**63rd Annual Meeting of the Illinois Council of Teachers of Mathematics**

October 19, 2013

Peoria, Illinois

**Alan Zollman**

Department of Mathematical Sciences

Northern Illinois University

Department of Mathematical Sciences

Northern Illinois University

**Bridging the Disconnect Between High School Achievement and College Expectations**

First Indicator of Student Success

7.8 % of high school students who take

ALGEBRA I get a college degree

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

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

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

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

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

Second Indicator of Student Success

Answers in the Tool Box, a study by U.S. Department of Education researcher Clifford Adelman, examined more than 20 variables--including high school courses, educational aspirations, race, socioeconomic status (SES), on-time versus late high school graduation, and parenthood prior to age 22--to determine what really influenced the college completion rates of over 10,000 students.

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 second most important indicator 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.

Specifically at

Northern Illinois University

Traditional measures of college readiness

courses taken in high school,

high school class rank,

grades received, and

ACT test scores

have not proven to be good measures.

The Mathematics Placement Exam purpose is to determine the mathematics course at NIU as the one a student will most likely benefit taking – matching the amount of mathematics one can actually use comfortably.

The Mathematics Placement Exam relates to a student’s future goals, seeking to determine what mathematics the student has mastered, not what was taken.

Since college mathematics classes are more intense than in high school (a 180-day high school course is completed at NIU in 45 lectures), NIU is cautious when placing students into college courses that superficially might appear similar to their high school classes.

A mixture of this score and the student’s ACT Math subscore determines a student’s placement into one of five categories: A, B, C, D or E.

“A placement” allows a student into Calculus 1 (MATH 229).

"B placement” allows a student into Business Calculus (MATH 211), Finite Math (MATH 210), Discrete Math (MATH 206), or Trigonometry (MATH 155).

“C placement” allows a student into College Algebra (MATH 110).

“D placement” does not allow a student into any NIU mathematics course, but lets the student into Kishwaukee Community College’s Intermediate Algebra (KCMA 098).

“E placement” only allows a student into Kishwaukee Community College’s Beginning Algebra (KCMA 096).

Approximately 60% of our beginning students are taking remedial courses.

Therefore NIU uses its own writing

and mathematics placement exams.

The following are suggested “take home messages”

based on what we have learned thus far:

2. There is a higher likelihood of success if mathematics is taken more closely to the time of enrollment in college mathematics.

It strongly is recommended that all high school students take math their senior year.

3. Content matters as well.

We found that taking higher-level mathematics courses in high school is related to higher placement and, in turn, more success in terms of retention.

4. The community colleges and NIU need to provide students with contextual skills and awareness.

Students need:

clear pathways to degrees;

information systems to see their progress to the degree;

required mandatory advising;

active, visible job placement assistance.

5. High schools, community colleges and NIU need to assist students attaining mature academic behaviors, skills and strategies to become self-regulated learners.

1. Rigor matters in high school courses.

It is recommended that high school math courses involve a high level of engaged instruction/learning that require students to engage in key cognitive process strategies such as communicating about math, making connections, representation, reasoning, and problem solving.

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

all learning is reflection

Fostering Self Determination

Cultivating Self Regulation

Capitalizing on Social Goals

Learning in the College Environment

Assisting students to become

self-regulated learners

All affective domains

motivation,

self-esteem,

self-confidence,

beliefs and

attitudes

are associated with personal identity strivings.

**Thank you!**

Dr. Alan Zollman

Department of Mathematical Sciences

Northern Illinois University

DeKalb, IL 60115-2888

zollman@math.niu.edu

http://www.math.niu.edu/~zollman

815/753-6733

Dr. Alan Zollman

Department of Mathematical Sciences

Northern Illinois University

DeKalb, IL 60115-2888

zollman@math.niu.edu

http://www.math.niu.edu/~zollman

815/753-6733

From the table at NIU after 8 semesters:

“A placement” student retention is over 70%, regardless of the major.

“B placement” student retention is over 60%, regardless of the major.

“C placement” student retention is just under 60%, regardless of the major.

“D placement” student retention is a little over 50%, regardless of the major.

“E placement” student retention is just under 50%, regardless of the major.

Number One Reason Students

Failed "Gateway" Courses at NIU

Students quit attending class

"Gateway Courses" GPA's (S'10)

JOUR 150 3.39

ENGL 103 2.74

CSCI 240 2.41

MATH 211 2.35

MUSC 220 2.35

COMS 100 2.33

HIST 260 2.27

MATH 229 2.16

PHIL 231 2.06

PHYS 253 2.03

CHEM 110 1.93

MATH 155 1.89

MATH 110 1.74

MATH 230 1.50

Fall 2000 (approx. 2000 students) A-14% B-37% C-35% D- 6% E- 8%

Fall 2010 (approx. 2000 students) A-10% B-26% C-28% D- 23% E-13%

Fall 2013 (approx. 2000 students) A-14% B-30% C- 30% D-18% E- 9%

Five constructs of Reflective Abstraction

interiorization

coordination

generalization

encapsulation

reversal (Dubinsky's refinement)

Instructor promoting reflective abstraction through:

self

instructor

curriculum and

peer initiates

References

Adelman, C. (2006). The Toolbox Revisited: Paths to degree completion from high school through college. Washington, D.C.:

U.S. Department of Education.

