*fraction*of a foot.

The conversation below occurs more or less as follows (at several different times, with the same student and with other students):

**Me:**How much of a foot is 4 inches?

**Student:**2.4

**Me:**No, just the 4 inches.

**Student:**Oh, 0.4.

*Me (thinking):**He's stuck in a base 10 metric system. Let's try something else.*

**Me:**OK. How many inches are in half a foot?

**Student:**50.

*Me (thinking):**What?!?! Oh! He means 50%.*

**Me:**How many

__inches__are in half a foot?

**Student:**50.

**Me:**No.

__Inches__. How many

**?**

__inches__**Student:**Oh, 0.5.

**Me:**You are thinking about 50% right? I am asking you how many

__inches__?

**Student:**50?

**Me:**OK. How many inches in a whole foot?

**Student:**What? Ummmm 12.

**Me:**Right! So how many inches in half a foot?

At this point, the student starts writing ratios to do a unit conversion, converting 4 inches to feet. Then he grabs his papers and says, "I'll work on it." But, he doesn't really work on it. Later when I drop by to see his progress, he has nothing. But he stills wants me to draw 2' 4" for him.

It is so much easier to just draw it for him and walk away. Miraculously, I resist this urge and proceed to have another conversation as before. This time, he gets it (at least he guessed the right answer) and he draws the square.

But now, there is a circle with radius 1' 3". Oh No!

Craig,

ReplyDeleteI've been meaning to comment on this post for a few days now. It sounds like that middle section is tough, as it is with most endeavours, where the early energy as warn off and the end isn't quite in sight yet for anyone. Keep plugging away and good that you're acknowledging the positive outcomes - blogging, reading, research, reflection. I can sense that we're going to make some really positive changes in the next year.

I think this post is a great example of two key components of the modelling approach that we will need to work on the next year. First, it illustrates the necessity to perfect socratic questioning techniques. It is a really challenging process to guide someone without giving the answer while helping them discover the next step or confront their misconceptions. Misconceptions play a huge role here and one of the recurring ones is that anything fractional must be on a base of 10. The same misconception comes up converting decimals to fractions. From the transcript you've done a good job of the verbal questioning, but I think the next piece that might make the difference is the second key component: developing a student toolbox of model approaches.

We need to find a way to help students identify the key models/skills/approaches they can use in solving problems. For instance, here a visual representation of 4" compared to 12" would probably have been really helpful in identifying that it clearly isn't half. I'm not sure how we do this exactly, but I do like David Wees' take on the idea in this blog post:

http://davidwees.com/content/toolkit-model-math-instruction

I think it starts with some activities like we've discussed where students see graphical representations, measurement, proportion, etc. as tools to solve problems. I also think the use of the whiteboards, collaborative problem-solving and communication back to the class will give us the opportunity to highlight successful strategies. However, I'm wondering if we could create a 1 page "Math/Science API Cheat Sheet" if you will (see David's post) that students could use to identify the strategies in their toolkit. Better yet they could work together to create the cheat sheet.

Some more fodder for our thinking. Thanks for continuing to post your thoughts regularly. I'm still here reading daily and doing my own thinking, just not getting it on the blog yet:(

@thinktoaction.com

ReplyDeleteI had a Math toolkit concept for my Yearlong courses years ago. The idea was to learn a mathematical tool and then put it into the toolkit once mastered. Then we could pull it out anytime.

There were 2 major problems. One, the "tools" were really just memorized procedures. Two, "mastering" the use of the tools was a challenge.

In the comments to David Wees blog post that you mention above, Whit Ford makes a point about avoiding memorization and seeking to understand the concepts first. The concepts become the "tools" and the procedures or algorithms are simply the implementation. Check out Whit's post here:

http://mathmaine.wordpress.com/2011/07/14/integrating-mathematics/

I know you are not saying that the tools are memorized procedures. I am just clarifying my own thinking here.

Thanks Blair.

I totally agree that we want to avoid memorisation and that this isn't necessarily a new strategy. I think the key point that Whit raises, which I was trying to get at as well is the importance of the communication piece. Justifying their answers using the whiteboards first in developing a solution with other members of their group then to the wider class. I'm under no illusions that this is going to magically be eloquent or well laid out to start with, heck it might not even be that way at the end of the semester, but the more they do it the better they'll get. I think this is where a large volume of WCYDWT/any questions type problems are needed to provide suitable contexts where memorisation of procedures is insufficient. This is actually one area I've been doing a fair amount of productive work over the summer developing an eye for shooting/seeing them, skills in editing videos using Adobe After Effects to add timers/slow footage down/etc. and finding other people locally who want to work on creating these types of problems. I'm thinking of hosting a monthly collaboration session to brainstorm and shoot footage. Working with others has been really helpful. It's not a massive collection, but it's a start. Most of the footage is raw, but I'll send you a link to some edited stuff soon.

ReplyDeleteI'm with you on having difficulty developing ideas around how to get them to learn and master the models/concepts. I'm thinking possibly some easier contextual problems early in the year where I'm explicit in how I want them to solve the problem (ie. Sketch a graph, use a proportion/scale/ratio...) followed by communication of their solution,and discussion of where this approach might be helpful, which would form the basis of what goes on the concept sheet.

More things for me to think about:) Thanks.

Blair