Monday, March 23, 2009

An "Average" Looking SAT-Type Problem for Middle Schoolers Too!

With juniors preparing for the May or June SAT (many already took it for the first time in March), I plan on having several posts dedicated to the kinds of questions one often encounters along with a discussion of math and test-taking strategies. The math content of many SAT questions is middle school level although the level of reasoning, the wording and the symbolism raise the bar higher.

Why so much focus on this standardized test? My contention has always been that these questions provide our math teachers with endless material for asking more higher-order questions, promoting reasoning and thinking 'outside the box'. They should not be thought of as 'taking away from the curriculum', rather they enhance the curriculum. Most importantly, questions like these should be integrated into regular textbook assignments. They need to be inside the assignment not placed at the end of the section or end of a chapter in a separate section (aka, Standardized Test Practice) or in a supplementary book.

The other central point about using these kinds of questions is to think of them as more than a warmup or SAT review before the test. Each of these questions can help students develop a deeper understanding of fundamental mathematics and therefore is integrally connected to the curriculum. Students often view these questions as something different from what they learn in school. Instead of applying the knowledge they've gained from the classroom, they abandon what they know. Test-taking strategies are fine but these should complement actual mathematics, not replace it.

The table below shows the relative population of students and average GPA by grade level at Standardized High School.

Avg. GPA

What was the average GPA for all students in the school?

Please click Read more to see the answer, suggested solution and more discussion.

Answer: Students must "grid in" 3.02 or 3.03

Suggested Solution:

"Weighted average" method: (0.28 x 2.85) + (0.24 x 2.74) + (0.22 x 3.34) +(0.26 x 3.21)

  • You might ask students why the process of adding the four GPAs in the table and dividing by four is incorrect here. Unfortunately, this incorrect method produces 3.035 and if the student doesn't round they would grid in 3.03 and receive credit! Hopefully, the testmaker would catch this and adjust the numbers slightly to catch this student error.
  • Would students have less difficulty with this question if the actual number of students in each grade level were given, rather than percents? Should this non-percent version be presented first when teaching this topic? I would think so. The problem in this post is not intended to be introductory. The instructor could deal with the percents by assuming a total school population of 100 students and proceed from there. The weighted average method shown above is more sophisticated. By the way, does it remind anyone of the concept of "expected value"? Make those connections! (when appropriate of course).
  • IMO, middle school students should be introduced to the ideas of weighted averages early on. Is this standard curriculum in 6th, 7th or 8th?
  • On the actual test the student might see many variations of this problem. For example, the data could be given in two separate displays: The % distribution of students by grade level could be presented in pie chart form and the other data in table form.
  • Can you think of variations on this problem? Do you have a good source of these? Are these questions designed primarily for the accelerated students? The Honors students? The Math Contest crowd?


jd2718 said...

So, I used to do these sorts of things. I would think, "28%, that could be anything from 27.5% up to but not including 28.5% - and I would run the numbrs multiple times, til I had minimized and maximized the answer.

Just one step up from updating stats from box scores (once upon a time)


Dave Marain said...

Interesting statistical take on the practical meaning of percents here, Jonathan. I like it. It adds a dimension of reality to the problem.

A couple of questions here...
(1) Never mind, middle schoolers, here. Do you believe most high schoolers would feel comfortable with the original question? Is the solution method I described specifically taught somewhere outside of Prob/Stat? Do you think juniors would perform well on this question on the SATs? Unfortunately, they do not...
(2) How would you develop this concept? Start with a discrete of number of students in each grade, rather than percents? Change the context completely to, say, average grade in Mr. Jones' period one class was 80 on a test, whereas the average was 90 in period two on the same test, etc.??

jd2718 said...

I have continued playing with some of your similar questions (bright 9th graders). I find that most need help the first time through, but maybe the third (and these pop up in odd places, it's not like there is a unit), many, perhaps most, are on board.

There are a few who don't mind per cents, but more find a good whole number and run with it. (Nb, we get hybrids - the kid who picked an okay number, but in the last line generates a fraction - he might tolerate working with a fraction for one step rather than go back to the start)

But, yes, short answer, it shows up, sprinkled through the curriculum in New York (and in other states where there is some integration) but odds are most kids won't see that question. Kids are more likely to learn to perform an exponential regression on the calculator than to take a weighted average. And no, they are not comfortable with it at first.