The Need to Rethink Placement in Mathematics Sheldon P. Gordon

advertisement
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.
Download