For example, for 6 = 3 , reversed to (satisfying the equality, finding points of x 5 intersection, etc.). Meanings 4 and 5 share a x5 function orientation in their way of treating the 6 = 3 , future teachers solved by saying “If my equation, emphasizing each part of the equation asnumber is 6 times bigger than x/6, then it is 6 times representing an image.bigger than 5/3”. Another way offered was to Many meanings also focus implicitly on conditional orientations, be it concerning theanalyze the ratio between each numerators and satisfaction of the equality or simply the possibility of finding a value for x. For example, in meaning 2apply it to denominators which had, in order to and 3, it is possible that no value of x is found and the same can be said for meaning 4, where it ismaintain the equality, to be of the same ratio: “If 6 possible that there be no point of intersection of the two equations or for meaning 5, where possibly nois the double of 3, then x is the double of 5 which is value of x could nullify the left side of the equationworth 10”. (e.g. x2 + 2 = 0 ). Without being explicit about it, these orientations represent a quest for finding aMeaning 7: Solving an algebraic equation is … possible value, a quest that can also be unsuccessful. This contrasts heavily with meaningfinding equivalent equations 6, because treating the equation as a ratio assumes or implies that a value of x exists. Meanings 1 and This meaning for solving the equation is 6 however do share something in common, which is related to an examination of relations betweenoriented toward obtaining other equivalent the algebraic unknown and the numbers in order to deduce the value of the algebraic unknown. Bothequations to the first one offered, in order to do not opt for a sequence of steps to undertake, but mainly for working with the equation as a wholeadvance toward an equation of the form (in global reading for meaning 1, in ratios for meaning 6). Meanings 3 and 7 share the fact that“x=something”. An example of such was done operations are conducted on the equation as a whole, be it through affecting both sides in thewhen solving 2 x = 1 , where some future teachers same way to keep the “balance” intact or to obtain 5 2 new equivalent equations. 4doubled the equation, obtaining 5 x =1 , which Finally, meanings 1, 2, 5 and 6 share the fact that they explicitly look for a number, wherewas simpler to read and then multiplied by 5/4 to the algebraic unknown is conceived as an unknown number that needs to be found; a significant issuearrive at x=5/4. to understand when solving algebraic equations (Bednarz & Janvier, 1992; Davis, 1975). Hence, be This is related to Arcavi’s (1994) notion of it through looking at which number could satisfy the equation (meaning 1), which number couldknowing that through transforming an algebraic nullify a part of it (meaning 5), which number satisfies the proportion (meaning 6) or which is theexpression to an equivalent one, it becomes number on which operations were conducted (meaning 2), all of them focus on x as being apossible to “read” information that was not visible number. Final remarks: On the potential of mentalin the original expression. Through these algebratransformations, the intention is not directly to This mental algebra activity, through theisolate x, but to find other equations, easier ones to 51read or make sense of, in order to find the value ofx.Similarities and differences in meaningsattributed to what solving an equationrepresents Albeit treated separately, these variedmeanings are not all different and some shareattributes. Therefore, in addition to the variety ofmeanings, significant links can be traced betweenthose, links that can deepen understandings aboutalgebraic equation solving; again, in view ofenriching mathematical experiences. For example,meanings 2 and 3 share an explicit orientationtoward isolating x, where others do not have thissalient preoccupation and focus on other aspects2 This is an avenue also reminiscent of arithmetical divisions,where equivalences are established: e.g. 5.08÷2.54 is equivalentto 508÷254, because 254 divides into 508 the same number oftimes as 2.54 into 5.08.Virginia Mathematics Teacher vol. 43, no. 2
