I’ve been writing a few pieces on and off, trying to get focused, and suddenly I thought of Darryl Yong. I’d forgotten his name, but I just googled “professor teaches high school math”.

Darryl Yong, a math professor at Harvey Mudd, decided to teach for a year. He didn’t teach calculus, he taught algebra and geometry, and he taught at a low income school worried about test scores and gangs.

You should read his entire excellent paper. He outlined four key lessons:

Lesson 1: Schools Are Complex Systems Involving People, Culture, and Policies

Lesson 2. Student Self-Concept Is the Best Explanatory Variable for Student Success

Lesson 3. Teaching Is a Far Less Respected Profession Than It Should Be

Lesson 4. It’s Not the Written Curriculum That Matters, It’s the Assessed Curriculum

Yong is writing for math professors, but his essay ought to be required reading by reformers and politicians all. I came into the game knowing 1 and 4 already. (Lesson 4, in particular, is something that no test prep instructor ever needs spelled out.) I’ve never felt disrespected as a teacher, so I can’t speak to lesson 3. His description of typical professional development is very similar to my experiences at my previous school. However, my first school and particularly my current school do a good job with PD. It’s not so much that I find it all useful, as it’s not a waste of time and it’s blissfully short of jargon. We are also given lots of department time. However, I don’t see why pointless PD has anything to do with respect or lack thereof. The administration gets mandates, it all rolls downhill.

But the first four or five times I read of his experiences, I growled when I got to Lesson 2. Self Concept, blah blah blah:

That’s the purview of happy talkers like Carol Dweck, I snarled mentally every time I read it previously.

So why, tonight, did I reread it? Couldn’t tell you, but for some reason I saw something I’d missed the first times I’d read Lesson 2. Yong gives an example of the need for “scaffolding” using factoring quadratics. It’s perfect. He gives a list of quadratics and points out that math professors (and many textbooks) think of all quadratics as roughly equivalent: easy to do, functionally indistinguishable. But to struggling algebra students, they are tremendously different activities. Hardest to factor are a>1 and b=0. (And then, after you beat that into their heads, they are suddenly stumped by c=0 cases—which they thought were easy before. Sigh.) He then goes on to describe a student who was stumped by solving simple equations but could do the same task if it was finding the x and y intercepts of a linear equation in standard form.

And I sat up and thought Hey.

I can’t even begin to tell you how many times I’ve expounded on this to my colleagues. I write about it, too, of how I redefined an algebra curriculum so that I could keep my weakest kids engaged and passing. In The Driftwood and the Vortex, I delineated the careful sequencing needed to keep struggling students engaged:

* I learned how long I could run an upfront discussion before their attention waned, carefully timing the moment when I moved them onto practice problems—which had to be carefully managed, too. Struggling students need to build momentum on a string of problems before they get to their first hesitation point. Hit that hesitation point too early and they “shut down”. They look away and find a more rewarding activity: talk to their neighbor, take a nap, turn up the volume on their iPod, sketch, tiptoe out of the room when I’m not looking, send objects airborne in pursuit of a target. Finding worksheets that started with problems simple enough to get them working and then built to more challenging work that wasn’t too hard took up a big chunk of my day. I’d spend hours looking through practice sets to be sure they didn’t leap to tough problems too soon, and often just wrote a dozen or more identical problems on the board, simply varying the numbers. Even with all that effort, some concepts were still too hard for some students, and I couldn’t always reach each one before he got pulled into a disruptive vortex. And so, from managing the math back to managing the students.*

I’ve also seen amazing things happen when I just let kids listen to poetry and think about it, rather than insist they read, understand, and analyze it as standards would dictate.

In the TEACH! documentary, Lindsay Chinn achieved improvement by teaching less, and giving her students a sense of success.

But read Dweck or others on “self-concept”, and they mean something quite different: If students believe that intelligence is malleable, their story goes, teachers can convince them to work harder.

Yong is not really talking about self-concept as it’s understood in the education policy world. And yet—he is. Which means that I, too, think that student self-concept is important, even though I’ve been sneering at the idea for the past two or three years. I just go about it, like Yong did, like Lindsay Chinn did, in an entirely different way than the one pushed by experts. I give my students the experience of success, of taking on a task *they find difficult* and then triumphing over it.

But you don’t achieve this by lying to students about intelligence, which is not terribly malleable.

The way to give students an improved self-concept in math is to make the math easier.

