Preparing Students for Boring Teachers

But they need to listen to lectures in college; don’t we need to prepare them for that too?

I’ve heard the argument plenty of times. Sure, all that fun and engaging stuff you’re doing is awesome, learning by doing, etc, but we need to make sure they’re ready to get lectured at in college. (When I taught middle school some people said the same thing about getting lectured at in high school.) This seems to particularly resonate with folks who aren’t teachers or who are new to teaching. Not sure why.

I disagree, pretty strongly. A few reasons:

  1. I’m skeptical there’s some special skills called “listening to a lecture” that students need to practice for a few hundred or thousand hours (by getting lectured at) in order to develop. Sure, we should think about students’ attention spans, but I’m skeptical lecturing at kids is the most useful way to improve a kid’s attention.
  2. If students are going to be lectured at in the future, it’s probably most important that they have a broad base of knowledge and skills so that the lecture makes sense and connects with their prior knowledge. I’m going to build that base of knowledge the best way I know how.
  3. If college is some dystopia of endless lectures, it underscores the urgency and importance of building students’ curiosity and love of learning so that they are in a better position to pursue their passions, regardless of the quality of some subset of their education.
  4. Colleges are changing faster than many folks think. Not that they’re changing very fast, but they’re changing. We’re talking about aiming behind a moving target. No sense preparing students for the past.

 

I was asked recently if I lecture. I don’t really know how to answer that question. Do I explain things, model ideas and strategies on the board, point out connections, and teach by telling? Absolutely. All the time. But I don’t think that has to be the defining characteristic of my teaching. In my class, students do math. I use that math to elicit what they know and don’t know, and based on that knowledge I may choose to deliver some explicit instruction. Maybe for two minutes at a time, maybe for twenty.

I don’t see a lot of value in talking in absolutes here. “Lecture” seems to imply that the teacher talks for the whole period, maybe bold students are willing to ask questions, and that’s the lesson. Proponents of lecturing seem to view the alternative as hippy-dippy projects, or aimless discovery learning. My teaching hangs out in the middle. Sometimes I deliver information. Sometimes I ask students to figure something out. Sometimes I ask students to practice a skill. This can happen in any order based on what tools I think are most useful that day. There’s no magic bullet, no one right answer. That intellectual work of figuring out what is going to work tomorrow for my students is probably my favorite part of the job.

Two Types of Teachers

Below is an outburst I made at a meeting yesterday in response to a comment from the facilitator about the afternoon being a hard time for people to pay attention. Not the most tactful thing I’ve done recently.

There are two types of teachers in the world. There are teachers who love to tell you all of their excuses for why kids can’t learn last period, or first period, or after lunch, or Friday, or Monday, or the week before Christmas, or the month of December, or the month of June. Then there are teachers who say my students have brains and can use them and we are going to learn today and I don’t give a shit what anyone says.

(I don’t mean to claim that all times are equally conducive to learning. Just that excuses don’t help, and my experience is that, with a handful of rare exceptions, any student can learn at any time if they have teachers who care about them and create opportunities for meaningful learning.)

Why I Don’t Try to Assess the Mathematical Practices

[A grade is] an inadequate report of an inaccurate judgment by a biased and variable judge of the extent to which a student has attained an undefined level of mastery of an unknown proportion of an indefinite material.

-Paul Dressel

The Standards for Mathematical Practice articulate eight very important skills that I hope my students are better able to exercise as a result of their time in my class. They are the goals I care most about in a given course. But I choose not to assess them, in the sense of assigning any type of grade based on students’ mastery of the practices. A few reasons why:

I’m Unreliable
I have all kinds of shortcomings when I try to assess what students know. I’m not terrible at deciding whether a student knows how to graph rational functions. But that unreliability is exacerbated when I try to assess something like “reason abstractly and quantitatively”. I’m going to end up inconsistent, I’m going to miss important information, and my assessment is difficult to attach to an objective reference point.

They’re Context-Dependent
“Mathematical modeling” is not one skill; a student could be great at modeling with linear functions and terrible at modeling with exponentials. This creates two problems — first, I’m making arbitrary decisions about which areas I want to prioritize in my assessment; and second, I’m giving an assessment that may not transfer to a new context. Doesn’t seem fair or helpful for students.

