TITE problem solving: Integrating computing and proving in secondary mathematics teacher education. S E RGE I ABRAMOVI C H S TATE UNIVE RS ITY OF NE W YORK AT POTS DAM MICHAE L L. CONNE LL UNIVE RS ITY OF HOUS TON – DOWNTOWN
Introduction The paper reflects on an earlier research on the use of technology in secondary mathematics teacher education through the lenses of newer digital tools (Wolfram Alpha, Maple), most recent standards for teaching mathematics, and recommendations for the preparation of schoolteachers. New ideas of technology integration into mathematics education courses have been proposed. The goal of the paper is to show how one can connect mathematical understanding, conceptual knowledge, procedural skills, and the modern day technological competence.
SITE: … supporting the advancement and
dissemination of the best practices of digital learning CAEP: EPPs should model best practices in digital learning and technology applications President’s Council of Advisors: “A growing need for new instructional materials … that are aligned with higher standards and provide much richer learning experience” CBMS: The need for instructional materials emphasizing the importance of “generalizing, finding common structures in theorems and proofs” . Research: Teacher candidates (while being “digital natives”) may not be prepared to use technology in the context of academic work.
Approach with ancient roots Archimedes: “Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said mechanical method did not furnish an actual demonstration. But it is of course easier, when the method has previously given us some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge”.
rd th From 3 century BC to 20 century to the
digital era
Piaget: “before being able to make a deduction the subject must observe it empirically in order to accept it as a true”. Nowadays, computers give new meaning to the classic ideas emphasizing the primordial nature of experimental evidence over formal mathematical thinking.
TITE Problems The goal of this paper is to demonstrate how certain properties of sequences of special form can be used as an entry point to a class of new problems referred to as TITE problems (Abramovich, 2014). Such problems are Technology Immune (TI), that is, they may not be solved automatically by software at the push of a button. At the same time, they are Technology Enabled (TE), that is, their solution and its demonstration can be significantly enhanced by the use of technology.
These are two such sequences of triangular numbers which will be referred to throughout this presentation. Sequence (1) 55 5050 500500 50005000 …
Sequence (2) 21 2211 222111 22221111 …
A Situational Context – Shaking Hands Sequence such as this can be generated in a number of concrete ways as illustrated in the famous “Handshake” problem. Two people – one handshake, three people – three handshakes (1 + 2), four people – six handshakes (1 + 2 + 3), and so on. So, the number of handshakes among seven people is 21, among eleven people – 55. In general, the sum 1 + 2 + 3 + … + n represents the number of handshakes among n + 1 people. Alternatively, the number of handshakes among n + 1 people can be represented by the fraction n(n + 1)/2 (e.g., counting the handshakes twice by using a tree diagram when each of the n + 1 stems supports n branches) from where the equality 1 + 2 + 3 + … + n = n(n + 1)/2 results. The tree diagram here shows the case for n = 4.
From Context to Concept Departing from the handshakes context back to the development of the concept of triangular numbers leads to the following question: How can one show that the terms of sequences such as 55, 5050, 500500, … and 21, 2211, 222111, 22221111, … are of the form n(n + 1)/2, that is, are half the product of two consecutive natural numbers? For example, factoring 55 yields 5 � 11 = 10 � 11⁄2 . In turn, solving the −1+ 1+8�55 2
quadratic equation n(n + 1)/2 = 55 yields 𝑛𝑛 = , implying that because 1 + 8 � 55 = 441 = 21 and (-1 + 21)/2 = 10, the number 55 is the triangular number of rank ten. Likewise, solving the equation n(n + 1)/2 = 21 −1+ 1+8�21 2
yields 𝑛𝑛 = number of rank six.
=
−1+13 2
= 6 implying that the number 21 is the triangular
Limitations of older technology tools In early research, in order to verify properties of such sequences, such as establishing the square root of eight times the tested number increased by one being an odd integer (known as a square test) holds true for other terms of sequences, a spreadsheet was suggested (Abramovich, Fujii, & Wilson, 1995). At the time, this was proposed to introduce a spreadsheet as a powerful computational tool that can motivate prospective secondary mathematics teachers to use technology in the classroom. However, a limitation of a spreadsheet is the number of digits that allow for correct computations - and this number is limited to 15. So, only two more terms of sequences initially explored could be correctly displayed by a spreadsheet. Furthermore, the spreadsheet is not able to support symbolic computations.
