Math Teaching, #5 out of 6

1. Jen

Jen made her bones as a math teacher at Roxprep.  She also has 2 kids same age as my 2 kids, so when she writes "Things were pretty busy this week at the Spencer house in the past few days" I have some idea of what she means.  Thanks Jen, for making some time for sharing your thoughts.

One perspective she brings builds on Sean's observation from yesterday: You Y'all We has higher upside but bigger downside. 

Jen worked in a typical Boston elementary as a math coach.  Here is her recollection using the district's assigned curriculum (called "Investigations" which is conceptual, uses manipulatives, etc).  With slight edits for clarity, Jen writes:

Straight up untrained, fully conceptual, little practice (Investigations) ends up with

-about a third (non-scientific analysis) of the kids completely comprehending a lot from the curriculum; they  can transfer easily to algorithms and efficiency.

-the second third of kids know that they should be getting it, but don’t, and are frustrated and don’t learn much because the teacher doesn’t know how to move them forward.

-the last third of kids are just playing with blocks and don’t get it at all.

(The Good) In my work with a self proclaimed math phobic elementary teacher in BPS, she said that teaching Investigations was the first time she ever understood math.   I think that is a win for the conceptual math people.

However, Investigations as a curriculum never gets to the point.   I think that is the hard thing for people who are teaching in a more conceptual approach.  The formalization.  The “Why ARE we playing with blocks?”  The “How do I do this quickly in 4th grade without having to draw pictures all the time?”  Way too much time spent on working on a problem with seemingly no point, no full class check ins, no guiding to very struggling students, no think-pair-share, no tickets to leave, lots of cutting and gluing and poster making, no class summary, no essential questions (or answers).

Putting all of this in the hands of any “non math” teacher is really difficult.  (And most elementary math teachers are "non math" teachers). 

Jen also makes this observation: that high-performing charters have both novice teachers and some expert teachers.  The expert teachers lead discussions particularly well.  They get visited not just by teachers in their school, but teachers from other schools.  Jen says:

When teachers get to visit experts (Thanks Paul for welcoming me last year!), and ask them to share their curricula, they are still  missing a really important piece -- the "documentation."  This is something I'm working on at Match.   Getting in writing what good teachers do (not just the lesson plan, or the problem sets) -- so another teacher(less skilled) can see what good discussion or questioning looks like for each math concept.  (MG: It's a variation on the purpose of Japanese Lesson Planning). 

Math teachers in general are not good discussion leaders. English teachers tend to be good at this.  But in all my conversations with ELA teachers about how to run a discussion, we have had a hard time figuring out how to transfer it to math well. We math teachers need to learn how to “get the kids there” with conceptual lessons and without giving away the "answer" -- really allowing kids to have a good struggle (Goldilocks style, just frustrating enough, but not too frustrating that they give up entirely…i.e., Zone of Proximal Development, Multiple Entry Points, insert your own jargon here). 

2. Dylan is a math major from Bowdoin who went through the Match Teacher Residency and is now a teacher at RoxPrep. 

He shared this:

My take on I/We/You vs You/Y'all/We: kids learn both ways, but there's more variance on the side of You/Y'all/We. The latter approach leads to deeper understanding when used appropriately (read: used most days, but not as the only teaching style) in the hands of a great teacher. I think it also leads to more challenges in classroom management, more struggles for kids with gaps in prior knowledge, and just a big ole mess in the hands of a novice.

I can attest to that last part, I've been there.

I/We/You is safer, can absolutely lead to excellent learning in the hands of an expert, but is, I think, less powerful than You/Y'all/We in terms of retention/real understanding.

I'd like to add a caveat to that last: I think what I said is much less true at the upper high school level, in particular Algebra II and beyond. If it's math that went undiscovered/unproven for thousands of years by smart people, it's going to be pretty tough to teach through inquiry. I'm looking at you, the quadratic formula, conic sections, and integration by parts. These topics exist at all levels, but there are a lot more of them in upper high school.

Anyway, I'm less interested in the argument between I/We/You and You/Y'all/We. I think it will go on, unresolved, forever. What I'm more interested in is the idea of students making sense of mathematics -- of believing that math consists of questions that can be answered by reasoning, experimenting, and applying prior knowledge; that it describes the world around us in ways that are useful, that math problems can be solved multiple ways and often have multiple answers. This can happen through I/We/You, You/Y'all/We, or through a number of other structures.

