Understanding of Rational Exponents - Mathematical Association of ...

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College Students' Understanding Rational Exponentsdigits of exponents from equations like "Solve for X: 5 X = 10". The procedure was practiced bythe students and made into their own system of notation and organization. Lists of decimal rootsof the base were produced by two students and systematically used to find all the digits throughdivision and multiplications. Other students carried out their calculations in a less organized way.One student used columns to organize his decimal calculations.The zero exponents were explained in relation to a situation with zero growth. The zerogrowth was then illustrated in both a graph (using shorter and shorter intervals) and in tables tocorrespond to a factor of multiplication of one (1). This teaching episode was not obvious to thestudents, but the emergence of the multiplication factor one (1) through higher and higherdecimal roots seemed to make an impression with the students. The place value system fordecimal exponents also offered a way to use zero exponents as actual place holders for factorsthat do not appear in a numeral and where the explanation "absence of factors" actually makessense.Negative exponents were treated as tools to undo a multiplication and the negative signwas linked to reverse movements on an (exponential) graph. The fact that numbers with negativesigns could indicate reverse movements or inverses in various contexts seemed to make it moreacceptable to students that exponents linked to reverse operations and movements naturally carrya sign to denote such movements.Research question 3: What is the impact of the teaching experiment on the knowledge ofthe participating students?The responses to the post-interview suggest that all the students made some progress inunderstanding rational and negative exponents. Not one student mentioned inconsistencies orlack of logic in their responses to the questions. Two of the five students produced an actual,Page 6 of 8
College Students' Understanding Rational Exponentsgeneralized notion of exponents as measures of how much multiplication of a certain base isused. Three students did not propose any generalized notion but they were able to talk about theprocess of finding a decimal exponent and they were able to explain every digit in the decimalform of an exponent and explain negative exponents using the movement metaphor.Final notesThe conjecture needs further polishing in the area of rates of growth and factors ofmultiplication. Students approached the multiplicative context with additive or linear notions.The procedure for calculating decimal exponents seemed to help students make sense of whatrational and decimal exponents stand for. The laws of exponents are re-established whenworking with rational and decimal exponents and may help some students to encapsulate thewhole process of mentally constructing the notion of rational exponents.ReferencesConfrey, J. & Lachance, A. (2000). Transformative teaching experiment throughconjecture driven research design. In A.E. Kelly& R.A. Lesh (Eds.) Handbook of researchdesign in mathematics and science education ( pp. 231-265). Mahwah: NJ. Lawrence ErlbaumAssociates, Publishers.Confrey, J. (1994). Splitting, similarity and rate of change: A new approach tomultiplication and exponential functions. In G.Harel & J. Confrey, (Eds.) The development ofmultiplicative reasoning in the learning of mathematics (pp. 291-330). Albany, NY: StateUniversity of New York.Confrey, J. & Smith. E. (1994). Exponential functions, rates of change, and themultiplicative unit. Educational Studies in Mathematics, 26, 135-164.Dubinsky, E. (1994. A theory and practice of learning college mathematics In A. H.Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 221-243). Hillsdale, NJ:Lawrence Erlbaum Associates, Publishers.Edwards, B. (1997). An undergraduate student’s understanding and use of mathematicaldefinitions in real analysis. In J.A. Dosey, J.O. Swafford, M. Parmantie & A.E. Dossey (Eds.).Proceedings of the19 th Annual Meeting of the North American Chapter of the InternationalPage 7 of 8
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