Guest Post by Gary Rubinstein

garyrubinstein.teachforus.org

‘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.

Reblogged this on Transparent Christina.

Great example of true thinking and exploring that cannot be uncovered in a standarized fill-in-the-bubble test. As it should be in young children’s art work, it is not the product but the process that is important.

Cool!

The problem that I see is that the (state/national) examiners need to set some “extending” questions to flush out the best students. The “extending” questions tend to be stuff taught at the next level e.g. stuff taught at college used to extend the best students in High School e.g. prove some simple gamma function equation.

Once it’s in the exam then teachers have to teach it because their students might meet it. So it beomes an arms race where the examiners keep looking for extension material and then teachers needing to teach it so their kids won’t come back at them “you failed us – you never taught us that”. Of course, since teachers can’t fit everything they start to go for breadth rather than depth and hope the good students will fill in the details.

Of couse this defeats the initial purpose – the kids who get the best scores are now the ones who have worked the hardest covering the material in depth rather than the ones we really want to identify i.e. the who can use their mathematical knowledge they have to work out unmet problems at a higher level.

I don’t see how the arms race for breadth will defeat the ideal of learning for depth while testing plays such an important part of the education landscape.

Based on the few samples revealed so far, common core assessments promise to make the arms race worse because of their increased persnicketty-ness over “academic” language, jargon and notation.

Here is a sample 4th grade math problem I found:

Students from three classes at Hudson Valley Elementary School are planning a boat trip. On the trip, there will be 20 students from each class, along with 11 teachers and 13 parents.

1)Write an equation that can be used to determine the number of boats, b, they will need on their trip if 10 people ride in each boat.

2)How many boats will be needed for the trip?

3) It will cost $35 to rent each boat. How much will it cost to rent all the boats needed for the trip?

Answers:

1) b = [20(3)+11+13] 10

2) The number of boats needed is 8 + 1 = 9 boats

3)Total cost = $35 × 9 = $315

Now, my kid’s class is all over this kind if math in a “get the answer any way you can sort of way” so Qs 2and 3 seem about right, but they have never been exposed to the language in in Q1. If anything their curriculum goes out of its way to avoid that sort of thing.

Providing the a more efficient method for solution set up as a single proper equation seems like a reasonable enough extension of what they are doing, but it doesn’t seem like something they could just intuit based on where they are conceptually.

Being comfortable with Q1 most likely indicates that a kid has had additional (or different) instruction. Its not next year’s work conceptually, its next year’s work in terms of its presentation. This is why NYC now has cram schools.

On some level this approach seems like a kind of mean spirited gotcha – a way for expert insiders to keep outsiders at bay. Its similar to when gun enthusiasts say you can’t participate in gun violence debates if you don’t know the difference between a clip and a magazine or when the IT Department laughs at you if you don’t identify your PC by its chip number and processor speed.

I can’t think of a more discouraging way to address students.

Sorry typed so fast. All over this kind _of_math.

I had hoped in 2008 that a new administration would take a fresh look at accountability and get the right kinds of metrics in place. I hoped that instead of the one-size-fits-all mentality of NCLB the Obama administration would institute something more akin to the comprehensive quality measures like those used to earn Baldridge Awards. They might benchmark individual student progress against something like the CCSS. Instead of using one-size-fits-all standardized test results as the primary means of “school quality” the Obama administration would develop questionnaires to ensure that schools were engaging parents, coordinating with social service agencies that support to students, and articulating with both pre-school programs and post-secondary institutions. Instead of using one-size-fits-all standardized tests to somehow render judgements about teachers they might work collaboratively with the NEA and AFT to develop the kinds of performance metrics used in the corporations deemed to be the “best places to work” by employees. In short, I hoped the Obama administration would seize the opportunity to get the metrics right. Instead, we have NCLB on steroids and a testing regimen that reinforces the sort-and-select factory schools instituted in the 1920s. Some reform, eh.

Screw the Common Core. Teach kids to think outside the box with creativity and joy, not fill in the circle with a mark heavy and dark.

Thankyou Gary, that is the coolest thing I’ve seen all year. Always loved the perfect triangles, http://en.wikipedia.org/wiki/File:PrimitivePythagoreanTriplesRev08.svg

How is it appropriate to take what was commonly done at the Senior level and move it down to the freshman level? Thereby pushing what was done at the freshman level down to the 7th grade level? Thereby pushing what was done at the 7th grade level down to the 5th grade level? Thereby pushing what was done at the 5th grade level down to the 3rd grade level? Thereby pushing what was done at the 3rd grade level down to the 1st grade level? Thereby pushing what was done at the 1st grade level down to preschool – which not every child has access to? All in order to push what is normally done as a freshman in college back to high school? Does no one else see the insanity in this?

One merely continues the madness that Jerry Weast worked upon Montgomery County (MD) Public Schools during his (too-long!) tenure: Increase kindergarten from half-day to full-day in the name of “academic rigor.” Realize, 2 years in, that 4YO’s aren’t cut out for 6 hours a day of seatwork. Raise the entrance age for kindergarten to 5YO BEFORE entry (not before January/during the school year). Voila! Kindergarten is now what first grade was only 2 years earlier!

As an added bonus, when the kids get to 5th grade and turn 11YO (or 12, if their parents red-shirted them), they are older than 5th-graders used to be and magically, your 5th-graders will be doing algebra and making you look like the bestest superintendent on the planet!

*gag*

Yes, I’m with you. As for the example in this post, I think it is great that 8th graders be exposed to it, but I wouldn’t expect everyone to get it. As for the pythagorean triangle, I’d hope that 95% of seniors could be taught to calculate the third leg given two legs using a calculator. And, in their head, if two legs are 3 and 4, what’s the hypotenuse? If two legs are 6 and 8, what’s h? If two legs are 5 and 12, what’s h? That is my definition of a common working knowledge of the Pythagorean right triangle.

It seems that you are objecting to the common core because it disrupts the current tracking system. What if you have put students in the wrong track? What if they can do as freshman what you thought they could only do as seniors?

No, I am objecting to the common core because it flies in the face of everything we know about developmental science and what is appropriate at various phases of development. I love how people immediately jump to “you are defending the status quo”. Yes, occassionally there is a freshman who can at that level things beyond the norm. That is the exception, not the rule. Tell me again how many years you spent in a public high school classroom teachineconomisot? During which years? In which state?

My post was based on the observation that over the last ten years or so 3 students in my local senior high school have taken graduate mathmatics classes while high school students. Many others have taken undergraduate mathmatics classes.

It might be just a handful in our school of 1,000 students, but in a school district the size of LA Unified there might be a significant number of students who could work at that level if givin the opportunity.

So accommodate that handful of gifted students. This has nothing to do with the developmental appropriateness of the CC for the vast majority of public school students, k-12. It also is only vaguely related to the practice of tracking. I am sure you understand the difference fully, but over the past few months I have observed that you seem to enjoy indulging in obfuscation and aren’t necessarily concerned with arguing apples and apples.

Part of my point is that defining what a student should be learning at each age is to define a track. (you are 12, so you will learn A,B,C, at 13 you learn D,E,F, etc)

I doubt there is any curriculum that would be appropriate for every student, and often argue that matching students to the best learning experience for them is important.