Assessment Creates Incentives
Whenever a grade is attached to anything, it creates incentives. I can live with creating incentives around learning logarithm rules. And, at some point, we will be finished learning about logarithm rules in my class, and I will feel comfortable attaching a grade to that learning. But students are never finished learning how to “construct viable arguments and critique the reasoning of others”, and attaching incentives to the learning of something as nebulous and challenging as argument is likely to incentivize shortcuts and create motivation problems.

Assessing Soft Skills is Time-Consuming
This by itself isn’t a reason not to assess the mathematical practices, but given everything else I could be spending my time on, if I’m not sure of the value of something, I’d love to put that time into finding and sequencing great tasks for my students.

Kids Prior Experiences Vary Enormously
Let’s pretend that I can assess the practice standards accurately. Say kids’ skill in “look for and make use of structure” varies from 1-10. Some kids are coming into my class at a 1, while others may come in at a 7. It’s pretty hard to move the needle on something as broad as the mathematical practices. Maybe I can move some kids a point or two, but if I’m being honest with myself, those gaps are likely to be impossible to close in a short year.

What I Am Interested In Instead

I don’t want to attach any type of formal assessment to the mathematical practices, but that doesn’t mean I don’t value them. Here’s what I do want to do.

Narrate the Practices As They Come Up
I want to make explicit whenever I see the practices in action, both to clarify what they are, and to model or use students as models of what it looks like to “attend to precision”. I want to make this an ongoing part of the language in my class, so kids are as comfortable talking about the practices as they are talking about content.

Student Reflection
I make the practices explicit, and I want to ask students to reflect on areas they feel comfortable in, and areas they’d like to improve. I have a one-pager that is meant to translate the practice standards into more kid-friendly language. I want students to spend time thinking, in a low-stakes context, about what the practices are and what they look like in math class.

Get Assessment Out of the Way
I have to assess students on something, so I’ll assess them on concrete, context-specific skills that reduce my inherent unreliability and bias and create a more level playing field. I also restrict grades to assessments on these skills so that for the majority of class time we can focus on doing math — on putting the practice standards into practice, reasoning, finding patterns, arguing, and figuring things out. That’s the work of mathematics I want to prioritize in my class.

Arithmetic is Hard

Annie Perkins posted a neat problem on Twitter yesterday, and I was hooked. Check it out:

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That was an email from a student. An 8th grader! So cool.

I had to do a little background on Lucas numbers and the golden ratio to appreciate the gravity of the conjecture. Whoa! That is surprising. Sounds like a problem worth solving.

Only one catch — I had no idea what Phi bar was. Turns out it’s pretty hard to google a Greek letter with a bar on top.

Ok, that’s fine. There’s enough information here to figure out what it is:

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I solved for Phi bar and got calculating, but didn’t end up with any Lucas numbers. Weird. I went back and realized I made a mistake finding Phi bar. Solved it again, and dove into the Lucas numbers. Again, no dice, I had made a different mistake. Went back a third time, managed to keep track of my negatives and subtract when I was supposed to subtract, and ended up with an elegant answer for Phi bar. (I’ll leave that as an exercise to the reader, for those out there who also aren’t familiar with that constant.)

Arithmetic is Hard 

I went on to have a great deal of fun with my new values and their relationship with the Lucas sequence. But this problem was a good reminder that I chose to persevere in large part because I have more mathematical maturity than almost all of my students. This is a situation I have put many students in, where I want them to work toward a certain realization or conceptual understanding, but challenges with arithmetic get them bogged down. For most students, that’s where they just turn off, and I lose the opportunity I was working toward. Worse, it’s only a subset of students who are getting bogged down and missing out on the realization — but those are the students for whom that type of mathematical thinking happens least, and who need it the most.

Arithmetic is hard. Algebra is hard. Lots of things in math are hard, and there are plenty of situations where that hard math stands between students and the big ideas that I want them to grapple with. It’s easy to forget, or to make excuses, or to just brush these issues under the rug. I don’t think I’m particularly good at this type of scaffolding, and this is a challenge that comes up day after day — for students, one that may have haunted them for years. And too many students stop asking questions to figure out where their mistakes are and just nod politely, copy something down from the board or from their neighbor, and let the mathematics pass them by.

Answering Questions

We must not fool ourselves, as for years I fooled myself, into thinking that guiding children to answers by carefully chosen leading questions is in any important respect different from just telling them the answers in the first place. Children who have been led up to answers by teachers’ questions are later helpless unless they can remember the questions, or ask themselves similar questions, and this is exactly what they cannot do. The only answer that really sticks in a child’s mind is the answer to a question that he asked or might ask of himself (199).

