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mislead a student for the sake of cranking out statistical data which, as a result of 'teaching to the test', no longer represents what it originally intended.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

R.I.P Mr. T^2

On October 15, 2008 the #1 teacher of mathematics in the United States of America, in fact to ever live, took the journey to be at the side of God. 

Mr. Toby H. Tovar Jr. left his mark upon the world through his family and thousands of students.

Mr. Tovar only days ago came to mind when, while I was teaching to young children, The director of a learning center said to me, “Angel, if I didn't witness you in action, I wouldn't believe a person that can teach the way you do exists!”  The only thing that came to mind that moment was, “You obviously never met Mr. Tovar!”  If I could ever only be 1/20th as good a teacher as Mr. Tovar was that would my greatest accomplishment and I would feel set for life.

The last time I saw Mr. T^2 weeks before his passing he gave me a copy of a short autobiographical sketch he wrote which I include below.  Those who new him know that the lines that are slant on that paper are not due to his penmanship(he had the best penmanship of anyone I've ever known), but rather my cheap scanner which I couldn't get to scan that page correctly.

And now in Mr. Toby H. Tovar’s own handwriting his autobiographical sketch. R.I.P. Mr. T we miss you!

Toby H. Tovar Jr. autobiography

Mr. Tovar first came into my life when as a freshman at El Paso High school I wanted to enroll in an Algebra I class, but the counselor wouldn’t let me because, “Mexicans can’t do math.”  He went with me to the counselors office, told me to wait, shut the door behind me, and after some yelling I could hear through the door he walked out with my new schedule that included his honors probability and statistics course, and his honors algebra course.  He believed in anyone and everyone. If the need arose he could teach calculus to a chimpanzee or even a rock.  When Mr. Tovar walked into a room you knew it even if you weren’t looking. The room lit up and you could sense something special in the atmosphere—his presence was certainly felt.
When ever I would praise him he replied that he was nothing special; that he couldn’t do what he did alone because frequently he felt he lacked the strength, and every day when he rose at 4:00 a.m. the first thing he would do is say this prayer by St. Ignatius of Loyola:

Lord, teach me to be generous.
Teach me to serve you as you deserve;
to give and not to count the cost,
to fight and not to heed the wounds,
to toil and not to seek for rest,
to labor and not to ask for reward,
save that of knowing that I do your will.

It was then and only then held by the hand of God that he could finally get up and do his will.

Shortly before his passing he said to me he had lived a long and wonderful sixty six years full of joy and packed with living. That he had no regrets and that he was ready for what God had in store for him next.  It was at that time that I first heard the following quote by Dr. Seuss.

“Don't cry because it's over, smile because it happened.”
― Dr. Seuss

R.I.P. Mr. Dr. Toby H. Tovar Jr.

Back in session . . .

We are once again at the beginning of a standard school year.  Today is the first day students return at our local school district.  This new school year brings to our school district a new superintendent; taking over, fresh from scandal, the fifth largest school district in Texas. Fortunately for him there is nowhere to go but up—so I hope.

Some changes I would like to see is unity among teachers and administrators.  For too long it has seemingly been an us v.s. them attitude among teachers and administrators.  However, I don’t believe either is at fault.  I feel that Principals have too much pressure from above to enforce ‘teaching to the test’, and other shady policies even if it means no longer exercising practical wisdom.

Teachers, it seems, want to teach and inspire children but are hindered by politics.  Administrators seemingly push teachers to ‘teach to the test’.  Don’t get me wrong, I believe there is value in repetition and by the numbers, but exclusively drilling students through algorithmic exercises preparing them to only excel at taking standardized multiple choice state exams fails to teach them critical thinking skills. Furthermore, they fail to engage their imagination and creativity.

The pressure our administrators are put through to achieve a certain score on standardized test, or receive compensation based on attendance is diverting them from what is truly important--Our children's education!  I am of course not suggesting that it is ok to not meet basic academic standards or that skipping school is ok.  On the contrary, I believe in the importance of a well rounded education and the expectation that children should attain a minimum level of mastery before they move on, as well as the importance of being in school where they may be guided by their teachers towards mastery.  Our school's leaders are being led, through pressure to crank out results via statistical data that has strayed from what it originally meant to represent, to put too much emphasis on ‘teaching to the test’.  Other predetermined statistical goals have lead to practical wisdom going out the window, e.g., when a child is absent due to serious illness, under pressure to perform at some level on whatever category attendance falls on, administrators encourage parents to home-school their children—even when they are honor roll students.

