[Peg Smith] Creating Opportunities for Students to Engage in Reasoning and Proof


When someone says “Yeahbut has a conference session ever actually caused you to change your practice?” I immediately point to Peg Smith. After catching her 5 Practices session two years ago at NCSM Indianapolis, strategies to facilitate productive classroom discussions replaced “Who wants to go next?”

No surprise I was looking forward to Dr. Smith’s NCTM Denver session. The description:

Although there is a growing consensus that the grades 7–12 curriculum needs greater emphasis on reasoning and proof, research shows that most textbooks offer limited opportunities to engage in these practices. We will focus on how to modify tasks to give students more opportunities to engage in reasoning and proving.

Smith began by contrasting modifying textbook tasks with Dan Meyer’s approach, which she described as rebuilding. Smith was highly complimentary of Dan’s work but the focus of her session would be on small, easy to implement changes; the problems themselves would remain similar to the original textbook versions.

Smith presented three tasks, each time giving us time to compare and talk about the before and after versions before she discussed them in depth.


Task A, making conjectures about the sum and product of any two odd numbers, received the least amount of modification. To my colleagues and me, adding “explain why” (with some language related to what it means to prove or disprove a conjecture) did move this task beyond fill in the blanks, but it just seemed like a given. We wondered about making other changes to the problem, such as removing the sentence stems, which are complete conjectures but for the last word.


The modifications to Task B, the geometric construction of a parallelogram, were more noticeable. In Task B’, three conjectures are possible whereas, in Task B, students are told the construction will produce a parallelogram. In Task B, reasoning and proof consists of stating a theorem found in the margins of their textbook. In Task B’, reasoning and proof consists of students themselves creating mathematical arguments. “Authority in math class should lie within the students and the math, not in the teacher or the text,” said Smith.


There were some significant improvements to the third task, a visual pattern involving trapezoids. In Task C’, students are first asked what they notice about the visual pattern. This is significant because these observations can be used later by students to prove their generalization. In Task C, the visual pattern serves no purpose other than to generate the sequence 5, 8, 11, 14 for students. Smith asked us to note the placement of the question about the perimeter of the 12th figure in each version of this task. In the textbook task, it follows the question about the formula; it serves to verify that the formula is correct. In the modified task, it comes before the question about the generalization; it encourages students to generalize.

Smith ended her session by asking “Do these modifications make a difference?” I think so. These modified tasks do give students more opportunities to engage in reasoning and proof. No question. However, I couldn’t help leave the session wondering if, by now, these modified tasks shouldn’t be our starting points.


Chris Hunter blogs at Reflections in the Why and tweets @ChrisHunter36.

2 thoughts on “[Peg Smith] Creating Opportunities for Students to Engage in Reasoning and Proof

  1. Pingback: Modifying textbook problems | Passion for Teaching

  2. Pingback: Pythagorean Exploration | Reflections in the Why

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