## A Case for Understanding

I’ve had several interesting conversations recently centered around decimal arithmetic and whether or not students should understand why the standard algorithms for decimal arithmetic produce correct answers.  For example, when multiplying .2 $\times$ 1.39 we should first think:

$2 \times 139 = 278$

and then think:

In .2 the decimal is moved over one spot.  In 1.39 the decimal is moved over two spots.  One plus two is three.  So we should move the decimal in our answer over three spots.  Our final answer is:

.278

Most of the objections to teachings why these algorithms work fall under a few (interconnected) umbrella reasons:

Reason 1:  It is too hard for many students to understand the why.  It is easier for them to simply memorize all of the rules (with no understanding informing their actions).

Reason 2:  There is simply no time to teach reasons why.  There is so much in the curriculum that teachers need to rush through things and the time necessary to explain why these algorithms work simply does not exist.

Reason 3:  We want students to do as well as possible on the end of year test (for NCLB) and to maximize our scores on this test we must focus on drilling the procedure rather than any sort of explanation.

There are VERY important reasons why we should want students to understand but what I want to focus on here is why I think teaching for understanding would like be  1)  Easier 2)  Faster  3) Would result in higher test scores.

Let us first be clear about why the standard procedures for decimal arithmetic work, and how this could be taught to students:  Decimals are a particular type of fraction, one whose denominator is a power of ten.  When viewed in this light, the “lining up of the decimal point” in decimal addition and subtraction is exactly finding a common denominator.  The “counting” procedure for decimal multiplication comes from multiplying the corresponding fractions and knowing that $(10^a) (10^b) = 10^{a+b}$.  The simplest way to teach students how to do decimal arithmetic is to simply tell them that they are a special kind of fraction.  Then when the student is told to compute $.2 \times 1.39$ they will do:  $\frac{2}{10} \times \frac{139}{100} = \frac{278}{1000} = .278$.   If students do decimal arithmetic in this fashion it is very likely that they will discover the shortcut (the standard procedure) themselves.  If they do not, they can be informed of the shortcut after they are comfortable with the conversion method.

Let us now discuss why this way of teaching (referring back to fractions) could very well be:

1)  easier for students

2) take less time in the classroom and

3)  result in higher test scores.

First, consider that with the new approach dispenses with three procedures (addition and subtraction of decimals, multiplication of decimals, and division of decimals) that students would otherwise need to learn.  This of course makes things easier for the student and frees up class time.  The learning of these three procedures is replaced with the the very easy skill of converting a decimal to fraction.  This is an easy task and a very important connection for the students to understand.  The two conversions that happen in each problem involving decimal arithmetic are a great opportunity to solidify the important connection between decimals and fractions.

Second, the new approach allows students to make sense of what is being done.  In addition to promoting a healthy conception of mathematics (things ought to make sense!), this greatly increases the likelyhood of students paying attention and thinking.  This actually makes things much easier for them and has the potential to raise test scores (remember that my focus is on these “practical matters” for this post).

Finally, teaching in this way provides the perfect opportunity to build multiple layers that will cement important knowledge in students minds.  First, students will get much more practice with fraction arithmetic and changing decimals to fractions, and they do this inside of a larger problem (this is good).  Second, and more importantly, the information about fraction arithmetic and the connection between decimals and fractions are crucially used in the subsequent demonstration (or discovery) of the shortcuts for decimal arithmetic.  It is a well known fact that when we use information to build further conceptual knowledge that uses that information, we are MUCH more likely to remember and correctly use that information later.  I think that this would have significant effects on students test scores for test questions that deal with either fractions or decimals (this is most decidedly NOT an exclusive or).

I recently had the opportunity to observe a 6th, 7th, and 8th grade classroom at a local middle school.  I looked in the 6th grade math book, the 7th grade math book, and the Pre-Algebra book.  In all three of the books, fraction arithmetic and decimal arithmetic were taught.  In ALL THREE OF THEM decimal arithmetic came FIRST.  I was astounded.  If one is following the textbook (and decide not to refer to what occurs in subsequent chapters of the book), one cannot teach decimal arithmetic in the way that I have outlined above (which seems to me the most mathematically natural way to teach it).

One last comment.  The state of Illinois has a very nice web tool called the Illinois Interactive Report Card.  In looking at several middle schools and the high schools that they feed into I noticed the following (disturbing) trend:  Since the year 2002 (the first available data), middle schools show a steady increase in the percentage of students that meet or exceed standards.  Over the same period of time, the high schools that these middle schools feed into show an unsteady decrease in the percentage of students that meet or exceed standards.  I propose the following explanation (this may not account for all of it, but I suspect it does account for some):  In middle school there are few enough things that simply memorizing the procedures for everything can be an effective strategy (if the end goal is to do well on the state test).  In high school things there are too many topics discussed (and they are complicated enough), that this strategy simply will not hold up as well for the general population.  Of course this strategy does hold up for those who deem it important enough to spend lots of time and effort on “studying” (unfortunately I have usually found that studying focuses much more on recall than on reason).

I suspect that if students in middle school were more accustomed to thinking (and less accustomed to recalling procedures), there would be a much greater chance of them thinking and understanding what was taught to them in their high school classes, and those test scores would also increase.