Wednesday, June 26, 2019

Discussion in Math Class: Part 1 -- Why Bother?

I just finished my second year teaching. Pretty much the whole time I've been thinking hard about class discussion in math class. My teacher prep program (Boston Teacher Residency) was BIG on class discussions, so from the beginning of my career it's been a big thing I've thought about. That's not to say it's been a big part of my practice. I've got a lot of questions I'm trying to work through, a few answers, some resolution, and a bunch of unresolved issues. I'll try to present not only my current conclusions, but some of the narrative as well. I'll try to break this reflection into chunks. This is Part 1.

To be clear, when I say "classroom discussion," I mean a class full of students, all talking about a topic, or series of topics, for a sustained time (like...more than a couple minutes?). This looks mostly like one student talking at a time, and everyone else listening. The teacher may or may not be involved, and there may or may not be any other participation structures.

Experiencing Classroom Discussion as a Student

In math class, the discourse was almost entirely centered around the teacher. I never found discussions particularly educational. I'm not sure who did. For the most part, the content seemed pretty cut-and-dry. Image of a traditional classroom, the teacher's modeling and lecture provided all the information. Any small details, confusion, or minor discovery came out in the problems sets completed alone at home. I never feel the desire or need to wrestle over the math in discussion.

In my smaller math seminars later in college, where the professor explicitly invited discussion, we were all so intimidated and self-conscious that we didn't want to participate for fear of revealing how little we understood. By "we," I mean my friends and I who bonded over how lost in the sauce we felt so often. I presume there was at least a handful of people who felt, as I did in high school, that discussion was unnecessary or uninteresting.

These experience as a student led me to form the following schema about classroom discussion:
It felt like there was a pretty narrow target of students who benefitted from discussion. I would say these students are in the "Zone of Utility," where classroom discussion provides some kind of utility or benefit. They are relatively small "Goldilocks" group where the content is neither so hard it's intimidating, or so easy it's uninteresting to discuss. One assumption of this model is that understanding is one-dimensional spectrum, which is an oversimplification for sure.

What They Taught Me In Grad School

What was told to me about classroom discussion in grad school:
  • It should be something that happens in almost every class, almost every day.
  • It is an effective way to make sure everyone engages with the lesson objective or Big Idea.
  • It is useful in an of itself, because it requires students to make sense of other people's thinking.
  • It is a form of communication that can honor the discourse culture of the individuals and whole (classroom) community.
This teacher-side experience of class discussion led me to develop another model for thinking about classroom discussion, which (noticeably) isn't incompatible with the model I had developed as a student:
  • Without discussion, many students can engage in a common math experience, but fail to arrive at the teacher's learning objective, these students are, as I described myself earlier, "Lost in the Sauce." Only a small subset of students managed to meet the teacher's learning objective.
  • With discussion, many students can engage in a common math experience. As shown in the "Without Discussion" model, probably not all students will arrive at the teacher's learning objective independently, so the discussion after the math experience serves as a way to "funnel" most students to the learning objective.
What I learned from experience about classroom discussion in grad school:
  • I didn't have a super clear idea about what a classroom discussion CAN do, and what ONLY classroom discussion can do.
  • Discussion for its own sake is a palpable waste of time.
  • Class discussions aren't "free learning." Just because students are talking about what they did or what someone else did doesn't mean the speaker or listener are learning from what's being said.
This discouraging experience led me to form the additional model about how my classroom discussions went (and how I thought they went down in many other classrooms):
  • With ineffective discussion, many students can engage in a common math experience, but fail to arrive at the teacher's learning objective independently. These students are, as I described myself earlier, "Lost in the Sauce." Only a small subset of students managed to meet the teacher's learning objective. I then manage to lead an ineffective classroom discussion, which at best serves to validate and confirm those few students who had pretty much met the learning objective already, and at worst further alienates students who felt they had not met the learning objective.
Why bother? During my residency year and the vast majority of my first year teaching, this is what it felt like I was doing. It felt like there was no new learning happening during the discussion. Or, that is, it felt like the thing that my students were learning was that discussions are not useful for learning math. So I would bail on them. Pretty much every time. I'd rather let students keep working right up until the end of class, because I felt it was more likely that they learn something during the independent-/group-work time than during a discussion.

My end-of-class discussions weren't going particularly well. Where any of my classroom discussions going well?

To be continued!

