How do we avoid the “show and tell” situation and instead keep students motivated and interested in wrestling with the mathematics in the task? Open-ended tasks that have a high level of cognitive demand have multiple entry points as well as multiple solution paths. As a result, strategies, different representations, and even more than one mathematical concept and the relationship between the concepts can be shared and discussed in a whole class discussion. Sometime the sharing might look like a show and tell instead of a true discussion and often students stop listening after one or two solution paths have been shared, or some students a busy mentally preparing their own presentations.
We suggest that in order to make the discussion engaging and deep, teacher use four strategies, that we call problematizing hooks because they focus student attention on important mathematical ideas. The hooks include:
These strategies can be used with students in any grade level. We will look at a discussion of an algebra pattern task.
When solving this task, algebra students are required to not only write equivalent expressions/equations to represent the total perimeter of the train number, n, but also connect each term in the expressions/equations to the visual representation. Many students and adults begin this task by setting up a table comparing the train number to the perimeter, they find the difference in the perimeters for every 1 difference in the train number and make that the slope, and then work backwards to find the y-intercept at the 0 stage train. In using a table to create an equation, a link between the terms of the equation and what they represent in the visual of the trains is yet another goal. As the 7th grade Expressions and Equations standard reads in the Common Core standards, “rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related” (CCSS 7.EE.A.2).
Read each problematizing hook and consider ways in which students will have opportunities to engage in making sense of the underlying mathematics of the task.
Examples of Problematizing “Hooks”
(Students in the class have all used 4x + 2 to solve the task.)
T: “It’s interesting that all of us wrote the same expression. But when we say “perimeter” and “hexagon”, I immediately think of 6 sides. I challenge you to write an expression that describes a way to figure out the perimeter of the hexagons when you begin by determining the perimeter of each individual hexagon first.”
Why use a “challenge”?
P = 6 sides on each hexagon, minus 2 vertical sides for each of the hexagons, plus two ends which were taken away are now added back in.
By challenging students to think of an alternative solution path to the 4x + 2 that they have all used the students will discover that there is more than one solution path but also that subtraction can be used to compensate for additional perimeters counted.
T: “How are Fabian’s and Layla’s solution paths similar? How are they different?”
Why use a “compare/contrast”?
This “compare/contrast” hook allows students to examine the solution path without having students discover both solution paths on their own. By putting two solution paths up at the same time and asking students if both solution paths describe the visual pattern and how are the two solution paths related, students explore the distributive property of multiplication over addition and explain why the two equations are equivalent.
T: “Jasmine claims that she knows what is happening with the visual pattern just by looking at the way addition and subtraction is used in the expression. For example, she said, ‘The perimeter of every hexagons is being counted and then subtraction is used to take away the inner perimeters that are touching and should not be counted when determining perimeter.’ What does Jasmine mean when she says the operation of subtraction tells her that some perimeters must be taken away?”
Why use a “claim”?
By inserting a claim, students have to take a position and agree or disagree and then defend their thinking.
T: “When I was solving this task, I first wrote 6 + 4x because I saw 6 in the first train and that I was adding 4 each time I added a train. It turns out that that is not an accurate expression. Why not?”
Why use a “counter example”?
By asking students why something does not work, it solidifies their understanding of what does work. A counterexample also serves the process of generalization about math: when does something work all the time versus when do we overgeneralize without thinking about exceptions.
Productive mathematical discussions are the crux of the math lesson; it is the time to leverage student thinking in order to explore and solidify students’ understanding of the deep mathematical concept. By problematizing the discussion, students have to engage in the math via making sense of others’ work, not just explain their own solution path.
Your turn – choose one of the hooks to start your next math discussion.
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