Friday, April 16, 2010

Brain Dump

Problems vs. Exercises
Today, I threw this problem up in front of my 8th graders and they looked like they'd seen a ghost.

I like this problem because today it was an opportunity for them to show their problem solving skills.  They have all the tools they need to solve this problem, but they need to figure out which tool to use and when. 
  • find a common denominator
  • add fractions
  • divide fractions by inverting and multiplying 
  • reducing fractions
Tomorrow it won't be a problem solving opportunity.  Tomorrow it becomes an exercise.  It may be a complex exercise, but it's an exercise nonetheless. 

Now that they've seen this problem and know how to work it, I have to view it differently.  It has to carry less weight if it finds its way on an assessment because now the problem provides its own context for them. 

I think that in order for a student to be considered an expert, she needs to demonstrate the ability to do something with the tools beyond what she's been shown.  When students encounter a problem on a test that makes them say, "but he didn't show us how to do that in class," that's a good thing.

It's true that standards based grading in math can be more than just reporting specific snippets of content knowledge(there ya go Matt) It has to be more than just skills because I don't have a problem posting my skills online with examples of how to perform the skill.  I don't mind showing them what they need to know and them assessing them on that content.  I do mind stopping there.  The skills are the floor, not the ceiling. 

How we best help students realize they need these skills is a great conversation that I'm looking forward to seeing fleshed out.  But I think that there is also a need to discuss not only what exactly we need to assess but how.  I'm not of the opinion that we should assess behaviors like organization, responsibility, etc.  I do think that we should assess skills that may be consistent throughout our content area but not limited to our specific course.

This leads me back to the assessment question I asked in my previous post. How do I assess this ability to know when and how to put the tools together?

Do I treat it as a skill/standard and allow the score to change as students demonstrate their ability throughout the year?

Is it enough to involve my students in activities that promote problem solving and simply grade them on what I observe in class?

Does it show up in a summative assessment testing multiple skills?  If so, do the other "skills" that are on the test tip the student off as to the context of the problem at hand?

Do we use projects for students to demonstrate the ability to put a string of skills together in order to create their own meaning? 

Alright, brain's empty.  Time to go home.


Malyn said...

I've struggled with similar issues as written in this blog post and more besides.

In our curriculum (NSW, Australia), we are encouraged to promote mathematical thinking (applied maths, problem-solving, logic - you get the picture). My colleagues and I agree that we often focus too much on the content (and there's so much!) that we forget or run out of time to actually do math'al thinking.

I will use your algebraic fraction tomorrow - first day back in Term 2. It'll give my year 8 students a bit of a shock but it's a good starting point to revise and extend what they already know PLUS, a good question for modeling mathematical thinking.

I'll put you in my blog roll.


grace said...

What's your opinion of giving assessment questions that haven't been taught? I could imagine something like this on a test along with any number of scaffolded questions that evaluate students' ability to independently problem-solve. For example, since we want students to be able to actually use the repertoire of strategies they've accumulated over the years, "generate a list of topics from this class that this problem reminds you of-- share why." Since we want students to develop thoughtful processes as opposed to just guessing and checking (although that's certainly appropriate sometimes too), "what would you try first, and why?" Since we want students to solve problems in novel contexts, throw this on an assessment that is only marginally related to fractions. Since we want students to evaluate the reasonableness of an answer, "what would you expect to see in your answer, and why? would make sense? why/why not?"

Just brainstorming. Very curious to hear whether this makes sense to you or sounds like incoherent rambling :)

David Cox said...

Hi Malyn
The content does seem to get in the way doesn't it? How'd your kids do with that fraction?

I have no problem with that as long as the skills necessary have been taught. My opinion is that once we have some sort of discussion on how to tackle a particular problem, then that problem "type" becomes a tool and is no longer an actual problem.

Nah, your ramble made perfect sense.:) I think that's what I'm after here. I see skill acquisition as being different than skill application. I'm beginning to think that I need to assess them separately as well. Another post! Dang!

Sameer said...

I'm not really responding to the issues/questions you brought up, but I wonder if you teach them to multiply the top and bottom by 60 to clear the fractions?


David Cox said...

To be honest, Sam, I didn't even consider doing it that way. Thanks. This is exactly what I like about problem solving.