually reasoning about these problems.

It is a worthy goal to show them your alternative approach. The more abstract method of applying algebraic laws formally is a very useful mathematical skill.

Just introduce it for what it is: an alternative method which will help them build some new skills. They may not be as comfortable with it at first, but promise (and enumerate) the rewards of this new and harder path. More importantly, explore the connections between the two approaches! Show how each line of algebra can be connected to a statement about areas, or to an aspect of the table.

Good luck!

• Do you use "area model" and "table method" with university students? – Rusty Core Oct 7 '19 at 17:00
• @RustyCore When I teach future elementary school and middle school teachers I do. – Steven Gubkin Oct 7 '19 at 19:45