This work was  presented at the ICTMT5 conference in Klagenfurt, Austria, August 9, 2001

Differentiating Mathematics Instruction through Technology:

Mapping (for) Personalized Learning

W O R K   I N   P R O G R E S S

Mara Alagic & Rebecca Langrall
Wichita State University
Wichita, KS 67260 – 0028, USA

 

Abstract

In the context of WHAT? - HOW? - WHO?, if the WHAT is a mathematics standards-based curriculum and the WHO? are learners, could the HOW explain our way of thinking, our teaching/learning/ reflecting philosophy, and/or our sense-making processes? Where is the place of Information and Computer technologies in these processes?

This pilot study attempts to describe the pedagogical content knowledge development of a group of teachers in a graduate course aimed at exploring the power of IC technologies to differentiate mathematics instruction for the individual learner. It poses questions for further research in the areas of students’ mental maps and teachers’ pedagogical content knowledge development of technology-enhanced instruction.

 

Introduction

One of the greatest challenges facing American teachers of the 21st century is a cost-effective way to design learning environments which address the needs of an increasingly complex student body -- complex not only from the standpoint of exceptionalities, cultural diversity, gender, differences in readiness, experience, interests, and learning preferences, but also because of recent findings from neuroscience which have enriched our understanding of the nature and development of long term memory (National Research Council, 2000; Wolf, 2001).

Is the current profusion of technological tools for learning a possible response to some of these complexities or just another complication? Can these tools help to map developments in students’ diverse skills and understandings while bringing them about? If our students’ personal experiences of learning content define its nature for them, are these tools changing the nature of subject matter itself? These are some of the questions we explored during a three weeks long graduate summer course titled "Technology in the Mathematics Classroom K-12."

Background

Teachers’ beliefs about teaching and learning shape their approach to planning and carrying out curricular activities in the classroom. These beliefs are lenses through which teachers make sense of the curriculum in the process of decision making about instructional approaches to be used.  Tomlinson (1999, p. 15) describes differentiation of instruction as "a teacher's response to learner's needs, guided by general principles of differentiation such as respectful tasks, flexible grouping, and ongoing assessment." Though it is accomplished through combinations of instructional and management strategies, differentiated instruction is a way of thinking that addresses variance in all the ways students differ, be it gender, culture, experience, readiness, interests, or learning profiles. In the differentiated classroom, individual student growth is emphasized over class rank. Using ongoing assessment of student readiness, interests, and learning profile teachers differentiate content, process, and/or products. Teachers and students learn collaboratively in the process of "linking learners with essential understandings and skills at appropriate levels of challenge and interest" (Tomlinson, 1999, p. 14).

Classroom environments for differentiated instruction in our understanding (Alagic & Emery, 2001) assume the following essential factors: (a) A classroom that promotes and nurtures understanding; (b) A reflective teacher who is aware that successful teaching requires student engagement and understanding; (c) What we teach is determined by standards-based curriculum and a teacher’s understanding of it; and (d) Whom we teach it to: Our students come to our classrooms with variety of life and educational experiences, capabilities, learning styles and modalities, multiple intelligences, and social contexts. They have different interests, understandings, life circumstances, and confidence in their ability to learn.

Research into technology-based instructional tools is in its infancy, yet preliminary findings suggest some promise for addressing many of the expectations placed on schools today. The following can summarize some of the technology capabilities for supporting learners’ internal negotiations and meaning making (Jonassen, 1999):

  1. Tools to support representing learners’ ideas, understanding, and beliefs,

  2. Information vehicles for exploring knowledge to support learning,

  3. Context for simulating meaningful real-world problems, situations and contexts,

  4. Social medium to support learning through conversation, and

  5. An intellectual partner to support learning-by-reflecting.

As one might expect, research studies provide some evidence of the positive impact of technology on student learning. Since these "research findings often reflect a narrow set of conditions, they require careful interpretation" (Kimble, 1999, p.1). There are examples of research available that show that impact of technology on students’ achievement depends on how the technology is used. For example, Harold Wenglinsky (1998) reports the grade appropriate use of computers was found to be more important in producing increased learning than the amount of time computers are used. Wenglinsky pointed out the need for high quality intensive and continuing professional development focused on teaching models that integrate higher-order skills, using computers for projects and problem solving that support the topics being introduced in the classroom

Oppenheimer (1997) voiced the need for different instructional processes and better preparation of teachers. He referenced an example where a highly skilled teacher facilitated use of appropriate processes to guide a high school class to create models for real world problem solving. Oppenheimer does caution that this instance does not reflect a typical level of teacher knowledge, but it does provide support for meaningful, ongoing professional development to ensure that high levels of teacher knowledge become the norm.

