- Title: Attention Contagion Online: An analysis of undergraduate core material in the light of hand-held computer algebra systems
- Authors: Pointon and Sangwin
- Access the original paper here
- Listen to a deep-dive podcast:
Paper summary
This 2003 study from the International Journal of Mathematical Education in Science and Technology analyzes the impact of hand-held computer algebra systems (CAS) on undergraduate mathematics coursework. The authors developed a question classification scheme to examine core mathematics courses, finding that a significant portion of tasks could be solved directly using CAS. Their research highlights the potential for surreptitious CAS use to undermine the educational intent of assessments and proposes the classification scheme as a tool for improving assessment design. The study also explores the capabilities of two specific hand-held CAS devices and examines the robustness of the classification scheme through inter-rater reliability testing. The findings reveal a heavy reliance on routine calculations in the assessed coursework and suggest potential adjustments to assessment strategies in light of readily available CAS technology.
What are the key implications for teachers in the classroom?
Teachers could use the following question taxonomy to promote deeper thinking and improve the validity of their checks for understanding.
1. Factual Recall: This category includes questions that primarily require students to recall information, often verbatim, without much processing or application.
Example: “State what is meant by saying that a set {v1, v2, . . . , vk} of k vectors in a real vector space V is (i) linearly dependent, (ii) linearly independent.”
2. Carry Out a Routine Calculation or Algorithm: This category encompasses questions that involve applying a known procedure or algorithm to solve a problem.
Example: “Find ∫(1/(2+13sin(x))dx from 1 to 12.”
3. Classify Some Mathematical Object: These questions require students to determine whether a given object meets the criteria of a specific definition or classification.
Example: “Let S be the linearly independent set {(1, 0, 0), (0, 1, 0)} in the real vector space R3. State, with justification, which of the following extensions of S form a basis of R3. (a) {(1, 0, 0), (0, 1, 0), (0, 0,−1)}, (b) {(1, 0, 0), (0, 1, 0), (1,−1, 0)}, (c) {(1, 0, 0), (0, 1, 0), (2,−7, 3)}.”
4. Interpret Situation or Answer: These questions involve translating a problem from a real-world context into a mathematical model, solving the mathematical problem, and then interpreting the solution back in the original context.
Example: “A cylindrical can without a top is made from thin metal to contain V cm3 of liquid. Find, in terms of V, the dimensions that will minimize the cost of metal to make the can.”
5. Proof, Show, Justify (General Argument): This category includes questions that require a general argument or proof, often involving abstract or general objects.
Example: “Prove that 1 is the unique multiplicative identity.”
6. Extend a Concept: These questions ask students to apply a concept they have learned in a new or unfamiliar situation.
Example: “Show that the set of polynomials in x is a vector space over the real numbers. Show that the process of formal differentiation defines a linear transformation on this vector space. [In the context of a course in which a limited number of examples of finite-dimensional vector spaces have been introduced.]”
7. Construct Example/Instance: This category involves creating a specific object that satisfies given conditions or properties.
Example: “For each of the linear transformations T : Rn → Rm, find (i) a basis for Ker(T) (if a basis exists) and (ii) a basis of Im(t). (a) n = m = 3, T(x1, x2, x3) := (x1 − 2×2 + x3,− 3×1 + 6×2 − x3, 2×1 − 4×2 + 2×3).”
8. Criticize a Fallacy: These questions present an incorrect argument or proof and ask students to identify the errors and explain why the reasoning is flawed.
Example: The sources provide a lengthy example involving a supposed proof that the determinant of a skew-symmetric matrix is 0. The student is asked to evaluate the acceptability of the given argument.
Quote
Taking the data set as a whole, the majority—three questions in five—require only routine calculations. It would be tempting to conclude that a student equipped with one of these hand-held CAS would thus be able to obtain very respectable marks with little or no mental effort.
