Jan 29 2010

What should common core math assessments look like?

‘Rigor’ is in, and the common core standards promise to raise the achievement in this country by raising expectations which students always rise to meet.

As a staunch “status-quo defender,” it might surprise ‘reformers’ that I have some pretty radical ideas about how I’d change the math curriculum in this country if I could.  While they tinker around with teacher evaluation formulas which could, at best, raise test scores by a little, I would like to see a complete overhaul of what we teach in math.

When I heard that the common core was going to address the problem that the math we teach is “a mile wide and an inch deep” and that we need to teach fewer things, but better, I thought that this was an excellent idea.  It was something I was thinking about for a while.  It is not possible to deeply teach too many topics in a year.  It would be like trying to have a class read fifty novels in a year in an English class.  It would not be possible to cram that much in and do it well.

As I learned more about the common core, I got concerned since they didn’t really seem to be removing many topics from the curriculum.  Instead math teachers are told to teach to a greater depth of understanding in the same time frame.  Another important issue is how students will be assessed to see if they have achieved a deeper understanding.

Since I teach at one of the top high schools in the country, Stuyvesant High School, and since I try to usually teach to encourage a deep understanding, I wanted to share with Diane Ravitch’s vast audience what a common core math activity could look like, what the assessment might be, and why the actual assessments will never accomplish what they were supposed to.

I chose 8th grade geometry standard 8.G.B.6 which states “Explain a proof of the Pythagorean Theorem and its converse.”

Now the Pythagorean is probably the most famous thing in all of math.  In Gilbert and Sullivan’s Pirates of Penzance they even refer to it in “I Am the Very Model of a Modern Major General”.

I’m very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About binomial theorem I’m teeming with a lot o’ news,
With many cheerful facts about the square of the hypotenuse.

At the end of ‘The Wizard Of Oz,” The Scarecrow, after receiving his ‘brain’ even takes a crack at it.

OK.  So after that gentle introduction, I’m going to remind anyone who might have forgotten that the Pythagorean Theorem describes a relationship between the lengths of the three sides of a right angled triangle.  Specifically, if you add together the squares of the two shorter sides it will equal the square of the longest side.

In the diagram below, AC is 5 units and BC is 12 units.  To use the Pythagorean Theorem to determine the length of AB, you would calculate 5*5=25 then add to that 12*12=144 to get 25+144=169.  Then you would have to find the square root of 169, which is 13 since 13*13=169.

Generally students are ‘told’ the Pythagorean Theorem.  Sometimes they are shown a formal proof of it using something called similar triangles, but that proof is not very convincing or memorable.

When I make a math lesson, my goal is for it to be thought provoking, relevant, or both.  So when I teach the Pythagorean Theorem in common core style, I’d want to get my students thinking about why the relationship is true, and a good way to do this is with some very cool geometric diagrams.  In the old days (300 B.C.) when people would say “In a right triangle the sum of the squares on the legs equals the square on the hypotenuse,” they meant it literally.  ‘Square’ did not mean to multiply something by itself, but the four sided shape that we learn about as toddlers.  So saying “a squared plus b squared equals c squared” in this context means that the combined areas of the yellow and blue squares are equal to the area of the orange square.

To get kids thinking about why this  might be true, I’d have them examine a few pictures.  Here’s one that should keep any curious person staring and thinking for at least ten minutes.

I’d ask students to try to justify why the five pieces that make up the big square are identical to the five pieces that make up the small and medium sized squares.

I’d then have them think about, and then discuss in pairs, this famous image.

My hope is that most of the class would be intrigued by this image to realize that since the left hand square is made up of four triangles and the orange square and the right hand square is made up of the same four triangles and the yellow and light blue squares, then the orange square must have the same area as the yellow and light blue combined.

For my ‘assessment’ which is also what would be natural on the common core, I’d present another picture kind of like these, only harder.

Now here’s where the common core assessments will break down.  As a teacher the way I’d assess my students would not just be if they “figured it out.”  While I’d be pretty happy if some students figured this one out, I could be satisfied if nobody figured it out.  If I saw my students concentrating on it, talking about it with their neighbors, thinking about it and not giving up for twenty minutes, making some progress, developing some theories and then testing those theories, smiling — enjoying this challenge.  That’s what I’d want to see and I seriously doubt that the common core will, or can, accomplish this.

By the way, in case you’ve been intrigued by this, I’m going to put the answer down so you have to scroll to it.























2 Responses

  1. Ted Cook

    Yes, that is the coolest thing I’ve ever seen, thanks. I like how in the second example you can see graphically that picture left,
    equals picture right looking only at the perimeter (a+b)^2
    which expands numerically to a^2+b^2+2ab, which you can see the expansion graphically
    and then setting them equal proves the theorem

    And then the third example picture left
    c^2-2ab (focus on the area of inner square)
    equals picture right upper left square
    which expands numerically to b^2+a^2-2ab, which one can visualize step by step graphically

    • Ted Cook

      What’s the graph of America’s favorite equation (difference of two squares) look like?

      a^2-b^2 = (a-b) (a+b)

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