Not easy. Not, as it is usually dismissed by politicians and reformers, by “dummying it down”. But by setting reasonable goals for the students you have.

Do you teach the math or teach the students? I’ve asked this before. It’s a fundamental question for teachers working with populations that so obsess education reformers. Yet reformers spew trite platitudes about “higher expectations”, as if teachers can eliminate struggles simply by superior pedagogy and refusal to tolerate failure.

I wish Yong had taken this issue on directly, rather than hinting at the problem but wrapping it in a popular buzzword that hid his message. Plenty of people read his work and think “ah, that’s the key! Get the kids to believe they can succeed at math!” when in fact, I think the message is closer to “give the kids mathematics tasks they can handle” which isn’t at all the same thing. I don’t ever let my kids think they are math rock stars. Many of them don’t, in fact, have the ability to learn the math necessary for advanced understanding of chemistry or engineering. But that doesn’t mean they shouldn’t be challenged, shouldn’t begin to understand the confidence necessary to dive in and give it their best shot.

But no one really dares advocate making math easier, particularly in the era of Common Core. Instead, we get platitudes like this paean to an old-school music teacher, advocating drill, failure, and, god save us, “grit”.

Both Yong and I are guilty of what education reformers everywhere decry as “the soft bigotry of low expectations”. It’s rhetorically convenient to ignore the fact that teachers lower expectations *because* they want to give their students the experience of struggling with intellectually challenging material.

Reading Yong again led me to realize that I need to start talking more about “self-concept”—not to dismiss it, but to redefine it. I care about my students’ self-concept. That’s exactly why I lower expectations, creating a rigorous yet achievable curriculum that dangles a reachable carrot in front of my students. In doing so, I get them to try.

This is all so tricky and confusing.

Dweck’s research doesn’t offer a quick-fix, though it’s often said to do just that. But despite so much of the silliness that gets tossed in these discussions, the core of Dweck’s research is pretty solid. The truth here about student motivation is going to be complex and subtle.

Lower standards are so often coupled with meaningless, intellectually empty activities for struggling students. Real learning inevitably means pushing students beyond what they’re currently capable of. But, and this is your point, material that is too difficult might as well be too easy, because unreasonable material offers no challenge to students.

What’s the solution? There is no solution, just the hard work of teachers trying to make the school year work for their kids. I’m sure that our testing regime isn’t making this hard work easier, and is often making this all impossible. But before we had a testing regime we had schools that gave up on so many students…

So complicated, that I can’t wrap my head around it.

As a postscript, I got barked at on my way out of my last job for making the curriculum easier for a student instead of failing them. That’s not relevant in anyway except…yeah, I know what you mean. This stuff is tough.

But, and this is your point, material that is too difficult might as well be too easy, because unreasonable material offers no challenge to students.Well, crap. I wish I’d said that. Nice.

One thing I’ve begun to realize is that, while I don’t think elementary teacher math skills are a problem so far as teaching, they may be a problem so far as developing curriculum. When faced with kids who can’t do the material, they don’t know how to adjust. So maybe we could get slightly more out of elementary school kids who struggle with math if their teachers were capable of slowing it down when faced with struggling students. That is, we don’t need more expert teachers to teach tougher math, but to slow down math instruction for the kids who need it. And even that might be a panacea.

However, do we really know that kids are being handed meaningless busy work?

I don’t know about Dweck, but I do think her research can only be relevant if the tasks are mentally relevant and reachable by the students–whereas people takeaway “effort leads to success”.

Thanks for keeping this dialog going.

I believe terms like “easy”, “complex”, “high cognitive demand” are all tricky to define. Also, it’s difficult for us educators to talk about “lowering expectations” without getting into trouble.

Consider these two staircases in the picture. Both represent instructional sequences of tasks where the “difficulty” or “complexity” of the first task and the last task are the same. So on the one hand you might say that both instructional sequences have the same high expectations of students because they both expect students to get to the top of the staircase.

However, if students don’t have the ability to climb large stairs, maybe you have to add more scaffolding. Is that lowering expectations? Perhaps, but I still expect my students to get to the same point. It might take more time, but my goal is for everyone to reach the top. And, I would argue that if my students needed more scaffolding and I taught using instructional sequence B, then fewer students would make it to the top of the staircase.

Of course, if students are lacking in preparation and experience, then you might need to start the staircase at a lower point. Maybe you don’t get as far on the staircase, but I would hope that we still expect great things of students with low proficiency in mathematics.

