1. Using Mathematical Practices to Promote Productive Disposition
Duane Graysay, Sara Jamshidi,
and Monica Smith Karunakaran
The Pennsylvania State University
Less than a minute. This is the same title as for PCTM. Duane will introduce us. I’m me, Sara’s Sara, we’re here to talk about a project that we conducted this past summer with our colleague, Monica S K.

2. Overview Students Course Upward Bound, 5-week program
College-like experiences for high school students
Course
Emulate mathematical research practices
Survey questions and interview protocols assessed
Productive disposition
Understandings of mathematics as a profession
Sara. 1-2

3. Motivation of the Project
Mathematics is taught as a Practical Tool
Concept
Procedures
Applications
Mathematics is also a Field of Inquiry
Development of…
Concepts
Often inspired by applications
Duane. 1-2 min. Talk about the two different visions of mathematics.

4. Specific Questions
How does engaging in inquiry projects impact students’...
understanding of what it means to “do math”?
perceptions of themselves as mathematically able?
productive and unproductive beliefs regarding math?
Duane. 2-3 min. We take for granted that it is a valuable experience to explore mathematics as a field of inquiry. So we tested how these projects impacted…
Productive disposition is the tendency to perceive mathematics as sensible, useful, and worthwhile; “to believe that steady effort in learning mathematics pays off”; and to perceive oneself as mathematically competent in learning and doing mathematics (NRC, 2001, p. 131). Mathematical inquiry projects are meant to engage students in activities that affect their perception of what it means to do math, which should interact with their perceptions of their own abilities to do math. Changing perceptions of what it means to do mathematics should also interact with their beliefs about mathematics, including productive and unproductive beliefs.

5. Students 21 students, selected by Upward Bound 11 female, 10 male
Students from underrepresented groups
Potential 1st-gen college students, and
Many needed additional college-prep experiences
Only one claimed to have done “write ups” before
Significant proportion reported they did not like math
Sara, 1-2 min.

6. Task Selection Accessibility Imaginable Mathematizable
Approachable (little prior knowledge)
Sara. >1 min.
Students need to have opportunities to solve challenging problems in order to gain confidence in themselves as mathematics learners (NRC, 2001, p. 131).

7. Task Selection - Types Type I: Solvable & Formalizable
Solvable: There should be a solution that can be found using problem-solving heuristics.
Formalizable: There must be an opportunity to formalize the solution.
Type II: Representative & Generalizable
Representative: The scenario must exemplify a generic type of problem.
Generalizable: Solving the scenario should afford a general understanding of solutions for the generic type.
Sara 2 mins
Mathematizing and formalizing are common activities for applied mathematicians, who try to apply mathematics to situations that are not yet recognized as mathematical. Exemplifying and generalizing are a second kind of activity that mathematicians experience. Type 1 operates on realistic situations, Type 2 tends to operate on mathematical objects.

8. Sample Task (Type I) Four Queens Problem Queens can move
horizontally
vertically
diagonally
A piece is “attacking” another if it is one move away.
How many ways can you arrange 4 queens on a 4x4 board so that no queen is attacking another?
Both. Duane introduces the activity, Sara introduces and manages the task.

10. Sample Task Process Exploration A Solution is Proposed!
Class Discussion
Remaining Solutions Found
Final Step: Justifying the Solutions
Sara, 1 min.

11. Activity Principles Introduce the problem Make sense of the problem
mathematical content is not clearly expressed
Make sense of the problem
use mathematics to model the problem
Arrive at a (partial) solution
discussion follows
Construct a viable argument
satisfy mathematical principles
Sara

12. Mathematical Practices (NGA Center & CCSSO, 2010)
Make sense of problems and persevere in solving them.
Construct viable arguments and critique the reasoning of others
Model with mathematics
Attend to precision (in communicating with others)
Look for and make use of structure
Duane, 1-2 minutes

