**Engergizing the Classroom**

National Council of Teachers of Mathematics Regional

Louisville, Kentucky

November 7, 2013

Dr. Alan Zollman

Mathematical Sciences

Northern Illinois University

**What happens**

when students

arrive at college?

when students

arrive at college?

600,000 students take

calculus each year

50% fail

The disconnect between

high school competence

& college readiness

Students have fairly strong

procedural knowledge

- but only with the calculator

Notation

Give six ideas of what x might mean to your students.

x first means multiplication

Terminology is contextual

example: "variable"

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

Function:

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

Infinity

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**

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

Continuity

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

**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

Department of Mathematical Sciences

Northern Illinois University

DeKalb, IL 60115-2888

zollman@math.niu.edu

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

815/753-6733

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

college students

take remedial courses

My graduate students' research with strong and weak students

calculus 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

self

instructor

curriculum and

peer initiates

**References**

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

Bezuidenhout, J. (2001). Limits and continuity: some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32(4), 487-500.

Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23 (3), 247-285.

Bressoud, D. M. (2004). The Changing Face of Calculus: First-Semester Calculus as a High School Course. FOCUS , 6-8.

Bridgers, L. C. (2007). Conceptions of continuity: An investigation of high school calculus teachers and their students. Retrieved from ProQuest Digital Dissertations. (AAT 3266284).

Cappetta, R. (2007). Reflective abstraction and the concept of limit: A quasi-experimental study to improve student performance in college calculus by promoting reflective abstraction through individual, peer, instructor and curriculum initiates. Unpublished doctoral dissertation, Northern Illinois University, DeKalb, IL.

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.

Castillo-Garsow, C. (2012). Continuous quantitative reasoning. In R. L. Mayes & L. L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context. (pp. 55-73). Laramie, WY: University of Wyoming.

Cipra, B. A. (1988). Calculus. Crisis looms in mathematics' future. Science, 239, 1491-1492.

Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly 88 (4), 286-290.

Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28(3), 258-277.

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

Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Boston: Kluwer Academic Publishers.

Davis, R. B. (1984). Learning mathematics: The cognitive approach to mathematics education. Norwood, NJ: Ablex Publishing Company.

Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303.

Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In G. Harel, & E. Dubinsky (Eds.), The concept of function: aspects of epistemology and pedagogy (pp. 85-106). USA: Mathematical Association of America.

Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005a). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics, 58(3), 335-359.

Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005b). Some historical issues and paradoxes regarding the concept of infinity: An APOS analysis: Part 2. Educational Studies in Mathematics, 60(2), 253-266.

Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17(1), 105-121.

Ferrini-Mundy, J., & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives, and integrals. In J. Kaput, & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning: Preliminary analyses and results (pp. 31-45). Washington DC: Mathematical Association of America.

Fischbein, E. (2001). Tacit models of infinity. Educational Studies in Mathematics, 48(2/3), 309-329.

Fischbein, E., Tirosh, D., & Hess, P. (1979). The intuition of infinity. Educational Studies in Mathematics, 10(1), 3-40.

Forscher, B. K. (1963). Chaos in the brickyard. Science, 142 (3590), 339-422.

Gulcer, B. (2012). Examining the discourse on the limit concept in a beginning-level calculus classroom. Educational Studies in Mathematics, 82 (3), 439-453.

Hart Research Associates, (2005). Rising to the challenge: Are high school graduates prepared for college and work? Washington, D.C.: Achieve, Inc.

Juter, K. (2005). Limits of functions: Traces of students' concept images. Nordic Studies in Mathematics Education (3-4), 65-82.

Markovits, Z., Eylon, B. S., & Bruckheimer, M. (1988). Difficulties students have with the function concept. In A. Coxford (Ed.), The ideas of algebra, K-12 (pp. 43-60). Reston VA: National Council of Teachers of Mathematics.

Monaghan, J. (1991). Problems with the language of limits. For the Learning of Mathematics, 11(3), 20-24.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.

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.

Peterson, I. (1986). The troubled state of calculus: a push to revitalize college calculus teaching has begun. Science News, 220-224.

Selden, A., Selden, J., Hauk, S., & Mason, A. (1999). Do calculus students eventually learn to solve nonroutine problems? Tennessee Technological University Mathematics Department Technical Report No. 1999-5.

Selden, J., Mason, A., & Selden, A. (1989). Can average calculus students solve nonroutine problems? Journal of Mathematical Behavior, 8 (2), 45–50.

Selden, J., Selden, A., & Mason, A. (1994). Even good calculus students can't solve non-routine problems. In J. J. Kaput, & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning: Preliminary analyses and results (pp. 19-26). Washington DC: Mathematical Association of America.

