Ask, Play, Argue

Ask we’re all packing things up to head home for the holidays, I thought I’d give a quick, fun reduction of the Common Core Standards for Mathematical Practice, courtesy of Christopher Danielson:

1: Ask questions. Ask why. Ask how. Ask whether your answer is right. Ask whether it makes sense. Ask what assumptions you have made, and whether an alternate set of assumptions might be warranted. Ask what if. Ask what if not.

2: Play. See what happens if you carry out the computation you have in mind, even if you are not sure it’s the right one. See what happens if you do it the other way around. Try to think like someone else would think. Tweak and see what happens.

3: Argue. Say why you think you are right. Say why you might be wrong. Try to understand how someone else sees things, and say why you think their perspective may be valid. Do not accept what others say is so, but listen carefully to it so that you can decide whether it is.

So go ahead and do it–ask, play, and argue all throughout break. The more we practice being mathematicians, the better we can teach other mathematicians.

(Also, hopefully you’re having an easier and easier time figuring out what it looks like for students to ask, play, and argue in your own classrooms. But if you want one more example, check out the “estimation wall.”)


What is “interest”?

I think one of the questions that plagues teachers, and maybe math teachers in particular, is how to get students interested in the material. When we’re handed a curriculum and told to teach, it can be hard to find ways to get students on the hook.

I came across a great blog post the other day about interest in math classes. If you hop over to that link, you can find a brief overview of the psychology of interest.

One of the most useful ideas is the description of the two “appraisals” we make that lead us to decide something is interesting:

The first appraisal is an evaluation of an event’s novelty-complexity, which refers to  evaluating an event as new, unexpected, complex, hard to process, surprising, mysterious, or obscure. . . The second, less obvious appraisal is an evaluation of an event’s comprehensibility. Appraisal theories would label this appraisal a coping-potential appraisal because it involves people considering whether they have the skills, knowledge, and resources to deal with an event. In the case of interest, people are “dealing with” an unexpected and complex event—they are trying to understand it. In short, if people appraise an event as new and comprehensible, they will find it interesting.

As we attempt to interest students, I think we often focus on the latter—making sure students feel the material is comprehensible.  This was a nice reminder, I think, that we need to balance that, and also make sure the material is new, complex, and puzzling at the same time.

If you’re interested in talking more about these ideas, please sign up for the next “How Students Learn” webinar on December 11!

Rigor and culture

Within Teach For America, we often talk about two different things: rigor and culture. In fact, you’ve probably heard from your MTLD something about these two. We’ve got a couple scales we use to consider what we see from students. In terms of rigor–which I prefer to think of as “student thinking”–we have five categories:

  1. No learning
  2. Passive and confused when presented with new content
  3. Factual recall and procedural thinking
  4. Analysis, explanation, and application
  5. Evaluation and synthesis

And in terms of culture we have five different labels:

  1. Destructive
  2. Apathetic or unruly
  3. Compliant and on-task
  4. Interested and hard-working
  5. Passionate, joyful, and urgent

Of course, if you ask most people, you might find that they don’t think these two things are quite so separate–the kind of “joy” students take in their classroom, for example, is going to have a lot to do with what kind of thinking they’re doing. Doing meaningful thinking is fun!

Which is why I was interested to come across this scale, with which you can consider how a learner is engaging with a given task:

  1. Disengaged: distracted by other things while doing the task;
  2. Compliant: completed the task but didn’t get much else out of it;
  3. Interest: being curious about what is presented;
  4. Engaging: wanting to be, and being involved in the task;
  5. Committing: developing a sense of responsibility towards the task;
  6. Internalizing: merging objective concepts (the task or what is to be learned) with subjective experience (what is already owned) resulting in understanding and therefore ownership, of new ideas;
  7. Interpreting: wanting and needing to communicate that understanding to others;
  8. Evaluating: wanting and being willing to put that understanding to the test.

This reminds me that ultimately, of course, student thinking and student “culture” are the very same thing–they’re all about how excited and invested a student is in the material that we put in front of them. If we want joyful students, and if we want students doing meaningful evaluation, we need to get materials in front of them that both pique their interest, and also allow them to evaluate some kind of thinking, and scaffolding materials in ways that encourage both those things.

So, for example, if we just tell students, “solve this equation for x,” of course we won’t get students being much more than disengaged. The question is–how do we take the materials we need to teach and modify them to get that kind of thinking.

Open Ended Questions!

