The Need to Rethink Placement in Mathematics Sheldon P. Gordon [email protected] Rethinking Placement Tests Two Types of Placement Tests: 1. National (standardized) tests Not much we can do about them. 2. Home-grown tests Rethinking Placement Tests Four scenarios: 1. Students come from traditional curriculum into traditional curriculum. 2. Students from Standards-based curriculum into traditional curriculum. 3. Students from traditional curriculum into reform curriculum. 4. Students from Standards-based curriculum into reform curriculum. A National Placement Test 1. 2. 3. 4. 5. 6. Square a binomial. Determine a quadratic function arising from a verbal description (e.g., area of a rectangle whose sides are both linear expressions in x). Simplify a rational expression. Confirm solutions to a quadratic function in factored form. Completely factor a polynomial. Solve a literal equation for a given unknown. A National Placement Test 7. 8. 9. 10. Solve a verbal problem involving percent. Simplify and combine like radicals. Simplify a complex fraction. Confirm the solution to two simultaneous linear equations. 11. Traditional verbal problem (e.g., age problem). 12. Graphs of linear inequalities. A Tale of Three Colleges in NYS 1. Totally traditional curriculum – developmental through calculus. 2. Traditional courses – developmental through college algebra, then reform in precalculus on up. 3. Totally reform – developmental through upper division offerings. All use the same national placement test. A Tale of Three Colleges in NYS BUT New York State has not offered the traditional Algebra I – Geometry – Algebra II – Trigonometry curriculum in over 20 years! Instead, there is an integrated curriculum that emphasize topics such as statistics and data analysis, probability, logic, etc. in addition to algebra and trigonometry. A Tale of Three Colleges in NYS So students are being placed one, two, and even three semesters below where they should be based on the amount of mathematics they have studied! And they are being punished: because of what is being assessed and what is not being assessed, because of what was stressed in high school and what was not stressed, because of what was taught, not what they learned or didn’t learn. A Modern High School Problem Given the complete 32-year set of monthly CO2 emission levels (a portion is shown below), create a mathematical model to fit the data. Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Avg 1968 322 323 324 325 325 325 324 322 320 320 320 322 323 1969 324 324 325 326 327 326 325 323 322 321 322 324 324 A Modern High School Problem 1. Students first do a vertical shift of about 300 ppm and then fit an exponential function to the transformed data to get: F (t ) 1.656e0.03923t 299.5 2. They then create a sinusoidal model to fit the monthly oscillatory behavior about the exponential curve 1 S t 3.5sin 2 t 0.5 24 3. They then combine the two components to get 1 0.03923 t C t F t S t 1.656e 3.5sin 2 t 299 24 4. They finally give interpretations of the various parameters and what each says about the increase in concentration and use the model to predict future or past concentration levels. Placement, Revisited Now picture an entering freshman who has taken high school courses with a focus on problems like the preceding one and who has developed an appreciation for the power of mathematics based on understanding the concepts and applying them to realistic situations. What happens when that student sits down to take a traditional placement test? Is it surprising that many such students end up being placed into developmental courses? Rethinking Placement Tests What Can Be Done: 1. Home-grown tests: Develop alternate versions that reflect both your curriculum AND the different curricula that your students have come through. 2. National (standardized) tests Contact the test-makers (Accuplacer – ETS and Compass – ACT) and lobby them to develop alternative versions of their tests to reflect both your curriculum and the different curricula that your students have come through.