If you're hungry for more I've put my favorites together on this page:

I looked at Dylan's page.  It's great.  If you are a math teacher you should visit.  Links to lots of terrific math teacher bloggers. 

3. Peter, a journalist, wrote

Hi Michael,

This discussion represents much of what baffles so many otherwise smart parents and educators.... I didn't count them, but mentions of "end goal" and "goal" here are numerous and mostly they involve variations of teaching kids "to think."  I thought the end goal was that kids learn math?

The big picture problem here -- and Elizabeth's nicely written piece is a perfect example -- is a mistaking-the-forest-for-the- trees problem.  The equivalent would be turning an English class into a study of  etymology or a history class into the study of historiography or a driving class into an engine repair class.

Parents want their kids to learn how to add and subtract, mulitply and divide -- and a little memorization doesn't hurt -- so they can go on to the harder stuff.

Most of this discussion is deep deep into the weeds and teachers should avoid pulling their kids in with them.  The goal of math education is learning math.  Feel free to post if you like.

My response to Peter:

Good point.  i think there is some language imprecision.   maybe another way of saying it would be this, let me try, and you react: 

A. let's say you have a goal - teach a kid to find area of shapes.  there are 3 levels of eventual understanding.

1. can't do it. 

2. can do it during class by pattern recognition - monday tue and weds we banged out a ton of length times width problems, so i can do it today thursday by remembering. 

but if 6 months from now you say "Hey we're going to put carpet into the basement, how much do we need?" they can't do it. 

many kids get stranded on this island. 

3. can find how much carpet - i.e., can apply the area concepts to all sorts of situations, grasp when to use it, etc. 

so math teachers are constantly struggling to get kids off the "can't do it in real life" island of "sort of knowing it." 

when they say "goal is to think" in part the more complete phrasing would be "goal is to it well enough to use it fluently, both in life and in tougher math, without using up any working memory." 

B. there's actually a second aspect of what math teachers mean by "think."  goes to willingham "memory is the residue of thought." 

the question is: during a math class, how much are kids watching a teacher show a pattern of 4 steps to do Z type of problem, then doing a few themselves where they're not thinking about the problem that much, they're thinking about the steps.  brain tax is low there on kids.  pretty easy. 


you could have them struggle with "how the heck do we even approach this problem?"  in this case, the "think" part is not the end.  it's the means -- the kids will remember it better, per willingham, if their brains sort of "chew on it"....even if they can't solve it themselves, the "right way" then is much more sticky in the brain when revealed. 

but that sort of thinking, per the book Thinking Fast And Slow, the brain does NOT like to do.  we trick ourselves to thinking that we like thinking.  but we really like puzzles that we can make steady progress on, not ones where we often get stuck. 

now in a typical high-poverty school (traditional or charter), or in a lowtracked class at a suburban high school, it's very hard for teacher to unilaterally build the positive class culture it takes to get kids to DO this thinking.  this is what Sizer wrote about - a cease fire treaty b/w teacher and students - "if you avoid obvious misbehavior, i won't ask you to think very hard." 

what teachers like Paul and Ryan are able to do, partly b/c they're part of an awesome team of teachers in outlier schools, is get kids to think that hard.  if you watched a typical USA math class and then watched either of these guys, you'd say "i'm impressed, particularly with the student effort happening here."  you'd probably guess "must be a good group of kids."  but if you then could watch video of these kids in their former schools, you'd see the exact same kids NOT trying hard at real math thinking (giving up when teacher says to do it), or the exact same kids in a class where they're simply copying the steps.


Wow.  I can't do justice to your wonderful elaboration, but I think Robert Pondiscio's post the other day was apt: we have to work with the teachers we have not the ones we don't have.

I believe that the teaching ranks are simply full of domain-deficient teachers.


Well, i agree we (USA) have a huge content knowledge problem.  however, i think it lives much more in the elementary teachers (who do a little math each day, much lower math apt) than the middle and high school math teachers. 

however, i think the teachers who are REALLY good at math content struggle much more with a different transactional problem -- they can't kids to want to try hard or participate in the "thinking part."  calling it "classroom management" captures some of it, i suppose, but people usually think that means "avoiding misbehavior."  there's a big second layer of classroom management which is "trying to think instead of comply" -- when the brain is wired to find this annoying.  so plenty of 800 Math SAT teachers can't get the non-honors kids to try.