Going into last summer I spent a lot of time reflecting on the failures and successes in my classroom. In particular, I focused on the classroom culture I created. I consider my classroom a high energy (albeit quirky), discussion based classroom. I absolutely LOVE seeing students on the edge of their seats, hands pointed skywards, desperate to volunteer some mathematical tidbit to the discussion. Unfortunately, this is a rare reaction in my room. Sure, every class will have 3-5 “high-achievers” who will raise their hand for anything- often before a question has even been posed. However, I have found that many students lurk in the background, unwilling to become involved. I had to find a way to get these students interested and confident with Algebra I.

This year, the strategy that has been met with the most success (in terms of student engagement, participation, and even math JOY) has been utilizing open ended questions. Open ended questions are questions in which their is no one answer. Sometimes there isn’t even a right or wrong answer. I have found these questions give students breathing room to get creative and helps dismiss their fear over being wrong. When the platform for a question is open all students feel confident that they can bring something to the table. In some classes open ended questions have plunged my room into mathematical chaos (one class applauded a girl who was able to rewrite 2(x + 3) as -2(-x – 3) who’d have thought? All the sudden math confidence and success was associated with being cool).

In this scenario, my question was represent the expression to the right in as many ways as you can-    2(x + 3)

Some answers included     2x + 6,   2(x) + 2(3),   2(3 + x),    (x + 3)(2),    x + x + 3 + 3    (that was my particular favorite)

Some incorrect answers included-    x(2 + 3),    3(x + 2),    2x + 3,    so on and so forth

I’ve found that I can address some very powerful foundational mathematical concepts by having the students explain why the incorrect answers are mathematically unsound. Some classes actually reveled in coming up with creative new ways to represent the expression. Bottom line- my students were engaged, unafraid to provide answers, eagerly discussion, and concretely understanding the distributive property.

Places to Improve- As much as I have loved using these open ended questions there have been areas that I would like to improve. For example, some students use this strategy as an out to pick the most obvious answer and then give up. I am brainstorming ways to continue to push the students to connect with the math on a deeper level. A second issue I have had is classroom management. Generally the students (and myself) are so eager to volunteer that the noise level can be downright deafening. If you have any cool ideas or anecdotes you could share to help me remedy these issues please let me know!

What is an “algebra”?

You learn something new every day.

Here’s how one teacher defines an “algebra”:

 An algebra is essentially a set of objects that can be both added and multiplied, with the two operations fitting together via the distributive property.

In other words, that means it’s a set of things where if you add (which really also includes subtraction) or multiply (which, ditto, includes division), the thing you get out is still in that set.  If you take a number, and add another number, you get out a number.

That property also holds true for: expressions with variables, including polynomials; matrices; functions; even transformations on the plane!

This allows me to see how some things we teach in algebra that sort of seem tangential–like matrices, for example, and also some of the geometry–make a whole lot more sense. We are figuring out what rules work on an algebraic system, starting with the system of real numbers that students have learned about all through elementary and middle school, and seeing how it also applies to other sets of mathematical objects.

Mr. Honner cues up a couple other “big ideas” in algebra. I’m rewriting those slightly, based on some feedback from Grant Wiggins, while trying to get somewhat student friendly:

  • Organizing sets into binary relationships allows us see when there is and is not equivalence (e.g., equality and inequality)
  • Some of those relationships can be matched perfectly to each other (e.g. functions, in a big simplification)
  • Those relationships can be mapped onto a simple coordinate plane (e.g. the Cartesian coordintes)

Thinking on the year-long level

Grant Wiggins, author of the classic Understanding by Design, and one of my favorite math bloggers right now, has a great post asking “what is a course?” In it, he brings up a lot of issues that we’ve been talking about lately in many of my workshops: are we making the content meaningful for students? Are we asking big, important questions?

Here’s his core point:

I am claiming that to be a valid course, there has to be more than just a list of valued stuff that we cover – even if that list seems vital to me, the teacher. Rather, a course must seem coherent and meaningful from the learner’s perspective. There must be a narrative, if you will; there must be a throughline; there must be engaging and stimulating inquiries and performances that provide direction, priorities, and incentives.

If you’re not there yet, he’s also got this list of prompts you can ask yourself to make sure your course is a course:

  • By the end of the year students should be able to…. and grasp that…
  • The course builds toward…
  • The recurring big ideas about which we will go into depth are…
  • The following chapters and sequence support my goal of…
  • Given my long-term priority goals, the assessments need to determine if students can…
  • Given my goals, the following activities need to build insight and incentive…
  • If I have been successful, students will be able to transfer their learning to… and avoid such common misconceptions and habits as…

How many ways can you look at this problem?

I recently found the webpage for Harvard Calculus, and one thing I think is very cool is this “rule of nine” that shows nine different ways we might approach a math problem:

algebraically, analytically, geometrically, historically, graphically, numerically, conceptually, psychologically, as well as experimentally.

