Orchestrating Mathematical Discussions: What is the Goal? Blake E. Peterson Brigham Young University Department of Mathematics Education

Professional Standards for Teaching Mathematics Select worthwhile tasks that require thinking and reasoning; Encourage student participation through questioning and deciding which ideas should be pursued; Monitor and organize students’ participation by maintaining an awareness of their thinking while seeking to involve all students.

Mathematics Teaching Today The role of the teacher in creating a learning environment is defined to include more than just the physical setting but “an intellectual environment in which serious engagement in mathematical thinking is the norm” (NCTM, 2007, p. 40).

Mathematics Teaching Today The teacher’s role is to provide an atmosphere that: ◦ values students’ ideas; ◦ allows time for students to grapple with significant mathematics; ◦ encourages learning as a collaborative practice in which students seek to clarify, justify and question the ideas being shared

How do you do it? The teacher asks open-ended questions to elicit student thinking and asks students to comment on one another’s work. Students answer the questions posed to them and voluntarily provide additional information about their thinking. (Level 2 of Hufferd-Ackles et al, 2004) The teacher facilitates the discussion by encouraging students to ask questions of one another to clarify ideas. Ideas from the community build on one another as students thoroughly explain their thinking and listen to the explanations of others. (Level 3 of Hufferd-Ackles et al, 2004)

Breaking away from Calculationally Oriented teaching Students who have come to view mathematics as “answer getting” not only will have difficulty focusing on their and others’ reasoning but may consider such a focus as being irrelevant to their images of what mathematics is about. (Thompson et al, 1994) A calculationally oriented teacher may believe that explaining the calculations one has performed is tantamount to explaining one’s reasoning (Cobb, Wood and Yackel as cited in Thompson et al, 1994)

How do you do it? Smith et al (2009) suggest: ◦ anticipating, ◦ monitoring, ◦ selecting, ◦ sequencing, ◦ connecting

What about the Unanticipated Thinking? Because of the uncertainty that is prevalent in varied student responses many teachers, both novice and veteran, may have misgivings about seeking to open up the mathematics to students in this way (Blanton et al., 2001; Borko & Mayfield, 1995; Franke et al., 2001).

How do we learn to do it? “Exhorting teachers to engage students in mathematical reasoning is inadequate as a support for their practice. Parsing the work of teaching makes instructional practice visible, and hence potentially learnable” (Ball, Lewis & Thames, 2008, p. 41).

Process of Using Students’ Mathematical Thinking 1. Listen to and understand our students’ thinking 2. Recognize the potential value of the student thinking 3. Use the thinking in a pedagogical and mathematical way

Recognize the potential value of the student thinking What has to be in place for us to recognize the value of the thinking? ◦ We must at least partially understand what the student is saying. ◦ The Mathematical Goal of the lesson MUST be clear in our mind!!! What gets in the way of our recognizing the value of student thinking? ◦ Assumption of understanding  Fill in the blanks  Simply remind

Use the thinking in a pedagogical and mathematical way What does it mean to use thinking in a pedagogical way? What does it mean to use thinking in a mathematical way?

Use the thinking in a pedagogical and mathematical way What are some examples of student thinking being used ineffectively? ◦ Do not know how ◦ Naïve use □ Student thinking as a triggertrigger □ Mere presence of the correct solution □ Mere presentation of multiple solutions ◦ Incomplete use

What kind of student thinking might emerge that could be used in a class discussion? Fill in the table, sketch the graph and write a symbolic rule for the situation. Rob is walking away from the motion detector at a constant rate. At 0 seconds he is 1 foot from the motion detector and at 2 seconds, he is 9 feet from the motion detector. Create a table, graph and equation describing the relationship between the time and Rob’s distance from the motion detector.

Video – Motion Detector Student Teacher M (9/14/06) What mathematical GOAL would you want students to reach from this discussion? Would you have asked different questions to reach that goal? Would you have asked them in different places to reach that goal?

What could have been done differently? Carefully select and sequence solutions so you know what you are getting? Don’t just ask what you did (calculational). As for the reasoning behind the procedure (conceptual). Don’t assume that the presence of the correct solution will clarify the erroneous thinking that is present.

Building Bridges

Questions to Accompany Bridge Task Create a table and graph of the number of pennies needed to break bridges that are 1, 2, 3, and 4 papers thick. Use this data to predict how many pennies would be required to break a bridge 2.5 layers or 6 layers thick. Do you think the relationship is linear or non-linear?

Sample Tables of Data How would you sequence the sharing of this data, what would your mathematical goals be, and what connections would you make?

What discussions could be had? What questions would you ask and what mathematics would you push on with regard to the “connecting the dots” thinking? What questions would you ask and what mathematics would you push on with regard to the different methods for predicting the number of pennies for a 6- layer bridge?

Orchestrating Mathematical Discussions: What is the Goal? You must have a clear mathematical goal in mind in order to better: ◦ Understand student thinking ◦ Recognize its value ◦ Know what to push on You must have a goal of understanding the reasoning behind the thinking not just a description of the thinking.