-John Holt, How Children Fail

I had a long stage of teaching where I avoided, as much as possible, answering student questions. I came across the article “Never Say Anything a Kid Can Say”, and it became a dogma for me. When a student asked a question, I would either ask them some clever leading questions to lead them to the answer, or (more often) tell them to “use their resources” or “figure it out”.

This didn’t work particularly well. Some kids were able to figure things out, but I left many more floundering, and created classroom management problems in the process. And it was the kids who were already struggling who benefited the least from this strategy, falling even further behind.

At some point I realized that this wasn’t working particularly well, but I still didn’t want to answer every question a student asked. My criteria became arbitrary; I met some student questions with my own questions, but I might answer a question if I was frustrated or the lesson wasn’t going as well as I wanted. I built up some intuition over time for what questions I thought I should answer and what questions I shouldn’t, but it was haphazard and I’m skeptical my questioning was particularly effective.

I recently read Peter Liljedahl’s “Building Thinking Classrooms”, in which he presents an alternate approach to answering questions:

Students only ask three types of questions: (1) proximity questions–asked when the teacher is close; (2) stop-thinking questions–most often of the form ‘is this right’l and (3) keep-thinking questions–questions that students ask so they can get back to work. Only the third of these types should be answered. The first two types need to be acknowledged but not answered (382).

I’m still sorting through what Liljedahl means by (1) and (2), but I want to focus on (3).

The idea of a “keep-thinking” question seems like a useful and practical criterion for answering questions. Will the answer to the question allow the student to keep doing mathematical thinking, or is the valuable mathematical thinking between the student and the answer they’re looking for?

When a student asks a question because they are stuck, and the answer to that question will allow them to keep thinking, that seems like a particularly useful moment for learning, and a moment where being helpful may be the best strategy.

The Common Core Way

Screenshot 2016-08-04 at 3.11.03 PM.pngI’ve heard the phrase the “Common Core way” several times this summer, as in “we used to teach math differently, but now we do it the Common Core way”. I want to unpack that idea.

What does the Common Core actually say about teaching and learning? Let’s look at the standards. There are exactly three points the standards make (pg. 3-8) before they dive into the content standards.

Toward Greater Focus and Coherence

To deliver on the promise of common standards, the standards must address the problem of a curriculum that is “a mile wide and an inch deep.”

We can argue about whether the standards actually achieve that goal, though I’m confident they are an improvement over the median state standards that preceded the Common Core.

Understanding Mathematics

One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y).

We want students to understand, and one tool to probe for that understanding is to ask kids to explain a piece of mathematics. Nothing groundbreaking here.

Standards for Mathematical Practice

The Standards for Mathematical Practice describe the 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.

The Standards for Mathematical Practice focus on what students should be able to do — to reason abstractly, to construct arguments, to model, to make use of structure. Good to know.

Key Shifts

We can also look, separate from the standards themselves, at the key shifts:

  • Greater focus on fewer topics
  • Coherence: Linking topics and thinking across grades
  • Rigor: Pursue conceptual understanding, procedural skills and fluency, and application with equal intensity

Nothing surprising here. Again, we can argue about whether the Common Core meets these goals, but these are goals worth working toward.

The Standards

Finally, there are the standards themselves. Does the content of the standards dictate instruction? Let’s look at the progression toward adding and subtracting fractions as an example.

3.NF: 

  • Develop understanding of fractions as numbers

4.NF:

  • Extend understanding of fraction equivalence and ordering
  • Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers
  • Understand decimal notation for fractions, and compare decimal fractions

5.NF:

  • Use equivalent fractions as a strategy to add and subtract fractions

These standards, like many other places in the Common Core, progress across several grades. First students should know what fractions are. The next year they should be able to compare fractions and think about them in multiple ways. Finally, they are ready to perform operations on them.

The standards also give an example of how students can link their previous knowledge to a new topic — using equivalent fractions (also called finding a common denominator) to add and subtract fractions.

This seems only to illuminate the point about coherence — a given grade is not an island, but a piece of a deliberate progression built around our best understandings of how children learn math.

The Common Core Way?

There are definitely differences here from standards that preceded the Common Core. My argument is not that everything is the same. Instead, I’m arguing that many teachers have always had these goals for students, and they represent common sense changes. Focus and coherence? Let’s do it. Standards that link content between grades? Awesome. Students who reason, argue, model? I’m in.

