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Not just mathematics, but mathematics problem solving!
Dr. Cristal JonesForum 2011 New YorkOctober 27, [email protected]
Not just mathematics….
• I am not a mathematics teacher…
• But a mathematics problem-solving teacher...
• Through teaching computer science content….
Do Now (Sponge Activity)
Solve:
13 + 8 / 2 + (7 + 16) =
* = Multiplication
/ = Division
Jot down how long it took to solve the problem. One minute? Longer?
Background
• New York Native
• Single parent home
• Attended P.S. 5, P.S. 233 in Brooklyn; Brian Piccolo I.S. 53; Far Rockaway High School and Baruch College
Background
• Moved to Atlanta, Georgia
• Taught computer science to African-American and Hispanic middle and high school students in face-to-face and online environments for 15 years
• Received Doctorate in Education at Georgia Southern University
Background
• I taught programming languages such as DOS , Q-BASIC (Quick Basic), Visual Basic, C++, HTML, XHTML, JavaScript and Java.
• Students can learn programming languages as early as third grade (Wiebe, 1991)
The Present of Education
Do Now:
Solve: A circle has a radius of 7 inches. Find the circumference of the circle.
Ask yourself: Describe your initial reaction to solving the problem. Write down your answer. How long did it take you to solve the problem?
The Future of Education
• We can identify the non-verbal and verbal behaviors of solving mathematics problems through metacognitive processes.
• Metacognitive framework handout (excerpt)
• A mixed-method research study was conducted to code metacognitive processes when solving precalculus and computer science problems.
The Future of Education
Polya’s 4 steps of problem solving
Understanding the problem
Devising a plan
Carrying out the plan
Looking Back
Broad categories of problem-solving
Orientation
Organization
Implementation
Verification
The Present of Education
• How do we build not just mathematics, but mathematical problem-solving activity for students to: – Understanding advanced content
– maintain interest in STEM careers?
• By building conceptual thinking
• By developing metacognitive ability.
• By engaging in problem-solving ability.
The Future of Education
• Teaching object-oriented programming concepts builds conceptual thinking and metacognitive thinking needed to gain mathematical problem-solving ability (Jones, 2010)
The Future of Education
The National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics (CESSM) termed how the mathematical problem solving should be the focus of a mathematics activity for the student:
The Future of Education
To develop such [problem-solving] abilities, students need to work on problems that may take hours, days, and even weeks to solve. Although some may be relatively simple exercises to be accomplished independently, others should involve small groups or an entire class working cooperatively. Some problems also should be open-ended with no right answer, and others need to be formulated (National Council of Teachers of Mathematics, 1989, p.1).
Object-Oriented Programming
Classes – classified information
Objects – a category of the class
Methods – characteristics of class
Animals
Cats, Dogs, Birds
Say Meow, Bark, Fly
public class Animals{
public static void main (String[] args){
System.out.println (“Meow”);}}
Object-Oriented Programming
Let’s solve both math problems using the Java programming language.
OOP and Mathematics
To tell the computer system to display string values to the computer
System.out.println ((13+8/2 + (7+16));
To define the method:
public static void main (String[] args){
To define the class:
public class OrderOfOperations{{
Research
• A classroom-based, mixed-method research study was conducted with African-American high school students who have taken or were jointly enrolled in Advanced Placement Computer Science and precalculus courses.
• Two male and three female high school students participated in the study
• One male high school graduate participated in the study.
PrecalculusProblems
1. State the number of complex roots of the equation 18x2
+ 3x – 1 = 0. Then find the roots (Holliday et. Al., 2001, p. 207, 221)
2. The sine of an acute <R of a right triangle is 3/7. Find the values of the reciprocal trigonometric ratios for this angle. (p. 289)
Precalculus Problems
Computer Science Problems
1. Create a Quadratic
Equation application
that gives the solution
to any quadratic
equation. The
application should
prompt the user for
values for a, b, and c
(ax2 + bx + c = 0) and
then display the roots,
if any. Use the
quadratic equation.
(Brown, 2005, p.127)
2. Create a TrigFunctions application that displays trigonometric and
reciprocal ratios given the following conditions: The sine of an acute <R
of a right triangle is 3/7. Find the values of the reciprocal trigonometric
ratios for this angle. The application should display output similar to:
The angle in degrees are: Sine: Cosine: Tangent:
The values in radians are:
The Math library (Java) provides methods for performing trigonometric
functions. Class Math (java.lang.Math)
Methods
sin (double angle) - returns the sine of angle, where angle is in radians.
cos (double angle) - returns the cos of angle, where angle is in radians.
