Teaching and Learning Mathematics in Technologically Intensive Classrooms MICHAEL L. CONNELL UNIVERSITY OF HOUSTON - DOWNTOWN SERGEI ABRAMOVICH STATE UNIVERSITY.

Teaching and Learning Mathematics in Technologically Intensive ClassroomsMICHAEL L . CONNELLUN IVERS ITY OF HOUSTON - DOWNTOWN SERGE I ABRAMOVICHSTATE UN IVERS ITY OF NEW YORK AT POTSDAM

Introduction Mathematics instruction is currently undergoing significant shifts concerning the nature of both content and the manner in which foundational understandings are to be developed.

This new focus includes an increased emphasis upon the dual nature of mathematics itself which must be understood if technology is to be used effectively. Basically, when viewed as a content area, mathematics has a bit of a split personality. To use an example from language, there are parts of mathematics that function very much like a noun (the concepts of mathematics), while others function more like a verb (procedures, which many think of as “actually ‘doing’ math”).

Content and Process For students to develop meaningful mathematical understandings, they should have many rich experiences in mathematics from these two markedly different perspectives. So, as we select appropriate technology we need to allow them to experience mathematical structures containing both concepts to think about – the “noun-like” content features, and processes to think with – the “verb-like” procedural features. Once a teacher can see this “dualism” about mathematics, it has major impacts on potential roles of technology in the mathematic classroom.

A Multiplication Example This can be shown very clearly when considering multiplication strategies. Multiplication is used to compute area, and area can be used to illustrate multiplication – so both the concept and procedure can be illustrated at once*. Here we see a rectangle being formed from placing representative tiles along two dimensions: X+1 in the vertical direction, and Y+2 in the horizontal direction. The resulting algebraic product is shown by the area itself. To fill this rectangle the student needs to use an XY piece, two X pieces, one Y piece, and two single squares. When this is written out in standard form it shows that (X+1)(Y+2) = XY + 2X + Y +2. In order to get to this point, however, students need to be able to utilize both the conceptual and procedural aspects of the representation created through interaction with this application.

Object Based Tools in Mathematics Teaching and Learning

An important distinction should be made at this point. Effective technology use does not involve simply visiting a webpage presenting information or a step-by-step demonstration of a process. Such information is important on occasion, but this does not constitute a particularly powerful understanding and does not take full benefit of the potential interactions between the student and technology.

Low Level Interaction An effective tool to think with in learning mathematics should encompass both the noun and the verb (i.e., both the conceptual and the procedural aspects of mathematics). Developing an understanding of basic multiplication will be used to model this process.

First, consider a very simple case. A technologically enabled Object simply presents static information – often in the form of facts to be memorized – to a student. Such a Low Level Interaction may be found at http://www.math2.org/math/general/multiplytable.htm and should be intimately familiar to most readers – the multiplication table.

A Calculator View – Medium Level Interactions

In a more powerful Medium Level Interaction, the student can act directly upon the technological object itself. In this case the understandings which emerge are created by the student who acts upon the technologically-enabled object (whose properties were both programmed and presented in a form allowing for easy manipulation by the student). At its most simple level, the students chooses to press certain keys to try for a particular result, and the device reacts. When a student uses a calculator, for example, this is typically the level of interaction they experience.

A Web-Based View – Medium Level Interactions

A multiplication based example of this Medium Level Interaction may be found at:

http://naturalmath.com/mult/mult2.html. This website provides an alternative view of our old friend, the multiplication table. This screenshot shows the display which is presented when the student uses this object to perform the action of 3 x 4 = □.

High Level Interaction This pair of screenshots shows a student’s interaction with yet another multiplication object (http://nlvm.usu.edu/en/nav/frames_asid_192_g_1_t_1.html?from=category_g_1_t_1.html ) to model the multiplication problem 23 x 11. In this case the object allows the student the ability to change a number of important aspects of the model. For example, it is possible to change from the Lattice representation shown in these diagrams, which should be familiar to Montessori teachers, to Grouping and Common models used in typical textbooks. Once a representation is chosen the associated records of activity and problem setting automatically change allowing the student to explore not just one, but many different ways of representing the problem.

Problems in Teaching and Learning in the Mathematics Classroom

Instruction typically emphasizes procedures, memorizing algorithms, and finding the "one right answer” at the fastest speed possible. Unfortunately, in such environments, reasoning, problem solving, and sensibility are rarely addressed if at all. Mathematics, as it is often presented in these settings, is not a subject open for discussion, debate, or creative thinking, nor are students encouraged to find alternative ways to solve a problem or different procedures for carrying out an operation (Abramovich & Connell, 2014).

To draw upon our earlier language example, these students become verb-strong, but they do not understand the nouns they are acting upon! Procedural and computational expertise can result from this, but little else. Such students are able to follow the algorithms necessary to solve a problem, but could not understand why or how those algorithms answer the question at hand. Given this, it is hardly surprising that many students became imbued with rigid mental representations of mathematical problems and lack any ability to apply metacognitive strategies (Campione, Brown & Connell, 1988). The consequences of this often escape the typical classroom teacher. This is not said to fault teachers but to draw attention to this problem.

Suggestions for Classroom Technology Use – Part One

1. Begin by developing basic numeracy. Students cannot effectively act upon numbers, regardless of technology, without a rich understanding of number concepts. Be sure that students recognize that numbers can serve as both a noun and verb and are able to provide examples of both situations.

2. Once numeracy, a deep understanding of the foundations of number and its uses, are in place, be certain that the basic operations are also thoroughly understood in both their noun and verb settings. It is far too easy for the correct answers provided by technology to hide student misconceptions and lack of foundational understandings.

3. Whenever possible use technology to confirm your thinking, not replace it. Perform a few sample calculations to test your ideas, use technology to check your calculations and confirm you are on the correct path, and then use the tools of technology to explore emerging ideas more efficiently.

4. Pick your technology to match the developmental needs and experiences of your students.

Suggestions for Classroom Technology Use – Part Two

5. Take advantage of “beneath the rules” moments. Often technology can provide important clues as to connections which may be built in the emerging mathematics.

6. Remember to use technology to pose problems, not just as a means to increase speed of solutions. The ability to pose questions, even when their immediate methods of solution are not readily apparent, is a major goal in mathematics education.

7. If one truly understands the questions that one is asking, it becomes possible to select an appropriate technology to help explore possible answers. If the questions being asked are not understood, then NO technology will be able of assistance. Numerical answers can be generated, but these numbers might not even relate to the questions being asked.

8. Remember that classroom technology is a moving target. By focusing upon the mathematics that is to be taught, teachers should able to adapt when a newer program, instructional package, or textbook is to be adopted. Teachers should not allow themselves to become so centered on how to enter the correct keystrokes that they forget why they are doing so!

References Abramovich, S., & Connell, M. L. (2014). Using technology in elementary teacher education: A sociocultural perspective. ISRN (International Scholarly Research Network) Education, Article ID 245146, 9 pages, doi: 10.1155/2014/345146.

Campione, J. C., Brown, A. L., & Connell, M. L. (1988). Metacognition: On the importance of understanding what you are doing. In R. I. Charles & E. Silver (Eds.), Teaching and assessing mathematical problem solving. Volume 3 (pp. 93–114). Reston, VA: National Council of Teachers of Mathematics.

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

 *National Library of Virtual Manipulatives. http://nlvm.usu.edu