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Today I had the pleasure of traveling up to Saint Ann’s School in Brooklyn (my first trip to to see a dear internet friend, Paul Salomon, and meet a few of his amazing colleagues.

One of the great things about visiting a school like St. Ann’s is just how different it is from the schools I’ve taught in and experienced—leafy campuses of hundreds of acres or more with a bounty of space (though we often still find ways to complain about a lack of space). Saint Ann’s packs an entire 4-12 program into a 13 story building with two elevators. Assemblies meet in odd classrooms and the lobby, and the entire building feels like a beehive of activity and engagement.

On of the very best things I saw is that every teacher in the department teaches in both the middle and upper school. It’s not at all uncommon for a teacher to teach 3rd grade math, 6th grade math, and functions (Algebra II). This seems hard wired in the DNA of the faculty, and as one teacher explained to me it comes from wanting students to be exposed to the entire intellectual enterprise of a department from the very beginning. Having visited and worked in an number of K-12 schools where it was difficult (if not impossible) to form inter-divisional connections, this seems like a pretty ideal solution, but it also clearly calls for a truly extraordinary skill set in in their teaching faculty and makes me wonder if this can be taught—can an extraordinary high school math teacher learn to 3rd grade with no previous experience? And vice versa?

## Paul Salomon’s Functions Class

My day started with Paul Salomon’s Functions class. He began by taking his students out to this incredible sculpture:

It’s a meta meta node, and this photo doesn’t even begin to do justice to the impressiveness of this sculpture and the effort that it took to produce it. So here’s a video from Justin Lanier:

Paul starts the day by asking his students to think about a few questions:

• How many nodes are in the sculpture?
• How many holes are in the sculpture?
• How many unfilled holes are in the sculpture?

And quickly students were off, calculating and asking questions of each other. I saw a number of different approaches, from taking a single plastic node and counting the holes on there, to more elaborate arguments based on the tiling of rectangles, triangles and pentagons that make up the nodes. After about five minutes of figuring, and students reaching some success with their calculations, Paul encouraged students to consider writing up their work for the end of the week.

For the remainder of class students worked collaboratively on a set of 6 problems from an NYC math tournament. In watching Paul’s students, I noticed that they jumped right into to problem solving. They didn’t seem to be daunted by the fact that the problems were unfamiliar, different from each other or part of some math contest. Paul was able to hang back for quite some time before students even began to ask him questions. I also noticed that Paul’s students weren’t the least bit daunted by vocabulary—some didn’t know the meaning of isosceles or what exactly a chord was, but Paul was quickly able to redirect the question to another student willing to take a stab at defining the term and students readily moved forward from there. And before you knew it, the bell was ringing and Paul closed out class by reminding students that if they found a particular problem interesting, to write up a solution and submit it to him.

After Paul’s class I ran into Justin Lanier, one of the Math Munch triumvirate, and we sat for a chat where I got to pepper him with all the “so…no grades” that all Saint Ann’s teachers must hear whenever they go out. Justin did share some interesting thoughts about how Saint Ann’s managed to transform from a heavily tracked math curriculum into the middle and high school to its present curriculum which emphasizes themes and differences in particular teachers’ preferences rather than particular ability groupings.

After meeting with Justin, I went to see Paul Lockhart teach a 6th grade math class. They were working on a series of Sangaku, small geometrical puzzles that students in the class had developed. When I came in, this drawing was on the board

After settling the class a bit, Paul called on a very enthusiastic student who wanted to present a discovery he’d made about this particular figure. The student drew a box around the triangle and then of dotted lines as shown below:

He cited the fact that the class had previously discovered that a triangle takes up half its box, and then began to count off the triangles, noting that each pair (1&2, etc) must have same area there were two boxes left over outside the the big equilateral triangle, and that these two boxes must have the same area as the inscribed square. “Genius!” the student said to the class and then he started to return to his seat.

Before the student sat down, Paul began to gently prompt the student to go back though his explanation making sure that every student agreed with each assertion in the student’s argument, asking the class and individual students “do we agree with this?”, while students offered their own arguments and explanations to support this student’s conclusion. And all the while, Paul was offering little narrations about the beauty of mathematics and the process they were following. Perhaps the most remarkable thing was that after about 10 minutes into this process, the students gently realized that the student’s result was non-obvious, and therefore interesting, Paul said, but also not really helpful in finding a complete solution to the the problem. Here’s where Paul helped the class to see that sometimes interesting results don’t lead to solutions, but that you can’t always see that from the beginning, and so you should always be willing to seek out these interesting results.

After that Paul brought up another student’s approach, which was to call the side of the triangle 1, and work to find an expression for the side length of the square, which they call $a$. The students seemed to know instantly from a previous investigation that the height of the equilateral triangle was $\sqrt{\left(\frac{3}{4}\right)}$ and a couple where tripping over their tongues to blurt out the answer in terms of the “square root of three fourths”, but Paul backed the class up and pointed out that “square roots of three fourths” sounds ugly, and a move mathematicians often make is just to give something ugly a name, like, $h$. It’s at this point that the class make’s a big insight and sees that the smaller equilateral triangle atop the square is just a scaled version of the larger triangle, and the scaling factor is $a$, and it was easy for the students to then see that the height of this smaller triangle must be $ah$ And from there, he asked the students to write out an expression for the the height of the larger triangle which they explained as just the height of the square plus the height of the smaller triangle, and so they could write as

$h=ah+a$

At this point, I was pretty stunned by how fluent these sixth graders were with fluent with scaling factors, how easily they could go from describing how to find the height of the triangle in words to writing an equation for it—it was some of the highest caliber mathematical thinking I’ve ever seen.

At this point, Paul is continuing to add thoughtful narration as the students work on solving the problem. He points out to them that they have now switched from being geometers who are trying to find relationships about these shapes, to algebraists who are are now trying to take an implicit relationship between a and h and “untangle” it to find an expression for h. And he turns the problem over to them again to think about how to go about solving this equation for h.

A student tries what seems like a solution derived from a previously learned mnemonic, and she writes up $h-a=ah$ on the board, and Paul helps the class to discuss how this is a perfectly legal algebraic move, but it doesn’t seem to be bringing the group closer to getting an expression for h alone.

He asks them what makes this equation hard to solve for a, and the students recognize that the two a’s in the equation make it difficult. He asks them to think of how you might combine the a’s, and launches into a mini discussion of what happens when you multiply two things together, like $\left(a+b+c\ldots\right)$and $\left(x+y+z\ldots\right)$ and the students quickly talk about writing out all the combinations, and are even able to say how many total combinations there will be. Paul then points out that sometimes we encounter a bunch of combinations, and if we look carefully, we can see how to re-write them as the product of simpler things. From there the students realize you have h a’s and are adding an a, so you could write it as:

$h=\left(h+1\right)a$

Never once did Paul or any of the students say factor, FOIL, distribute or any other math procedure laden term.

From there students worked to solve the equation and come away with with

$a=\frac{h}{h+1}$

Narrating again, Paul notes that our work as algebraists is complete, and we can switch to being arithmeticians, now that we’ve found a beautiful, simple expression for the length of a side.

Students were quick to say that

$a=\frac{\sqrt{\frac{3}{4}}}{\sqrt{\frac{3}{4}}+1}$

Students then get into a discussion about how you might be able to write this in different ways, as $\frac{\sqrt{3}}{\sqrt{3}+2}$, and wether you could ever write it in such a way that no square roots appeared. It lead to a good discussion about why the answer had to be an irrational number.

This is getting long, so I won’t get into the amazing discussion at the end of class that the students had about another student’s Sangaku:

Incidentally, I just discovered Sangaku for myself in writing this blog post. These were sacred mathematical tablets created during the Edo period Japan (1700s) that presented a sort of geometrical puzzle, and were solved by entirely different, non-western mathematical techniques, since the Japanese did had not seen calculus. I discovered that a Princeton Physics Professor is collecting them and published them in a book.

What do you remember from sixth grade math? I don’t remember much. Maybe I was introduced to irrational numbers then. If so, I’m sure that soon after it was drilled into my head to always rationalize my denominators to get rid of those radicals in the denominator (though I never knew why, and better math teachers that me now say this practice is dumb and outdated). But I bet if I had been introduced to sacred Japanese math puzzles in 6th grade, I wouldn’t have forgotten them, and I would have developed a much greater appreciation for the beauty of mathematics and the uniqueness of culture at a much younger age.

## Mathematical Art

The last class I visited was Paul Salomon’s Mathematics and Art class were he did a very cool exploration of Platonic solids, why there are only 5 of them, and then a cool exploration of the various Archimedean solids, prisms and anti-prisms (the Freedom Tower is a great example of an anti-prism). None of these topics were ever in my high school math curriculum, and Paul’s students are now working to use everyday objects to build Archemdian solids of their own.

## Conclusion

Overall, I was wowed by how comfortable Saint Ann’s students seemed with struggle and how immersed they are in the world of mathematical thinking. I can immediately see two major contributing factors to this. First, Saint Ann’s has no grades, and I can only imagine how empowering this must be for helping students to take risks, to learn from feedback from the teachers and peers without having to worry about the judgment of grades and ranking.

The other factor might be that St. Ann’s Math department has developed a K-12 vision of what Mathematics education should look like (it is beautiful-jump to page 36 to read it), and then taken it upon themselves to see that every member of the department has a role in making that vision a reality at every level. It’s simply not possible for a teacher to just teach high school and grouse about how the teachers in the lower levels aren’t doing X or Y to prepare kids for high school math.

It is incredible just how present this vision of the beauty of math was in each go the classrooms I visited, and I could easily see how this atmosphere could lead to creating something like MArTH Madness MArTH Madness.

Many thanks to Paul Salomon, Justin Lanier, Paul Lockhart, and all the rest of the Saint Ann’s teachers who gave generously of their time to allow me to visit.

1. March 19, 2013 11:47 pm

SO glad you came. Thanks for the kind words, John!