variety of strategies but mostly of meanings that other mathematical topics!). However, already thisemerges, shows promise for algebra teaching and emerging variety of meanings through mentallearning. These emerging meanings are significant, algebra shows important promise for enrichingbecause they offer different entry paths into the algebraic experiences of future teachers, andtasks of solving algebraic equations and do not possibly their students!restrict to a single view of how this can be done. Aswell, this variety of meanings for what solving an Figure 1. Mental process of preservice teachersalgebraic equation can be offers significantreinvestment opportunities for pushing further the Referencesunderstanding of algebraic equation solving in Arcavi, A. (1994). Symbol sense: informal sense-mathematics teaching. Issues of conditionalequations, of deconstructing an equation regarding making in formal mathematics. For theoperations done on a number, of maintaining the learning of Mathematics, 14(3), 24-35.balance, of finding equivalent equations, of seeing Bednarz, N. (2001). Didactique des mathématiquesan equation as a system of equations, and so forth, et formation des enseignants. Canadianoffered varied ways of conceiving an equation and Journal of Science, Mathematics andof solving it. It opened various paths of Technology Education, 1(1), 61- 80.understanding. Bednarz, N. & Janvier, B. (1992). L’enseignement de l’algèbre au secondaire. Proc. Did. des I have not offered details about the math. et formation des enseignants (pp. 21-discussions that ensued each sharing of strategies, 40). ENS-Marrakech.those being contextual and solvers’ understanding Davis, R.B. (1975). Cognitive processes involvedrelated. But, suffice to say that discussions about in solving simple algebraic equations.and explanations of the strategies took an important Journal of Children’s Mathematicalpart of the session: only about 6 tasks were given/ Behavior, 1(3), 7-35.solved in 2 hours! Hence, most of the work in the Filloy, E. & Rojano, T. (1989). Solving equations:session revolved around explaining, justifying, the transition from arithmetic to algebra.contrasting and exploring the strength, meaning For the Learning of Mathematics, 9(2), 19-and relevance of each strategy developed to solve 25.the equations. Nathan, M.J. & Koedinger, K.R. (2000). Teachers’ and researchers’ beliefs about the In addition, it is through the mental algebra development of algebraic reasoning.activity that all this emerged, and not through theexplicit teaching of these strategies: strategies and 52meanings became relevant in the need for solvingthe equations and these meanings were directlyconnected to those equations. This makes theactivity more about the exploration ofmathematical ways of meaning, and less about theteaching of explicit strategies for solving. Ofcourse, outcomes will probably vary from onegroup to other, sometimes offering more,sometimes less. But, in all, it is in the practice offinding ways of solving that these meaningsemerged, and their discussion, and it is through thispractice that mathematical experiences wereenriched and enlarged in relation to ways ofsolving algebraic equations. Obviously, this is just one example, andmuch more is to be explored along these lines with(future) secondary mathematics teachers (and withVirginia Mathematics Teacher vol. 43, no. 2
Journal for Res. in Math. Education, 31(2), Jérôme Proulx 168-190 ProfessorReys, R.E. & Nohda, N. (Eds.) (1994). University de Québec Computational alternatives for the 21st [email protected] century: Cross-cultural perspectives from Japan and the United States. Reston, VA: NCTM.Schoen, H.L. & Zweng, M.J. (Eds.) (1986). Estimation and mental computation. 1986 NCTM Yearbook. Reston, VA: NCTMRecursion in Secondary Mathematics Classrooms Nicole Fratrik and Joe Garofalo Recursive functions are advocated by both fractions.the National Council of Teachers of Mathematics Continued Fractions(NCTM) and the Virginia Department of Education(VDOE). NCTM goes as far to state, “In grades 9 – Evaluating continued fractions such as this12, students should encounter a wide variety of one can be done in a variety of ways; hence taskssituations that can be modeled recursively, such as can be designed for students in different coursesinterest-rate problems or situations involving the (see Figure 1). Pre-algebra and algebra students canlogistic equation for growth. The study of recursive use their basic knowledge of rational numbers topatterns should build during the years from ninth create a numerical sequence, with fractions ofthrough twelfth grade…Recursively defined increasing complexity, look for a pattern, and thenfunctions offer students a natural way to express approximate the value of the continued fraction.these relationships and to see how some functions This type of task can be done with or withoutcan be defined recursively as well as technology, depending on what students areexplicitly” (NCTM, 2000). The VDOE Algebra II expected to do. Such an exploration can be used toCurriculum Framework includes “Essential introduce students to ideas about irrational numbersUnderstanding: Sequences can be defined and limits. Students in courses beyond a firstexplicitly and recursively” and “Essential course in