tan (double angle) - returns the tan of angle, where angle is in radians.to Radians (double deg) converts degrees to radians p.130
0
2
4
6
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12
A1 A2 A5 A6 A7 B1 B2 B3 B4 B5 B6 C1 C2 C3 C4 C5 C6 C7 D1 D2 D4 D7 D8
Occ
uren
ces
Problem-Solving Strategies
Comparison of Jay's Precalculus and Computer Science Problems
PreCalc 1
PreCalc 2
CompSci 1
CompSci 2
Jay's Pre-Calculus Problem 1 & 2
Analytic Scoring Scale Results
Pre-Calculus Problems 1 & 2
P2P2P1P1
Sco
res
8
6
4
2
0
Jay's Computer Science Problem 1 & 2
Analytic Scoring Scale Results
Computer Science Problems 1 & 2
P2P2P1P1
Sco
res
8
6
4
2
0
Jay exhibited a high frequency of problem-solving strategies in the understanding the plan phase when solving precalculus problem and implementation phase when solving computer science problems. Jay averaged scores indicated average levels of problem-solving ability when solving precalculus problems; low level of problem-solving ability when solving computer science problems.
Jay's Precalculus Problem 1
Time Series of Problem-Solving Strategies
1:00 2:00 3:00 4:00 5:00
Time
A1
A2
A5
A6
B3
B4
B5
C2
Strat
egies
W
W
W
W
W
W
W
W
W
W
W
Jay's Precalculus Problem 2
Time Series of Problem-Solving Strategies
0:00 2:00 4:00 6:00 8:00 10:00 12:00
Time
A1
A2
A5
A6
A8
B1
B2
B3
B4
B5
C1
C2
C3
D1
D2
Strate
gies
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
0:00 15:00 30:00 45:00 60:00 75:00 90:00
Time
A1
A2
A5
A8
B1
B3
B4
B6
C1
C3
C4
C5
C6
C7
D2
D7
D8
Strate
gies
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
W
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W
Time Series of Problem-Solving Strategies
Jay's Computer Science Problem 1
0:00 2:00 4:00 6:00 8:00 10:00 12:00 14:00
Time
A1
A2
B3
C1
C3
C4
C6
D4
Strate
gies
W
W
W
W
W
W
W
W
W
W
W
Jay's Computer Science Problem 2
Time Series of Problem-Solving Strategies
What is metacognition?
• Metacognitive is thinking about thinking.
• There are two facets to metacognition:– 1) Knowledge of cognition – knowledge of facts and
beliefs
– 2) Regulation of cognition – effective decision-making strategy choices
The Future of Education
Research study revealed that computer science content (i.e. object-oriented programming languages) builds knowledge of cognition such as conceptual thinking and analytic thinking needed to develop mathematical problem-solving activitybased on the high frequency of the implementation phase needed to solve a computer science problem.
The Future of Education
• We need longitudinal research with large student populations to determine metacognitive processes and mathematics problem-solving activity.
• We need to focus on building the levels of problem-solving ability in students, not solely on obtaining the solution to the problem.
The Future of Education
• The National Council of Teachers of Mathematics (1989) Curriculum and Evaluation Standards for School Mathematics (CESSM) in 1989 regarded mathematical problem solving among high school students (grades 9-12) as learning that should be mature and extend through real-world applications in areas such as business, engineering, physical sciences to further connect mathematical theory (National Council of Teachers of Mathematics, 1989).
The Future of Education
Strategies to increase mathematics problem-solving ability1. Offer PD in computer science2. Develop departments with mathematics and
computer science teachers3. Conduct research, collect and analyze data for
computer science as an alternative curriculum4. Keep mathematics content in ‘tact’; strategies
are needed to enhance mathematics learning, while keeping mathematics knowledge ‘pure’.
The Future of Education
Computer science is the laboratory of mathematics problem-solving activity.
Questions?
Contact Information:
Dr. Cristal Jones
5095 Roswell Road Suite 308
Atlanta, GA 30342
404-402-6677
ReferencesReferences:
Far Rockaway High School Image http://schools.nyc.gov/SchoolPortals/27/Q465/default.htmBaruch College image http://aux.zicklin.baruch.cuny.edu/yue/Yu_Ryan_Yues_Acaemdic_Website/Welcome_files/shapeimage_2.pngAtlanta, Georgia image http://www.tripadvisor.com/Tourism-g60898-Atlanta_Georgia-Vacations.htmlProgramming languages. White Fang. http://www.whitefang.com/programming-languages.htmlTop 10 Lego Creations http://www.techeblog.com/index.php/tech-gadget/feature-top-10-lego-mindstorms-creationsWiebe, J. (1991). At-computer programming success of third-grade students. 24 (2) p.214+. Journal of Research on Computing in Education. Retrieved February 27, 2004, from Professional Development Collection Database.National Council of Teachers of Mathematics. (1989). Introduction. In 1989 Curriculum and Evaluation. Retrieved July 2, 2007, from http://my.nctm.org/index.htmJones, C. (2010). Understanding African-American students’ problem-solving ability in the precalculusmathematics and advanced placement computer science classroom.
