epistemological discontinuities and cognitive

EPISTEMOLOGICAL DISCONTINUITIES AND COGNITIVE HIERARCHIES IN TECHNOLOGY-BASED ALGEBRA LEARNING Michal Yerushalmy University of Haifa There is speculation about the degree to which new technologies will lead to replacement of current curricula with new content. How does the use of new curriculum that is based upon new epistemological assumptions change our capability to anticipate students' difficulties and strengths? Taken from a series of studies carried out as design experiments in algebra classrooms over the last decade, I will present examples where students’ performance in a technology-supported curriculum is different from the performance one might have predicted for students learning this content in a non-technology supported environment. These examples suggest that technology can transform student learning. The technology and the sequence help to bridge known transitions that assumed difficult to students to be 'natural' (for example: Modeling, recursive thinking, solving unfamiliar equations or visualizing equation in 2 unknowns in 3D). However, the transitions between fundamental concepts or operations remained the difficult and non-trivial parts. In doing such analysis, I suggest that identifying critical discontinuities is an important research tool for studying students' construction of knowledge and for analyzing classroom guided inquiry supported by technology. School algebra reform In popular culture, school algebra, has a negative reputation. It is believed that school algebra classroom involves students in the application of unjustified methods to classes of problems, and that this is problematic because it does not get students involved in creative thought that is representative of mathematical work. Studies of educators and mathematicians concerning the ways in which the discipline is portrayed to students have not demonstrated interest in algebra, probably because it presents students with limited opportunities to engage in something like the solving of problems in ways that reflect hallmark of mathematical thought. Indeed, school algebra has been described as a part of the curriculum "overly focused on “meaningless manipulation.” memorized rules focusing on specific strategies for specific types of problems. As part of the larger reform movement across the world, during the last fifteen years, mathematics educators have tried to work to reshape school algebra and respond to the above mentioned criticisms. There have been calls for more contextually-based problems that would allow algebra to emerge from quantitative situations in the lives of students. There have been calls to integrate attention to justification and proof throughout the school curriculum, including school algebra. And, influenced by the increasing availability of computer and calculator technology that supports multiple representations of functions, there have been moves to use this technology to change the nature of the school algebra curriculum (e.g, Heid et al., 1995). Algebra reform for algebra beginners has taken several forms, some of them called functions approach to algebra. Although there are important differences between them, these new forms organize the algebra curriculum around the concept of function, emphasize and support concrete representations, and base learning on situations that appear realistic and are centered on mathematization in the form of modeling and of abstractions at different levels (Kieran & Yerushalmy 2004). With

respect to equations, the function-based proposals suggest that an equation is a question of comparison between two functions (in the spirit of Freudenthal’s, 1973, notation). Graphing technology and tabular representations of function values can suggest the number of solutions to any equation and can approximate their values. Learning of this type requires the active participation of the students: making conjectures and performing actions with physical tools and representations in ways that were not possible in traditional algebra. Discontinuities in long term learning Technology interacts in important ways with the desires of mathematics educators to work for what Tall (2002) calls “long term learning.” Noss (2001) suggests that what is natural to express and how one might do so both may change as a function of the expressive tools available to a culture at a given time. This view of the power of tools and media underlies much of the enthusiasm in mathematics education for technological tools. Perhaps technology might upset the hierarchies of prerequisite skills that often seem to dictate the practice of mathematics in schools. Perhaps technology might aid teachers in making accessible to learners' powerful mathematical ideas in a different sequence and rate than has been traditionally deemed feasible. But technology is only one aspect of the reformed constructions of long term learning. In Tall’s terms, when learners in school need to reconstruct their knowledge in order to make progress, then they have experienced a discontinuity in the curriculum. He suggests that discontinuities cannot be made disappear and argues that they are a crucial matter of curricular design and understanding. Tall exemplifies places where cognitive reconstruction is called for, even if learners do not find this reconstruction difficult and asserts that such discontinuities are inevitably part of mathematics teaching and learning and should be explored, rather than avoided. In a recent paper Yerushalmy & Chazan (to appear) point on the need that mathematics educators develop better descriptions of how curricula differ and what bets (cognitive, pedagogical, and technological) particular curricular make. They argue that it is less important to be able to make statements about the effectiveness of a particular curriculum or the use of a particular technological innovation and more important to be able to associate evaluation data about a curriculum with key aspects of the design of the curriculum. We argue that a key aspect that would allow evaluation data to accumulate across different curriculum development projects is the discontinuities students will encounter. Using this notion of discontinuities I will examine the design of an Israeli algebra curriculum (Visual Math 1995) at two points and the discontinuities they entail. Visual Math - An example of a long-term learning sequence