I don’t see any evidence that the Common Core is telling me how to teach. It’s not completely agnostic, but I see no evidence of a “Common Core Way”. The big difference I notice is when, as Dan suggests, I watch the verbs.

The Common Core asks students to understand, to explain, to reason, to argue, to model, to build on their knowledge from year to year, and to make sense of mathematics. It’s not just about what they know, it’s about what students are doing in math class. It’s about the verbs.

Maybe that’s what folks mean by the “Common Core way”. But the standards are still a political lightning rod, and the more they are framed as prescribing a new and different way of teaching math, the more polarizing they will be. Instead, I’d love to focus on the fact that the Common Core names some very reasonable goals for students, that none of these goals are particularly surprising, and that if anything we are raising the bar for what students are able to do in math class. With this focus, I think the Common Core will find more support and less polarization.

There are lots of tools to move students toward these goals. Some of them will continue to frustrate parents, as parents will naturally be frustrated when their children have a tough time with something new. But I think that if we say, “oh, well that’s just the Common Core way,” we’re setting the standards up for failure. Instead, we can frame it as, “I have ambitious goals for your daughter or son — to engage her or him in mathematical thinking, reasoning, and sense-making. It’s going to be hard sometimes. That’s fine — it’s hard because it’s worth doing.” That’s a message that is more likely to resonate with parents, and is faithful to the spirit of the Common Core.

We send a message, which is also my firm belief, that the Common Core is not some directive we received from on high to change everything. Instead, it’s one step forward, and the best tool we’ve got to reach the potential for teaching and learning mathematics for all students.

Thinking Classrooms and Formative Assessment

I recently read Peter Liljedahl’s paper “Building Thinking Classrooms: Conditions for Problem Solving” (available for free on ResearchGate!). It’s a pretty quick read, and has been really thought-provoking for me. I had been familiar with Peter’s work primarily through Alex Overwijk’s writing and presenting on vertical  non-permanent surfaces and visibly random groupings. I thought that was all there was to it — vertical non-permanent surfaces were a useful tool to get students collaborating on problems, and visibly random groupings were a useful tool to break down status barriers. Cool. Done.

But Peter’s work is much less interested in the specific tools of vertical non-permanent surfaces or visibly random groupings than some broader goals around what a math class should look like.

I wanted to build, what I now call, a thinking classroom–a classroom that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals thinking collectively, learning together and constructing knowledge and understanding through activity and discussion (362).

Peter’s article outlines how his work led to nine principles of thinking classrooms. They are divided into three stages in this table:

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I have questions about several of these elements that I want to explore in the future — the diagram is certainly insufficient to describe the practices that lead to a thinking classroom. Check out pages 381-382 in the article for more detail if you are interestd. That said, I find this a really useful framework in that it sets a goal — a thinking classroom — and uses structures to move toward that goal, rather than treating each structure as an end by itself.

Inputs vs Outputs

I notice that these elements seem to be focused on teacher inputs. The way students are grouped, the way they do their work, the way questions are answered, the moments chosen for instruction. These are all teacher moves to encourage thinking. But there is little attention paid to the learning outputs — the quality of thinking and learning that is happening. Assessment falls in that category, but Peter frames it in his paper as:

Assessment in a thinking classroom needs to be mostly about the involvement of students in the learning process through efforts to communicate with them where they are and where they are going in their learning. It needs to honour the activities of a thinking classroom through a focus on the processes of learning more so than the products and it needs to include both group work and individual work (382).

I think I agree with that statement, but it doesn’t provide much specificity. When do I figure out if kids are learning? How does responsive teaching fit into a thinking classroom?

I’m excited to try out some elements of the thinking classroom this coming year, and hopefully flesh out what they look like and how they work in my classroom. But I don’t want to do so with an exclusive focus on the inputs. If I execute these aspects of a thinking classroom, I will create an environment where students become more likely to engage in thinking and reasoning in my class. But I don’t want to act as if, just because I’m using some effective teacher inputs, my students are certain to learn.

I would conjecture that, if the principles of a thinking classroom are useful to occasion thinking, these structures have to be complemented with a system of formative assessment focused on the outputs and the products of that thinking, so that I can make sure each student is actually learning, and learning what I think they’re learning. And if they’re not, I need systems to be able to responsive and provide new learning opportunities.

I’m starting to stack up more goals than are reasonable for this year, and I’m not sure where this one will fall. But here goes: work to promote the principles of a thinking classroom for my students, and match that with equal energy and attention to whether students are actually learning and figuring out how I can respond to what I learn about their learning.