Sunday, March 29, 2015

Math Teachers at Play 84

Welcome to the 84th Math Teachers at Play Blog Carnival!

84 is a portentous number. It's the sum of twin primes (What's the previous sum of twin primes? Next?). It's thrice perfect, twice everything.  It's positively Orwellian. It's even a town in Pennsylvania. It's a game (actually a variant of the domino game 42 if you play with 2 sets.) It is the #edtech that shall not be named.

(click for full size image)
84 puzzler 1:
Number the intersections of these five circles with the integers 1 to 20 so that the points on each circle sum to the same.

An exciting development this week was Jed Butler unveiling the Math Twitter Blog-o-Sphere Directory. It will be a great resource, but only if you add your information! Maybe you blog, maybe you tweet, maybe not... but you are a mathy type on the interwebs or you wouldn't be reading this!

What to Read?
It was a good month for math reading related posts.

84 Puzzler 2
84 is a side length in the smallest integer perfect tetrahedron, and is tetrahedral number to boot. Which triangular pyramid has 84 points? How many points in the nth pyramid?
(Image made in GeoGebra 3D)

Tool Tips


84 Puzzler 3
What is the odd pattern that produces these multiples of 84? Highlight for hint: 7 is involved.
0, 2184, 78120, 823526, 4782960, ...

Why Do We Do the Things We Do?

Lesson Lab

That concludes the carnival; 84 cheers! Remember, you can submit posts to the next carnival via Denise's MTaP form. If you didn't see last month's Carnival 83, it was at CavMaths. We're looking for a host for Carnival 85 - can we come over to play at your blog? Email the Founder of This Here Shindig to give the all clear. Thanks to everyone who pitched in with submissions!

Number fact references: Archimedes Lab, MathWorld and Wikipedia.

GeoGebra sketch of the tetrahedral numbers.

Thursday, March 26, 2015

What Is It Good For?

This is a thinking out loud post. Anyone who can push my thinking, I'd be glad for it, in comments or on Twitter. Names omitted to at least partially protect well meaning folk from my ignorance. I may be more wrong in this post than I have ever been before. (Which would take some doing.)

I was at a conference recently where I got to hear someone speak whom I respect a lot. We have our students read their work, and good things come of it. The subject of the talk was long running research on teacher preparation. This was prescient, as teacher education is being heavily questioned, there are proposals for ridiculous restrictions and regulations, and we have our first competition ever from alternative certification programs.

So how would you show teacher preparation is effective? Maybe compare the learning of the students of teachers who were prepared vs students of teachers who weren't. Oh, there aren't any of those. Many of those, anyway. Also, hard to compare different programs. OK. Evaluate your program. Hmm, but there's even a lot of variation within a school. Well, we could make everyone teach exactly the same. And always the same topics. We'll need detailed lesson plans.

That's what they did. To test effectiveness, then, they kept track of a random sample of students. (It's possible the randomness was simply which students they could track.) Then they tested them on the content and teaching of three topics from their freshman level teacher preparation course. For comparison, they also tested them on one topic not from the course. What would you expect they would find? What would you hope they would find?

They found the teachers did better on the topics that were taught. And how much better was proportional to how much emphasis they put on those topics. This was true for the students as seniors, first year of teaching, second year of teaching and so on.

Even after being tested on the un-taught topic, the teachers realizing they don't know it as well as the others, the next year found the same results. The lack of transfer, it was said, proves the effectiveness of teacher preparation. If they could transfer, all we'd have to do is teach them one thing.

What about the practices? (Which would have been the processes or proficiencies back when this research was started.) Those are taught implicitly. Because you have to do them to get at the content, you know. My experience, and that of my betters at GVSU is that you cannot teach these implicitly, or even as an add on. Front and center and even then good luck.

For the capper, it was stated that the state might only certify teacher education programs that covered all of the common core standards. This was obviously foolish to everyone in the room. There's so many! No one could cover them all! Almost no one seemed struck by the irony that one of our leaders may have just proved that it is necessary.

It was a bitter pill.

It also seemed to go with one of the themes in the Twitter week for me: teacher-proof curriculum or curriculum-proof teachers? If you have a moment, take 5 minutes to listen to Megan Taylor's (@ilovemath11) Incite.

The researchers I'm talking about, and math ed folks in general, are strongly tempted by the teacher-proof curriculum. I know curriculum is important - I was a part of an effort to move a large district to a problem solving approach when they still had a traditional curriculum. (Turns out, teachers do not have time to gather the materials themselves.) But you could hand teachers the lessons from the Teacher's Edition of The Book (stealing from Erdös*) and it won't make as much difference as what Megan is talking about.

I get to teach with David Coffey (@delta_dc) and it is a humbling experience. He's great. But one of his most frequent dictums is that he doesn't want a lot of little Dave Coffeys running around. He's not trying to turn his students into him. The wisdom of this is now something I accept. Teaching is so deeply personal, it just will not work if it is not authentic. (Belief, not research.)

I tried to suss it out with my preservice high school teachers this week.  I have 2*2*14*(55/60) hours with them in a semester. I cannot cover all of their content, nor all of their crucial content, nor all of the most difficult content to teach.

Just like they will not have enough time with their students. What are they to do? What am I to do?

Use the opportunities we do have to teach our learners how to learn on their own. The math is important, the practices are important, but this is most important. My university teaching of teachers is best when it's holistic and I am doing what I want them to do. Math is such an amazing context for doing this; we are truly blessed.

Disclaimer: I have a rather unfortunate bias against a lot of education research as it is. Maybe it's a remnant of having been a mathematician, and the culture of the hard sciences looking slightly askance at social science research. Maybe it's a function of how often education research results are abused. "There's one study that didn't find class size to matter for test scores. Let's pack 50 kids in every class!" Research is vitally important, but teaching is so hard, and there are so many variables, that I just think it's very difficult to extend results. As a consequence, I tend to value story and qualitative research more than quantitative.

I'll still read the research, looking for techniques, ideas, and inspiration. (Read a great bit from Ilana Horn and Sara Campbell just today!) But we can't start thinking it's the answer in itself. (Maybe learning is solvable?) Just because we can make something testable, doesn't mean we should. I think what I want most is integration of researchers with practicioners, working on the problem together. And then conversation with the researchers, not a lecture. Didn't someone prove those don't work?

End rant.

*Erdös: "Even at this early point in his career, Erdős had definite ideas about mathematical elegance. He believed that God, whom he affectionately called the S.F. or Supreme Fascist, had a transfinite book (“transfinite” being a mathematical concept for something larger than infinity) that contained the shortest, most beautiful proof for every conceivable mathematical problem. The highest compliment he could pay to a colleague’s work was to say, 'That’s straight from The Book.' " from the Encyclopedia Brittanica