A sample TITE problem in MAPLE Because triangular numbers have the form n(n + 1)/2, one can check to see that the general term of sequence (1), 5 00 … 0 5 00 … 0, 𝑘𝑘
𝑘𝑘
where k = 0, 1, 2, …, can be presented as half the product of two consecutive integers. This demonstration is immune from the use of symbolic computation and therefore it can be characterized as a TI part of the task of exploring sequence (1). We have 5 00 … 0 5 00 … 0 = 5 � 1 00 … 0 1 00 … 0 = 5 � (102𝑘𝑘+1+10𝑘𝑘 ) 𝑘𝑘
𝑘𝑘 10𝑘𝑘 (10𝑘𝑘+1 +1)
𝑘𝑘 𝑘𝑘 1 = � 10𝑘𝑘+1 (10𝑘𝑘+1+1), 2 𝑘𝑘+1 10 + 1 are consecutive
=5� where 10𝑘𝑘+1 and integers. That is, the kth term of sequence (1) is the triangular number of rank 10𝑘𝑘+1 , k = 0, 1, 2, … and the terms 𝑥𝑥𝑘𝑘 of sequence (1) can be generated through the formula 1 𝑥𝑥𝑘𝑘 = � 10𝑘𝑘+1 10𝑘𝑘+1 + 1 , 𝑘𝑘 = 0, 1, 2, … . 2
From symbolic to numeric using Maple. This illustration shows the generation of the first ten terms of a triangular sequence.
TITE explorations in recursion using Wolfram Alpha – Part 1 This demonstration is a combination of TE and TI sub-problems. Entering the command
Inputting recurrence
“solve x(k+2)=110x(k+1)-1000x(k)” (recursive representation of sequence (1) found by entering it’s first few terms in OEIS® - Online Encyclopedia or Integer Sequences) into the input box of Wolfram Alpha yields the solution x(k) = 10k(c210k+c1) which is required to find the values of c1 and c2.
Obtaining a general solution
TITE explorations in recursion using Wolfram Alpha – Part 2 In order to complete the solution, one has to solve the system of equations 100c2+10c1=55, 10000c2+100c1=5050, something that, once again, can be addressed using Wolfram Alpha. As these illustrations shows, the result which confirms the formula for the k-th term: 1 𝑥𝑥𝑘𝑘 = 2 � 10𝑘𝑘+1 10𝑘𝑘+1 + 1 , 𝑘𝑘 = 0, 1, 2, … .
Finding a solution using initial data.
TITE explorations in recursion using Wolfram Alpha – Part 3 So, when c1 = c2 = 1/2, the formula x(k) = 10k(c210k+c1) coincides with the initial closed formula for the sequence 55, 5050, 500500, … . In this demonstration, establishing the equivalence of these formulas can be considered being technology enabled.
Summary
When used in technology-enhanced courses for secondary mathematics teacher candidates, activities such as these allow the connection of procedural and conceptual knowledge in a technological paradigm. Conceptualization of an algorithm, which is an element of procedural knowledge in mathematics, is needed for enabling an algorithm to work with computational technology. For example, as is well known, both formulas 𝑥𝑥𝑛𝑛 = 𝑥𝑥𝑛𝑛−1 + 𝑛𝑛, 𝑥𝑥1 = 1 and 𝑥𝑥𝑛𝑛 = 𝑛𝑛(𝑛𝑛 + 1)/2, recursive and closed, respectively, generate the sequence of consecutive triangular numbers starting from one. They can be used effectively with different digital tools. But the rules that generate sequences so that a computer can understand those rules are much more complicated. Finding such rules brings about a TITE problem. It has a TI (technology immune) component for it requires to create an algorithm through which sequences can be generated and has a TE (technology enabled) component as the algorithm has to be verified computationally.
