Understanding African-American Students’ Problem-Solving Ability in the Precalculus and Advanced Placement Computer Science Classroom

Abstract

This study was conducted to assess African-American student’s problem-solving strategies and solutions between similar mathematics and computer science tasks. Six African-American participants comprised of five high school students and one high school graduate who had taken or jointly enrolled in precalculus and AP computer science courses participated in the study. Data collected were precalculus and computer science problem solutions, think-aloud and retrospective interviews, problem-solving strategies used to solve problems, and analytic scoring rubric scale scores. Student problem-solving strategies when engaged in solving precalculus and computer science problems were coded by the researcher and co-rater to determine inter-rater agreement. Student precalculus and computer science solutions were graded using an analytic scoring rubric scale to determine levels of problem-solving ability. Results found that students did not exhibit the same problem-solving strategies in both contexts. Implications of this finding between mathematical and computer science problem-solving are presented.

Appendices

Appendix 1

Precalculus problems	Computer science problems
1. State the number of complex roots of the equation 18x2 + 3x – 1 = 0. Then find the roots (Holliday et al., 2001, p. 207)	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. The application output should look similar to: Enter value for a: Enter value for b: Enter value for c: The roots are: (Brown, 2007, p. 126)
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 (Holliday et al., 2001, p. 289)	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: 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 (Brown, 2007, p. 128)

Appendix 2

Episode 1 strategies/indicators (A) [reading and understanding phase]	Episode 2 strategies/indicators (B) [planning the process]
1. Reading the problem silent or aloud 2. Restating the problem in his or her own words/reminding himself or herself of the requirements of the problem 3. Asking for clarification of the meaning of the problem 4. Stating or asking whether he or she has done a similar problem in the past/knowledge of a similar problem 5. Representing the problem by drawing a picture, writing key facts, or making a table, diagram, or list 6. Representing the problem by assigning variables or using symbolic notation 7. Says that he/she doesn’t understand problem 8. Other	1. Describing an approach that he or she intends to use to solve the problem (steps to be taken or a general strategy to be used) 2. Using deductive or inductive reasoning 3. Synthesizing (creating) 4. Stating operative proposition (theorem, pattern search, equation, algorithm, etc. such as Pythagorean theorem, Gauss theorem, system of equations, percentage formula, pattern recognition, factoring, summation formula, ratios, computation, probability knowledge, algebra, counting) 5. Using a calculator 6. Stating that he/she has forgotten procedure stating that he/she has forgotten how to solve 7. Stating that he/she will try random trial and error 8. Other
Episode 3 strategies/indicators (C) [implementing the plan]	Episode 4 strategies/indicators (D) [verifying the outcomes of the plan]
1. Using successive approximations (using trial and error) 2. Engaged in an orderly, coherent, and well-structured series of calculations/uses algorithm 3. Stop working to see what has been done and where it is leading 4. Reviews solution 5. Checks that all hypothesis have been used or checks solution 6. Corrects any errors 7. Says he/she cannot remember formula, algorithm, etc. 8. Other	Obtaining an intermediate correct or incorrect solution by: 1. Checking the solution by substitution, retracing steps, or if the solution makes sense 2. Checking that the solution satisfies conditions Obtaining a final correct or incorrect solution by: 3. Questioning uniqueness of solution 4. Expresses liking for problem 5. Solving problem by alternate method 6. Attempting to simplify solution Reaching an impasse by: 7. Expressing uncertainty about solution shows concerns for performance and admits confusion 8. Other

1. NOTE: Non-verbal strategies are shown in plain text. Verbal strategies are shown in italics text. Verbal strategies will be coded for the retrospective interview. Non-verbal strategies are coded for the think-aloud protocol sessions

Appendix 3

Understanding the problem phase [Episode 1] 2: Complete understanding of the problem is illustrated by choice of models or diagrams to reframe problem 1: Part of the problem misunderstood—weak choice of way to represent the problem 0: Little evidence of understanding

Planning a solution (choosing a strategy) phase [Episode 2] 2: Chooses a correct strategy that could lead to a correct solution 1: Chooses a strategy that could possibly lead to a solution, but route has many pitfalls or is inefficient 0: Inappropriate strategy chosen

Execution of the solution (carrying out the plan) phase [Episode 3] 2: Implements a correct strategy with minor errors or no errors 1: Implements a partially correct strategy, or chooses a correct strategy but implements it poorly 0: Poor strategy with poor implementation, or correct strategy with no implementation

Looking back phase [Episode 4] 2: Checks solution for accuracy. Solution is correct and has correct label for the answer 1: Checks solution for accuracy, but there is a computational error or partial answer for a problem with multiple answers 0: Student does not utilize heuristics of checking for accuracy. There is no answer or wrong answer based on an inappropriate plan

Appendix 4

Retrospective Interview Questions adapted from Malloy (1994).

• 1. What were you thinking when you first read the problem?

• 2. Explain the precalculus (or computer-programming problem) in your own words.

• 3. Was there anything that you did not understand about the problem?

• 4. Did you understand the problem right away?

• 5. Have you ever solved the other problems like this before?

• 6. If you drew a picture or diagram, ask: Can you show me your diagram or picture and tell me about it?

• 7. Tell me about your thoughts as you solved the problem. What steps or algorithm do you have to solve the problem?

• 8. How did you feel about solving the problem?

• 9. Could you have found the answer to the problem another way?

• 10. How did you decide to solve the problem the way you did?

• 11. For computer science: Did you program compile and execute at the first attempt? If not, what did you do to see if the program generated the correct output?