These are only some of the issues I am counting on our superintendent to tackle.

Though I now slightly digress, let me tell you, I too have experienced the wrath of a principal under pressure to crank out what I believe are now meaningless scores and statistics.  On one occasion a principal tried to throw my son out of his school after a brief hospitalization for an appendectomy—that didn't turn out so bad, but on a separate occasion things became so tense with my daughter’s principal that I was moved to write the letter you see below to the entire school board, E.P.I.S.D. administrators and members of the local media! She just wanted my daughter out of her school although she had straight A’s, because she is a severe asthmatic.  Even though she didn't know me, she even once told me,  “If you listen to me and do exactly what I tell you to do, and how I tell you to do it, your daughter could be the first person in your family to graduate from high school!”  Is my lack of education written on my face?  Do I give away the fact I have a low IQ by the clothes I wear?  Anyhow, if you are curious as to what I wrote in response read the letter below; otherwise just skip it.

School Board Letter by AngelAguero

To the district’s credit, I received a phone call from the campus director, although on personal leave because of a death in her immediate family, at 7:00 a.m. the next morning with an apology and all the other formalities.  By 10:00 a.m. the assistant superintendent in charge of special education was knocking on my door.  On top of that, my daughter’s principal and I went on to have a wonderful professional relationship and still do to this day.  I found out what a wonderful lady she really is and that deep down she really cares for the children in her school.

Once again, it is only the beginning of the school year and teachers and administrators seem optimistic about the new year and the new superintendent.  What I gather from teachers and administrators I’ve spoken to is he cares about the children—we’ll see.  Most importantly I hope he creates an environment in which teachers and administrators are once  again on the same side. And after that tackles the challenge of educating our children by encouraging critical thinking, imagination, and creativity by letting teachers teach, the administrators once again worry about the children and exercise their  practical wisdom, and finally realizing that mastery of the ideas will lead to mastery of the test not the other way around.

“The most erroneous assumption is to the effect that the aim of public education is to fill the young of the species with knowledge and awaken their intelligence, and so make them fit to discharge the duties of citizenship in an enlightened and independent manner. Nothing could be further from the truth. The aim of public education is not to spread enlightenment at all; it is simply to reduce as many individuals as possible to the same safe level, to breed and train a standardized citizenry, to put down dissent and originality. That is its aim in the United States, whatever the pretensions of politicians, pedagogues and other such mountebanks, and that is its aim everywhere else.”

― H.L. Mencken, 1924

Once more unto the cold night . . .

It is all to easy to create a blog, but to decide what to include in it is quite the puzzler.  Should I talk about birds, cars , games, or the stars?  Should my audience include painters, athletes, or laborers? Should it specialize, generalize, or be one big incoherent mess? In any case, here I am scribbling away my fragmented thoughts with no goal towards achieving clarity or harmony.  In the end these are my thoughts, for me if you will, or those tired of reading all the great and interesting things out there and now turn to the worst so that they may experience it all--the good, the great, the bad, and the ugly.  I shouldn't be expected to focus on just a few things. Robert A. Heinlein said,

A human being should be able to change a diaper, plan an invasion, butcher a hog, con a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.

I wholeheartedly agree.  Hence, my focus shall be on nothing.  I will let my fragmented thoughts flow free caring not whether there is a connection from one post to the next, or even if I have made any sense within a single post.  I will take no care not to go off on tangents and share with you the delightful fact that pi can be written as,

$$\pi = \int_{0}^{1} \frac{4}{1+x^{2}}$$
or perhaps even spontaneously link to what I may feel at the moment is a delightful image... 

In the end I may be making this journey alone.  My audience consisting of only myself.  Comforting my solitude with my own words; my own fragmented thoughts and ideas.  That's alright as I may be the only one who can make sense of the disorder within my own head.  Not turmoil, but rather a sort of organized chaos nourished with dreams and making sense only to the imagination of that one individual who gave birth to the ideas.

“Be who you are and say what you feel, because those who mind don't matter, and those who matter don't mind.”

― Bernard M. Baruch

Test^2

Consider, $$ \begin{equation}\label{eq:gravt} F=G\frac{mM}{r^2} \end{equation} $$ and, $$ \begin{equation}\label{eq:gravo} F=mg \end{equation} $$ Combining ($ \ref{eq:gravt} $) and ($ \ref{eq:gravo} $) yield, $$ g=G\frac{M}{r^2} $$