I know this analogy isn’t perfect, but it helps me to think about it in this way. I think the key is to have a high expectation for student growth, but at the same time to adjust instructional sequences of tasks so that students can reach those expectations.

Darryl, thanks so much for your comment–and thanks so much for taking the time to teach math for a year. Your experiences rang very true for me.

Of course, I agree that saying that we “lowered expectations” is politically problematic. But in fact we are actually *raising* expectations for those students who would otherwise tune out or give up. we are moving them from no mental effort to significant, meaningful mental effort. The problem is the awareness gap between the teachers and those who demand “high expectations”.

In your picture, of course, the real issue is time. I have a colleague, an experienced math teacher who came to our school because he wanted a block schedule and year-round algebra I to really focus in on teaching kids well. He is a bit discouraged because at least half the class is still lost. I’ve told him to consider slowing it down even further, but he says that they won’t get through the course in the year–even given double time. But is it more important to get through the course or keep the kids learning math? (And that’s leaving aside the issue of what to do when half the class needs more time but the rest is ready to progress.)

So the real question is what if some kids have to start at a lower rung, one that isn’t on your picture, and end on one at the bottom of your picture? They’ve still made progress. We’ve still gotten them challenged and learning math. But at a certain point, we’re just not telling the truth if we say we are teaching them algebra. Yet we have no other choice.

I’ve been teaching for five years, and I mostly work with kids who have survived algebra I, in theory. Not top kids, not the kids who take Algebra II in freshman year, but the kids who barely made it through algebra I by freshman year. I’m teaching them in subsequent subjects, but in most cases, their understanding of algebra I is extremely sketchy, and their understanding of those subsequent subjects practically non-existent. So I end up reteaching algebra I while covering the concepts in the new course as well. I can’t go nearly as deep into the new material.

So at what point does it make sense to limit the material and change the course name? Because as you probably know, many math teachers are determined to teach the course material, even if they leave half the kids behind—and since they are actually teaching the course on the transcript, it’s hard to fault them.

That’s why I wish you’d been more frank about the issue, even as I understand the problematic nature of being frank. You were only in it for a year, but can you see how, if we simplify the material to give the struggling kids the scaffolding they need, they won’t get to the same place in a year? And will they ever be at the place–given the same time–as kids who don’t struggle?

Sorry for the long comment–thanks again for responding!

JMK–I still feel that careful scaffolding can address both lack of instructional time/too much to “cover” and low proficiency. It won’t and can’t solve everything, but it can help, and that is all I was trying to assert in my article.

By scaffolding carefully, students can develop mathematical proficiency and feel more successful. That leads them to be willing to take larger and larger “steps”, which can help to accelerate one’s instructional sequences.

Though it’s difficult, I still believe that it’s important to believe that all students can reach high levels of proficiency. Whether you can do it in the amount of time that you have is another thing. Take for example, the Alg 1 standards as defined by the current CA Math Framework–there are way too many things in there in my opinion. That year I taught Alg 1, I made the decision to do a better job at some of the topics and to skim other topics. I wouldn’t characterize this decision as lowering expectations, but instead instructional choices about what’s important given limited time. My rationale is that students would be better served by better understanding what I felt were the central ideas in Alg 1 rather than to have a poor or merely algorithmic understanding of everything in the Alg 1 standards. Maybe in this way we would have to do less reteaching from year to year?

I wouldn’t characterize this decision as lowering expectations, but instead instructional choices about what’s important given limited time.I wouldn’t either, but that’s how it is characterized nonetheless. That’s the insight I had—that we reduce the amount of new material we give our students because learning more slowly gives them a greater sense of accomplishment, and self-concept. When education reformers are looking at test scores, though, they don’t see the self-concept. They see “lowered expectations”.

And ultimately, if you slow down instruction year after year, you run out of time. There’s no question that kids who learn more slowly, who need more time and scaffolding to learn the same material, will eventually run out of time in high school to learn the material that kids who learn more quickly have mastered. I believe that getting the middle third of high school students properly through second year algebra would be sufficient. We could properly teach them math, give them a rigorous, positive experience, and they would be ready for trigonometry or pre-calc in college.

But again, there’s no earthly way that this wouldn’t be described as lowered expectations.

So my question to you, as a highly respected math professor, coming from a lowly math teacher who isn’t even teaching her strongest subject, is this: how do we communicate the fact that many students are better off learning math more slowly than others? How do we tell them that holding students to high standards and rigorous math may mean teaching less math? Because trust me, that’s not a message they want to hear.