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mislead a student for the sake of cranking out statistical data which, as a result of 'teaching to the test', no longer represents what it originally intended.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

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

In an effort to improve education, we have entered an era of accountability via high stakes testing.  What does this mean to mathematics education, and what impact does this accountability have on learning?

Both as a private math instructor, and a math learning center instructor, I work with students of all levels and ages. One observation I have made is that the math homework assigned to most of my public school students consists almost entirely of standardized tests worksheets.  My students really know how to take a test.  They know a great number of 'tricks', even some that don't require them to understand, or even read the question completely!  Apparently,  they spend a large percentage of class time every day learning test taking 'techniques' and working problems almost strictly from these practice worksheets--time that, I believe, can be better spent actually learning the material.

When I first begin working with some of these students, and I assign them some straight forward problem to work out, I usually get blank stares.  Some even ask me, "What are the choices?"  Of course I explain to them I would like them to try and solve the problem by themselves as much as they can.  Frequently, thirty seconds later I get a remark saying, " I still don't get it," or "It's still too hard." At this point I ask the student to read the question to me, and surprisingly about 70% of the time they are able to solve the problem after just reading it, or actually writing some work down and trying after reading it. It turns out all they had to do was actually read the problem thoroughly, or actually attempt the problem by writing out and organizing their ideas.

What happens when the problems aren't so straightforward though? In my experience they seem to give up after about two minutes if they can't arrive at an answer.  An observation I don't experience very often with their private school counterparts.

Conditioning our students to solve math problems this way cheats them from understanding what math really is, and how beautiful it can be. In order to solve more interesting problems in mathematics a student must be willing to invest the time and effort, as well as apply a bit of ingenuity and imagination.  Sometimes patterns are straightforward, other times this is far from the case.

Take for example the following sequence:

1, 2, 3, 4, 5, 6, 7, 8, 9, ...  

What is the next number in this sequence? If you said, "10" that's great. You saw a pattern that works, in particular, that the nth term is n.

Now, how about this sequence:

2, 4, 6, 8, 10, 12, 14, 16, 18, ...

What is the next number in this sequence? If you said, "20" great again. Once again, you saw a pattern that works, this time however, that the nth term is 2*n.

While these are acceptable solutions to questions regarding these sequences, a natural question to ask is, "Are these the only ones?" It turns out they are not. A particularly curious and creative student may say that given,

1, 2, 3, 4, 5, 6, 7, 8, 9, ...  ,

the next number in the sequence is 0 if the sequence represents the product of the digits of natural numbers, or if this instead represents the  sequence of palindromes then next number  in the sequence is 11. These rules work, hence, they are also correct answers.

This leads me to my next observation.  These standardized testing practice worksheets are filled with questions that aren't even well posed(some samples and explanations in a future post). Yet we continue to give our students these poorly structured questions, and grind into their brains not only what test makers expected them to answer, but that there is no other way. We are, as Sir Ken Robinson describes, "Killing our children's creativity." Instead we should be nurturing our students curiosity and creativity, after all, they are more essential to learning and understanding.

The point is, there is much more to math than we lead our students to believe, and only by not crippling them, because we care more about how we look statistically on paper, will we be able to hand down the values and skills we already should be handing down--especially since they empowered us and trust us to do so.

What then is the problem? Should we give up standardized tests altogether.  The answer to this is of course no. We need to hold our educators accountable, and to some extent standardized tests can help. The problem, however, is with implementation. First we need to encourage critical thinking, imagination, and creativity by letting teachers teach, and realizing that mastery of the ideas will lead to mastery of the test, not the other way around.

"Good, he did not have enough imagination to become a mathematician." 

—David Hilbert's response upon hearing that one of his students had dropped mathematics to study poetry.

Part II here: 'Mathematics Problems in an Age of Accountability and High Stakes Testing II'