What types of teacher knowledge are needed to use technology-based instructional tools effectively? As with non-technology enhanced instruction, we posit that a key understanding is an appreciation for the unique mental maps student create when learning something new. Downs and Stea (1973) formally define this process of cognitive mapping as: "a process composed of a series of psychological transformations by which an individual acquires, codes, stores, recalls, and decodes information about the relative locations and attributes of phenomena in their everyday spatial environment. According to Sanford (1985) mapping in problem solving can be described as a sequence:

  1. A mapping between a problem - statement and relevant schemata in long term memory

  2. A manipulation of this mapping in working memory, and

  3. If an information-state developed in working memory matches a structure in long-term memory, a new structure is stored in long term memory.

In problem solving with abduction, the new knowledge-state that a solved problem represents can often be achieved by introducing information from source external to the problem-statement.

The individualistic nature of cognitive maps comes from the fact that how the observer interprets and organizes a common exterior form is unique, which governs how the observer directs his attention. From research findings in neuroscience we know that emotion drives attention which drives learning (National Research Coucil, 2000), thus cognitive maps are both highly individualistic and emotion-based. Key concepts employed in studying cognitive mapping, therefore, are representation and environment.

How is the information processed once it has been perceived and has entered the cognitive system? The answer to this question depends on the way information is represented in the system. Understanding this constitutes a second important piece of teacher knowledge for designing an effective technology-enhanced learning environment. Some types of knowledge representation preserve much of the structure of the original perceptual experience. Those are called perception-based representations. Our minds have an ability to best remember what is most important. Meaning-based representations are quite abstracted from the perceptual details and incorporate the meaning of the experience (Anderson, 2000). Representations can be a process and a concept; a tool for thinking and a finished product. They are observable both externally and internally (NCTM, 2000). There are many ways of knowing through representations: examples, models, demonstrations, simulations, analogies, and metaphors.

Teachers who can represent a concept in a variety of ways provide a vehicle for all students to grasp the concept and make connections to previous and future understandings. Teachers' use of multiple representations can supply a rich repertoire of access points for accommodating the different ways students have been found to learn (Fischer,1980, and Bidell & Fischer, 1992, as cited in Fink, 1993), provided such representations are already familiar to students (Janvier, 1987, p. 102-103; Dufour-Janvier, Bednarz, & Belanger, 1987).

A third key piece of teacher knowledge for building a technology-based learning environment is how to teach for transfer. Teaching practices congruent with a metacognitive approach to learning include those that focus on sense-making, self-assessment, and reflection on what worked and what needs improving. These practices have been shown to increase the degree to which students transfer their learning to new settings and events (Schoenfeld, 1983, 1991). Multiple representations for certain concepts have been linked with greater flexibility in student thinking (Ohlsson, 1987, as cited in Leinhardt, et. al. 1991). Such flexibility, in turn, has also been associated with better transfer of learning into the ill-structured domains typical of the real world (Spiro, Vispoel, Schmitz, Samarapungavan, Boerger, 1987). As an arena in which teachers' creativity can come to the fore, instructional representations provide a temporary context for incubating student understanding. By blending familiarity and challenge to stimulate development, they are akin to Papert's "microworlds" (1980), Schoenfeld's "reference worlds" (1986, as cited in Leinhardt, et. al., 1991) and Kegan's (1982) "holding environments." When teachers end an activity with reflective questions, students become more aware of their own thinking/metacognition and at the same time build new or revisit their own representations. Students make their own sense of facts, concepts and principles and develop more effective cognitive strategies for accomplishing different tasks. We are describing these connections as what occurs when a pebble hits quiet waters.

An early stage of knowledge development for designing a technology-enhanced learning environment might entail an understanding that some representations are better than others for portraying particular aspects of a situation, knowledge of a range of representations, as well as a developmental approach to the ways teachers and students can use them (Vergnaud, 1987). With the increased availability of IC technology to provide students with easy access to a range of representational formats, explicit instruction in crossing between symbol and referent, as well as how certain representations convey mathematical content more efficiently than others is now being seen as a crucial aspect of mathematics education (Kaput, 1987).

