Monday, October 30, 2017

Books 2017: Quarter 3

Friends, it is fall. And fall is definitely the best time for talking about books.

As you probably already know, I've been reading a classic a month for the last two years. It started as a one-year project in 2014, but I've enjoyed it enough to keep going with it & will probably continue until it starts to feel like a chore. You can find my past reviews by clicking on the "books" tag at the end of this post, or be my friend on Goodreads. (You can also just go to the site & hunt down my review feed without being my friend, if that's more your speed.)

ICYMI, the classics I selected to read in 2017 are here.

On to the reviews!

Wednesday, October 4, 2017

SMPs for Parents and Other Civilians: MP 1 (Part 2)



This is the second follow-up post to a previous one aimed at parents and other non-math teachers, The Real Power of Studying Mathematics, about the significance of the Standards for Mathematical practice.

For the first follow-up post, see MP 1, Part 1: Making Sense of Problems.
* * *

In our last post, we talked about the first chunk of MP 1--what it means to make sense of a mathematics problem. Today, we'll continue with the second chunk of Practice 1: Persevering in solving problems. What does it mean to persevere in the context of a mathematics problem? What does perseverance look like? What does it not look like? And what can parents do to support students?

At the most literal level, perseverance just means to keep going and not give up. Sounds simple, right? Alas, for many students, especially as they begin to experience rich, challenging mathematics for the first time, it's not always so simple.

Wednesday, September 27, 2017

SMPs for Parents and Other Civilians: MP 1 (Part 1)

This is the first follow-up post to a previous one aimed at parents and other non-math teachers, The Real Power of Studying Mathematics, about the significance of the Standards for Mathematical practice.

I was talking to someone once about this practice (MP 1, "Make sense of problems and persevere in solving them") and she asked me, "Isn't that just, like, doing math?"

Well, yes and no.

Yes, in that if you can't make sense of a problem, you're probably not going to get very far toward solving it. And if you can't persevere, you probably won't get to a solution even if you can make sense of the problem. In some ways, MP 1 is a kind of "Gateway Practice"; if you struggle to make sense and persevere, it's difficult to get much of any math done at all.

No, in that MP 1 is more of a starting point, a 'necessary' condition for doing powerful mathematics, but it is far from sufficient. You made sense of a problem enough to get started? Great! You're hanging in there and persevering? Also great! But there is a lot more to be said regarding how students are able to make sense of problems, and a lot to be said about how effective and efficient our perseverance is, and more yet to be said about what we do once we have solved a problem. All of that is what we get from the other seven practices.

Let us break MP 1 down into its two constituent parts:

    1) Make sense of problems.

    2) Persevere in solving problems.

Today we'll focus on what it means to make sense of problems; in the next post we'll come back to what it means to persevere in solving them.

So...what does it mean to make sense of a problem?

Ask any math teacher; if s/he had a nickel for every time a kid read a problem and threw up their hand to proclaim "I don't get it!," s/he'd have a whole lot of nickels.

To make sense of a problem means to read or look at a problem, comprehend the information that's being communicated, and be able to ask yourself, "What mathematics is involved in solving this problem? How will I know when I have an answer, and what will the answer mean?"

Wednesday, September 20, 2017

The Real Power of Studying Mathematics

Hint: It's not knowing 17 different ways to solve a quadratic equation.

As the wee progeny of many of my friends and family began to wind their way through elementary (and middle???) school, I've had a number of conversations lately with adults who work outside of education about what it means to have a "good" math education and/or a "good" math teacher and in particular what should they look for in a school or classroom to know that their child is going to get a decent mathematical education.

It's a very tricky, very loaded question that I always struggle to answer in any helpful way. I mean, I can certainly list any number of elements I would hope to see in any math classroom my child were going to spend time in. For example:

  • A focused, coherent, and rigorous curriculum that aligns at least pretty well with the Common Core
  • A balance of procedural, conceptual, and problem solving/application elements
  • A focus on making sense of mathematics rather than on "answer-getting"
  • Rich, open-ended opportunities for tinkering, discovery, and general "mathematical play"
  • Procedural fluency that is built on conceptual understanding and robust number sense rather than rote memorization
  • De-emphasizing speed, but without altogether abandoning the idea of "mathematical efficiency"
  • Explicit discussion of the myriad ways of accessing and making sense of mathematics (multiple representations, multiple solutions, etc.)
  • Explicit discussion of the brain science that underlies fixed vs. growth mindsets
  • Celebration of mistakes as a necessary part of the learning process that enriches and enhances our understanding
  • Opportunities for collaboration with diverse peers (and I mean diverse in every sense of the word)

...Etc.

...Etc.

...Etc.

Wednesday, July 12, 2017

Books 2017: Quarter 2

Guys. Where is the year going. July?? Crazy pants.

As you probably already know, I've been reading a classic a month for the last two years. It started as a one-year project in 2014, but I've enjoyed it enough to keep going with it & will probably continue until it starts to feel like a chore. You can find my past reviews by clicking on the "books" tag at the end of this post, or be my friend on Goodreads. (You can also just go to the site & hunt down my review feed without being my friend, if that's more your speed.)

ICYMI, the classics I selected to read in 2017 are here.

On to the reviews!

Friday, May 5, 2017

Dividing small by big (fraction division & pattern blocks, part 2)

Last time, I related the tale of how, with a single fifth grade arithmetic problem, Cathy Humphreys shook my confidence in my math abilities to the core and then rebuilt it again from the rubble, better, faster, stronger, because that's how she rolls.

(Do you know why 1 ÷ 2/3 = 3/2? Are you sure?

Are you?

ARE

YOU

?)

1 ÷ 2/3 is tricky because, unlike, for example, 3/2 ÷ 1/4, the divisor does not fit evenly into the dividend. But once you understand the nature of the problem--what fraction of the unit in question comprises the leftover bit?--you can probably more or less make your way through most problems where the divisor is at least smaller than the dividend.

So hold onto your pantaloons, mateys; we're about to go off the map a bit. Here there be dragons, ie, problems where we are asked to divide a SMALL fraction by a BIGGER fraction.


CRAZY TALK.

Tuesday, May 2, 2017

That time Cathy Humphreys taught me to divide fractions

Despite my background as a high school teacher, I've gotten deeply interested in grade 3-5 math in the past few years, particularly all the bits related to number, operations, & algebraic thinking and how they weave together to create the ramp that ultimately gives kids access to formal algebra.

But it was not always this way! As a college math major filling out applications to masters & secondary credential programs, I definitely saw myself as a high school teacher, much more interested in the complexity and rich structure of Algebra II and trigonometry and calculus than in the usual middle school topics. And I *certainly* had never gone back to closely examine my own conceptual understanding of the foundational mathematics we learn in elementary school. Who wants to teach fractions and decimals when you could be initiating kids into the wonders of trigonometric functions??

BO.

RING.

So, I got into a secondary program, started student teaching Algebra I, & learning all the magical things they teach you in Curriculum & Instruction (ie, "methods") class about how kids make sense of ideas like variables and functions and data analysis over time and what it really means to understand all these things anyway. It was mystifying and terrifying and amazing, and in addition to learning how to teach, those experiences also unlocked for me an entirely new dimension of understanding. It was exhilarating ("Who knew math could be even MORE AMAZING??) but also a bit panic-inducing ("How the HECK am I supposed to get kids to understand it THIS way?!?"). All in all, though, I was starting to feel pretty darn good about my content knowledge.

And then, one day, Cathy Humphreys came to class.

She came to teach us about fractions.