Schoenfeld, A. H., & Arcavi, A. (1988). On the meaning of variable. In B. Moses (Ed.), Algebraic thinking, Grades K-12: Readings from the NCTM’s school based journals and other publications (pp. 150-156). Reston, VA: National Council of Teachers of Mathematics.

Shemale, S., & Zollman, A. (2010, November 4th). An assessment of error patterns of college students in trigonometry. Presented at the 109th Annual Convention of the School Science and Mathematics Association. Ft. Myers, FL.

Shumway, R. (Ed.) (1980). Research in mathematics education. Reston, VA: National Council of Teachers of Mathematics.

Szydlik, J. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31 (3), 258-276.

Tall, D. (1990). Inconsistencies in the learning of calculus and analysis. Focus on Learning Problems in Mathematics, 12(3&4), 49-63.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169.

Thompson, P. W. (1994) The development of the concept of speed and its relationship to concepts of rate, in Harel, G. and Confrey, J. (eds.), The development of multiplicative reasoning in the learning of mathematics, SUNY Press, Albany, NY, pp. 181–234.

Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In S. Chamberlin, L. L. Hatfield, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education. Laramie, WY: University of Wyoming.

Usiskin, Z. (1988). Conceptions of school algebra and uses of variable. In A. F. Coxford, & A. P. Shulte (Eds.). The ideas of algebra, K-12 (pp. 8-19). Reston, VA: National Council of Teachers of Mathematics.

Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematics Education in Science and Technology, 14, 293-305.

Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366.

Wagner, S. & Parker, S. (1993). Advancing algebra. In Research ideas for the classroom: High school mathematics (pp. 119-139). Reston, VA: National Council of Teachers of Mathematics.

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.

Williams, S. (1991). Models of limit held by calculus students. Journal for Research in Mathematics Education, 22(3), 219-236.

Zollman, A. (2010a). Defining college readiness. School Science and Mathematics Association Math-Science Connector Newsletter, 5-6.

Zollman, A. (2010b). Problem solving for the 21st century commentary: Back to the future or forward to the past: Mathematics education for the 21st Century. In Sriraman, B., & English, L., (Eds.) Theories of Mathematics Education: Seeking New Frontiers. (pp. 297-301). New York, NY: Springer.

Zollman, A. (2012). Learning for STEM literacy: STEM literacy for learning. School Science and Mathematics, 112 (1), 12-19.

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

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

Bezuidenhout, J. (2001). Limits and continuity: some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32(4), 487-500.

Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23 (3), 247-285.

Bressoud, D. M. (2004). The Changing Face of Calculus: First-Semester Calculus as a High School Course. FOCUS , 6-8.

Bridgers, L. C. (2007). Conceptions of continuity: An investigation of high school calculus teachers and their students. Retrieved from ProQuest Digital Dissertations. (AAT 3266284).

Cappetta, R. (2007). Reflective abstraction and the concept of limit: A quasi-experimental study to improve student performance in college calculus by promoting reflective abstraction through individual, peer, instructor and curriculum initiates. Unpublished doctoral dissertation, Northern Illinois University, DeKalb, IL.

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.

Castillo-Garsow, C. (2012). Continuous quantitative reasoning. In R. L. Mayes & L. L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context. (pp. 55-73). Laramie, WY: University of Wyoming.

Cipra, B. A. (1988). Calculus. Crisis looms in mathematics' future. Science, 239, 1491-1492.

Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly 88 (4), 286-290.

Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28(3), 258-277.

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

Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Boston: Kluwer Academic Publishers.

Davis, R. B. (1984). Learning mathematics: The cognitive approach to mathematics education. Norwood, NJ: Ablex Publishing Company.

Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303.

Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In G. Harel, & E. Dubinsky (Eds.), The concept of function: aspects of epistemology and pedagogy (pp. 85-106). USA: Mathematical Association of America.

Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005a). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 1. Educational Studies in Mathematics, 58(3), 335-359.

Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005b). Some historical issues and paradoxes regarding the concept of infinity: An APOS analysis: Part 2. Educational Studies in Mathematics, 60(2), 253-266.

Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17(1), 105-121.

Ferrini-Mundy, J., & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives, and integrals. In J. Kaput, & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning: Preliminary analyses and results (pp. 31-45). Washington DC: Mathematical Association of America.

Fischbein, E. (2001). Tacit models of infinity. Educational Studies in Mathematics, 48(2/3), 309-329.

Fischbein, E., Tirosh, D., & Hess, P. (1979). The intuition of infinity. Educational Studies in Mathematics, 10(1), 3-40.

Forscher, B. K. (1963). Chaos in the brickyard. Science, 142 (3590), 339-422.

