Algebraic symbolism conveyed by ICT

1. Algebraic symbolism conveyed by ICT Taking an epistemological lens to better apprehend students' use and understanding of algebraic symbolism. Caroline BARDINI Université Montpellier 2 ICTMT 9 - July 6th 2009

2. René Magritte, The human condition I (1933) Background • Frailty with which some students carry out algebraic tasks as factoring A taskaccomplishedblindly, mostoftenwithoutquestionning the rulesthattakeplace; transforming-remaining- meaninglessalgebraic expressions. • How do studentspercieve the expressions theymanipulate? How do theyperceiveitsdifferent constituent? Whatishidden, for students, behind a symbol? • Whatalgebraicsymbolism do ICT convey? • What impact on students’ use and understanding of symbols?

3. What ICT? Symbolism? • CAS • Microworlds specifically designed for teaching specific aspects of algebra • Algebrista • Alnuset • Aplusix • Expression • FeliX1D • T algebra • … • The use of letters in algebraic expressions

4. Handbook of research on PME (1976-2006) [A. Guitierrez & P. Boero (Eds.), 2006] The role and uses of technologies for the teaching and learning of algebra and calculus [Ferrara, Pratt, Robutti]: • Technology is allowing students to explore the symbolic language as a computational tool • Algebra’s symbol system is being linked more powerfully to the tabular, geometric and graphical contexts Changes in the perspectives from which a concept can be seen Interactivity and dynamicity variables and expressions End of 80’s: research on relevance of multiple representations

5. Research interest in programming as a medium for learning about expressionswaned Logo, Pascal, Basic… • Increasing development and experimentation of such microworlds supported the bridge of the gap between manipulation skills and abstract reasoning with algebraic symbols • Studies focused on students’ appreciation of: • formal algebraic notations • generalization • abstraction processes • meaningful construction of symbolic language for expressing mathematical ideas

6. Algebraic notation : variable, unknowns and parameters “The variable is, from the formal standpoint, the characteristic notion of Mathematics” [RUSSEL, The principles of mathematics (1903)] “(…) the variable is a very complicated logical entity, by no means easy to analyze correctly. “ 1. Students’ struggle with variables, unknown and indeterminate objects Historical and epistemological pillars of the “indeterminate” 2. Variables and ICT

7. Not an actual number Particular number Variables, unknown and parameters as indeterminateobjects Unknown Variable Not-knownnumbersand samesyntacticoperationscanbecarried out on them Indeterminateobjects Historically 1st to have a symbolic representation Viète : letters to designate a fixed, yetarbitrarynumber A letter to designate what has been “looked for” Focus of attention is not on findingnumbers, but on the variable as such Existsonly as designation of a numberwhichidentitywillbedisclosedat the end Parameter Indeterminate but fixedelement of the values taken by a variable. Not subject to an inquisitorial procedurethatwouldrevealitshiddennumericidentity

8. Viète: a pioneer? Before Viète: Several solutions to problems where given and required magnitudes where asked other than within geometrical framework. Manuscripts in whichapproacheswhereunfolded by the means of symbolism Algebra Bombelli (1572) 2x2 + 12x = 32 Analytical approach Solution within symbolic register

9. Symbolic expressions Solve one equation this Solution similar to Viète’s one to solving problems involving known and not-known magnitudes Bombelli Given: Objects explicitely given Not a procedure for solving an equation: for solving an instance of an equationinstead

10. T E X T I N T E R P R E T. Given and unknown in mathematics Before Viète: Problems dealing with given and unknowns Universal feature (arbitrary) Geometryonlytool to develop a rigorous proof Given: explicitely given; known (by all) Computation, with all its advantages (automaticity…) Viète: Introduction, for the very first time, of a symbolic representation, other than numerical, for the “given” in mathematical text. Since then, “given” acquired within the symbolic register as well, a universal feature “any” Given no longer explicit Despite contradiction Given to be also considered as arbitrary

11. Originally, no doubt, the variable was conceived dynamically, assomething which changed with the lapse of time, or, as is said, as something which successively assumed all values of a certain class. This view cannot be too soon dismissed. If a theorem is proved concerning n, it must not be supposed that n is a kind of arithmetical Proteus, which is 1 on Sundays and 2 on Mondays, and so on. Nor must it be supposed that n simultaneously assumes all its values. If n stands for any integer, we cannot say that n is 1, nor yet that it is 2, nor yet that it is any other particular number. In fact, n just denotes any number, and this is something quite distinct from each and all of the numbers. It is not true that whatever holds of any number holds of 1. The variable, in short, requires the indefinable notion of any which was explained in Chapter V.” [Bertrand Russel, The principles of mathematics, (London:1903), reedited (Norton: New York and London, 1986). Chapter VIII, The variable, p. 90]

12. Meaningsstudentsattribute to variables Previousresearch: Bednardz, Kieran & Lee (1996) MacGregor & Stacey (1993) Trigueros & Ursini (1999) Variable as an indeterminatenumber of a specifickind. Not a numberin itsown Temporallyindeterminatenumberwhichfateis to becomedeterminateat a certain point. Letters as ‘n’ in ‘2n+1’ considered as potentialnumbers. Letters as index (Radford, 2003): signindicating the place that an actualnumberwilloccupy in a process (Sfard, 1991) temporarily in abeyance.

