Sam Shah’s been writing a lot of thoughtful material about calculus instruction lately, including this piece on related rates.
He includes a worksheet with that post and two items struck me. One, this is a pretty charming illustration of a rocketship climbing into space.
Two, it asks students to climb down, not up, the ladder of abstraction. Check it out. It asks students to calculate a table of values for the rocket …
… then it asks for a prediction about the graph.
It asks students to calculate the instantaneous rate of change …
… and then make a prediction about the instantaneous rate of change.
Calculation is something you can do once you’ve ascended the ladder and turned a concrete situation (a rocketship lifting off) into an equation (h = 50t2). Prediction is something students can do while they mill around at the bottom of the ladder and it’ll make their eventual ascent up the ladder easier.
So I’m here, again, wondering what would happen if the worksheet had asked the prediction questions first and then moved on to calculation. Would the students be more successful? Would they have enjoyed the work more?
2014 Feb 24. Sam Shah updates us:
Yup. I introduced the rocket problem this year and I had each group make guesses for what the three graphs were going to look like. I loved hearing their conversation and their incorrect thinking for some of them. Tomorrow they are going to do the calculations and see what they got right and what they got wrong…
Thanks for pushing back in this good way. I’m glad I remembered to go back and reread this this year!
14 Comments
Jason Dyer
March 13, 2013 - 8:00 am -I have had a couple times in class where prediction did not help and probably made students enjoy things less (for example, predicting a percent where they clearly are shooting completely random or picking their favorite number like 69%). These were cases where there wasn’t enough of a basis to make a logical prediction — that is, they had very little intuition to work with.
So the question with this situation is: would the students know enough to have a prediction anywhere in the ballpark, or would it be a complete guess? (Calculus class I would say “yes”, Algebra 1 would be “maybe”.)
Don
March 13, 2013 - 8:03 am -I wonder if that image has a little too much already in it, but that might reflect the aspect that’s been on my mind lately. I have a terrier who loves any reason to run and chase things, and one of the ways I keep him exercised is with a laser pointer.
I stand in one place in the backyard and use the pointer. He wants to run at a constant pace and I want it to remain entertaining to him, so I try to keep the dot a constant distance in front of him. The same problem the camera operator has in the image, though my dog’s trajectory is less consistent :)
Perhaps I’m giving this too much weight for a calc-level student. After all, this is a concept that Benny Hill was mining for jokes with soldiers marching in a line. But I think I’d have iterated that image and left off the angle indicator and dotted line, at least in the beginning.
Stebbo
March 13, 2013 - 12:40 pm -Why are students so adverse to making predictions / guesses?
Is it because they are so scared of being wrong or because they have no idea? I think the former.
Cheers,
Chris.
Jeff Layman
March 13, 2013 - 9:49 pm -Am I the only one that’s willing to bet that rockets don’t go straight up? Maybe that can be Sam’s “Sequel,” tilting the rocket to more accurately reflect it’s trajectory.
Jim Doherty
March 14, 2013 - 4:23 am -@Stebbo
I think you’ve hit the nail directly on the head. Students don’t want to be wrong – especially out loud and ESPECIALLY in a math class where many of them have become convinced that there is 1 right answer and that’s it. Of course, in many cases there is one right conclusion you;d like to reach but the journey there should be much more interesting than it is for many.
The timing of this post is perfect as I am using Sam’s worksheet that preceded the rocket one in my class today.
David Cox
March 14, 2013 - 6:17 am -Are we to assume that climbing down the ladder of abstraction means students are moving from a task with more cognitive demand to one with less?
Andrew
March 14, 2013 - 7:02 am -@Stebbo I think that there is an answer to your question in Jason Dyer (#1)’s comment also. I think that Jim makes a good point, but I also think that there are time when we are asking our students to make predictions about situations where there isn’t enough intuitive background knowledge for a student to be able to make a decent prediction.
The students know when that’s the case. I have seen it in geometry (especially with angle measures). Until the students learn enough about what a degree is as a unit, and how to make some general connection in their minds about the look of an angle and its corresponding degree measure, there is very little value in having them predict… and the students are reluctant to do so.
William
March 14, 2013 - 7:38 am -David:
[quotes]Are we to assume that climbing down the ladder of abstraction means students are moving from a task with more cognitive demand to one with less?[/quote]
I’d say no. Looking at the common core that Dan referenced a couple of posts ago, I’d say that
[quote]interpreting the results of the mathematics in terms of the original situation,
validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable,
reporting on the conclusions and the reasoning behind them.[/quote]
are all moving down the ladder of abstraction. They require different skills (i.e. synthesis instead of analysis) than moving up the ladder, but they’re just as cognitively demanding.
Sam Shah
March 14, 2013 - 11:29 am -I’m liking this post and am excited about reading people’s comments/thoughts. I hope I don’t forget to chime in later (spring break starts earlier) once I’ve had some time to think.
I did want to give credit to where credit is due for the Rocket problem and image… Bowman Dickson [bowmandickson.com] posted it on Geogebra tube [http://www.geogebratube.org/material/show/id/2187]
It’s exactly this sort of dynamic visualization, this idea of getting a concrete sense of what’s going on in a situation, and THEN going forward to mathematize that situation… this is where I think related rates can be less stupid. (Because right now I’m still on the fence about them.)
As I pined on my blog a while ago:
“The more I mull it over, the more I think that geogebra has to be central to my approach next year… teaching students to make sliders to change one parameter, and having them develop something that dynamically illustrates how a number of other things change. And then analyzing how those things change graphically and algebraically.
(A simple example: Have a rectangle where the diagonal changes length… what gets affected? The sides, the angle between the diagonal and the sides of the rectangle, the area, the perimeter, etc. How do each of these things get affected as the diagonal changes?)”
Sam
Dan Meyer
March 18, 2013 - 6:25 am -David Cox:
Not necessarily. Those two spectrums (“cognitive demand” and “the ladder of abstraction”) don’t correlate exactly. For just one example, if someone spends too much time just working on one level of abstraction, they can show symptoms of what Hayakawa called “dead-level abstracting” which would make a downward move on the ladder very demanding work.
Sam Shah
March 21, 2013 - 12:19 pm -So I’ve thought a little more about this. And I have come down on the side that I do think having more and more “what do you think will happen?” would have been way better to lead into this unit. Like an entire day of that. The day just has to be carefully designed, because I honestly think I’m afraid that if not, the kids won’t find the picture/drawing engaging/interesting. And so I compensate by overstructuring and over-scaffolding. I’m scared to really let go. But that’s my fear that holds me back.
But a set of well-designed situations, some which are intuitive and some which are counterintuitive (like the rocket graph of angle vs. time), could possibly be the trick to getting them to engage/care. I need to scaffold less and let them play around more, and related rates is a perfect place for them to just play. I did something like that last year and this year to lead into optimization which went over well, and they were overall engaged [http://samjshah.com/2012/03/15/optimization-an-introductory-activity-project/].
Thanks for getting me think about my practice. As always.
Sam
Sam Shah
February 24, 2014 - 6:09 pm -Yup. I introduced the rocket problem this year and I had each group make guesses for what the three graphs were going to look like. I loved hearing their conversation and their incorrect thinking for some of them. Tomorrow they are going to do the calculations and see what they got right and what they got wrong…
Thanks for pushing back in this good way. I’m glad I remembered to go back and reread this this year!