Gulcer, B. (2012). Examining the discourse on the limit concept in a beginning-level calculus classroom. Educational Studies in Mathematics, 82 (3), 439-453.

Hart Research Associates, (2005). Rising to the challenge: Are high school graduates prepared for college and work? Washington, D.C.: Achieve, Inc.

Juter, K. (2005). Limits of functions: Traces of students' concept images. Nordic Studies in Mathematics Education (3-4), 65-82.

Markovits, Z., Eylon, B. S., & Bruckheimer, M. (1988). Difficulties students have with the function concept. In A. Coxford (Ed.), The ideas of algebra, K-12 (pp. 43-60). Reston VA: National Council of Teachers of Mathematics.

Monaghan, J. (1991). Problems with the language of limits. For the Learning of Mathematics, 11(3), 20-24.

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.

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.

Peterson, I. (1986). The troubled state of calculus: a push to revitalize college calculus teaching has begun. Science News, 220-224.

Selden, A., Selden, J., Hauk, S., & Mason, A. (1999). Do calculus students eventually learn to solve nonroutine problems? Tennessee Technological University Mathematics Department Technical Report No. 1999-5.

Selden, J., Mason, A., & Selden, A. (1989). Can average calculus students solve nonroutine problems? Journal of Mathematical Behavior, 8 (2), 45–50.

Selden, J., Selden, A., & Mason, A. (1994). Even good calculus students can't solve non-routine problems. In J. J. Kaput, & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning: Preliminary analyses and results (pp. 19-26). Washington DC: Mathematical Association of America.

Schoenfeld, A. H., & Arcavi, A. (1988). On the meaning of variable. In B. Moses (Ed.), Algebraic thinking, Grades K-12: Readings from the NCTM’s school based journals and other publications (pp. 150-156). Reston, VA: National Council of Teachers of Mathematics.

Shemale, S., & Zollman, A. (2010, November 4th). An assessment of error patterns of college students in trigonometry. Presented at the 109th Annual Convention of the School Science and Mathematics Association. Ft. Myers, FL.

Shumway, R. (Ed.) (1980). Research in mathematics education. Reston, VA: National Council of Teachers of Mathematics.

Szydlik, J. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31 (3), 258-276.

Tall, D. (1990). Inconsistencies in the learning of calculus and analysis. Focus on Learning Problems in Mathematics, 12(3&4), 49-63.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169.

Thompson, P. W. (1994) The development of the concept of speed and its relationship to concepts of rate, in Harel, G. and Confrey, J. (eds.), The development of multiplicative reasoning in the learning of mathematics, SUNY Press, Albany, NY, pp. 181–234.

Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In S. Chamberlin, L. L. Hatfield, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education. Laramie, WY: University of Wyoming.

Usiskin, Z. (1988). Conceptions of school algebra and uses of variable. In A. F. Coxford, & A. P. Shulte (Eds.). The ideas of algebra, K-12 (pp. 8-19). Reston, VA: National Council of Teachers of Mathematics.

Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematics Education in Science and Technology, 14, 293-305.

Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366.

Wagner, S. & Parker, S. (1993). Advancing algebra. In Research ideas for the classroom: High school mathematics (pp. 119-139). Reston, VA: National Council of Teachers of Mathematics.

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.

Williams, S. (1991). Models of limit held by calculus students. Journal for Research in Mathematics Education, 22(3), 219-236.

Zollman, A. (2010a). Defining college readiness. School Science and Mathematics Association Math-Science Connector Newsletter, 5-6.

Zollman, A. (2010b). Problem solving for the 21st century commentary: Back to the future or forward to the past: Mathematics education for the 21st Century. In Sriraman, B., & English, L., (Eds.) Theories of Mathematics Education: Seeking New Frontiers. (pp. 297-301). New York, NY: Springer.

Zollman, A. (2012). Learning for STEM literacy: STEM literacy for learning. School Science and Mathematics, 112 (1), 12-19.

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

Why is Illinois famous?

Neither Lincoln nor Obama were born in Illinois...

... it's our famous

state employees!

George Ryan

Governor 1999-2003

Federal Prison 2007-13

Rod Blagojevich

Governor 2003-209

Federal Prison 2012-?

Dan Walker

Governor 1973-77

Federal Prison 1987-88

Otto Kerner

Governor 1961-68

Federal Prison 1973-75

… that's less than if

you became

Governor of Illinois

**Helping students succeed**

on the Affected Domain

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 scientists

- more educated population

8 + 9 = ? – 6

action vs. equivalence

x is an unknown

solve for x

horizontal axis

(the abscissa)

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

**YOUR HOMEWORK**

Let me know your thoughts on my

letter to incoming students

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

Multiple Representations:

Algebraic

Numeric

Graphic

Applications

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