Mathematics Problems in an Age of Accountability and High Stakes Testing II

This is the second part of my previous post, 'Mathematics Problems in an Age of Accountability and High Stakes Testing'.  Here I will address the issue of what constitutes an ill posed math question and what we should be doing about it.  Also, although I may include some, not necessarily straightforward mathematical content, you may readily skip over it without loss of content or continuity--The more mathematically inclined are encouraged to verify my statements and calculations. 

Let's examine our first example of an ill posed problem. As you may recall from my previous post, we have to be careful with problems involving sequences. Not too long ago a student brought me one of her STAAR test  practice worksheets. In particular, she needed help with one problem that read:

Given the sequence 2, 8, 18, 32, 50, . . .  What is the next term in the sequence?

1. 60
2. 72
3. 120
4. 192 

The student had selected 'c' for her answer, but her teacher marked it wrong, and circled the correct answer as 'b'.

Lets think about this for a second, and try to work out an answer. With a little effort we can come up with say,
$$f(n)=(-\frac{1}{10})n^{5}+\frac{3}{2}n^{4}-\frac{17}{2}n^{3}+\frac{49}{2}n^{2}-(\frac{137}{5})n+12$$
which results in
$$ {2,8,18,32,50,60, . . . } $$

But,
$$f(n)=2n^2$$
gives
$$ {2,8,18,32,50,72, . . . } $$

However,
$$f(n)=n^5-15n^4+85n^3-223n^2+274n-120$$
gives,
$$ {2,8,18,32,50,192, . . . } $$

But then there is,
$$f(n)=(\frac{2}{5})n^{5}-6n^{4}+34n^{3}-88n^{2}+(\frac{548}{5})n-48$$
that gives,
$$ {2,8,18,32,50,120, . . . } $$

It turns out ANY of those choice of answers are a possible continuation of the given sequence! They all work(check for yourself). Thus, they are all correct.

How is this possible?  It turns out that given any finite set of numbers, the set does not necessarily define any single sequence. How then can we in good conscience count our students' answer against them when we have this situation, or how can we in good conscience mislead a student for the sake of cranking out statistical data which, as a result of 'teaching to the test', no longer represents what it originally intended.  We can't!

We have to rethink how we implement these accountability tools so that our children's learning is not crippled in the process. The data collected with these tools was once meant to represent how well our students were prepared academically, now it has simply been reduced to how well a student can fill in a bubble sheet.

Let's contemplate the example above once again.  A problem like this provides a great opportunity for deep thought.  This seemingly boring and straight forward problem is of great investigative value, because with a little contemplation, investigation, and ingenuity it becomes a fascinating and exciting project.  Most people would be satisfied after finding one solution, but as educators we can lead our students into using their imagination and creativity to ask, "Are there any others?"  As you recall I posted three more. Then we ask, "Can we find more?" And the answer to this is a big YES! We can take it further in this day and age of technology, and use a computer to investigate its behavior, generate neat diagrams, and find other interesting solutions. In the process our students learn to really use the technology they have grown accustomed to, while at the same time, gaining an understanding, and appreciation for mathematics. After gaining the deep understanding that comes from hard work and discovery, questions from the STAAR test will become a piece of cake. After all, investigating a problem like this one, as deeply as I describe touches upon most of the content knowledge required to master about 80% of what is on the test.  Once again I stress that mastery of the material will result in mastery of the test and not the other way around.

I close with another ill posed problem, though this one probably intentionally.  This one reminded me of the types of problems I've seen on standardized test, and has recently, in one variation or another, been making waves over the internet. The problem is the following simple sequence asking for the value of the '?',

8 = 56 

7 = 42 

6 = 30 

5 = 20 

3 = ?

A great many even displayed an irrational rage when the solutions of others did not agree with theirs. It turns out this once again is a poorly posed question open to interpretation. In the link below I discuss this sequence assuming they are using the relation operator for assignment rather than equality, as the former has more interesting consequences.  The layman can skip the mathematical details without loss of content or continuity.

Elementary Sequences by AngelAguero