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Using Applications to Enhance Student Interest and Achievement in Mathematics:

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.

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Using Applications to Enhance Student Interest and Achievement in Mathematics:

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  1. 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

  2. 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.

  3. 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.

  4. 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.

  5. 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. • Biology students have frequently viewed mathematics as a hurdle rather than as a significant contribution to their education in the field of their choice. • Future business leaders are given little opportunity to see the value of their high school mathematics in contexts that inspire them in their fieId. • 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.

  6. Biological and environmental contextsversions 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)

  7. 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; trees Quadratic 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.

  8. 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 asa 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 asd→cVertical 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 →infinityas x → c, for some constant c.

  9. 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 + Cfrom discrete data. • Note end behavior.

  10. 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 days Lead in a child’s bones: half-life about 19 yrs. Cadmium in adult body: half-life ranges from 9 to 47 years.

  11. Coral Populations Short-term models of growth and decay (warm waters, healthy viruses) • Quadratic functions; interpretation of • positive and negative slopes, • y-intercept, • x-intercept • turning point

  12. Fish PopulationsStudying 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

  13. 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.

  14. Business contexts and Social Sciencesversions available from (1) www.uvi.edu/sites/uvi/pages/imsa-home.aspxusername:imsa-uvipassword:mathincontext(2)Comap, in Consortium Pull-out sections(3) http://www9.georgetown.edu/faculty/sandefur/handsonmath/(4) rdance@uvi.edu (on request)

  15. Raising and lowering prices; effect on demand • Quadratic functions • Understanding factors and zeros of polynomials. See “Herbal Business” IMSA-UVI

  16. 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

  17. Arms races. Models of World War I and Cold War. • Linear functions. • Discrete processes. • Equilibrium values See Consortium website, www.comap.com/product

  18. Physics contextSpeed 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

  19. 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.

  20. Sense of community,Atmosphere of challenge Research suggests that these two, in combination, are powerful contributors to student persistence: • students’ desire to learn mathematics and • motivation to stick with it long enough to achieve their own goals.

  21. 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.

  22. 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.

  23. 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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