Abstract
This chapter demonstrates how the appropriate use of commonly available digital technology tools can motivate and support problem-posing activities. Informed by the authors’ work with teacher candidates, the chapter underscores the importance of theoretical considerations associated with the use of computers in problem posing. The theory is illustrated by cases in elementary and secondary teacher education contexts. These cases vary in complexity from the formulation of problems for an elementary classroom context to discovering new knowledge within a familiar secondary education context. The use of graphing software as a medium for reciprocal problem posing is shown to be conducive for developing rather sophisticated questions about algebraic equations with parameters. As many traditional problems can be solved effectively by modern technology, modifying such problems to be not directly solvable by technology would open the whole new avenue for problem posing in the technological paradigm.
In the real world, most of the time, an answer is easier than defining the question
(Dyson, 2012, p. 163)
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Notes
- 1.
There are mathematical methods, not studied at the pre-college level, allowing one to answer the question “how many?” without actually finding all solutions. One such method is to calculate the value of D(n; a 1 a 2, …, a k ) referred to in Comtet (1974) as the denumerant of n with respect to the sequence a 1, a 2, …, a k . In the case of Problem 4, one has to calculate D(20; 1, 5, 10). Another method is to find the coefficient of x 20 in the expansion of the product \( {\displaystyle \sum}_{i=0}^{\infty }{x}^i\times {\displaystyle \sum}_{i=0}^{\infty }{x}^{5i}\times {\displaystyle \sum}_{i=0}^{\infty }{x}^{10i} \). For more information on the use of technology in calculating denumerants or coefficients in the expansion of the products of geometric series, see Abramovich and Brouwer (2003).
- 2.
A solution strategy that can be introduced to preservice teachers in the context of Problem 4 is to reduce it to three simpler problems each of which depends on the quantity of $10 bills used. As shown in Figure 4.4, the range for $10 bills is [0, 2] (row 3); the range for $5 bills is [0, 4] (column C); the numbers below and to the right of these ranges represent the corresponding quantities of $1 bills. For example, using two $10 bills yields no possibilities for other bills; using one $10 bill yields three possibilities for other bills: ten $1 bills, one $5 bill, and five $1 bills, or two $5 dollar bills. The number of possibilities to use these $1 and $5 bills increases as the number of $10 bills used to pay $20 decreases. The spreadsheet of Figure 4.4 shows how the number of $1 bills decreases by five vertically (counting by five) and by ten horizontally (counting by ten).
- 3.
For example, in programming the spreadsheets shown in Figures 4.1, 4.2, 4.3, and 4.4, the following problem can be posed: How can one make the ranges in row 3 and column C dependent on the number in cell A1? To answer this application-oriented question, one has to use algebraic inequalities for which, thereby, the need to construct a computational environment serves as an agency. For example, when making the 24-cent postage (Figure 4.2, cell A1) the largest quantity of 6-cent and 4-cent stamps that one can use is four and six stamps, respectively. Therefore, the spreadsheet is designed not to generate numbers greater than four in row 3 and greater than six in column C. For more information on spreadsheet modeling as an agency for posing problems leading to the use of algebraic inequalities see Abramovich (2006).
- 4.
Note that preservice teachers had experience in exploring equations with parameters as described by Abramovich and Norton (2006).
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Abramovich, S., Cho, E.K. (2015). Using Digital Technology for Mathematical Problem Posing. In: Singer, F., F. Ellerton, N., Cai, J. (eds) Mathematical Problem Posing. Research in Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6258-3_4
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