The Visual Math curriculum (1995) is an algebra, pre-calculus and calculus curriculum where technology is being used to help learners develop knowledge from their perceptions of the world and to develop conceptual understanding of symbols. The growing research in the field of embodied cognition suggest the idea that bodily activities are centrally involved in conceptualization of mathematics and that important parts of Algebra and Calculus are understood via conceptual metaphors in term of more concrete concepts. In particular the notion of continuous functions and directed graphs are viewed as mathematical concepts developed through human motion experience. The learning of algebra in Visual Math is preceded by semiqualitative modeling. Another challenge of the innovative development is rooted in

the symbolic world where technology that supports multi representations of functions allows students to develop symbolic understanding using the feedback from graphs or table of values, generating and viewing a rich repertoire of non-prototypic examples. In general, an important goal of this curriculum is to help students develop strong symbolic skills and to learn to do a variety of standard algebraic manipulations. But, the curriculum is aimed at helping students learn to do such manipulations with an understanding of the graphical and tabular meanings of these manipulations, as well as a sense of the purposes for which such manipulations are useful. We will look at how such new epistemological structures afforded by digital technology impact the cognitive hierarchies; resolve or change the nature of known transitions or mark new critical discontinuities in the curricula. Algebra and Mathematical Modeling

Using technology such as simulations' software, MBL or other modeling tools that include dynamic forms of representations of computational processes, it is now possible to construct graphical models without first writing symbolic expressions with x’s and y’s. Several studies suggest that such emphasize on modeling offers students means and tools to reason about differences and variations (rate of change). The Visual Math curriculum is an example of an algebra curriculum where the learning of algebra is preceded by such semi-qualitative modeling. Graphs and what we will call staircases (a graphical depiction of differences in y value for a set change in x Schwartz and Yerushalmy 1995) emerged as models of situations, and also as models for reasoning about mathematical concepts. Using the grammar of objects and the operations on them, young students constructed complex mathematical models, based on qualitative analysis of variation. Thus, if students become familiar first with ideas of continuous change and finite differences, explicit closed forms may become less natural aspect of expressing phenomenon. For example: in Yerushalmy (2005) I describe students whose earlier experience with ideas of differences complicated their use of explicit forms. They were seeking help in understanding why two numerical phenomena they identified as appropriate: the linearly increasing differences and the squaring describe the same quadratic phenomenon. In other words: why does the solution of the difference equation f(x+1) –f(x)= ax+b is of the type: f(x)=x^2. This complication might not have arisen if they had not had earlier support for recursive reasoning. Thus the affordances of technology that made the recursive thinking the natural way to think about a phenomenon and to symbolize it in a model is viewed as strength however, it challenges thinking about teaching that can support the transition to algebraic close rules. Meaningful Manipulations The lion’s share of the early parts of the Visual Math Algebra curriculum focuses on functions of one variable and equations of one variable conceptualized as the comparison of two functions of one variable. With this way of thinking about equations, students acquire alongside the algebraic procedures alternative methods to solve equations. Prior to and while using symbolic manipulations students are encouraged to use systematic guessing, intuitive numerical and graphical analysis strategies to narrow down the search interval. Research suggests that mature problem solvers of word problems in algebra devote a substantial portion of their work to representation of the problem at the situational level. Forming the situational model is a necessary stage in understanding the story of the problem and is a major component of model-based reasoning. Recently Gilead & Yerushalmy (2001) studied Visual

