Discussion: What are the obstacles to teaching math well?
This week in the science of learning, we look at one important question
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Readers discuss: Teaching math
I thought I’d try something different this week, and instead of links and ideas, I’d ask you, dear readers, some questions related to Friday’s post on instructional coach Neily Boyd’s experience helping teachers with math. Boyd says when it comes to math, we’re getting too hung up on terminology and not paying enough attention to how concepts and procedures get taught to students incrementally over time, especially in the crucial elementary school years.
I’d love to know what you think, and what your experience has been. Comments are open!
Question 1: What do teachers do when they’ve got a class behind grade level in math?
Here’s an excerpt of Friday’s conversation:
The Bell Ringer: A fourth grade math teacher I spoke with recently said the kids who have come to her this year are way behind in math. She’s a good, experienced teacher, but she was saying, ‘Now I have to figure out, how do I keep the fourth grade curriculum moving and fill in all the gaps at the same time?’
Neily Boyd: When the gaps are as significant as they are, then you need teacher expertise in essentially planning and adapting their own lesson. You've got a set of materials from your scripted curriculum to start with, but you're not guaranteed that's the right fit, so a lot of teacher planning is still required.
Question 2: Has the kind of heavily teacher-guided inquiry method that Boyd describes in the story gotten watered down until it’s not effective?
Again, from Friday:
The Bell Ringer: Just talked with a parent the other day who had moved her kids to a different school, and she was just so relieved they were showing them how to do math problems. According to this mom, what her kids were doing before in some classes was watching a video and then doing a worksheet. So that's not inquiry teaching, and that's not direct instruction—that's actually not anything, there's no teaching.
Neily Boyd: So when several charter schools started doing this guided inquiry method of math instruction—not only Brooke, but Achievement First, DSST in Denver was doing something similar—people like me at other charter schools took notice, because it was working. But you can watch how, over the progression of a decade and as its spread to schools without robust teaching coaching structures, it has often become such a watered-down, ineffective version of inquiry.
Teachers have to have enough judgment to translate the curriculum for themselves, depending on their students. And sometimes what happens is they might say, to be honest, ‘I don't know how to teach this, so I'm going to put this video on and kids will watch this,’ like the classroom you referenced.
Question 3: What are the keys to helping more elementary school kids solidify foundational skills? What’s missing from classrooms now?
From Friday:
The Bell Ringer: What I want to understand from your point of view as an instructional coach, is how do we start to fix math teaching so more students can learn math?
Neily Boyd: If I was a superintendent of a school district, what would I do right now? I can't change what's happening in teacher ed programs and I can't change the teacher pipeline that I have. I would focus our math resources on getting pre-K, K, first and second grade right. But we would do guided inquiry like Brooke, and I would train every person touching it, every principal who's observing it. I would make the problem so small that I could fully wrap my arms around it, and I would say: this is what we're going after. In order to get math in the upper grades right, we have to get foundational math right first.
Looking forward to hearing from you!
The video. I don’t know what the situation is in this particular school - you say they get good results - but I would not recommend doing this at all.
This class previously spent a week doing similar activities to work out things like ¾ x 6=8. Now they are spending a long painful class trying to figure out ¾ x 9 in the same way. Why? What next? Another week working out 2/17 x 5? Yes, I know this is supposed to be “understanding.” No, it’s just a complicated way of working out a problem using pictures.
You can demonstrate some simple examples for students using pictures to get the point across, but this doesn’t need to go on for so long. We should be trying to make things less complicated for students, not more complicated.
Some points:
a) This approach is confusing and not all students will be able to follow this. Even in this class, not all students are participating. Note the one boy (~2:30) who answers something incorrectly (the teacher doesn’t correct him?) and then puts his head down. I highly doubt this is going to work out well in your average class of 5th graders. Some students get to be the stars that come up with all the ideas; other students just feel stupid.
b) This is incredibly time consuming. If there were huge benefits to this, that may be reasonable. I’m not convinced. Students would be better off spending more time practicing fraction arithmetic.
c) Students who have been taught to rely on intuitive arguments like this often make mistakes (e.g. the student who tries to use chocolate bar explanations to convince themselves that 0/0=0; I’ve literally seen this sort of nonsense from my students in 1st year calculus)
d) The idea that students who have spent loads of time on these intuitive picture explanations will be better at fraction arithmetic is absurd. Fraction arithmetic needs to be practiced so much it’s automatic. Any student who resorts to pictures to figure out why you need to flip and multiply in the middle of an algebra problem is not fluent with fraction arithmetic and is bound to struggle. So, this begs the question, why spend so much time on chocolate bar fraction explanations?
e) Regarding point d), I’ve seen this on university entrance exams. Some students have spent more time on picture explanations than actually practicing math and will draw tick marks and circle groupings to figure out 24/8. Believe me, they’re spending time on this in school. I see the results.
f) Many elementary teachers would find this lesson challenging to give. I suspect both the teacher and students would be confused. It’s also time consuming to prepare a lesson like this and think of all the possibilities, etc. Then the teachers get blamed when it doesn’t work out. No, it doesn’t need to be this complicated in the first place. I think scripted lessons are the best way to go in elementary school.
g) I suppose we’re supposed to believe this is “understanding.” However, this is simply intuitive reasoning using pictures that works only in certain situations. If we are really serious about understanding then we’d have to discuss the algebraic reasoning, but for some reason that never makes it into the conversation.
I’m not a math teacher, but I found the video lesson very interesting. A lot seemed useful to me but also somewhat inefficient. I wonder how this concept is taught in Direct Instruction programs.