Using Applications to Enhance Student Interest and Achievement in

Mathematics:

Examples, Rationale, and Evidence

Rosalie A. Dance, University of the Virgin Islands

rdance@uvi.edu

Mathematicians and mathematics educators love mathematics for the intrinsic beauty

of its logic and structure..

We easily succeed in teaching two kinds of students:

• those who are just like us, and• those who see the need for mathematical

competence and are blessed with supreme dedication and ability to persist against all odds.

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A huge third category of students needs a different mathematical

classroom culture:

• those who neither fall under the thrall of the beauty of mathematics, nor recognize their need for mathematical competence, but who are competent students when motivated.

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Such students need to know that the mathematics they are learning now has relevance to the real world as they experience it.

• They need to see its usefulness in relation to their own intellectual interests.

• Students value incidental learning.

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With opportunity to learn mathematics through

mathematical models of their world, students enjoy the learning of mathematics more, they increase their knowledge of the phenomena we

model.

o Biology students have frequently viewed mathematics as a hurdle rather than as a significant contribution to their education in the field of their choice.

o Future business leaders are given little opportunity to see the value of their high school mathematics in contexts that inspire them in their fieId.

o Students of history rarely see how mathematics can model their areas of interest.

If teachers provide mathematics investigations in a variety of contexts in schools, students are steeped in the relevance of mathematics to their own intellectual pursuits.

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Biological and environmental contexts

versions available from (1) Comap in the Consortium Pull-out sections(2) http://www9.georgetown.edu/faculty/sandefur/handsonmath/ (3) www.uvi.edu/sites/uvi/pages/imsa-home.aspx

username: imsa-uvi password: mathincontext(4) rdance@uvi.edu (on request)

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Sickle cell anemia and malaria. Intermediate Algebra, Pre-calculus or Discrete Mathematics.

Probability• Genetics simulation: fixed proportion of sickle cell alleles vs. normal alleles in a

population; fixed proportion of deaths due to (a) sickle cell anemia in SS population and (b) malaria in NN population. Survival rate of whole population.

• Probability models: area diagrams; treesQuadratic functions• building a function to represent the fraction of births that survive to adulthood; • finding and interpreting the meaning of the zeros of a function using factors; • determining the domain of a function in context; • finding the maximum value of a quadratic function using its symmetry;• interpreting the maximum point of a function in context; • analyzing the direction of the slope of a function;• analyzing the effect of a parameter on a family of quadratics.Recursion equations and equilibrium• Proportion of N alleles in population after n generations approaches convergence.• Mathematics uncovers the reason the sickle cell allele thrives in populations

where malaria is a killer. • Indicates that prevention and cure for malaria could eventually banish sickle cell

anemia. 7

Alcohol in the Bloodstream Pre-calculus: Rational functions.

• The proportion of alcohol eliminated from the body per hour depends on the amount present.

c/(k+a) → 0 as a Horizontal asymptote• The amount of alcohol eliminated from the body per hour tends to a

constant: ca/(k+a)→ c as a Horizontal asymptote

• Inverse functions: drinking rate (g/hr) is a function of amount of alcohol present in the body, d = f(a); its inverse gives amount of alcohol present as a function of drinking rate, a = g(d).

• The amount of alcohol present a=kd/(c-d)→infinity as d→c Vertical asymptote.

Effective investigation of rates of change. Students see that a horizontal asymptote occurs where

dy/dx → 0 as x →infinity ; a vertical asymptote occurs where dy/dx →infinity as x → c, for some constant c.

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Caffeine and Medicines in the Bloodstream Pre-calculus:

Exponential functions; piecewise defined functions.

– Two 8oz. cups of brewed coffee at 8am, then no more caffeine all day: f(t) = 260(0.87)^t

– Coffee at 8, a coke at 9:30, stronger coffee at 3 o’clock:

130(0.87)^t, t Є [0, 1.5)f(t) = 145(0.87)^(t-1.5), t Є [1.5, 7)

233(0.87)^(t-7), t Є [7, 24)

• Develop g(t) = Ar^t + C from discrete data.• Note end behavior.

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Heavy metals in the environment: children, adults Variable level: Percents; exponential decay

• Modeling elimination from the body leads to exponential decay functions similar to those for caffeine.

• Half-life exploration. Lead in child’s bloodstream: half-life ~45 daysLead in a child’s bones: half-life about 19 yrs.Cadmium in adult body: half-life ranges from

9 to 47 years.

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Coral Populations Short-term models of growth and decay

(warm waters, healthy viruses)

• Quadratic functions; interpretation of positive and negative slopes,y-intercept,x-interceptturning point

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Fish Populations Studying harvesting techniques.

• Growth rates (r) as a function of population size, p r = ap + gi, where a and gi, are determined from data.

• Quadratic function gives population growth, g, as a function of the size of the population, p: g = pr = p(ap + gi) ◊ Determine population carrying capacity ◊ Determine what value of p maximizes population growth

• Analyzing the effect of fishing ◊ Where harvest size is a linear function of population size, h(p),

determine h(p) – g(p) from a graph of the functions. ◊ Determine harvest size that maintains population size ◊ Analyze effects of varying harvest rates on population size

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Diet and Exercise Calories burned daily in routine living depends on height, weight & age Calories burned during physical workouts depend

on weight and intensity of exercise.