Sunday, June 16, 2019

Proposal: A Less-Sequential High School Math Experience

How can we design a high school math department (and graduation requirements) so that we are accurately reflecting the vastness, diversity, and non-linearity of mathematics? Consider this representation of a hypothetical high school math department (Image here):


Understanding the Map
  • Each course is a semester-long math course
  • If there are two separate arrows from A to C, and B to C, then that means either A or B are required for C, not necessarily both. If the arrows meet before C, then that means that A and B are both required for C. Consider this helpful diagram (Image here):


Graduation Requirements
To go with this more open curriculum offering, is a graduation requirement system that tries to honor both the openness of the program, and the necessity of certain math topics in high school.
  • You must pass at least eight (8) distinct courses, including:
    • Proportionality, Linearity, Financial Literacy, Probability, Correlation & Causation
  • There are four Fields of mathematics, and students must take the appropriate number of courses in each field. Some courses satisfy multiple requirements (Image here.).
    • (A) Algebra (take at least 4)
    • (G) Geometry (take at least 2)
    • (P) Applied Math (take at least 1)
    • (U) Pure Math (take at least 1)

  • You must pass any and all prerequisite courses before taking a course.
  • Students are permitted to "test out" of prerequisites provided they demonstrate sufficient proficiency, as determined by the math department.
    • If a student tests out of a course, they are not awarded the credit for that course, nor does that satisfy any Field requirements.
Logistical Comments I Have
  • The standard pushback here is that this kind of system would require teachers to teach lots of different courses in a year. More preps makes things more difficult for teachers.
  • Lots of students might be forced to take certain courses simply based on the fact that it was the only course available at a certain time in a certain semester. The bigger the school, the better the school's capacity to support all these different courses.
  • It might make more sense to default all 9th graders to starting in "Linearity," and only walk them back to "Proportionality" if necessary.
  • It would take a TON more resources to make this work if you have special populations of students who might need to be grouped (students with learning disabilities, students learning English as a second language, etc.)
  • This would work best if the other departments (ELA, History, Science, etc.) used a similar system.
Questions I Have for Y'all
  • What courses would YOU make required?
    • I thought Proportionality, Linearity, Financial Literacy, Probability, Correlation & Causation were the most important, recognizing that most high schoolers would already have proportionality from middle school, and some might already have linearity from middle school.
      • Those five courses I identified that everyone had to take are based solely upon my own value judgements of what math content I think every person should know before leaving high school. They definitely need to do more math than just those courses, but they definitely have to do at least those courses.
  • What other courses would you add?
    • These were all the courses/topics I could think of that I've either taught, seen, or felt spanned the traditional high school math curriculum.
      • Some courses I would need to figure out how to add are all the AP's and Computer Science.
      • I'd also like to add some kind of geometry-centered course where students are getting lots of geometry measurement and whatnot. Like, where in this course offering are students going to think hard about the volume of geometric solids (except calculus)?
      • I think Economics might better be split into Micro and Macro? Is one a prereq for the other?
      • If other departments get in on this, then that opens the door for requirements to be met across entire departments. For example, lots of schools consider Econ to be a social studies class. Next year we are teaching Mathematical Biology, which could count as a science credit.
    • Best case scenario, we think of the biggest reasonable course map possible, with all the appropriate relationships thought out, and then schools would pick some subset of that map.
  • Do these course descriptions make sense?
    • Some of them I don't even know how to describe (e.g., Proportionality)
    • Some of them I'm not sure are a whole semester of high school level work (e.g., Linear Algebra, Periodicity)
  • What do you think about the Fields?
    • I made the Field requirements so that they were roughly proportional to the course offerings. That is, there are way more Algebra classes in the map, so I required way more Algebra classes. But this is a reflection of my own limited ability to create course offerings. If I could think of more geometry courses, then I could make one.
    • I'm lumping statistics in with Applied Math, which would include Econ and Financial Literacy. The implication of this is that kids could take Projectile Motion or Financial Literacy and never HAVE to take any other stat classes. Except that I've already said that half of the Applied Math Field is a requirement on its own. I guess that says something about my own feelings about Applied Math in high school?
  • How does this gel with our understanding of intellectual development of kids?
    • By this model, you would have 9th graders taking Probability. You could also have 10th graders taking Linear Algebra. Is there a reason this couldn't be the case? It certainly feels a little weird, expecting 9th graders to do content that is typically assigned to 11th and 12th graders.
      • Is that because they are literally not OLD enough to do that kind of math? At which point, we should place another pre-requisite there, to slow them down? Or have an age requirement? But that feels wrong.
      • Or is that because we typically haven't taught them "enough math" until they are older? At which point, we should place another pre-requisite there, to make sure they get all the required math.
    • This would put kids of all grades/ages in the same classes. Lots of classes might be mostly grade-grouped--Linearity and Proportionality would have mostly 9th graders. Polynomials and Solving Triangles would have mostly 11th and 12th graders.
  • Where does solving equations go?
    • I think that for the most part, most of the big first "this is how you solve equations" would come out in "linearity" maybe? And then as appropriate, as you run into different contexts for solving equations, you learn new methods for solving equations?
  • How can you make sure that students are making informed decisions?
    • They would have a good idea about what each course is about, so that they can pick the classes that are the most interesting to them.
    • They would have to have a good understanding of the requirements system overall, so that they could "see" the paths and scheduling required to get to certain outcomes. Which is complicated, for sure. I mean, it's like trying to plan a major for college--only you have a major for math, science, history, ELA, and whatever else.
  • What do you think?