Another early stage of knowledge development for designing a technology-enhanced learning environment might involve the awareness that students, faced with multiple representations for the same concept, often learn how to make one correspond with another using the syntactic rules of math, but without developing a sense of the underlying concept being represented. Premature or inappropriate use of representations can cause frustration and misconceptions in children and place undue focus on the representation at the expense of the target concept; thus, effective representations for the younger student must be based on students' own drawings and codes (Dufour-Janvier, et. al., 1987).

Yet another stage of development might include the recognition that with today’s emerging technologies the very nature of the problems that can be solved and methods used in the process are changing: Performing calculations; collecting, analyzing, and representing numeric information; creating and using models and simulations; representational scaffolding higher levels of abstraction, solving problems with mathematical premises. The hands-on, minds-on learning experiences fostered through today’s interactive technology applications empower students with a level of mathematical power they cannot achieve without technology. (Potential for stimulating higher order thinking when freed from the mechanics of calculating.)

A more advanced stage could involve the understanding of the dialectic between perception and conceptualization. For example, accessing geometrical knowledge is more often presented as resulting from the ability to rely efficiently both on spatial and geometrical competencies, as opposed to resulting from rejection of some perceptive apprehension of geometrical objects. Fostering the dialectic interplay between these differing competencies through more emphasis on the relevant traits within problems and situations leads to the development of geometrical expertise (Hoyles & Keith 1998, Laborde 1998, as reported in Lagrange 2001).

The "conceptual contexts" in which students develop ideas are shaped by such factors as teachers' expectations, the types of students who take particular subjects, policies affecting curriculum and assessment, and the teaching and learning environment (Stodolsky & Grossman, 1995). Students’ conceptual contexts are also heavily influenced by the degree of their teacher’s subject matter knowledge.

Because of the historical emphasis on didactic forms of instruction, even teachers with adequate subject matter knowledge need intentional training in recognizing the way students think, in order to know how to unpack disciplinary knowledge into content and processes learners find meaningful (Grossman, 1989, 1990; Hollen, Roth and Anderson, 1991; Lampert, 1989; Peterson, Fennema and Carpenter, 1991). However, adequate subject matter knowledge plus training may not yield adequate insights into students’ thinking if a teacher's epistemological assumptions (Peterson et. al, 1991), values (Anderson & Roth, 1989), or lack of conceptually oriented materials stand in the way (Hollon, Roth & Anderson, 1991).

Mathematics teachers need opportunities to experience and do mathematics in environments supported by diverse technologies (Dreyfus and Eisenberg, 1996). Understanding, using and appreciating mathematics are essential components of the development of mathematical power. Empowering teachers through the use of technology in mathematics exploration, open-ended problem solving, interpreting mathematics, developing conceptual understandings and communicating about mathematics is in the heart of professional development and teacher education (Bransford, et al, 1996; Schoenfeld, 1982, 1992; Silver, 1987).

The Study

Teachers take a three-week summer course called "Technology in Mathematics Classroom K-12" either as a part of their requirements for graduate coursework or from an interest in advancing their knowledge of integrating technology in their classroom. Typically, all four grade bands are represented (preK-2, 3-5, 6-8, 9-12) and an occasional advanced undergraduate signs for this course. All groups were represented this summer, with 19 participants. From the start, this entire group was motivated to try possibilities and share experiences with what IC technologies can offer to mathematics teaching.

Previous experiences in teaching with IC technologies of this group of teachers vary extremely. Depending on resources available in their schools all of them have tried in some way or another to utilize IC technologies, but only 9 have used it in the teaching of mathematics concepts. The computer lab used during the course has state-of-the-art equipment, including wireless technology and most of the mathematics software available on the market. During class time, the facilitator also had technical support.

The content of the course included (a) Changes in the teaching of the school mathematics brought both by current school reform for standards-based teaching that supports integration of technology (NCTM, 2000) and by the development of the IC technology; (b) Variety of computer programs for learning and doing mathematics; and (c) Web resources and possibilities for the developing of mathematics teaching and learning. The final product included an integrated unit plan that teacher-participant will use during their fall instruction. Some of the other elements of the course were daily e-mail reflection exchanges with the facilitator of the course, daily assignment that varies between accomplishing mathematical tasks with appropriate technology to reporting on teaching strategies that should be effective in mathematics classroom that integrates technology. A significant amount of time was spent on critical evaluations of available software and Web resources in terms of their effectiveness in learning, based on existing evaluative resources and teachers’ experiences.

Findings

Intent: As stated above, the majority of teachers in this study gave many reasons why mathematics teachers should use new IC technology in their mathematics lessons. They are also trying to enhance their pedagogical content knowledge of technology enriched mathematics.

