There is no alternative answer for 4 2/3 x 3 5/8

My 13-year-old son had given me reason to believe the results from his latest Math quiz were not going to be good. They were not. I began to review the test with him and nearly fell out of my chair. Not because of the work my son had done, but because of the test itself.

Several of the questions read, "answer the following questions using a complete sentence". I had to check again - yes this was a Math test. When I got to the fourth question I really couldn't believe it. The question can be summarized as follows:

Tommy computes 4 2/3 x 3 5/8 and gets 12 10/24 as an answer. Mary is correcting Tommy's paper. What should Mary discuss with Tommy?

The question seems to beg for wisecrack remarks. Perhaps Mary wants to ask Tommy to the prom. Or maybe Tommy should consider a career in art? I asked my son to compute the answer. He did not do it correctly. On his test his rather garbled response had been credited with 5 points. I would have given him a zero for what he actually computed in front of me.

Forgetting about the test itself for a minute I began to work with my son on the actual material on the test. He actually knew a good bit of it - knowing how to reduce fractions and convert mixed numbers into an equivalent improper fraction. But he did get mixed up in several areas. So I have no quarrel with the school system about my son's grade, but the wording of the quiz really bothered me.

I wrote a letter to the teacher explaining my objections. By asking for complete sentences I thought she was penalizing students who may be good in Math but not good at expressing their thoughts accurately in sentences. And with only words to grade, the teacher cannot correct in the student's work precisely where the student may have made an error. It is this personal feedback (seeing your mistake circled in red with a -10 on it exactly where the error is) that makes the test a learning experience.

Even more objectionable is the fact that the student could possibly do quite well on this question (repeating some words he or she remembers hearing in class like "convert to improper fractions") and not actually be able to do the problem. I went all the way through advanced calculus in my days at school, and many things that seemed clear in the class when the teacher was explaining them suddenly were not so clear when I had to do a problem myself. That is where the real learning process began. To be able to say something in a sentence about a problem is not anywhere near the same thing as being able to do the problem and get the correct answer.

Certainly this type of question does not do the teacher any favors either. I have seen some of the sentences that middle school kids write nowadays - they are not very often good examples of clear concise expression of ideas. So how do you correct a test question like this?

In response to my letter, the teacher informed me that this question was part of a program existing in Fulton county (and many counties around the country) called alternative assessment. In this program, only a limited amount of any test can be to ask students a "simple" question. An example of a simple question is: "compute 4 2/3 x 3 5/8 ". But by wording the question as she did, she gives the students an alternative way to express their understanding of math - besides simply computing the answer. I was told 20% of any test in my county must be alternative assessment questions. Until we are sure our kids can get the simple answers correct, I wish we would stick to the simple questions! There are no alternative answers for 4 2/3 x 3 5/8!

There is a campaign in education circles to use alternative assessment techniques in schools everywhere. I was told Fulton county has been using it since 1991.

The Ph.D. in Math at my county's board of education assured me that although this was not the way our generation was taught, there was ample evidence in education research supporting this concept. I remain skeptical. The best way to measure a child's ability to multiply mixed numbers is to have them do it.

I agree that children learning math must be taught in such a way that math problems are learned conceptually - so the subject does not seem to be a myriad of confusing rules, formulas and solution procedures. The subject of mathematics is really simple logic - and it is a shame that so many children just never see that. But the solution to this problem is not to change the test so kids simply attempt to write this logic in words.

In real life there is no substitute for getting the correct answer. That is the true measure of how well we have learned the logic. In our schools we should demand nothing less than a correct answer from our children.

Stacy Loesch
stacyloesch@alum.mit.edu

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