What the heck does that mean? This may require some calc knowledge, but for something like finding the derivative of sin(x) at x=pi/3, you could think in these ways:

algebraically we know sin’=cos and cos(pi/3)=1/2
analytically we know sin(x) = x-x^3/3! + x^5/5! – … and so sin'(x) = 1-x^2/2! + x^4/4!-… = cos(x) so that sin'(pi/3)=1/2.
geometrically sin(x) is the height of a right angle triangle with hypotenuse of length 1. The rate of change of this length in dependence of the angle can be seen geometrically.
historically the derivative can be derived from Euler’s formula exp(i x) = cos(x) + i sin(x) which has the derivative i exp(i x) = i cos(x) – sin(x). Comparing real and complex parts shows cos’=-sin, sin’=cos.
graphically draw the graph of sin(x) and determine the slope at x=pi/3
numerically take a small number 1/1000 and compute 1000 sin(pi/3+1/1000)-sin(pi/3)) which gives 0.499567.
conceptually since sin(x) increases for increasing for acute angles, the result is positive.
psychologically my teacher does not like to assign problems with irrational numbers as answers. The result should be a simple rational number. Because 0 and 1 are out of question, the next reasonable result is 1/2 ……
experimentally here is an esoteric experiment: why not use Fourier series and differentiate that series. To make it interesting, take f(x) = |sin(x)| since sin(x)=|sin(x)| around pi/3 …

One thing they note, though, is that if you try to approach a problem from too many ways at once, you’re going to overwhelm yourself before you even understand anything. So we should pick three, maybe four approaches, to make sure we really get something without going overboard.

Jo Boaler (yes, I know, I keep going on about her) has adapted this for algebra, and suggests that for algebra problem, we should have students try to see if they can express it in four of the five following ways:

in words, in pictures and diagrams, in tables, in graphs, and in symbols.

How many ways can you think through this problem?


Two kinds of algebra

I’m working through a free online class from Jo Boaler, author of What’s Math Got to Do with It?, a recommended read, (registration may be open still, and the further in the class I get, the more I recommend it). In the session I finished yesterday–“Appreciating Algebra”–she differentiated between two different kinds of algebraic thinking:

First, there’s procedural algebra. That’s generally when a variable stands for just one number, and we ask students to find that number. You know:

2x – 7 = 7. Find x.

But there’s also structural algebra. This is when we use variables to abstract and generalize–where rather than having just one value, a variable is used because it allows for flexibility, so that we can apply the same rule various circumstances. So, for example, we could build a rule that says in the nth case with this pattern, there will be 1 + 2(n – 1) blocks.

That’s powerful for various reasons. One big one is that it allows the algebra to rest on top of other ways of understanding. Even before students wrap their head around variables, they can see what the pattern is here: that in each new instance, two new blocks are added. They can see that in the first case, there was just one block. We can move from that intuitive understanding to an idea of how algebra and variables allow us to generalize–and not just extend this pattern to the fifth, sixth, seventh case, but tell us exactly what these shapes will look like in the 100th case, or in the 1000th case.

One study showed, though, that 73% of 13-year-olds believed variables could only have one value. That is, they could only understand procedural algebra. That’s because we spend so long at the beginning of algebra classes (or units, in earlier grade levels) doing that kind of work.

So how do we fix that? Research shows that an introduction to algebra should start with exploring patterns–without any algebraic symbols. Rather than asking closed questions (“how many blocks will be in the 5th shape?” “how many blocks will be in the hundredth shape?”) ask open questions (“how do you see this pattern growing?”, which in visual terms could have multiple correct answers), that help them understand that this is not a matter of right and wrong answers–it’s a matter of thinking and visualizing. This can be done with interesting, real world problems–and we should ask students to show their thinking with diagrams and tables of values. Then work up to doing structural algebra, representing the general case with variables that can take on multiple values. Only after that should students see structural algebra.

What do you think of all that? How does it shape the way you’ll approach your first units? Share in the comments, or in the Facebook group!

Oh those test scores

If you follow education news at all, you know that New York’s latest test scores just came out. And they ain’t pretty. It’s a big deal, because New York moved to Common Core testing this year, and the unveiling of these scores tells us a lot about how people in other states (like ours) are going to react as the shift happens.

What I don’t think it tells us, though, is much about how students are really doing. People are making a big deal about the drop in students scoring proficient, but that, as always, is a subjective thing. Politicians got to set the proficiency rating wherever they wanted. The only way we can use these scores meaningful is to compare place to place across New York. With the switch, any kind of historical analysis about student performance is moot.

I’ve been reading a lot about these scores, and here’s one take that’s stuck with me. While I disagree (as I’ve said before) with the claim here that Common Core has little worth, I agree that the numbers out there today simply don’t mean much.