## The Right Answer is Not the Thing

This is not another post on assessment, though the content is related.  The central theme in this post is faculty being wise about the process of helping students navigate through mathematics in an efficient manner (something we might call “learning mathematics”) 🙂 .

As context, I want to share part of a lesson from our math literacy course.  Like many such courses, we both use accessible situations and recognizing patterns in the learning process.  This particular lesson uses interest (simple and compound) with these basic steps in the process.

1. We deposit $500 in an account that pays 6% interest each year, on that$500.  Find the interest earned in the first 4 years by competing the table.  [The table shows a row for each year.]
Find the total money by adding the interest and the original $500. 2. We deposit$500 in an account that pays 6% interest each year, on the current balance (including prior interest).  Find the interest and current balance for the first 4 years by completing the table.  What is the total money for this account?
3. Which account results in “more money” for us?
4. We found the current balance by calculating “0.06 × 500 + 500”.  Is there a way to simplify this calculation so there is only one multiplication and no addition?

Of course, much time is used in the first two steps.  Students often have misunderstandings about percents, but these are motivational questions … as is #3.  However, the learning in the problem is all about the fourth step, which is looking for “1.06 × 500”.

Many teachers will present the 4th question in a manner that defeats the purpose of the question … “we added 6% to 100%; what do we get?”  This approach ‘works’ in that many students will see how we got the 1.06, and we feel good that they got the right answer.  Unfortunately, we just avoided all of the meaningful learning in this context.

First of all, students need to really know that percents do not have any meaning by themselves.  When we say “added 6% to 100%”, we have reinforced the basic misunderstanding that percents work like decimals in all situations.  It’s easy to determine if students have this misunderstanding by asking a variation of the classic question:

We had a 10% decrease in pay last year, and this year we got a 15% increase in pay.  Our current pay is what percent increase or decrease compared to the pay before the decrease?

This problem is tough for students because it does not explicitly state the core situation … that the base for each percent is the current pay … and we might think that this is the main reason we get the wrong answer “5% increase”. However, even when this fact about the base is pointed out, students continue to add the percents.

Secondly, the “we added 6% to 100%; what do we get?” question divorces the situation from the algebraic reasoning.  We’ve done adding of fractions, where a common base is required.  Somehow, with percents, we are comfortable leaving the base out of the problem when this produces more ‘right answers’.  Each of those percents has a base, which happens to be the same number in this ‘interest’ situation.  A more appropriate instructional move is to provide a little scaffolding:

Let’s write 0.06 × 500 + 500 this way:  (0.06 × 500) + (1.00 × 500)

Remember how we added 4x + 2x?  We got 6x.

Does that suggest how we might do the adding first?

Now, this instructional move will not make the problem easy.  The goal with this move is to connect the new problem to something fundamental in mathematics:  “like” things can sometimes be added.  Having the right answer without applying this concept is not learning any mathematics.

In our Math Lit course, this lesson introduces the concept of ‘growth factor’ which is then used as we identify sequences that are linear versus exponential.  That discrimination in sequences can get quite sophisticated, though we generally keep the level reasonable for the needs of the course and students.  The phrase ‘growth factor’ is used temporarily until we consider declining situations … however, this “adding to get one multiplication” is behind all exponential models.

Unrelated to the main point of this post, it’s interesting that many of us think of the number ‘e’ when exponential models are being discussed.  There are, of course, very good reasons why that is the most commonly used base in mathematics; unfortunately for the learning process, using base ‘e’ presents a disguise of the direct process involved in the situation … a multiplicative factor based on a percent increase or decrease.  I don’t see using ‘e’ prior to a pre-calculus course, in terms of helping students.

Back to the main point … whether you are teaching Math Literacy, Algebraic Literacy, or even the old-fashioned courses, “right answers” are a poor measure of the quality of learning.  The learning process itself needs to be richer and more valid than using a measure known to have limited validity.

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### 1 Comment

• By schremmer, September 26, 2016 @ 9:20 am

A succession of very good points.

The ultimate point, though, is that most of us think locally instead of globally while the point of a given course, if thought of in mathematical terms rather than in educando, will pretty much dictate how to deal with any particular issue.

Re e: I had always thought that the story of the inventor of chess was the most striking way to introduce exponential growth: when the king realized he would never be able to make good on his promise of 1 grain of wheat on the first square, 2 on the second square, 4 on the third square, …, the inventor got his head cut off.

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