Final journal entry for my current math course. Lots to think about looking back.

  • Comment on our discussion of point gathering vs data gathering.

Short version: every teacher (in Peter’s words) is either doing one of these two things, or trying to mush them together. Point gathering is what it sounds like: a gradebook where everything gets points and you just add the points up at the end to get a grade. Data gathering is where you collect data that you can look over holistically and read a narrative from that tells you whether someone’s learned. That data narrative can then turn into a grade based on professional judgment and/or by translating it into scores to add up / average, but then you’re scoring based on learning, not based on specific assessment events.

This is more-or-less the idea behind Standards-Based Grading (SBG) which the math blog world already got on board with heavily around the time I was starting teaching. It’s also shown up in various other names, with different twists and spins on how it’s organized, from other assessment gurus and researchers. Whatever the specifics of the system, though, it’s fundamentally different than point-adding because we’re using assessment events as data to generate a measure of learning, rather than letting specific events (eg. old, early quizzes back when someone was just forming new ideas) dictate the grade.

So, yeah, I’m on board with this, no doubt.

  • Re-respond to the following questions…

What is mathematics?

Mathematics is the practice of finding logical patterns and exploring them logically. It is the community of mathematics practitioners and their shared wealth of understanding of the world’s mathematical patterns. It is the language in which we communicate those ideas to each other.

How does someone best learn mathematics?

They best learn math by being given resources and an environment in which to do mathematics that are on the edge of their current understanding. Those resources can include peers, technological tools; the environment ought to be social, supportive, safe and guided.

How does someone best teach mathematics?

By providing an effective, efficient environment in which to think mathematically while also giving well-designed challenges to the learner that afford mathematically ‘flow’ in the direction of new ideas and understandings.

  • Read your entire journal and comment on what has changed the most for you?

Mostly, I’ve gotten what I wanted to get, which was to be supported in implementing the Thinking Classroom approach to math class. I’m more ready to just dive in and do it; I’ve got a better theoretical and data-driven background in why these techniques work and what they’re meant to improve. I’ve got a much greater appreciation for the nuance and skills that this style of teaching requires, and know I’m both on the right track and also have a long road ahead to work on improving this for myself.

I also have more questions about designing good tasks than I had when I first started. I feel like there’s a lot of room to research and investigate how to design a good task, rather than stumble across one, and how to pass those design strategies on to teachers for their own work.

So during this class, we discussed homework. The research on who actually does it (not the kids who need it), the effect it has on inequality in education (makes it way worse), and generally why it should be knocked off its high pedestal.

Now, honestly, a lot of us teachers kind-of-sort-of know this but then assign “homework” anyway, but give time in class to complete it. Does this fix the problem? Well … there’s the matter of what students actually *do* when you assign homework.

Pie charts showing what students do when assigned either marked or unmarked homework
This, uh, this ain’t that great

If homework is “marked” in any way – including a simple check-for-completion! – you get the graph on the left (based on Peter’s research observations in multiple classrooms; image grabbed from a ppt from the course). ‘Getting Help’ means that someone like a tutor or a parent did the work ‘with’ them – and they probably can’t do it on their own afterwards.

If you remove homework marks completely, the cheating almost disappears and is split roughly evenly between Did It and Didn’t Do It. So it’s an improvement … that’s still only helping around 1/3 of your students. That’s, uh, not great numbers.

So what’s the fix? For starters, recognize that there really is enough time in-class for students to learn things. You might need to recover some of that time from things that aren’t students-doing-the-thinking, but it’s possible.

The next step is to replace homework with “check your understanding” opportunities … wait, no, bear with me, yes I know your textbook calls its question sets “check your understanding” and that hasn’t helped, just hold on before you leave. It’s not about replacing the questions or just what we call them – it’s about replacing the mentality and the motivation for doing them.

So the example we discussed was, a set of CYU questions are put out for students to work on together, labelled as to what topic they’re checking and/or what difficulty. Students are not required to complete them all. They’re there solely as a way for students to check whether they can do them. They’re there to give some indication of where the student is going and where they’re currently at.

Now, a weird effect that’s been observed is that even though textbooks have stuff you can use for this, you almost definitely are better off not giving students something in a textbook format – like not even a photocopy of a textbook page. Textbooks have too strong a “complete this busywork” association tied to them already, such that it poisons the well.

The where they are and where they are going guideposts are the most important part to any approach to this. Homework as ‘practice’ more-or-less doesn’t work: those who need it, don’t get it (see above), and unlike physical practice where you actually can get even better at something like weightlifting by continuing to lift over and over and over, math skills are not that transferable. Once you know how to factor quadratics, you reeeeeeally don’t need to do 50 more. Like, okay, maybe a few. But not 50.

But if someone isn’t yet there, like maybe they can factor the easy ones but don’t know how to handle the harder ones, they need some way to know that. Solving 50 easy ones won’t actually build up mental “muscles” that suddenly enable them to solve hard ones! That’s nonsense. They need to know that there’s more to learn and roughly how to get there.

  • Comment on our discussion on numeracy in general and our discussion on the relationship between numeracy and mathematics in particular. (and etc on numeracy tasks)

Okay, so Prof. Liljedahl has a very particular idea of what numeracy is and what it isn’t. I’m convinced, but the challenge will be whether the rest of the world chooses to understand the term this way.

What numeracy isn’t: being able to add, subtract, multiply and divide. knowing your multiplication tables.

What numeracy is: stepping up with whatever mathematical tools you’ve got and getting the job done.

Pithy, but requires unpacking.

The ‘job’ in this case would be, any kind of messy situation where math may be useful. Like are you willing and able to take this weird scheduling situation and apply some mathematical reasoning to it? Juggle the numbers around and make sense of it? Even when there isn’t an easy “right” answer?

Peter has a set of “numeracy tasks” available on his website, and they all tend to have a few things in common:

  1. Low floor, high ceiling.
  2. HUGE degrees of freedom.
  3. Fixed point – an “obstacle”.
  4. Intentional ambiguity.

The obstacle helps rein in the huge degrees of freedom by giving a creative constraint. The ambiguity and the degrees of freedom will, in some problems, draw out questions of personal choice and value judgments into the solutions. For example, we worked on one problem to do with how fundraising dollars were split among students who had done uneven amounts of fundraising, and whose ski trips would cost varying amounts. Questions of individual responsibility, offsetting the expenses for the underprivileged or those new to the sport, and generally trying to make sure no one would be left feeling taken advantage of were all themes that came up in our solutions. Just by asking us to sort out how to split some numbers fairly!

So, okay. Back to the definition of “numeracy”. This isn’t how everyone uses the word. Often ‘innumeracy’ is associated with poor estimation skills, inability to mentally process differences in large numbers, or simply being unable to multiply. Peter’s take on those cases is that they’re not about numeracy, they’re about “number-acy”. Which, okay, also probably not a word that everyone else will buy into, but it gets the point across. Being numerate is more than arithmetic in the same way that being literate is about more than spelling or grammar quizzes. You have to be willing to engage with the world through the medium of language to be literate. Likewise, you have to be willing to engage with the world through the medium of, and with the toolset of, mathematics to be numerate. If you aren’t willing and able to grapple with life using the mathematical tools you’ve got, that’s akin to being unable to handle situations in life that require reading.