• Develop linear equations in n variables by using n+1 data points: c=6.55w+6.50h-7.06a+980.9 (women, age > 15) c=9.3w+19h-10.2a+105.5 (men, age > 15)

• Piecewise defined functions naturally appear in data students collect themselves as they increase the intensity of the exercise on a treadmill, for example.

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Business contexts and Social Sciences

versions available from (1) www.uvi.edu/sites/uvi/pages/imsa-home.aspx username: imsa-uvi password: mathincontext(2)Comap, in Consortium Pull-out sections(3) http://www9.georgetown.edu/faculty/sandefur/handsonmath/

(4) rdance@uvi.edu (on request)

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Raising and lowering prices; effect on demand

• Quadratic functions• Understanding factors and zeros of

polynomials. See “Herbal Business” IMSA-UVI

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Life expectancy over short terms (50 years)

• Linear functions; fitting lines to data• Solution of systems of linear equations; • Indications of non-linearity; • Recognition of historic events in data.

See “How long can we expect to live?” at http://www.uvi.edu/sites/uvi/Documents/SciMath/IMSA-RDance/20.pdf

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Arms races. Models of World War I and Cold War.

• Linear functions. • Discrete processes. • Equilibrium values

See Consortium website, www.comap.com/product

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Physics context Speed of light in water.

• Pythagorean theorem. Opportunity to review history of this theorem before the

Greek era.• Solving equations involving radicals.

See http://www9.georgetown.edu/faculty/sandefur/handsonmath/

and for “looking at an iguana vs. looking at a fish” context,

http://www.uvi.edu/sites/uvi/Documents/SciMath/IMSA-RDance/22.pdf

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Mathematics Classroom Culture

With contexts of interest to students and a mathematics classroom culture that supports the development of a learning community, we can supply two critical factors that support mathematics learning in traditionally underserved populations of students:

◊ A sense of community,◊ An atmosphere of

challenge.

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Sense of community,Atmosphere of challenge

Research suggests that these two, in combination, are powerful contributors to student persistence: students’ desire to learn mathematics andmotivation to stick with it long enough to

achieve their own goals.

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Who says so?• Anderson, J.R., Reder, L.M. & Simon, H.A. (1996). Situated learning and

education. Educational Researcher, 25(4), p5-11.

• Cobb, P. & Bowers, J.. (1999) Cognitive and situated learning perspectives in theory and practice. Educational Researcher, 28( 2), p4-15 .

• Dance, R., (1997) A Characterization of the Culture of a Successful Inner City Mathematics Classroom, Ann Arbor: UMI Dissertation Services

• Dance, R., Wingfield, K. & Davidson, N. (2000). A high level of challenge

in a collaborative setting: enhancing the chance of success in mathematics for African-American students. In M. Strutchens, M. Johnson, and W. Tate, Changing the Faces of Mathematics: Perspectives on African Americans, Reston, VA, National Council of Teachers of Mathematics.

• Doerr, H. & Lesh, R. (2002). Beyond constructivism: A models and modeling perspective on mathematics problem solving, learning and teaching. Hillsdale, NJ: Lawrence Erlbaum Associates.

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And?

• Kastner, Bernice. Evaluation of NSF Teacher Leadership project in the Washington, DC metro area. Summary at http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=9554939

• Schoenfeld, A.H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J.F. Voss, D.N. Perkins, & J.W. Segal (Eds), Informal reasoning and education (pp. 311-343). Hillsdale, NJ: Erlbaum.

• Stodolsky, S. (1988). The subject matters: Classroom activity in

mathematics and social studies. Chicago, IL: University of Chicago Press. • Vygotsky, L. S. (1978) Mind in Society: The development of higher

mental process. Cambridge, MA: Harvard University Press.

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Who else says so?Albury, A. (1992). Social orientations, learning conditions and learning outcomes among

low-income Black and White school children. Unpublished doctoral dissertation. Howard University, Washington, DC.

Boaler, J. (2002). Experiencing school mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.

Cobb, P., Yackel, E. & McClain, K. (1999). Symbolizing and communicating in mathematics classrooms. Hillsdale, NJ: Lawrence Erlbaum Associates.

Heath, S.B. (1981). Questioning at home and at school: A comparative study. In G. Spindles (Ed.), Doing ethnography: Educational anthropology in action. New York: Holt, Rinehart & Winston.

Mehan, H. (1979). What time is it, Denise? Asking known information questions in classroom practice. Theory into Practice, 18(4), 285-294.

Piaget, J. (1952). The origins of intelligence in children. New York: International Universities Press.

Sinclair, J. & Coulthard, M. (1975). Towards an analysis of discourse: The language of teachers and pupils. London: Oxford University Press.

Treisman, P.U. (1992). Studying students studying calculus: A look at the lives of minority students in college. The College Mathematics Journal, 23 (5), 362.

Sandefur, J. and Dance, R. Hands-on Activities for Algebra at College. http://www9.georgetown.edu/faculty/sandefur/handsonmath/

Kaahwa, Janet. The role of culture in mathematics teaching and learning. In press.

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