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)




As long as we're making wish lists, I'd also want a teacher with solid content knowledge, including MKT, SCK, and all the rest. I'd want a teacher who's well-versed in how to encourage a growth mindset, who believes that all students can learn mathematics at a high level, and who has spent some serious time wrestling with questions of mathematical access and equity. I'd want a teacher who actively strives to improve his or her practice and has opportunities to develop professionally, who has the enthusiastic support of his or her department and administrators in doing so, including the time and resources to do it. Unrealistic, you say? Hey, it's my wish list!

The trouble with this sort of wish list is that it doesn't translate very well into something a parent with no training in mathematics education can walk into a classroom for an hour and determine. Heck, it's not even something someone with LOTS of mathematical training can determine in your typical, say, one-hour (or less?) classroom visit on one day. We do studies where we agonize over the best ways to characterize the quality of mathematics instruction that is taking place in a particular classroom and/or school and/or districts, run by experts with decades and decades of experience between them as teachers and administrators and coaches and teacher educators and researchers, and it is still very, very non-trivial.

Which is all to say, being a parent is hard enough without suddenly having to become an expert in every academic subject that is taught in school. And I feel bad that I don't have a whole lot to share, a lot of the time, that makes it much easier. In a way, it's a little like the philosophical discussion my doctor friends have sometimes about whether someone who is not themselves a doctor can ever truly give informed consent.

In these conversations, though, the Common Core comes up a lot (as you might imagine), including lots of questions about what it even is and how it's different from "regular math" and why the heck is this happening anyway. And that inevitably leads to the Standards for Mathematical Practice, the part of the Common Core that seems to get the least air time outside of mathematical circles. And interestingly, these discussions seem to often be the ones parents find most useful.

At some point I promised someone I'd try to explain them in civilian adult-friendly terms, along with why exactly they are such a big deal and why it is so, so important that we have them and are teaching them well, so here we are.

Content Standards vs. Practice Standards

The first thing to know about the Common Core math standards (which, these days, we just refer to in California as 'The California Math Standards') is that they have two parts.

The first part is what we refer to as the Content Standards. I am not going to get much into the structure of the Content Standards because that's a whole other post and about a million people have already written beautifully about it. (Who knows, I might even update this post with links to them!)

For the purposes of this post, let us just say that the content standards are what people usually think of when they think about math standards; the content standards are where you'll find the parts that say "You have to know how to multiply fractions" and "You have to know how to solve equations" and "You have to know scientific notation" etc. etc. etc.

"But wait a moment," I hear you non-math teachers asking. "What else do we even need in the math standards?"

Friends, I am glad you asked.

What's All This About Practice Standards?

What are the practices? Well, to quote the CCSSM...

    "The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years." (p. 8)

Essentially, as in many fields, people who do math in really productive, effective ways have some particular "habits of mind" that serve them well when they try to solve mathematical problems or even just make sense of mathematical ideas. The idea of the Practices was to try to take all those habits and ways of working and distill them down to a manageable number of "practices" that teachers could try to foster in their students in order to help them be successful in mathematics (where "success," at least in my opinion, means not only "able to solve lots of hard problems efficiently" but also "appreciates the power and elegance of mathematics and at least kinda-sorta enjoys it at least some of the time").

Source: Math Coach's Corner

I feel like for years math teachers have had these sort of "underground" conversations about the concept of mathematical practices, whether we called them "mathematical habits of mind" or "habits of powerful math learners" or whatever. The point being that, it's all very well to teach kids scientific notation or the quadratic formula or how to divide fractions, but the content knowledge alone only really goes so far in terms of helping them extend their learning to an unfamiliar context or gnaw their way into a novel type of problem or see deep connections between what they learned in 3rd grade and 6th grade and Algebra II.

Frankly, it's not enough to know how to plug numbers into the quadratic formula and memorize what the type of answer means. It's not even enough to know how to derive it and why the type of answer means what it means. The question is, how did I derive it? What skills did I use to reason about this question, based on what I already knew? How did I think about this problem? How did I communicate about it? How did I relate it to problems I've solved in the past or mathematics I already understand? What questions did I ask myself to move forward? What did I do when I got stuck?

The Practices are the real answer to the age-old question, "When am I ever going to use this, and don't say if I become a math teacher or rocket scientist." With some exceptions (and excepting obvious professions like math teacher and rocket scientist), the real value of studying mathematics is less being able to solve a quadratic equation and declare how many solutions it has and of what type and more being able to reason, strategize, persevere, and communicate effectively in a quantitative situation.

As a math teacher, I do care that you can solve a multi-step algebraic equation and tell me something about the solution and its different representations and the different meanings it could have, because you need to be able to do that to be successful in your next math course and graduate with decent marks and choose your best, most satisfying path forward in life, etc. etc. But other than that, I don't really care about you being able to solve the equation.

That's right; I said it.

The thing is, if you can't solve it, you might be struggling with quantitative reasoning, or with thinking about problems in different ways, or with considering the structure of a problem, or with thinking back to how you've solved related problems in the past, and those are struggles that may close doors for you later in life. That's what I care about.

What I care about is that you can reason your way through the process, that you can take what you understand about numbers and operations and see how they apply in a particular type of problem, that you can communicate all of it effectively (in words, equations, graphs, tables, whatever), that you can see how the meaning of the answer changes depending on the question that's asked.