algebra can be asked to develop aKnowledge and Skills: Generalize patterns in a recursive function to determine the value of thesequence using explicit and recursive fraction and even asked to derive the fraction valueformulas” (VDOE, 2009). Recursive functions can algebraically. These three ways to evaluate thebe used to explore sequences and limits, look at continued fraction in Figure 1 are shown below.connections between rational and irrationalnumbers, model growth patterns, compute and Figure 1: An infinite continued fractionanalyze compound interest and mortgages, andlearn some basic computer programming. Numerical strategyFurthermore, tasks involving recursive functions Many students start by generating acan provide opportunities for students to linkmultiple representations of functions – numerical, 53algebraic and graphical. One way to introducerecursive thinking is through the use of continuedVirginia Mathematics Teacher vol. 43, no. 2
sequence with the first few iterations of the fraction With this approach of course it is importantby hand (e.g., 3, .75…). However, use of a for the teacher to ensure that students understandcalculator allows them to proceed in a more what they are doing, asking students to explainefficient way using the home screen of their what each entry signifies and what is happeningcalculator. Using this method, students can more each time they ENTER.easily generate a sequence and notice that it is Recursive function strategy“converging.” This can be done by entering a firstiteration (e.g. 3) into their calculator, and then A second strategy involves generating andusing the form of the continued fraction to generate using a recursive function. With this methodsubsequent iterations (see Figure 2 for screenshots students first analyze the relationship betweenof early and later iterations). By setting up the successive iterations, generated either by hand orrecursive process in this way, students can press with a calculator, and use this relationship to deriveENTER for the next iteration, and keep pressing it an appropriate recursive function and determine anuntil they notice a pattern or limit. appropriate starting point. For example, one preservice student teacher said, “I tried to take it Figure 2: Screenshots showing early and later one step at a time, starting with the first term of the iterations. sequence… it finally clicked that I was entering 3 divided by 1 plus the previous answer every time. Figure 3 a, b, c: Using the sequence mode of a TI-84 graphing It was at this point that I realized the pattern for the calculator to generate representations. sequence.” From this realization he was able to create an equation that modeled the continuedVirginia Mathematics Teacher vol. 43, no. 2 fraction (see Figure 3a). Students can enter their recursive function u (n) = 3/(1 + u(n-1)) , the beginning step number nMin=1, and the beginning value at that step u (nMin)={3} into the sequence mode editor of a TI- 84 graphing calculator as shown in Figure 3a, to generate a numerical sequence (Figure 3b) and generate a graphical representation (Figure 3c) of the sequence, both of which can help students visualize what is happening at each step of the process. Linking these representations can help students understand the mathematics more deeply. Quadratic equation strategy A third strategy for evaluating this continued fraction is to derive its value by connecting the recursive equation to a quadratic equation. Some students realize, often after working with one of the strategies shown above, that in the long run, the terms and approach a limit, and hence treat these two terms as equal, resulting in the revised equation: . (Note: we have seen few students use that limit, say L, in the revised equation). From this point students can use their algebraic manipulation skills to write this as the quadratic equation + – 3 = 0, and then use their knowledge of quadratic formula to 54
find that = , which is Recursive functions can also be used to spark discussions about numbers. For example,approximately 1.302775. when first being shown the infinite continued fraction above, one student asked, “Is that even aDiscussion number?” This led to a short discussion on the relationship between rational and irrationalThis continued fraction task is a great numbers.introduction to recursive thinking. There are many For more information on continued fractions, see Olds (1963) and Khinchin (1997).entry points and possible extensions. As shownabove, students can use their knowledge of rationalnumbers to generate patterns that will eventuallylead to the ability to write recursive equations.Another pre-service teacher wrote, “I really had no Referencesclue how to start this, but I started with what