I grow weary writing about Math 8 all the time, so I'm going to devote today's post to Geometry.
Last night, Michael Pershan (@mpershan), Geoff Krall (@emergentmath) and I had a discussion about the difference between procedural knowledge and conceptual knowledge. It was a very lengthy conversation with increasing frustration for me because I was having trouble making my view clear in 140 characters.
My stance is that conceptual knowledge is what is used to set up a problem and procedural knowledge is used to solve it. Even this explanation doesn't adequately state what I want, so instead, I'll use the geometry class to illustrate.
Over the last three days, we have been doing geometric constructions. They have been using compass and straightedge to find the incenter, circumcenter, orthcenter and centroid of triangles. They had to define those terms and tell me what lines or segments were used to find them.
"Find the incenter of the triangle" is a task that requires procedural knowledge.
Today, I handed them a map of Los Angeles.
Me: "You and two of your friends have graduated from college and been offered jobs in Los Angeles. It's an expensive city so you've decided to get a place together. What kinds of things do you need to think about?"
S: "Expenses, rent, groceries."
Me: "What else?"
S: "How far we are from our jobs."
Me: "Why would that matter?"
S: "It wouldn't be fair if one person was next door to their job and the others have to go across the city."
I gave them the addresses of their jobs and told them to find a place to live.
I didn't give them any other criteria or directions.
S: "Will we need a compass for this?"
Me: "If you choose a method that requires a compass, I suppose you will. If not, no."
S: "What method are we supposed to use?"
Me: "I don't care. I care about results and reasons. Be able to justify your choice."
What I was hoping for was "Our apartment will be here because this point is of equal distance from all of the jobs. We found the circumcenter of the triangle formed by the addresses because the circumcenter is equidistant from the vertices."
I happily accepted several different answers and reasons as long as they were justified.
This is the kind of problem that I consider as one that requires conceptual knowledge. Yes, they would need procedural knowledge to find the circumcenter, but to know what they were looking for is my goal.
I don't want my students to ask "when am I ever going to use this." I worry that we focus heavily on procedure to the exclusion of the conceptual knowledge. It's great to know how to find the circumcenter, but it's better to know when you would need to and why.
In my opinion, procedural knowledge is the how, where conceptual knowledge is the when and why.
Conceptual knowledge and understanding is where the fun and beauty of mathematics lives.
We did similar activity with the proposed aquatic "center" for 3 comprehensive high schools in my district. It was helpful to compare definition of "middle" from 2 points, to 3 points.ReplyDelete
I think I prefer your approach with the apartment hunting. It feels easier to relate to, like the students can imagine this is something they would actually do (possibly multiple times). I'd love to see a deeper performance task with this criteria and other stuff that you hint at. Develop a cost of living with data (gas, grocery, parking, rent, etc.) and have weighted zones. Costs (x) amount of money to live in this area.
I'm also a big fan of Voronoi Pizza from NCTM illuminations: http://illuminations.nctm.org/Lesson.aspx?id=2688
More about dividing equal regions with various spreads of points.
When you talk about conceptual vs procedural I often don't add knowledge to both. I'd see it more as conceptual knowledge, procedural skill. I can understand how a concept works in the overview (big picture) and still get lost in details along the way with the processes that get carried out within a concept (small picture).
There is a balance, and I've always felt that stronger conceptual understanding leads to more notice and wondering, a better awareness of the relationships that arise throughout the math. From a classroom teaching and learning perspective, I feel that conceptual understanding often benefits from explorations and iterations of procedure / practice. Much like leading into the general abstract after starting with the specific concrete.
These are all good thoughts and practices you share Justin. Thanks for keeping these conversations going with your blog and tweets.
Would you say that being able to "discriminate" knowledge is the same as building conceptual knowledge? (For example, much like the "discriminant" tells you how many real/complex roots you will have in a quadratic equation, so also tasks that ask you choose a method build conceptual knowledge?)ReplyDelete