A Status Report

The 21^st Century and Math in the United States

Michael Round

May 22, 2009

 The May 19, 2009 Boston Globe reported the following:

"Nearly three-quarters of the aspiring elementary school teachers who took the state's licensing exam this year failed the new math section, according to results being released today that focus on the subject for the first time."

What are your first thoughts regarding this story?  Anger?  Puzzlement?  The former, perhaps, in response to the sentiment, "How can so many adults not pass a math licensing exam for elementary schools?", the latter, maybe surprisingly, with the same question!

What is going on here?

Before we rush to judgment, ridiculing the many aspiring teachers failing the exam, let's take a look at what the state licensing exam consisted of. As this was an exam certifying teachers to teach elementary school, my thought is the test would consist of elementary-level math.  Let's see.

Sample Question #1

In the number 2010, the value represented by the digit 1 is what fraction of the value represented by the digit 2?

My first inclination is to say 1/2.  Then, I realize the question is talking about the place value of '1' versus the place value '2'.  The '1' is in the 10s place, the '2' in the thousands".  A second guess is, then 10/1000, but I quickly see each of the answers has a '2' in it.  The answer must be 10/2000 = 1/200.  It's a  good thing this problem was multiple choice!

But who asks a question like this?  What "fraction of the value represented by the digit"?  Who uses these words?

OK - maybe I'm 1 for 1.  I think I'll take credit for that one, though I'm pretty sure I would have flagged the question to "come back to it later".

Sample Question #2

If P is a positive integer, which of the following must also be a positive integer?

This one is merely trial-and-error on the four answers provided on the test. There's no "problem-solving" here at all.  I just try many examples to see if a pattern emerges.  One does.  Hardly a proof, but good enough for me - at least now!  Answer:  If P is a positive integer, then P² is a positive integer. 

As I moved on, I realized these two problems were two of the easier ones!

Sample Question #3

The measurements in this diagram are shown rounded to the nearest whole number.  Which of the following is a possible value of A, the area of the rectangle?

My initial answer to the area is 8.  Obviously, it's not right.  There is no '8' to choose from.  Oh, that's right.  There can be rounding, so the "real width" can be as low as 1.51, and as high as 2.49.  Similarly, for the length, the minimum width is 3.51, the maximum 4.49.  Using these, I can calculate the minimum and maximum areas of the rectangle:

Once again, I know the right answer is 5.5 square inches - only because the other three choices (5.0, 11.5, 12.0) are outside my range.

Problem after problem is like this.  I'm not sure of the answer.  The only way to get an answer is to look at the given answers.  It's awful.

Let's try a couple more, quickly.

Sample Question #4

A book distributor is trying to divide an order of textbooks into equally sized groups for shipping in cartons.  The textbooks can be divided into groups of 12, groups of 15, or groups of 18, with no books left over.  Which of the following inequalities is satisfied if N is the smallest possible total number of textbooks?

I'm honestly still trying to figure out what this question means.  Even the four answers provide me little clue:

A.  100 <= N < 150

B.  150 <= N < 200

C.  200 <= N < 250

D.  250 <= N < 300

Sample Question #5

The prime factorization of a natural number n can be written as n = pr² where p and r are distinct prime numbers.  How many factors does n have, including 1 and itself?

I get the right answer - eventually - by doing a few examples, but hardly anything approaching an answer I'm entirely comfortable with.  I hope I get the answer right - 4.

Then, I come upon a gimme - the type of problem I'm great at:

Sample Question #6

Given that 100 milliliters is equal to approximately 0.4 cup, 205 milliliters is equal to approximately how many cups? 

I love this type of problem, because I can structure it so I'm certain things work out right - because I'm always canceling units.  The answer is easy:  0.82 cups.

This is but one of several methods ... there's also one of proportions, where I could do the following and solve for x, which, of course, gets me the same result:

However, when I go to find the right answer, I see the problem was not even asking for an answer!  It was asking "which expression models the solution to the problem?"  The possible answers are:

A.     (100 - 0.4)(205)

B.     105% of 0.4

C.     (205 - 100)(0.4)

D.     205% of 0.4

What is this?  Where's (205)(0.4)/100?  It's not there!  I've got the answer plus two beautiful methods, and none of this appears anywhere!

A Little Bit of Empathy

My initial reaction regarding the headline, one of incredulity about the ignorance of adults not being able to do elementary math, has been replaced with the emotion of anger at what the test actually was.  What was it testing? 

Certainly not much related to what the average adult would be considered important regarding math and elementary-aged students!

Instead of anger, I feel empathy - perhaps even shedding a mathematical tear - towards the real victims in this case:  the teachers - and yes, the eventual students.

We see too, with this small review, better teachers and more money would solve nothing.  Like the miller's daughter in Rumplestiltskin, if this tested material at all mimics the actual elementary curriculum, teachers are being asked to spin straw into gold.

But the test was intentional.  Someone wrote the questions.  Who?  Why? To what end?  So too is the underlying curriculum.  Where does this come from?  

And what has this to do with the problems we've seen in math over the past 1/2 century, continue to see today, and likely will see into the future?

Unless something can be done.

It can!  It must!  It will!

Stay tuned for Part II!