I care that you're in the habit of asking yourself, "Is this the most efficient way? Is there a cleaner way?," and of thinking, "Usually I'd do this, but because of this particular aspect of this particular problem, this other way makes more sense" and "Hm, I'm not sure about that answer, let me think back through my process and see where it doesn't make sense."

I want you think, "This worked in all these situations, I wonder if it always work? Why might it not work?" and "This problem seems a lot like this other one where I did ____, I wonder if that might work here" and "Wow, s/he's working mighty hard to convince me his/her solution is right, but does it really make sense to me?"

That's the real value of studying mathematics, in my opinion. That's the power of the Practices.

So what are these much-vaunted math-doing habits?

For those of you who don't quite have them tatooed on your brain yet (non-math teachers, you are forgiven):

(Non-teachers and teachers alike -- It's okay if you don't yet totally understand what they mean. We'll get to that.)

As you can imagine, much blood ink was spilled and many garments rent in the process of determining exactly how to slice and dice really effective mathematical ways of thinking and working into a finite number of coherent practices. (This is what I've been told, anyway.) How to characterize exactly how the best math thinkers think?? And how to make them general enough that they apply universally to the entire discipline, while also being concrete enough to see how they might apply in a particular grade level?? And how to word them juuuust right, and are these two too similar, and should this one actually be split into two separate practices?? etc. etc. etc.

You can certainly make the argument that they're not perfect (and people do), but personally, I think they're a pretty darn functional place to start and we could do much, much worse in terms of a framework for teaching kids not just the content but how to engage with it most effectively.

Next: Let's tackle these Practices one at a time in non-math teacher-friendly ways.

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?




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.


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??



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.

Wednesday, April 12, 2017

My NCSM & NCTM 2017 Highlights

Hi friends,

I spent this past week in San Antonio attending the annual conferences of the National Council of Supervisors of Mathematics and the National Council of Teachers of Mathematics as I do most years, and as always, I had a fantastic time and learned a ton.

Just a quick overview of some of the highlights:

Learning, Planning, and Teaching Together: Facilitating Job-Embedded Professional Development, Kendra Lomax and Becca Lewis (University of Washington). I feel incredibly strongly about practice-based PD, and since I also happened to know that Kendra got rave reviews from a group of K-2 teachers she'd recently been doing some practice-based/job-embedded PD with, I was excited to hear what she and her colleague had to say on the topic. We heard about how they've been exploring the idea of rehearsal as a way of teachers preparing to teach a lesson, where the rehearsing involves very focused and thoughtful real-time coaching from a teacher educator. They shared some interesting videos, and I look forward to exploring more of their work on on the TEDD (Teacher Education by Design) website.

Tuesday, April 4, 2017

Books: 2017 Quarter 1

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!

Tuesday, January 10, 2017

Books: The 2017 Classics

BEHOLD! The classic novels I'll be reading in 2017:

JANUARY: The Forever War, by Joe Haldeman. "The Earth's leaders have drawn a line in the interstellar sand, despite the fact that the fierce alien enemy that they would oppose is inscrutable, unconquerable, and very far away. Private William Mandella has been propelled through space and time to fight in the distant thousand-year conflict; to perform his duties without rancor and even rise up through military ranks. He's willing to do whatever it takes to survive the ordeal and return home. But 'home' may be even more terrifying than battle, because, thanks to the time dilation caused by space travel, Mandella is aging months while the Earth he left behind is aging centuries." I like to do one true sci fi classic each year & this one got a lot of votes.

FEBRUARY: Midnight in the Garden of Good and Evil, by John Berendt. "Voodoo. Decadent socialites packing Lugars. Cotillions. With towns like Savannah, Georgia, who needs Fellini? Midnight in the Garden of Good and Evil takes two narrative strands--each worthy of its own book--and weaves them together to make a single fascinating tale. The first is author John Berendt's loving depiction of the characters and rascals that prowled Savannah in the eight years it was his home-away-from-home. Then, on May 2, 1981, the book's second story line commences, when Jim Williams, a wealthy antique dealer and Savannah's host with the most, kills his "friend" Danny Hansford. (If those quotes make you suspect something, you should.) Was it self-defense, as Williams claimed--or murder? The book sketches four separate trials, during which the dark side of this genteel party town is well and truly plumbed."

Wednesday, December 30, 2015

Books Update: Quarter 4

Congrats on making it through another year, you crazy diamond, you! This year I read 80 books & over 32,000 pages. We're now at the tail-end of quarter 4, so let's talk about what I've been reading over the last three months.

As you may or may not 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.

These were my last three classics of the year:

October: The Picture of Dorian Gray, by Oliver Wilde (1890, 166 pages). 4 stars. This was my spooooky Halloween read! Young, beautiful, innocent, naive Dorian Gray sits for a portrait for an artist friend, who proclaims the painting his best work. Dorian is suddenly struck by the horrible thought that he will age and lose his beauty while his portrait will remain beautiful and youthful forever. In a fit of panic he desperately prays that his and the painting's roles should be reversed, so that the painting ages while he stays young and beautiful. Not only does his wish come true, but the portrait also begins to reflect the condition of his soul. Creepiness and philosophy ensue. The writing is clever and gorgeous, and Wilde is a master of dry wit & witty repartee, plus it's less than 200 pages so pretty easy to knock out on a plane ride or similar. But, it is worth mentioning that it's still not a light read. Some parts of it are quite dense and heavily philosophical (I found myself carefully re-reading many sections because I didn't catch or process it all the first or second time around), so definitely something to save for when you are fully alert and not distracted.