13. Figure 2 Figure 3 Figure 4 Figure 1 Figure 2 Figure 3 Figure 1 Task (pattern generalization – grade 11) Problem 1 – “Original pattern” - First figures were drawn - Find out the number of toothpicks for specific figures • Write an algebraic formula to calculate the number of toothpicks in figure ‘number n’ Problem 2 – “Mireille’s pattern” - First figures drawn - Mireille begun her pattern at the 4th “spot” (place) of the original pattern Problem 4 – “Shawn’s pattern” • Shawn begun his pattern at the “spot m” of the original pattern • Provide an algebraic formula, in terms of m, that indicates the number of toothpicks in figure number 1 of Shawn’s pattern

14. Figure 2 Figure 3 Figure 4 Figure 1 Students’ use and understanding of letters Write an algebraic formula to find the number of toothpicks in fig. ‘number n’ Index: something indicating a place that will be occupied by a number Amount of toothpicks n : designates the schema ‘2x+1’ x : temporarily indeterminate number n not a genuine algebraic variable “Potentially determinate” number (as soon as ‘x’ takes on its numerical value)

15. Researcher: It starts at figure ‘m’ […] How many toothpicks will its first figure have? Sam: Yeah, well we don’t know this. Daniel: Well, that’s what we have to find out. [...] Daniel: His 1, his 1, where is it located according to this (pointing the “original” pattern). (...) Where is the ‘m’ according to this? [...] Denise: So, if you want to find the amount of toothpicks in his pattern (sic), if you had the number of the figure you could do it, but we don’t have it. That’s the only thing I don’t know how to do. The effect of the indeterminate origin on using a schematic formula Finding an algebraic expression for the number of toothpicks in a figure located at a place which is : • Temporally indeterminate Previous (fig. number 1, number 25, number ‘n’) Formulaic schema made sense insofar as considered as a process in abeyance • Indeterminate as such Shawn’s pattern (figure number ‘m’) Parameter: indeterminate but fixed element. Despite its indeterminacy it makes sense to think about it and of the figures at that place Even if no numerical values attributed Layer of math generality: “Existence” of the object does not depend on numerical determinacy (actual or potential).

16. The effect of the indeterminateorigin on using a schematic formula Daniel: We don’t have Shawn’s pattern. [...] We don’t know where it starts at and where it ends... We can almost not do it […] Sam: I’m going insane.[...] We have nothing... Acceptance of indeterminacyis a real obstacle. Need to attribute a numerical value in order to progress in the mathematicalactivity. Able to produce a formula and manipulate it, formula is still seen as a process and not yet as an object (Sfard, 1991) Daniel: We just don’t know how to find ‘m’. [...] What did you say? Denise: x = 2m+1. Researcher: Do you agree with that? [...] Sam: Yeah, but it takes you nowhere. It’s nice to have a formula, but you have to get a number. Researcher: We don’t have to have a number! Denise: We have nothing. Studentsacceptdealingwith the indeterminacy, but only for a while, for the formulas have to provide a result. Daniel: Yeah, this would work, yeah, it’s just m that we don’t know how to find. Denise: We don’t know how to find it. Yeah, that’s the thing.

17. Shawn’s Pattern First problems A few remarks Students have succefullyconceptualizedletters as variables and meaningfullyproduce and manipulate formulas Semioticproblem of indeterminacyreveals the frailty of students’ understanding of algebraic formulas Highlightsfrailty of perceiving formulas as schemas, putting intoevidencetheirlimited scope. Making students consider the figures not necessarily characterized by actual or potential numbers but as genuine conceptual objects, that can only be referred to through signs. Generalization of patterns There is layer of generality in which mathematical objects can only be referred to symbolically, detached in a significant manner from space and time. Students need to learn to cope with the kind of indeterminacy that constitutes a central element of the concepts of variable and parameter. Although one may very well be asked to begin from “nothing” (cf. Sam) there is no reason to go insane: one still can go somewhere else to symbolic algebra.

18. Variables and ICT – “Solve” command • Unknown and parameters (Bills, 2001) : Command highlights the often implicit notion of “unknown to be found”. • Equation solved with respect to an unknown Need to specify the unknown in the syntax: Solve (x2+bx+1=0, x) • Can be used to express one of the variables in a parameterized equation with respect to other variables • Expression vs. equation (Drijvers, 2005) : Raises awareness between expression and equation Multiple representation: Helps students perceive formula as object that represents the solution

19. Variables and ICT • Computer-based models favoring multiple representations • Display representations • Allow for actions on those representations • Dynamic view of algebra flourished (software & games designed to favor it) Dynamic media inherently make variation easier to achieve [Kaput, 1992] Multiple representation Helps students perceive formula as object that represents the solution

20. Impact on ICT development • Attempt to improve the conceptual understanding of the use of letters in algebraic expression and equations Wide diffusion of “generic organizers” softwares “Microworld which enables learner to manipulate examples of a concept. The term “generic” means that the learner’s attention is directed at certain aspects of the examples which embody a more abstract concept” [Tall, 1985] • Studies looked at cognitive aspects of abstraction and generalization process in learning environments supporting different types of algebraic notations Flourishing microworlds • Algebraic Patterns (Yerushalmy & Shternberg, 1994) • Mathsticks (Healy & Hoyles, 1996) • L’ Algebrista (Mariotti & Cerulli, 2001)

21. Differences not to be underestimated ! • Heck (2001): Differences between algebraic representations found in computer algebra environment and in traditional mathematics Graphing calculators • Variable always has a value • Can play more than one role in a single statement • Rules for manipulating differ from mathematical manipulation rules (eg. order in statements) • Some do not exist in maths (‘ANS’) CAS • Variable always point to a value • Manipulation with its own rules (Maple: 2(x+3) automatic: 2x+6) • Expression both object and process • Focus on generic problems (special values of parameters not taken into account) • Variables in spreadsheets?…

22. Concluding remarks / perspectives • Is the curricular role of elementary algebra changing as a result of the availability of new technology? • Impact of research on expression and variables on design of technology ? • ICT eliminating need for symbols: • Data capture and directly transferred to computer • Data entry by students while experimenting