Researchers in this pilot study were aware of a substantial number of factors that integration of technology brings into intricate complexity of mathematics learning processes via multiple representations. Their understanding of how people learn identifies differentiated instruction as a crucial shared component of their teaching philosophies. These two dimensions, integrating technology and differentiating instruction, brought an enormous number of questions in the Technology in the Mathematics Classroom learning environment. That is why this study is more about prioritizing questions than searching for answers.

Motivated by the necessity to differentiate instruction in this diverse classroom, by the help of this study, researchers attempted

    1. to clarify those motives on which these teachers base their decision concerning to what extent, if at all, they are willing to differentiate instruction through the use of technology in their own classroom

    2. to find out How these teachers perceive their own newly developed conceptual understanding of integration of IC technologies into their classroom environment?

Both quantitative and qualitative data was gathered for the study.

What Did Teachers Say: Support and Impediments

Much evidence exist to support the view that a paradigm shift in teaching and learning mathematics with use of IC technologies is taking place. Increasingly efficient IC tools and services support the view that the learning environment in school mathematics is changing in more technological one. These teachers are aware of current changes and involved in the processes of these changes in their schools.

Inadequate technological resources, absence of appropriate curriculum materials and limited time are quoted as perennial barriers to major changes. Although very enthusiastic when preparing their units for the coming fall, they talk about day-to-day obstacles, realities of class management, and "covering material" which are signs of struggle between appearance of acceptance of "changes for better" and everyday realities.

All the participants estimated afterwards that both their skills for utilizing IC technologies and their conceptual understanding of mathematical representations had improved. However four teachers identified their confidence level in teaching mathematics with technology lower than at the beginning of the semester. To quote on of those, "I see how little I know about enhancing mathematics with technology".

High school teachers that have had experiences with graphing calculators in their classrooms shared a number of successful stories and ideas for they further use. The issue of "memorizing multiplication table" was not missed, as usual.

The attitudes toward differentiating instruction through the use of technology were positive, although many obstacles have been noted. Some teachers had explicit plans for implementing what they learned by adapting the unit that they had been working on during this course. A very lively discussion occurred a few times in terms of teacher as "all knowing" and new generations being "experts" around IC technologies. When probed further about mathematics knowledge versus technological skills, almost everybody agreed that pupils are just "more comfortable around technology tools than we are."

Future Imperative: More Research

An attempt in the foundation of our research is to make a case for establishing research questions and appropriate methods for investigations about changing learning opportunities for differentiated instruction and personalized learning in the presence of technology in mathematics classrooms. For example, if the recent profusion of technology-based tools for learning can offer some solutions,

At the time of writing this paper, new group of teachers is taking this course. Both quantitative and qualitative data is being gathered for the study. It is our plan to continue collecting data more closely trough online communication, as they start new school year and try their newly developed experiences and ideas in their classrooms.

References:

Alagic, M. (2000). Structures: Categorical and Cognitive, Conference Proceedings, Bridges - Mathematical Connections in Art, Music, and Science, Winfield, Kansas, 379-387.

Alagic, M. & Emery, S. (2001). How one mathematics standard can make a difference in the way we differentiate instruction for personalized learning. Manuscript submitted for publication.

Anderson, J. R. (2000). Cognitive psychology and its applications (5th ed.). Worth Publishers: NY.

Downs, R. M. & Stea, D. (1973). Cognitive Maps and Spatial Behavior. Process and Products. In Downs, R. M. & Stea, D. (Eds.), Image and Environment, (Aldine Publishing Co., Chicago, pp 8-26

Dufour-Janvier, B. Bednarz, N., and Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 109-122). Hillsdale, NJ: Erlbaum.

CEO Forum on Education and Technology (1999). Professional Development: A Link to Better Learning. Washington, DC: CEO Forum.

Fink, R. (1993). How successful dyslexics learn to read. Teaching Thinking and Problem Solving, 15(5), 1, 3-6.

Grossman, P. L. (1990). The making of a teacher: Teacher knowledge and teacher education. New York: Teachers College Press.

Grossman, P.L. (1989). A study in contrast: Sources of pedagogical content knowledge for secondary English. Journal of Teacher Education, 40(5), 25-31.

Goldin, G. A. & Janvier, C., Eds. (1998) Representations and the psychology of Mathematics Education: Parts I and II. Special issues of the Journal of Mathematical Behavior 17 (1) and (2).