Beth, E. and Piaget, J. (1966). Mathematical epistemology and psychology. D. Reidel, Dordrecht, The Netherlands: D. Reidel.

Capetta, R.W., and Zollman, A. (in press). Agents of Change in Promoting Reflective Abstraction: A Quasi-Experimental Study

on Limits in College Calculus. REDIMAT – Journal of Research in Mathematics Education

Cappetta, R.W., & Zollman, A. (2009). Creating a discourse-rich classroom on the concept of limits in calculus: initiating shifts

in discourse to promote reflective abstraction. In Knott, L., (Ed.) The role of mathematics discourse in producing leaders of

discourse. (pp. 17-39). Charlotte, NC: Information Age Publishing.

Conley, D. T. (2007). Toward a more comprehensive conception of college readiness. Eugene, OR: Education Policy

Improvement Center.

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical

thinking. (pp. 95-126). Boston: Kluwer. Erikson, E. H. (1968). Identity: Youth and crisis. New York: Norton.

Mid-continent Research for Education and Learning (McREL). (2010). College Readiness. Denver, CO: McREL.Nakkula, M.

(2008). Identity and Possibility: Adolescent Development and the Potential of Schools. In M. Sadowski (Ed.), Adolescents at

school: Perspectives on youth, identity and education. (pp. 11-22). Cambridge, MA: Harvard Education Press.

Patel, R, McCombs, P., & Zollman, A. (in press). Metaphor clusters: A study characterizing instructor metaphorical reasoning on

limit concepts in calculus. School Science and Mathematics.

Piaget, J. (1967). Genetic epistemology, a series of lectures delivered by Piaget at Columbia University. New York: Columbia

University Press.

Wangle, J. (2013, March 3rd). Calculus Student Understanding of Continuity. Presented at the 40th Annual Meeting of the

Research Council on Mathematics Learning. Tulsa, OK.

Zollman, A., Smith, M C., Reisdorf, P. (2011). Identity development: Critical components for learning in mathematics. In D.

Brahier, D., (Ed.) Motivation and Disposition: Pathways to Learning Mathematics. Seventy-Third National Council of

Teachers of Mathematics Yearbook. (pp. 43-53). Reston, VA: National Council of Teachers of Mathematics.

Zollman, A. (in press). Bricks in a field: Research on the learning of calculus. In Berlin, D. F., & White, A. L. (Eds.). Initiatives in

Mathematics and Science Education with Global Implications. Columbus, OH: International Consortium for Research in Science and Mathematics Education.

Zollman, A., (2010, Summer/Fall). Defining college readiness. School Science and Mathematics Association Math-Science

Connector Newsletter, 5-6.

Zollman, A., (2007). NIU letter to students. Fall Orientation Packet. DeKalb, IL: Northern Illinois University.

Neither Lincoln nor Obama were born in Illinois...

Actually it's because of our famous state employees

Dan Walker

Governor 1973-77

Federal Prison 1987-88

George Ryan

Governor 1999-2003

Federal Prison 2007-?

Rod Blagojevich

Governor 2003-209

Federal Prison 2012-?

Otto Kerner

Governor 1961-68

Federal Prison 1973-75

**Why is Illinois famous?**

Why do some students succeed in

college mathematics courses while other fail?

Students have a fairly strong procedural knowledge, but a fairly weak conceptual knowledge of mathematics....

and most students only act on their strengths, not their weaknesses

My graduate students and I have investigated our beginning calculus students. Here are nine examples of the differences between strong and weak students.

1. Notation: Same name to different things --

example: "x"

2. Terminology: example, what does "variable" mean – label, object, unknown, varying quantity, constant, parameter, placeholder, generalized number, or abstract symbol?

2. con't: Label vs. Variable: example 3 feet is 1 yard, but the equation 3f=1y is incorrect (plug in 9 for feet and you get 27 yards)

3. Functions: f(x) ≠ f (and neither does y)

example, algebraic expression for a function vs. algebraic expression, f(x+h) ≠ f(x) + h

3. con't: Weak students: Functions must be a formula (that you can use the process of substitution) –

f(x) = 5 is not a function

a piecewise function is not a function

4. Infinity: Potential Infinity vs. Actual Infinity

process that continues indefinitely

vs. completion of an infinite process

5. Limit: Strictly Dynamic vs. Static End State

work graphically vs. algebraic

with contradictory answers

6. Continuity: continuous is defined

vs. discontinuous is not defined;

connectedness of a graph;

limit existing

(concept image vs. concept definition)

7. Image of Change: Chunky vs. Smooth Reasoning of Covariance

countable vs. continually changing (exponents, logarithms)

65 mph: distance traveled vs. instantaneous velocity7.

8. Multiple Representations: numerical, algebraic,

graphical, applications

– compartmentalize thinking vs.

making connections

9. Composition and Decomposition: Procedures,

Definitions, Theorems

– Beyond "Happy Exercise Doers"

Our university mathematics courses need to follow the Common Core State Standards and become more focused, coherent and rigorous –

"a mile deep and an inch wide"

Our students need to begin with their strengths but venture into their weaknesses while reflecting on the mathematics content. Our student need to decode, conceptualize connect and apply their knowledge beyond a purely process level.

Placement Exam Results

**Retention Results**

**Homework:**

help me edit my letter to incoming NIU students

help me edit my letter to incoming NIU students

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

... that's less than if

you became

Governor of Illinois