The idea is becoming more commonplace: giving math students regular chances to be creative in the process of solving problem has many benefits to their learning.
Less common is the connected idea: giving students regular chances to be creative in the process of solving problems has many benefits to MY learning.
Case in point: Today, my students worked on the Scale Model Floor Plan Problem, which required them to create a scale drawing on the classroom. Fairly simple instructions, right?
But teachers out there, do yourselves a favor: Don’t give them any more instructions than what is on the handout (linked above). Just stay involved, wander around and continue to answer their questions with, “Hmm… I don’t know. Do what you think is right.”
And watch what they show you.
Right away, they all reached for the yard and meter sticks, despite the fact that our chairs and desks are all sitting on one-square-foot tiles. Few groups thought to use them as a reference. The “hold-your-finger-there-while-I-move-the-yard-stick” method was far more common.
Cinder blocks create grids on the walls. Counting the blocks seems more accurate and easier on the back and knees.
Please understand, my goal isn’t to poke fun at these students. But let’s examine this. What would make a group of students choose a more difficult method of solving a problem?
I don’t think it is a lack of concern for quality. The drawings they produced suggest that quality matters to them.
I don’t think it is poor math skills. On the contrary, there are some very successful and highly-achieving students who worked on these scale drawings.
Seems like I should yield the floor to Dr. Jo Boaler (@joboaler) who said:
“There is a pervasive public view that different teaching pedagogies influence the amount of mathematics knowledge students develop. But students do not only learn knowledge in mathematics classrooms, they learn a set of practices and these come to define their knowledge. If students ever reproduce standard methods they have been shown, then most of them will only learn that particular practice of procedural repetition, which has limited use outside the mathematics classroom” (pg. 126).
I unpack this idea in slightly more detail in a previous post, but bottom line, the hidden curricula of our math classes seems a likely culprit.
According to most math curricula, this task requires multiplication, use of a ruler, and some understanding of proportional reasoning.
But, it also requires flexibility, resourcefulness, the ability to predict, and the ability to consider multiple methods. Which unit did we put those skills in? Which prerequisite course did we expect the students to pick up those practices?
Bottom line: We need to think beyond what math content we are expecting and consider the type of experiences we are giving our students with the content.
Open, flexible, contextual problems yield knowledge that can be used in open, flexible, contextual situations.
Procedural, specialized, algorithmic problems yield knowledge that can be used in procedural, specialized, algorithmic situations.
What kind of students do we want walking across the stage at graduation?
Boaler, Jo (2001). “Mathematical Modelling and New Theories of Learning.”Teaching Mathematics and Its Applications. Vol. 20. No. 3, p 121-128.