Math students and equations' based algebra students solving linear motion problems. 87.3% of the 196 solutions given by the Visual Math students for problems in context included graphical description of the situation which formed a situation model either using a sketch (73%) or an accurate graph (14.3%). Investigation of the comparable algebra students suggest that the students who were the more successful students of a traditional algebra sequence which focus on unknowns and stress manipulations of equations were substantially less capable to solve the same problems that were part of their curricular sequence as well. While 90.8% of the solutions of the Visual Math were correct solutions only 57.2% correct solutions were given by the equations' based approach students. The function's approach to algebra that we took stressed the algebra signs and symbols and the expressions and equations using these symbols to be a meaningful language to express ideas. One of the ways to observe meaning is to view equations as describing situations out of the mathematics and as graphical models. This habit has been proved to support students learning when solving constant rate problems. However, this same view of equation had probably leaded to a discontinuity while approaching equations in two variables. In typical algebra instruction, solving an equation in two variables and then a system of equations requires to shift from a non explicit form (x+y=2) to an explicit function form: y=2-x. One then substitute and use similar solution techniques as in a single variable case. While technique does not require dramatic change the shift from an equation in a single variable to two variables requires a shift in understanding the nature of the solution: from a single definite solution to a set of solutions. For the Visual Math function approach students the equal sign of the equation represents a symmetric comparison sign and the function equal sign represents an a symmetric assigning sign. Taking this view the nature of the solution remains the intersection values of the intersection of the two functions. But the graphical and tabular representations of functions of two variables must be developed in order to help students see their connections to the ways of representing functions of one variable with which they are familiar. Thus the transition remained critical; either one has to develop new ideas about presentations of function in two variables or one has to rewrite the equation in a way that violets the distinction between function and equation. It required acknowledging that simple algebraic technique can change the mathematical objects in hand: from equation in two variables to a function in a single variable. Discontinuities- a tool for curriculum analysis

The underlying assumption is that discontinuities exist in any long term learning process and they cannot be made disappear. I demonstrated how the use of technology in function's based school algebra has the potential to introduce strengths that looked as if the "unnatural" became "natural". Using two examples; about the transition between open and close sentences and the transition from one variable equations to equations in two unknowns I argue that often discontinuities are stable – probably because they point on, or a result of epistemological gaps. However, I suggested that we should learn to remap the algebra with technology terrain in a way that would help us to identify new or what appear as new discontinuities. An important question for educators, teachers of new learning approaches and designers of reformed curricula is whether it is possible to expect specific discontinuities. The two examples I presented were expected when we planed the Visual Math sequence. Knowing that they are not the optimal ones and that there could be other choices Visual Math bets on the

pedagogical values of these choices. As analyzed in Yerushalmy & Chazan (to appear) function-based approaches to school algebra are not univocal. We suggested that the study of discontinuities – whether stable or new – should be considered as an important challenge for curricular design and evaluation. REFERENCES Freudenthal, H. (1973). Mathematics as an Educational Task. Dordrecht, Reidel. Gilead, S. & Yerushalmy, M. (2001). Deep structures of algebra word problems: Is it approach (in)dependent? In the Proceedings of the 25th annual meeting of the International Group for the Psychology of Mathematics Education, (PME) vol.3, pp. 41-48. The Netherlands. Heid, M. K., Choate, J., Sheets, C., and Zbiek, R. M. (1995). Algebra in a technological world. Reston, VA: NCTM. Kieran, C, Yerushalmy, M. (2004). Research on the role of technological environments in algebra learning and teaching. In K. Stacey, H. Shick & M. Kendal (Eds.) The Future of the Teaching and Learning of Algebra. The 12th ICMI Study. New ICMI (International Commission on Mathematical Instruction) Study Series, Vol. 8. pp.99-152. Kluwer Academic Publishers. Noss, R.(2001). For a learnable mathematics in the digital culture, Educational Studies in Mathematics 48(1),21-46. Schwartz, J.L., Yerushalmy, M. (1995) On the need for a Bridging Language for Mathematical Modeling. For the Learning of Mathematics, 15 (2), 29-35. Tall, D. (2002). Continuities and Discontinuities in Long-Term Learning Schemas. In D. Tall and M. Thomas (Eds), Intelligence, Learning and Understanding – A Tribute to Richard Skemp ( pp. 151–177), Brisbane, Australia: PostPressed. Visual Math (1995) Algebra and functions, (In Hebrew) Centre for Educational Technology, Tel-Aviv. Yerushalmy, M. (2005). Challenging known transitions: Learning and teaching algebra with technology. For the Learning of Mathematics. 25 (3), 37-42. Yerushalmy, M. & Chazan, D. Technology and Curriculum Design: The Ordering of Discontinuities in School Algebra. To appear in L. English et al. (Eds.) Handbook of International Research in Mathematics Education (2nd edition). Mahwah, NJ: Lawrence Erlbaum.