Hollon, R. E., Roth, K. J. & Anderson, C. W. (1991) Science teachers' conceptions of teaching and learning. In J. Brophy (Ed.), Advances in research on teaching (Volume 2, pp. 145-185). Greenwich, CT: JAI Press.

Jonassen, D., Peck, K. & Wilson, B. (1999). Learning with Technology: A constructivist Perspective. Upper Saddle River, NJ: Prentice Hall.

Kaput, J.J. (1987). Representation systems and mathematics. In C. Jonvier, (Ed.), Problems of Representation in the Teaching and Learning of Mathematics, Hillsdale, NJ: Erlbaum.

Kegan, R. (1993). Minding the curriculum: Of student epistemology and faculty conspiracy. In Andrew Garrod (Ed.), Approaches to moral development: New research and emergent themes (pp. 72-88). New York: Teachers College Press.

Kimble, C. (1999) The Impact of Technology on Learning: making Sense of the Research. Policy Brief. Mid-Continent Regional Educational Laboratory. Aurora, CO. p.1-7

Kuipers, B. (1983) The cognitive map: Could it be any other way. In Pick, H. L. & Acredolo, L. P. (Eds.), In Spatial Orientation: Theory, research and application, Plenium Press, New York. pp 345-360.

Leinhardt, G., Putnam, R. T., Stein, M.K., and Baxter, J. (1991). Where subject knowledge matters. In Jere Brophy (Ed.), Advances in research in teaching (Volume 2, pp. 87-113). Greenwich, CT: JAI Press, Inc.

March, L. & Steadman, P. (1971). The Geometry Of Environment. London: Methuen.

National Research Council (2000). How people learn: Brain, kind, experience, and school. Washington, DC: National Academy Press.

National Council of Teachers of Mathematics. (2000). Principles and standards of school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Neisse, U. (1976). Cognition and reality. WH Freemn, San Francisco.

Todd Openheimer, "The Computer Delusion," Atlantic Monthly, July 1997.

Papert, S. 1980 Mindstorms: Computers, Children, and Powerful Ideas. New York: Basic Books.

Peterson, P. L., Fennema, E., & Carpenter, T. P. (1991). Teachers' knowledge of students' mathematics problem-solving knowledge. In Jere Brophy (Ed.), Advances in research in teaching (Volume 2, pp. 49-86). Greenwich, CT: JAI Press, Inc.

Sanford, A.J. (1985). Cognition and Cognitive Psychology. London.

Schoenfeld, A.H. (1983). Problem solving in the mathematics curriculum: A report, recommendation and an annotated bibliography. Mathematical Association of America Notes, No. 1.

Schoenfeld, A.H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. Pp. 311-343 in Informal Reasoning and Education, J.F. Voss, D.N. Perkins, and J.W. Segal, eds. Hillsdale, NJ: Erlbaum.

Spiro, R. J., Vispoel, W. P., Schmitz, J.G., Samarapungavan, A., & Boerger, A.E., (1987). Knowledge acquisition for application: cognitive flexibility and transfer in complex content domains. In B.K. Pritton & S. M. Glynn (Eds.), Executive Control Processes in Reading (pp. 177- 199). Hillsdale, NJ: Lawrence Erlbaum Associates.

Tomlinson, C. A. (1999). The differentiated classroom: Responding to the needs of all learners. Alexandria, VA: Association for Supervision and Curriculum Development.

Vergnaud, (1987). Conclusion. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 227-232). Hillsdale, NJ: Erlbaum.

Wenglinsky, H. (1998). Does it compute? The relationship between educational technology and student achievement in mathematics. Princeton, NJ: Policy Information Center, Educational Testing Service.

Wolfe, P. (2001). Brain Matters: Translating Research into Classroom Practice.

APPENDIX:

Question 1: How would you rate your confidence and abilities at using appropriate technology as you teach mathematics (or when you begin teaching) in your own classroom? On a scale of 1 to 10 with 10 as high. (Graph attached: 15 students felt more confident, 4 felt less confident; mean1 = 3.7; mean2 = 5.4; )

Question 2: If in the fall you had everything you wanted in IC technology to enhance teaching of mathematics, what other obstacles would you need to overcome, if any? (categories found: lack of time , knowledge of appropriate technology, need for technical support, understanding how to adapt lesson plans & teaching). After focusing on the order in which these obstacles have been mentioned, we find "lack of time" in 48% In the first category (first "obstacle" to be mention)

Question 3: How are you planning to differentiate instruction in your classroom as a result of experiences in this class? (categories found: