I periodically send update e-mails to our supporters.  Here’s the latest:

Dear friends of SPMPS,

What happens when almost 40 middle school students from low-income backgrounds get to learn mathematics topics like Logic, Number Theory, and Combinatorics?  Amazing, amazing things.

In Group Theory, instructor Ben-Blum Smith invented a dance.  As Ben played on his guitar, the students responded to calls like “East-West Switch”, “LAX to JFK”, and “Jam in Place” to learn about symmetry groups.  In Proofs, Shelley taught her students about the pigeon hole principle and how to phrase a formal mathematical proof.  In Numbers, Sets, and Mappings, Marcus helped students prove that the quantity of natural numbers is the same as the quantity of even natural numbers, but that the real numbers are a higher order of infinity.  In Circuit Design, Sage helped her students construct circuits to add two numbers in binary.  In Digital Communications, Taylor’s students designed their own methods of sending images across a room using nothing but sound.

Thirty-nine students came to the program and got their first entry point to advanced mathematics.  Of course, the summer is just the launching-off point: over the coming year we will be connecting students with selective high schools and summer programs, the New York Math Circle, math contests of all kinds, and other opportunities that will open a new world up to them.

We’re still crunching the data, but here are a few outcomes that we’re already seeing:

• Students took the AMC-8, a contest given to advanced students nationwide.  By the end of the program, the average of our students’ nationwide rankings grew by 21 percentile points!  Six of our students, exposed to serious mathematical study for the first time, saw their scores rise by 40 percentile points or more.
• Students reported huge changes in their dispositions towards mathematics and challenges.  97% agreed that the program “showed me that I can learn more than I thought I could”, and 82% strongly agreed.  Students also learned the importance of hard work (not just natural brilliance) and of frustration and challenges: every student agreed that “Working can improve one’s ability in mathematics”, and 95% agreed that “time used to investigate why a solution to a math problem works is time well spent”.
• Students raved about the program in their summer evaluations.  “A life-changing thing,” wrote Edson.  Math is “the best subject in the world” wrote Faith, who also said that she learned to “stop being shy”.  “I find math as a more common language than the language I speak”, said Seth.  Tiffany summed up something that many students said: “I want from liking math to loving it.”

Next year, SPMPS is looking to grow to a second campus.  We’ll be able to serve twice as many NYC-area students.  This expansion is possible thanks to a three-year grant from the Jack Kent Cooke Foundation which expires after next summer.  We’re fundraising now to raise the difference to make the second site possible next year, and also to help us transition smoothly as this grant concludes.  If you are able to contribute or you can make an introduction to someone who might be able to, please do let me know!  We are working to raise an additional \$150,000 over the next twelve months to help support summer 2014 and summer 2015.

There are two other ways you can help us out:

• We’re seeking meeting space in NYC on evenings and weekends.  This year, we will be greatly increasing our year-round programming to help students enter future programs for advanced study, and space costs may become an issue.  If you have access to free or low-cost space, please get in touch.
• We are also looking for someone to help redesign our website to match our growing profile.  Please drop me a line with any connections!

http://www.bard.edu/news/news.php?id=76

Thank you so much for all your help and support with the program!  Until next time,

Dan

There are lots of exciting things in the coming year with much stronger year-round programming and connections to other mathematical offerings.  (What this post doesn’t tell you, for example, is that we’ve seen an explosion in SPMPS alumni registering for the NY Math Circle.)  This is a very exciting time for SPMPS!

The Many Uses of =

The equals sign is a sophisticated, subtle tool.  You may not think of it as such, but it is deep.  It is hard to learn and use and understand.

I first understood the challenge while teaching a calculus class.  I had two expressions, something like (x – 2)(x + 3)(x – 7) and (x – 2)^3 – x + 50, and our goal was to show that they were equal.  I took one, and I simplified it to x^3 – 6x^2 – 13x + 42, a chain of equals signs spread across the blackboard.  I took the other, and simplified it to the same thing.  So I proudly declared that they were equal, and thanks to the magic of my boardwork, the two identical expressions had ended up next to each other.   I drew an equals sign between them, creating a giant chain of equals signs connecting the two original expressions.  I thought this was a very clever presentation.

They didn’t get it.

I’d heard before that this might be a challenge.  They’d all been drilled in elementary school on problems like 5 + 2 = ?.  They thought of equals as meaning the outcome of an operation, which is why they felt comfortable using it sloppily.  (You know what this looks like: asked to compute “five squared plus two”, they write 5^2 = 25 + 2 = 27.)

I thought, then, that it was time to correct this misunderstanding.  So I speechified at length about how they had been tragically mis-taught about the equals sign.  It actually means that two things are the same thing!  I pointed to (x – 2)(x + 3)(x – 7) = (x – 2)(x^2 – 4x – 21) and said “these two things are the same thing!”  Then I pointed to (x – 2)(x^2 – 4x – 21) = x(x^2 – 4x – 21) – 2(x^2 – 4x – 21) and proclaimed, “these two things are the same thing too!”  Since the first expression was the same as the second, and the second was the same as the third, the first and third must be the same, and a pointed and said it.  I went on, through all the equals signs, until I had discovered that the two things on opposite sides of equals signs were the same thing.  I discussed why I didn’t like the way they did a problem like “five squares plus two”.  I was proud of my improvisation, but somehow, it didn’t seem like the students had the great moment of realization (and repentance for their mathematical sins!) that I wanted.

I say it again: the equals sign is a sophisticated, subtle tool.  Despite my oratorical skill, I had failed to bring together a deeper understanding of all the possible uses of equals.  There are at least four:

• As the outcome of an operation (used in school, but mathematically the wrong way to look at it).
• To declare that two things are equal all the time, as in 4(x + 2) = 4x + 8.  This is an equvalence.  There is a hidden universal quantifier.
• To declare that two things will be equal for the right value of x, as in 4x + 2 = 3x + 9.  (This is disambiguated from the previous case by explicitly saying, “Find x so that…”  To a mathematically sophisticated reader, it is clear that this is the same = sign, just with an existential quantifier first.  This is not clear to someone just learning mathematics.)
• As a definition, such as f(x) = 3x^2 – 4x.

Without any explicit guidance, we expect students to recognize these different situations.  To recognize quantifiers and definitions without any discussion of what “equals” really means.  All this while learning how to “solve” these problems.  Also without discussions of the other subtleties of equals: symmetry, transitivity, reflexivity.

Embedded here is also the understanding that two things can look completely different and can be the same thing, can be equal.  30/3 and 10 are different representations of the same number.  7 + 3 and 5 + 5 are all different representations of that same number.  All of these things are equal!  They are also equal to 2x when x = 5.  The mathematical symbols alone are not enough; the words nearby must be used to interpret them.

Worse, uses of the equals sign get conflated in different settings.  For example, suppose that you are trying to find an x so that 4x + 2 = (x + 2)^2.  You might write that (x + 2)^2 = x^2 + 4x + 4.  One of your equals signs is only true for a few values of x; the other equals sign is true for all values of x, all in the same math problem.  Imagine how confusing it would be if you wrote 4x + 2 = (x + 2)^2 = x^2 + 4x + 4!

The same thing can happen when you are defining a function.  You might say “Let f(x) = (x + 2)^2 = x^2 + 4x + 4”.  In one line, two different uses of the equals sign.

To me personally, these uses of = are all the same thing, stemming from the same definition.  To a student just learning algebra, however, there are layers upon layers of subtlety, and they must be addressed explicitly or students will not truly understand the mathematics they are doing.