Using Mathematical Practices to Promote

Productive DispositionDuane Graysay, Sara Jamshidi, and Monica Smith Karunakaran

The Pennsylvania State University

Overview

● Studentso Upward Bound, 5-week programo College-like experiences for high school students

● Courseo Emulate mathematical research practiceso Survey questions and interview protocols assessed

Productive disposition Understandings of mathematics as a profession

Motivation of the Project

● Mathematics is taught as a Practical Toolo Concepto Procedureso Applications

● Mathematics is also a Field of Inquiryo Development of…

Concepts Procedures

o Often inspired by applications

Specific Questions

● How does engaging in inquiry projects impact students’... o understanding of what it means to “do math”?o perceptions of themselves as mathematically able?o productive and unproductive beliefs regarding math?

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

o Only one claimed to have done “write ups” before

o Significant proportion reported they did not like math

Task Selection

● Accessibilityo Imaginable o Mathematizableo Approachable (little prior knowledge)

Task Selection - Types

● Type I: Solvable & Formalizableo Solvable: There should be a solution that can be found using

problem-solving heuristics. o Formalizable: There must be an opportunity to formalize the solution.

● Type II: Representative & Generalizableo Representative: The scenario must exemplify a generic type of

problem.o Generalizable: Solving the scenario should afford a general

understanding of solutions for the generic type.

Sample Task (Type I)

● Four Queens Problemo Queens can move

horizontally

vertically

diagonally

o A piece is “attacking” another if it is one move away.o How many ways can you arrange 4 queens on a

4x4 board so that no queen is attacking another?

Sample Task Process

● Exploration ● A Solution is Proposed!● Class Discussion● Remaining Solutions Found ● Final Step: Justifying the Solutions

Activity Principles

1. Introduce the problemo mathematical content is not clearly expressed

2. Make sense of the problemo use mathematics to model the problem

3. Arrive at a (partial) solutiono discussion follows

4. Construct a viable argumento satisfy mathematical principles

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

● 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)

Outcomes

● Students tended to disagree with unproductive beliefs from the beginningo 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.

Observed Outcomes● “Students can discover math on their own.”

o Slight movement toward agreement● “Students learn better when they work together.”

o Movement toward agreement and strong agreement● “Knowing how to perform a mathematical procedure is more

important than understanding why the procedure works.”o Movement toward disagreement (5 switched to disagree; 9

maintained disagreement) ● “The teacher should do most of the talking in the classroom.”

o Much movement toward disagreement and strong disagreement.o (9 switched from agreement to disagreement on this)

● “Math is easy for me to do.”o Students became more moderate about this

statement● “I feel confident in my ability to help my peers.”

o Slight movement toward disagreement

Observed Outcomes (cont.)

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.

Dana’s Responses (cont.)

Understanding of Mathematics

“How would you describe math to someone?”

BEFORE

1. “equations to solve problems”

2. “It's not a good time, but it is very important”

3. “Lots and lots of numbers and letters”

AFTER

1. “involved logical and critical thinking”

2. “A problem with many routes to the answer”

3. “math is using logic to systematically break down problems using numbers and letters to solve for the bigger problem”

Understanding of Mathematics

“What is the job of a Mathematician?”

BEFORE

1. “different jobs teach research”

2. “to find the measurements of everything they want”

3. “Teach others the use of the numbers and how they can work together”

AFTER

1. “They try to come up with new formulas and solutions to problems.”

2. “Use the things we do everyday and apply math to make it much easier”

3. “Trying to solve hard problems and explaining them specifically.”

Understanding of Mathematics

“What is required to be successful at math?”

BEFORE

1. “It is required that you know you numbers and be able to think a problem through.”

2. “to be successful at math it is required that you know to multiply, divide, add and subtract”

3. “understanding of the basics”

AFTER

1. “The capability to think logically and have determination in order to solve the problem”

2. “you need to have a flexible mind”

3. “critical thinking and focus”

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.”

Summary

● These inquiry projects, under the set activity principles, appeared to…o Maintain existing productive beliefso Promote a more productive understanding of the

nature of mathematicso Promote more productive perspectives on

collaboration and active participation