Sometimes when I tutor I tell students that they are “hanging the in Math Cafe.”
I explain that when I tutor, I like to offer students a “menu of math options.”
Instead of showing students just one way to work a math problem, I try to present a “menu” of approaches, and I tell students that they need to listen so they can decide which approach works best for them.
Students tend to perk up when they hear that they are getting a variety of options. Since they will get a chance to choose the approach that best suits them, they become more interested in what I’m saying. Getting this choice boosts student interest and involvement, I’ve found.
Offering a variety of approaches helps students in other ways, too. For one, students start to learn that each approach has advantages and disadvantages. So students start to see that it might make sense to use Approach #1 in one situation, but better sense to use Approach #2 in a different problem. Encouraging children to evaluate the relative merits of different approaches engages them in higher-order thinking skills. And that in itself is a good thing.
What is more, encouraging flexibility of thought helps students develop other skills that serve them well not only in math, but in other subject areas, too. It could be argued that this helps them in life in general. Finally, presenting a variety of approaches helps more students find success with the math skill. That is because when students have a menu of options, they choose the option that works best for them. As a result, more students will do the math skill successfully.
I try not to worry about why one approach works better for a particular student than other approaches, although I am confident that this would be interesting to study in a controlled way somehow. I just am glad that when I present a variety of approaches, more students succeed in solving the problem.
I’d like to present one example of what it looks like to present a menu of math options. So here’s a simple example of three options for teaching students how to subtract two-digit numbers, with the possibility of re-gropuing.
OPTION 1) First I teach the “standard approach,” in which students re-group, trading one group of 10 for 10 ones. Then I show students how to move those 10 ones to the 1s place, and then subtract. This is the way I learned to subtract with re-grouping (called “borrowing,” back then in the ‘old days’).
For example, in a problem such as 42 – 27, students first demote the 4 of 42 into a 3, and then transfer 10 to the 2 of 42, making the 2 into a 12. Students then subtract, first getting: 12 – 7 = 5. Then, after that they subtract in the 10s place, getting: 3 – 2 = 1. Joining the 1 to the 5, they get their answer: 15.
OPTION 2) Subtract by “adding up.” So in a problem like 42 – 27, I teach students to think of the problem as representing a journey along the number line, with the numbers that end in 10 serving as “towns” along the road where they can stop and get a refreshment. In the problem of 42 – 27, students first travel 3 miles, from 27 to 30, where they get a tall drink. Then they travel 10 miles, from 30 to 40, where they eat lunch. Then they travel 2 more miles, from 40 to 42, at which point they reach their destination. Students just add up the three distances traveled: 3 + 10 + 2, and they get their answer of 15. This approach lends itself much more to “mental math” —doing the problem mentally, in one’s mind, without paper and pencil.
OPTION 3) For my third menu option, I have children subtract from left to right instead of the usual right to left approach. I also have students use negative numbers in a basic way. First students subtract in the 10s place: 40 – 20 = 20. Then they subtract in the 1s place: 2 – 7 = – 5. Putting the two partial answers together, they get 20 and – 5. They learn that this means that they should subtract 5 from 20, to get 15, the answer.
Some people might think that elementary students are too young to work with negative numbers. For the record, I have found that many elementary students are quite ready to work with negative numbers. I don’t even have to explain much about these numbers. I just tell them that when we take more than what we have, the answer is “negative.” I’ve been rather amazed at how many students seem to intuitively grasp this notion.
In any case, those are the three approaches that I usually present for this kind of problem. Of course, there are still other ways to subtract these numbers from each other, and all ways that work are valid.
The thought I’d like to leave you with here is that presenting a menu of options — a range of strategies — can really help students find math class more interesting. Give it a try if you have not yet done so. You might want to start with two options. See how it goes. As you continue with this approach, you will find more and more ways to teach the same math skill to your students, and your repertoire of approaches will grow.