Khinchin, A. Y. (1997). Continued fractions (3rdseemed to be the simplest case and built it from ed.). New York: Dover Publications.there (see Fig. 4). I realized it’s actually quite NCTM. (2000). In Carpenter J., Gorg S. (Eds.),simple: start with 3. Add 1 and make it a new 3 Principles and standards for schooldivided by that 1+3 we now have.” This young mathematics. Reston, VA: The Nationalman went back to his knowledge of rational Council of Teachers of Mathematics, Inc.numbers in order to make sense of the continued Olds, C. D. (1963). Continued fractions (Vol. 18).fraction. New York: Random House. VDOE. (2009). Curriculum framework: Algebra II. Retrieved from http:// www.doe.virginia.gov/testing/sol/ frameworks/mathematics_framewks/2009/ framewk_algebra2.pdf Figure 4. First steps done by a pre-service teacher Nicole Fratrik Doctoral Fellow Recursive thinking is advocated by NCTM University of Virginiaand VDOE in the high school curriculum. The [email protected] shown above and other continuedfractions can be a way to initially engage students *A photos of the other author wasin recursive thinking. The three strategies shown not providedabove for the infinite continued fraction lendthemselves to Algebra II and PreCalculus. ForAlgebra I, one could also use finite continuedfractions that converge to rational numbers (seeFigure 5). Figure 5. Finite continued fractions 55Virginia Mathematics Teacher vol. 43, no. 2
Summer 2016 STEM Camps Camp Age Level Date Location Cost Contact/WebsiteGovernor’s STEM Gifted students of Dates vary by See website for more Price varies http://www.doe.virginia.gov/ Academies instruction/career_technical/ any age location details gov_academies/academies/ index.shtmliD Tech Camps 7-12; Weekly Camps William & Mary, See website for https://www.idtech.com/ Mathtree 13-17 details locations/virginia-summer- June-July Williamsburg camps/williamsburg/id-tech- 5-15 college-of-william-and-mary/ June through Au- Varies $380-$465 http://www.mathtree.com/ gust, depending on varies by mt_FindCamp_VA_0.aspx program locationVT Imagination Rising 7th-8th July 10-14 Virginia Tech, $150 www.eng.vt.edu Grades July 17-21 Blacksburg info email: imagina- (financial aidCamp STEM-WVU Grades 9-12 June 18-23 West Virginia available) [email protected] University, $350 Morgantown www.wvutech.edu (financial aid Kimberlyn Gray,STEM Summer Academy Grades 9-12 June 28-July 3 West Virginia available) WVUtechSTEM- for Girls University, [email protected] Morgantown $350 www.wvutech.edu/girlsinstemCamp Invention Grades k-6 June 19-23 Radford University (financial aid wvutech.edu/girlsinstemSummer Bridge available) Register at: $225 http://campinvention.org/2016 (other times availa- (other locations -program/ ble) available) Rising 10-12th July 9-14 Radford University $400 before Contact David Horton grade girls Scholarships [email protected] Math JokesMathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.\"What's your favorite thing about mathematics?\" \"Knot theory.\" \"Yeah, me neither.\"How do you prove in three steps that a sheet of paper is a lazy dog?1. A sheet of paper is an ink-lined plane.2. An inclined plane is a slope up.3. A slow pup is a lazy dog.There really are only two types of people in the world, those that don’t do math, and those thattake care of them.Virginia Mathematics Teacher vol. 43, no. 2 56
Virginia Mathematics Teacher vol. 43, no. 2 57
Virginia Mathematics Teacher vol. 43, no. 2 58
The PuzzlemakerDr. David Shoenthal Rules:Professor, Chair 1. The digits 1 through 9 appear in each row and each column exactly once.Longwood University 2. Digits can appear in a shaded region multiple times as long as Rule 1 isn’[email protected] broken. 3. Digits in each region must either add or multiply to the indicated number. Both the operation and the sum/product are indicated in each region. 4. Squares with multiple colors (for example, the square in row 1, column 3) contain a number that’s used in the operation for adjacent regions of each of those colors. 5. The puzzles are rotationally symmetric--the board can be rotated by 90 degrees and the design remains the same.Virginia Mathematics Teacher vol. 43, no. 2 59
Conferences of Interest Virginia Council of Teachers of Mathematics | www.vctm.org 2017 Regional NCTM Conferences: Orlando, FL: October 18-20, 2017Chicago, IL: November 29-December 1, 2017 MAA: Math Fest Chicago, IL: July 26-29, 2017 2017 Annual PME-NA Conference Indianapolis, IN: October 5-8, 2017 2017 Annual PME International Conference Singapore, July 17-22, 2017 2017 ATCM Conference Chungli, Taiwan: December 15-19, 2017 2018 AMTE Conference Houston, Texas: February 8-10, 2018Virginia Mathematics Teacher vol. 43, no. 2 ISSN 2474-8749 60
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