13. Math can be creative (P)
Getting answers correct is more important than understanding why the answer is correct (U)
Most math problems have only one way to solve them (U)
Knowing how to perform a procedure is more important than understanding why it works (U)
Students can discover math without it being shown to them (P)
Students learn math better when they work together (P)
Students should be able to figure out for themselves whether answers are correct (P)
I am confident in my ability to help my peers (P)
It is important for me to learn mathematics (P)
Duane. To understand how this affected students, we asked them to express their agreement or disagreement with some belief statements, either “productive beliefs” or “unproductive beliefs” about mathematics; about learning mathematics; about themselves as people who do mathematics.
Productive disposition is the tendency to perceive mathematics as sensible, useful, and worthwhile; “to believe that steady effort in learning mathematics pays off”; and to perceive oneself as mathematically competent in learning and doing mathematics (NRC, 2001, p. 131). Mathematical inquiry projects are meant to engage students in activities that affect their perception of what it means to do math, which should interact with their perceptions of their own abilities to do math. Changing perceptions of what it means to do mathematics should also interact with their beliefs about mathematics, including productive and unproductive beliefs.

14. Outcomes
Students tended to disagree with unproductive beliefs from the beginning
Exceptions: They tended to agree that . . .
Knowing how to perform a mathematical procedure is more important than understanding why the procedure works.
The teacher should do most of the talking in the classroom.
Duane
If we have time, give an example of a statement and explain what we mean by “tended to disagree with”.

15. Observed Outcomes “Students can discover math on their own.”
Slight movement toward agreement
“Students learn better when they work together.”
Movement toward agreement and strong agreement
“Knowing how to perform a mathematical procedure is more important than understanding why the procedure works.”
Movement toward disagreement (5 switched to disagree; 9 maintained disagreement)
“The teacher should do most of the talking in the classroom.”
Much movement toward disagreement and strong disagreement.
(9 switched from agreement to disagreement on this)
Sara

16. Observed Outcomes (cont.)
“Math is easy for me to do.”
Students became more moderate about this statement
“I feel confident in my ability to help my peers.”
Slight movement toward disagreement

17. Dana’s polar shift on 6 of 13 statements
After the course, she disagreed that:
Math is mostly facts and procedures to memorize,
It is important for her to learn math,
Math is easy for her to do,
The teacher should do most of the talking in the classroom, and agreed that
Students should be able to figure out whether an answer is reasonable.
Duane

18. Dana’s Responses (cont.)

19. Understanding of Mathematics
“How would you describe math to someone?”
BEFORE
“equations to solve problems”
“It's not a good time, but it is very important”
“Lots and lots of numbers and letters”
AFTER
“involved logical and critical thinking”
“A problem with many routes to the answer”
“math is using logic to systematically break down problems using numbers and letters to solve for the bigger problem”
Sara.

20. Understanding of Mathematics
“What is the job of a Mathematician?”
BEFORE
“different jobs teach research”
“to find the measurements of everything they want”
“Teach others the use of the numbers and how they can work together”
AFTER
“They try to come up with new formulas and solutions to problems.”
“Use the things we do everyday and apply math to make it much easier”
“Trying to solve hard problems and explaining them specifically.”
Sara

21. Understanding of Mathematics
“What is required to be successful at math?”
BEFORE
“It is required that you know you numbers and be able to think a problem through.”
“to be successful at math it is required that you know to multiply, divide, add and subtract”
“understanding of the basics”
AFTER
“The capability to think logically and have determination in order to solve the problem”
“you need to have a flexible mind”
“critical thinking and focus”
Sara

22. How did the course affect students?
“I think through the class, the way that the problems were set up they didn’t seem like mathematical problems. They were problems that you might run into in everyday situations that’s also part of the reason why I liked the class
My [original description of what mathematicians do was] ‘all they do is math, they just solve math problems’ I didn’t talk at all about how they use question given to create a logical answer for it.”
Duane

23. Summary
These inquiry projects, under the set activity principles, appeared to…
Maintain existing productive beliefs
Promote a more productive understanding of the nature of mathematics
Promote more productive perspectives on collaboration and active participation
Sara