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National Council of Teachers of Mathematics
Transcript of National Council of Teachers of Mathematics
National Council of Teachers of Mathematics Regional
November 7, 2013
Dr. Alan Zollman
Northern Illinois University
arrive at college?
600,000 students take
calculus each year
The disconnect between
high school competence
& college readiness
Students have fairly strong
- but only with the calculator
Give six ideas of what x might mean to your students.
x first means multiplication
Terminology is contextual
in science it means?
in mathematics it means?
Variable is used as labels, objects, unknowns, varying quantities, constants, parameters, generalized numbers, placeholders, arguments, and abstract symbols.
Strong students know the difference between labels and variables
difference between the algebraic expression for a function and an algebraic expression, f(x) ≠ f
Operations does not overrule the concept
f(x+h) ≠ f(x) + h
f(2x) ≠ 2f(x)
but sin (-x) = -sin x???
Most students believe a function must be a formula thus connecting function with the process of substitution
Many students stick to the belief that “plugging in” one x to get one y must occur for a relationship to be function
Potential infinity arises from thinking about processes that continue indefinitely...
Actual infinity has to do with conceiving of infinite sets existing. One thinks of the completion of an infinite process.
Use f for feet and y for yards in "3 feet equals 1 yard"
Most students write: 3f = 1y
Put in 6 for feet -- do you get 2 yards
-- or 18?
in terms of variables:
“three times the number of yards is equivalent to the number of feet” or 3y=1f
This difference between labels and variables confuses many students, as the reverse procedure does work:
given the mathematical equation 12i=f, one correctly can directly translate as 12 times i equals f.
limit as a number (or end state)
v. limit as a process
Process: idea of substituting a finite number of values for x to determine the limit of a function
Students reason with graphical and algebraic representations in contradictory ways with limits – unaware of any contradictions by compartmentalizing their thinking
The Good News - Part 2
Of all the high school indicators of academic preparation, the one that is the strongest is taking rigorous and intense courses in high school
Taking rigorous and intense high school courses has a greater impact on African-American and Latino students than on white students.
Mathematics taken is the most important for college success. The odds that a student who enters college will complete a bachelor's degree more than doubles if that student completed a mathematics course beyond Algebra II (e.g., trigonometry or pre-calculus) while in high school
Socioeconomic status had some impact (but it was minimal after the first year of college), and race did not have a statistically significant impact at all
Dr. Alan Zollman
Department of Mathematical Sciences
Northern Illinois University
DeKalb, IL 60115-2888
If you grow up in South Central Los Angeles,
it is said that you have a one-in-three
chance of ending up in prison
The Good News
7.8 % of high school students who take
ALGEBRA I get a college degree
23.1% of high school students who take
GEOMETRY get a college degree
39.5% of high school students who take
ALGEBRA II get a college degree
62.2% of high school students who take
TRIGONOMETRY get a college degree
74.3% of high school students who take
PRE CALCULUS get a college degree
79.8% of high school students who take CALCULUS get a college degree
40% of beginning
take remedial courses
My graduate students' research with strong and weak students
25 Things Students Must Know To Succeed in College Mathematics
2. Cultivating Self Regulation
Self-regulation skills and strategies for learning help students learn how to identify and set goals, select among a repertoire of learning strategies, actively monitor and evaluate their efforts at goal attainment, and redirect their behaviors when they fall short of a goal
Also, a hoped-for self that is concrete, realistic, and detailed— and that invokes the necessary strategies for achieving a desired goal— will influence student behavior, producing the intended results over time thereby serving a self-regulatory role for learners
3. Capitalizing on Social Goals
Students spend much of their time in school socializing with friends and classmates and observing and learning from one another
Increasing student belongingness and academic performance is possible through cooperative learning, group projects and study cliques (e.g., Women in Calculus)
1. Fostering Self Determination
According to self-determination theory (Deci and Ryan 1985;), three basic human needs require nurturing for people to be psychologically healthy
First, students need to feel autonomous, that is, to have control and agency in their lives and actions
Second, they must become competent in the activities in which they engage
Lastly, they need emotional connection to and support from others, as well as the ability to participate in mutually satisfying relationships
According to Piaget,
all learning is reflection
Fostering Self Determination
Cultivating Self Regulation
Capitalizing on Social Goals
Learning in the College Environment
Instructor promoting reflective abstraction through
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Why is Illinois famous?
Neither Lincoln nor Obama were born in Illinois...
... it's our famous
Federal Prison 2007-13
Federal Prison 2012-?
Federal Prison 1987-88
Federal Prison 1973-75
… that's less than if
Governor of Illinois
Helping students succeed
on the Affected Domain
J. H. Poincare said,
“Mathematics is the art of giving the same name to different things.”
Calculus is a gateway course in college for STEM
- more educated population
8 + 9 = ? – 6
action vs. equivalence
x is an unknown
solve for x
covariance between x and a function
I got the problem wrong
elements of the domain vs. operations on the range
a constant function, e.g., f(x)=5, cannot be a function
nor can a non-continuous function, e.g., a piecewise function, cannot be a function
example: Does the set of positive integers have the same number of elements as the set of perfect squares 1, 4, 9, 16, etc.?
context: for 8x, we do not mean to substitute 7 for x, as 8x does not mean 87, but 8 times 7
Let me know your thoughts on my
letter to incoming students
Continuity and discontinuity are related only to the graph of the function
Weak students confuse continuity with "connectedness" of a graph --
the pen test
Students confuse continuity with the limit existing and
with the function being defined
As with limits, weak students reason with graphical and algebraic representations in contradictory ways
Images of Change
Smooth vs. Chunky Reasoning
Chunky images of change are based in countable and completed amounts, whereas smooth images of change are based in imagining a continually changing experience
What does the rate 65 mph mean to students?
Is only thought of as going 65 miles in one hour?
Plotting points vs.
movement on the horizontal and vertical
Most students persist with only the graphical representation or, to a lesser extent, with the algebraic representation, but never see that these are different representations of the same mathematical concept.
They compartmentalize their thinking and therefore stay with procedures, even when these procedures give illogical or contradictory results.
example: Drawing three squares (with side of the radius) inside a circle approximates the area of the circle, e.g. A=πr2