Concluding Comments TITE problems in preparation of secondary mathematics teacher candidates lead to an interaction between two major approaches to mathematical explorations: an approach supported by pure argument (a TI part immune from technology) and approach supported by computation (a TE part enabled by technology). The later approach, known from the time of Archimedes, is based nowadays on the power of computers equipped with software capable of sophisticated symbolic computations. The successful interaction between the two approaches requires from teacher candidates possessing good “understanding of the knowledge of technology and the knowledge of their subject area” (Niess, 2005, p. 520), something that is critical for the concept of technological pedagogical content knowledge (TPCK). Connecting TITE and TPCK concepts in the area of mathematics has great potential to enrich the best practices of mathematical learning in the digital age.
Selected References
Abramovich, S., Fujii, T., & Wilson, J. (1995). Multiple-application medium for the study of polygonal numbers. Journal of Computers in Mathematics and Science Teaching, 14(4), 521-557. Abramovich, S. (2014). Computational Experiment Approach to Advanced Secondary Mathematics Curriculum. Dordrecht, The Netherlands: Springer. Abramovich, S. (2015). Mathematical problem posing as a link between algorithmic thinking and conceptual knowledge. The Teaching of Mathematics, 18(2), 45-60.
Abramovich, S., & Connell, M. L. (2015). Digital fabrication and hidden inequalities: connecting procedural, factual, and conceptual knowledge. International Journal of Technology for Teaching and Learning, 11(2), 76-89. Archimedes. (1912). The Method of Archimedes. Edited by T. L. Heath. Cambridge, England: Cambridge University Press. Beiler, A. H. (1966). Recreations in the Theory of Numbers. New York: Dover. Common Core State Standards. (2010). Common Core Standards Initiative: Preparing America’s Students for College and Career [On-line materials]. Available at: http://www.corestandards.org. Council for the Accreditation of Educator Preparation. (2013). CAEP Accreditation Standards and Evidence: Aspirations for Educator Preparation. Recommendations from the CAEP Commission on Standards and Performance Reporting to the CAEP Board of Directors. Washington, DC: Author. Dahlstrom, E., & Bichsel, J. (2014). ECAR Study of Undergraduate Students and Information Technology, 2014. Research report. Louisville, CO: ECAR, October 2014. Available at http://www.educause.edu/ecar. Kennedy, G. E., Judd, T. S., Churchward, A., Gray, K., & Krause, K. L. (2008). First year students’ experiences with technology: Are they really digital natives? Australasian Journal of Educational Technology, 24(1), 108-122. Niess, M. L. (2005). Preparing teachers to teach science and mathematics with technology: developing a technology pedagogical content knowledge. Teaching and Teacher Education, 21(5), 509523. Piaget, J. (1966). Psychological problems of “pure” thought. In E. W. Beth & J. Piaget (Eds), Mathematical Epistemology and Psychology. Dordrecht, The Netherlands: Reidel. Prensky, M. (2001). Digital natives, digital immigrants. On the Horizon, 9(5), 1–6. President’s Council of Advisors on Science and Technology. (2010). Prepare and Inspire: K-12 Education in Science, Technology, Engineering and Math (STEM) for America’s Future. Washington, DC. Author. Available at: https://www.whitehouse.gov/sites/default/files/microsites/ostp/pcast-stemed-report.pdf
Contact Information for Sergei Sergei Abramovich, Ph.D. Professor, Mathematics Education State University of New York Potsdam, NY 13676
[email protected] http://www2.potsdam.edu/abramovs
Dr. Abramovich has over 40 years of research and teaching experience in mathematics (differential equations, control theory) and mathematics education. His current scholarly interests are in the use of technology at all educational levels, K-16. His most recent book is titled ”Diversifying mathematics teaching: Advanced educational content and methods for prospective elementary teachers” (Singapore, World Scientific, 2017).
Contact Information for Michael Michael L. Connell, Ph.D. Professor, Mathematics Education University of Houston-Downtown One Main Street, Suite C440H Commerce Building Houston, TX 77002-1001
[email protected] or
[email protected] Dr. Connell has over 30 years of mathematics education experience at both graduate and undergraduate levels working with students in field based teacher certification programs. His research interests lie at the intersection between educational technology, learning theory, and mathematics education. He has written, presented and published extensively in these areas.
