Assessing Conceptual Understanding in a CAS Environment

1. Assessing Conceptual Understanding in a CAS Environment David DriverBrisbane State High School ddriv1@eq.edu.au

2. Background • All assessment in Queensland is school based – there are no central examinations. • Students are assessed on three equally weighted criteria: • Knowledge and Procedures; • Modelling and Problem Solving; • Communication and Justification. • Assessment must include examinations and “assignments”. • Individual schools choose what technology to use.

3. Rationale If a Computer Algebra System is used by the student in the classroom, it should also be used when completing assessment tasks. Rationale

4. What are we trying to assess? CAS v’s non-CAS environments • If the assessment task is the same as was used in a non-CAS environment, • it will provide less information about a student’s understanding, and • the opportunity to obtain information about the student’s manipulative ability will be lost. • Conversely, we don’t want to simply assess whether or not students have memorised the syntax or the sequence of key strokes required to perform a procedure.

5. What is important? Our Objective Understanding the mathematics is at least as important as being able to manipulate the mathematics.

6. Test Item Development • The fundamental task for assessment task developers is to write items that give students opportunity to show what they know. • The temptation when writing test items in a CAS environment is to make the task more difficult, e.g. by using parameters to remove the advantage of having the CAS available to “do” the mathematics. • Although this is valid, it is not the only approach available in a CAS environment.

7. A sample task The equation can be solved in at least half a dozen ways. Describe at least three distinct methods for solving this equation. (Your description should say what you do and how the method works, without actually solving the equation.) For three of the methods you described, show full working for the solution. For each method, indicate whether the solutions are likely to be exact or approximate and indicate the circumstances in which you are likely to use it. (This could include an example of an equation that would be best solved using the particular method.)

8. Integral Calculus Compare the following non-CAS and CAS versions of essentially the same problem. non-CAS Find the area bounded by the curve and the x-axis.

9. CAS A student calculated the area bound by the curve and the x‑axis and was surprised that the area was a whole number and exactly two‑thirds of the area of the smallest possible rectangle enclosing the shape. Show that the area bounded by the x-axis and thecurve [where p and q are positive real numbers] is exactly 2/3 of the area of the rectangle formed by the x-axis, the horizontal line through the vertex of the curve and the vertical lines through the x-intercepts. Solution 1 Solution 2 Classpad

10. Task Analysis If a student knows (and understands) that the area bounded by a curve and the x-axis is the definite integral with lower and upper bounds of the x-intercepts and that the x-intercepts of a curve are the zeros of the function, then conceptually, this question is no more difficult than: Find the area bounded by the curve and the x-axis. Solution 3 Classpad Solution 4

11. The Traditional Item Find the area bounded by the curve and the x-axis.

12. The parametric problem

13. Calculus – Analytic Geometry Compare the following non-CAS (graphing calculator) and CAS versions of the same problem. non-CAS: Tangents are drawn to the curve at the points where the curve crosses the x-axis. Find the coordinates of the point of intersection of the tangents. CAS Tangents are drawn to the curve at the points where the curve crosses the x-axis. Find the coordinates of the point of intersection of the tangents.

14. Conjecture The CAS enabled problem is of a similar difficulty to the non-CAS problem and requires the same knowledge and procedures, but gives greater insight into the students understanding of the mathematics involved.

15. A novel test item Consider this unconventional method for solving quadratic equations: Solve Solution: Solution: Solve Use any suitable method to show that the solutions to the two equations above are correct. Use the method illustrated in the examples to solve the equation Explain either why this method works or how you select the factors of the constant on the right-hand side of the equation.

16. A model solution • If x = 5, • If x = -15, • If x = 8, • If x = -9, • (Note: other methods are acceptable.)

18. The two factors of the constant on the right-hand-side of the equation must differ by the same amount as the two factors on the left-hand-side of the equation. (i.e. the constants in the two factors on the left-hand-side of the equation must differ by the same amount as the two factors on the right-hand-side of the equation.) The factors must be arranged in the same order (e.g. the smaller of each pair of factors must be first.)

19. Suppose the left-hand-side can be expressed as: and the right-hand-side can be expressed as: (i.e. the two factors on the left differ by the same amount, d as the two factors on the right.) Then the original equation, becomes so Sample responses

20. Discussion • With some ingenuity, it is possible to devise assessment tasks which provide students with the opportunity to demonstrate their knowledge, understanding and ability to perform mathematical procedures when access to a CAS is available, without making the question trivial or more difficult than a similar question in a non-CAS environment. • In doing so, it is important to analyse a typical response in detail, to determine which procedures can be performed by the student using their CAS.