Preservice Performance Assessment, standard B2c, question 1:
Does the candidate appropriately balance activities for developing conceptual and procedural learning activities to understand mathematics?
Does anyone? At the end of a year of student teaching, I know a lot of the ways I want to be better, and I’ve seen (or read! yay blogosphere!) what better looks like, so I at least know which direction to go. But procedure vs. concept is still chicken and egg. Which comes first?1
Item: My students can approximate distance traveled from a velocity table. They can approximate the area under a velocity curve. But they don’t recognize these as the same computation.
Item: My students can describe why their paper boxes that are too shallow or too tall hold fewer jelly beans. They can also tell me that to find a critical point, take the derivative, set it to zero, and solve for x. But they can’t get from one to the other.
I’ve lectured. I’ve modeled. I’ve Socratic-ed. I’ve left them alone in groups with thought-provoking questions. I’ve worked one-on-one. What I consistently see is that “what am I supposed to do?” [with the symbols] lives in a different neighborhood of their brains than does “what is this about?” If I make the connection, they don’t come with me. If I leave them to make the connection, they don’t. Is this the dark side of playing to multiple learning styles? If kids do a physical or manipulative exercise and then do an analytical exercise, but don’t make the connection, then I wonder if the mode-switching just confuses them.
So it seems there are two big design challenges in a lesson. First of course, and biggest, what’s the genuine problem that leads to the burning question?
But next, what has to happen so they see their question in its symbolic setup? All my students could tell me that they’re looking for the peak of a graph and that the slope is zero at the peak. They could all tell me that the derivative measures the slope. But fewer than 1 in 10 could go from “where is the peak?” to “f’(x)=0; solve for x.” That’s where I lose them. It happened in harmonic motion, related rates, optimization, integration. Is there a design principle for getting this right?
1I’m fully bought in to context before content. The procedure-and-concept issue arises as we move into content.
Right – you do it for them, you might as well not bother. If you leave them to do it and they’re not used to doing this (which is most students), they won’t.
Here’s the lesson of my experience, for what it’s worth:
You have to make it their job to do it. Repeatedly. Connecting the rich, concrete, instantiated problem/process/idea with the symbols is something that needs to be practiced, with repeated practice. In fact, maybe more than anything else.
So, they’ve estimated distance from the table and looked at area under a curve for velocity. Give them a table and graph with matching data. Have them do the two computations.
I bet kids will comment (annoyedly) in the second problem that they’re doing the same thing again. If so, skip to (2). If not:
(1) You: Why did the answer come out the same??
Communicate (with tone of voice, body language, etc.; out loud if need be) that class will not move on if you don’t get a satisfying answer, and that this is definitely not one of those times they’re going to get you to bail them out by acting helpless enough.
Them: We did the same arithmetic.
(2) You: Why did you do the same arithmetic?
Again, communicate (body language, tone of voice, and out loud if you have to) that class isn’t moving on without a good answer here. This is the key to everything. They have to know that if they don’t turn their brain back on, everybody is going to sit there awkwardly till they do. (This can be tricky if there are management issues, but these need to be addressed on their own terms. If the kids can’t maintain decorum in the face of a thought-provoking question that they don’t know how to answer at first, the cognitive demand on them is not where the problem lies.)
Once they have given you a satisfying answer, make them write it down. (Full sentences; readable by a parent or even better a younger sibling.) Then, 3 weeks later, or however long, make them write about it again. Essay: how is estimating distance related to area under a velocity curve? Why? Fully justify all claims. This assignment can extend the original idea: for example, how can this connection be used to compute the exact value of a definite integral? (Here, I have in mind a particular not-especially-rigorous-but-very-conceptually-powerful way of deducing the Fundamental Theorem: definite integral = area under velocity curve = distance covered in a certain amount of time = difference between position at end and beginning = antiderivative(b) – antiderivative(a). Assume nonnegative velocity for this.) But whether it extends it or not, it needs to force them to walk through the sequence of mental connections again and write it all down. The connections don’t stick if they don’t get practiced and they don’t get practiced unless you insist on them being practiced.
Just my 2 cents; hope it’s helpful. Keep your head up – you’re in the right struggle.
Very helpful – thanks, Ben!
[7/1 – so helpful, in fact, it provided the fuel for my next post.
Provocative stuff here, Dan.
I’ve had similar difficulties connecting conceptual development to skill practice and I wonder if asking students to convert a skill practice problem back into a conceptual development problem wouldn’t be an effective design pattern, an effective mediator between two extremes.
For example, we give students constraints on the jelly bean box, against which they theorize about shallow and tall boxes and then develop an equation for the volume from which they find the maximum.
Then we ask “What kind of situation would yield the volume function V(x) = x(10-2x)^2?” to help students move freely between conceptual and practical problems.
I like this because it requires mapping ingredients of the equation onto real things – the first x on the RHS means one thing, the (10-2x) means another, and within that, the 10 comes from something and the 2 comes from something…. One of the zillion articles that passed beneath my nose this year in ed school was about how kids can write (tell stories) before they can read, so teachers can use that by scribing and then reading what kids say; kids then learn to see their thoughts in the symbols. If I can find the article I’ll post something more about this. You and Ben are both advocating that if they have a hard time getting over a bridge, spend deliberate time going back and forth over the bridge. So if we have context (a problem), concept (an approach for finding a solution), and “skills” (in this case, practice with symbolic manipulation that codifes the concept), in lesson planning we’ll need at least four kinds of lessons: one each focused on those three, plus one more focused on the bridge between the concept and the skill.
damn, I thought my student teaching year went well. It sounds like you are a straight up champion. Context and content in your student teaching? damn.
How goes the job hunt? Are there math jobs in boston? What would you say was the hardiest and/or easiest part of student teaching? Do you think your teacher credentialing program was useful?
Just looking for a east costal perspective.
You are hitting on big issues, Dan, and not easily solved; further, they present themselves year after year, so must be solved repeatedly in new contexts.
Several comments to “soft skills” posts already refer to Carol Dweck’s “Mindset”, but it’s worth repeating, and, as I have found, worth rereading sections of the book regularly. Moving from a fixed mindset to a growth mindset is not easy, and there is ample opportunity for backsliding. Practice is essential, even after understanding occurs, in order to move the learning from short term to long term memory. See the brainology.us website which is mindset for junior high. My high school remedial students balk at the juvenile presentation – cartoons! – at first, but come to recognize that the material is serious, thought-provoking, and the ideas and vocabulary of mindset are going to be a regular part of our work for the year. (And a regular part of my work as well; the site helps by offering numerous video clips of growth mindset at work.) It helps to end the “I just can’t do math” talk, and helps them see how their own habits of mind have them trapped in a “can’t do” mental loop. The same comments come up in algebra 2, precalc and calc. (I teach in a small rural school and have all levels, freshmen to seniors).
I have seen references, in some blogs, to Jo Boaler’s book, What’s Math Got To Do With It, as well, and would add Elizabeth Cohen’s Designing Groupwork (Teachers College Press) to your summer PLC discussion list. Many of the problems you have referenced make for engaging group work where a student doesn’t have to feel alone, but can find the emotional support and collaborative effort of peers to be tremendously helpful. Group work can be a terrific addition to a classroom in which some students comfortably hide from the challenge of thinking through problems.
Let me just add a thank you to the people who launched this virtual conference on soft skills, and to those who are commenting and sharing.
@Anne: I just talk a good game. Context-before-content is easy to agree with but I’m just beginning to learn how to pull it off.
@Ed: Thanks – I think doing explicit mindset-based models with students will be an important part of this year (“math ability is a function of … what?”)