Duane Graysay Sara Jamshidi and Monica Smith Karunakaran The Pennsylvania State University Overview Students Upward Bound 5week program Collegelike experiences for high school ID: 373665
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Using Mathematical Practices to Promote Productive Disposition
Duane Graysay, Sara Jamshidi, and Monica Smith KarunakaranThe Pennsylvania State UniversitySlide2
Overview
StudentsUpward Bound, 5-week programCollege-like experiences for high school students
Course
Emulate mathematical research practices
Survey questions and interview protocols assessedProductive dispositionUnderstandings of mathematics as a professionSlide3
Motivation of the Project
Mathematics is taught as a Practical ToolConceptProceduresApplications
Mathematics is
also
a Field of InquiryDevelopment of…Concepts
Procedures
Often
inspired by applicationsSlide4
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?Slide5
Students
21 students, selected by Upward Bound11 female, 10 male
Stu
d
ents from underrepresented groupsPotential 1st-gen college students, andMany needed additional college-prep experiences
Only one claimed to have done “write ups
”
before
Significant proportion reported they did
not
like mathSlide6
Task Selection
AccessibilityImaginable MathematizableApproachable (
little prior knowledge
)Slide7
Task Selection - Types
Type I: Solvable & FormalizableSolvable: 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.Slide8
Sample Task (Type I)
Four Queens ProblemQueens 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?Slide9Slide10
Sample Task Process
Exploration A Solution is Proposed!Class Discussion
Remaining Solutions Found
Final Step: Justifying the SolutionsSlide11
Activity Principles
Introduce the problemmathematical content is not clearly expressedMake sense of the problem
use mathematics to model the problem
Arrive at a (partial) solution
discussion followsConstruct a viable argumentsatisfy mathematical principlesSlide12
Mathematical Practices (NGA Center & CCSSO, 2010)
Make sense of problems and persevere in solving them.Construct viable arguments and critique the reasoning of othersModel with mathematics
Attend to precision (in communicating with others)
Look for and make use of structureSlide13
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)Slide14
Outcomes
Students tended to disagree with unproductive beliefs from the beginningExceptions: 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.Slide15
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)Slide16
“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
Observed
Outcomes
(cont.)Slide17
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.Slide18
Dana’s
Responses (cont.)Slide19
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”Slide20
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.”Slide21
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”Slide22
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.”Slide23
Summary
These inquiry projects, under the set activity principles, appeared to…Maintain existing productive beliefsPromote a more productive understanding of the nature of mathematicsPromote more productive perspectives on collaboration and active participation