Peg SmithNorth Carolina Council of Teachers of

Mathematics2012 State Mathematics Conference

Creating Opportunities For Students to Engage in Reasoning and Proving:

Modifying Existing Tasks

What is Reasoning-and-Proving?

By focusing primarily on the final product - that is, the proof - students are not afforded the same level of scaffolding used by professional users of mathematics to establish mathematical truth.

Therefore, reasoning-and-proving should be defined to encompass the breadth of activity associated with:identifying patternsmaking conjecturesproviding proofs, andproviding non-proof arguments.

Stylianides, 2008

Why Reasoning-and-Proving?

There is a growing consensus that high school mathematics programs need to include a greater emphasis on reasoning and proof. Reasoning and proving are central to the mathematical practices identified in CCSS (2010).

Practice 2: Reason abstractly and quantitatively Practice 3: Construct viable arguments and critique the reasoning of others Practice 7: Look for and make use of structure Practice 8: Look for and express regularity in repeated reasoning

NCTM in their most recent policy document, argues that reasoning and sense making “should occur in every mathematics classroom everyday” (2009, p.5).

Math Class Needs a MakeoverDan Meyer

You Are Doing Math Reasoning Wrong If... Students don't self-start. You finish your lecture block and immediately you have five hands going up asking you to re-explain the entire thing at their desk;Students lack perseverance; They lack retention; you find yourself re-explaining concepts three months later, wholesale; and There's an aversion to word problems, which describes 99 percent of my students. And then the other one percent are eagerly looking for the formula to apply in that situation. This is really destructive.

Comparing Two Versions of a TaskCompare each task to its modified version

(A to A’, B to B’, C to C’)Determine how each pair of tasks is the same and

how it is differentLook and consider:

what the modifications in the tasks were trying to accomplish

what modification principles can be generalized whether the differences between a task and its

adaptation matter

Comparing Two Versions of a Task:How are they the same and how are they different?

TASK A

MAKING COJECTURES Complete the

conjecture based on the pattern you observe

in the specific cases.29. Conjecture: The sum of any

two odd numbers is ______?

1 + 1 = 2 7 + 11 = 181 + 3 = 4 13 + 19 = 323 + 5 = 8 201 + 305 = 506

30. Conjecture: The product of any two odd numbers is ____?

1 x 1 = 1 7 x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = 15 201 x 305 = 61,305

TASK A’For problems 29 and 30, complete the conjecture based on the pattern you observe in the examples. Then explain why the conjecture is always true or show a case in which it is not true.

MAKING CONJECTURES Complete the conjecture based on the pattern you observe in the specific cases.

29. Conjecture: The sum of any two odd numbers is ______?

1 + 1 = 2 7 + 11 = 181 + 3 = 4 13 + 19 = 323 + 5 = 8 201 + 305 = 506

30. Conjecture: The product of any two odd numbers is ____?

1 x 1 = 1 7 x 11 = 771 x 3 = 3 13 x 19 = 2473 x 5 = 15 201 x 305 =

61,305

Comparing Tasks A and A’SimilarBoth ask students to

complete a conjecture about odd numbers based on a set of finite examples that are provided

DifferentTask A’ asks students to develop an argument that explains why the conjecture is always true (or not)Task A can be completed with limited effort; Task A’ requires considerable effort – students need to figure out why this conjecture holds up

Ways to Modify Tasks

1. Engage students in investigation and conjecture instead of just giving answers.

2. Provide all students with access to a task by first making observations about a situation before moving on to more focused work.

3. Require students to provide a mathematical argument, proof, or explanation.

4. Take away unnecessary scaffolding.5. Ask students to explore a situation by generating

empirical examples and looking for patterns.

What Were the Modifications Trying to Accomplish?

Press students to do more reasoning and justifying than the original versions of the task.

Give students more access to the task.Give students the opportunity to do more

investigation and less of just giving answers.Engage students in PROOF (without actually saying

PROOF).

Donald’s Task (D and D’)TASK D

a. Simplify each expression. (-2)2 (-2)3 (-2)4 (-2)5 (-3)2 (-3)3 (-3)4 (-3)5

b. Make a Conjecture Do you think a negative number raised to an even power will be positive or negative? Explain.

c. Do you think a negative number raised to an odd power will be positive or negative? Explain.

Task D’1. Solve the following examples. (-2)2 (-2)3 (-2)4 (-2)5 (-3)2 (-3)3 (-3)4 (-3)5

2. Make some observations about any patterns that you notice.3a. Using what you notice about the examples above, make a conjecture about negative numbers to an even power.3b. How do you know that this will be true for all negative numbers?4a. Using what you notice about the examples above, make a conjecture about negative numbers to an odd power.4b. How do you know that this will be true for all negative numbers?

Comparing Tasks D and D’SimilarBoth tasks ask students to calculate the answer when -2 and -3 are raised to odd and even powersBoth tasks ask students to consider the sign of the products when odd and even integers are raised odd and even powers

DifferentTask D states the two options for the conjecture while Task D’ asks students to make the conjectureTask D’ invites students to first make observations about what it occurringTask D’ moves beyond the examples by asking how they will know that the conjecture will always be true

Challenges

Being clear about the goal for the lesson and how the modification helps accomplish it

Making sure the content of the modification matches or exceeds the original content

Making sure that the modification is not just a surface level change that doesn’t really increase opportunities to engage in reasoning-and-proof

Ensuring that there is sufficient scaffolding for students to actually do what is being suggested

Let’s Try Modifying:Read problem #27 from p. 298 of our current

Algebra I text.(a close read)

What would you do to modify the task?

What would make it more “Common Core friendly”?