Each teacher has different ideas about using materials and technology in the classroom to help students learn better. When planning lesson planning, as well as when evaluating mathematics programs, it is important to consider all of the available resources. If a tool or resource is not available, can it be developed? Can a teacher make effective use of a computer, calculator, textbook, generator, spreadsheet, or library to enhance instruction?
Mathematical learning tools can be traditional, technological, or social. The most frequently employed tools are traditional, which include physical objects or manipulatives (e.g., cubes), visualization tools (e.g., function diagrams), and paper-and-pencil tasks (e.g., producing a table of values). Technological tools, such as calculators (i.e., algebraic and graphic) and computers (e.g., computation and multiple-representation software), have gained attention because they can extend learning in different ways. Social tools, such as small-group discussions where students interact with one another to share and challenge ideas, can be considered a third type of learning tool. These three tools can be used independently or conjointly, depending on the type of learning that is intended.
Learning Tools in Mathematics
A learning tool can be as simple as an image or as complex as a computer-based environment designed to improve mathematical understanding. The key characteristic of a learning tool is that it supports learners in some manner. For example, a tool can aid memory, help students to review their problem-solving processes, or allow students to compare their performance with that of others, thereby supporting self-assessment. Learning tools can represent mathematical ideas in multiple ways, providing flexible alternatives for individuals who differ in terms of learner characteristics. For example, learners who have difficulty understanding the statistical ideas of the arithmetic mean (center) and variance (spread) may be assisted through interactive displays that change as data points are manipulated by the learner. A mathematical learning tool can scaffold the learner by performing computations, providing more time for students to test mathematical hypotheses that require reasoning. In the statistics example, learners can focus on why changes to certain parameters affect data–and in what ways, rather than spending all their time calculating measures.
Traditional Tools. Traditional tools are best suited for facilitating students’ learning of basic knowledge and skills. Objects that can be manipulated, such as cubes, reduce the abstract nature of concepts, such as numbers, thereby making them real and tangible, particularly for younger children. Such tools support the development of children’s understanding of arithmetic by serving as a foundation for learning more complex concepts. Visualization tools, such as graphs, can support data interpretation, while paper-and-pencil tools that provide practice of computational skills can support memory for procedures and an ability to manipulate symbols. Combining physical tools with visualization tools can substantively increase students’ conceptual knowledge. Dice and spinners, for example, can be used to support elementary school students in creating graphs of probability distributions, helping them develop an understanding of central tendency.
Technological Tools. Technological tools are most effective in facilitating students’ understanding of complex concepts and principles. Computations and graphs can be produced quickly, giving students more time to consider why a particular result was obtained. This support allows students to think more deeply about the mathematics they are learning. Electronic tools are necessary in mathematics because they support the following processes: (a) conjectures–which provide access to more examples and representational formats than is possible by hand; (b) visual reasoning–which provides access to powerful visual models that students often do not create for themselves; (c) conceptualization and modeling–which provide quick and efficient execution of procedures; and (d) flexible thinking–which support the presentation of multiple perspectives.
Spreadsheets, calculators, and dynamic environments are sophisticated learning tools. These tools support interpretation and the rapid testing of conjectures. Technology enables students to focus on the structure of the data and to think about what the data mean, thereby facilitating an overall understanding of a concept (e.g., function). The graphics calculator supports procedures involving functions and students’ ability to translate and understand the relationship between numeric, algebraic, and graphical representations. Transforming graphical information in different ways focuses attention on scale changes and can help students see relationships if the appropriate viewing dimensions are used. Computers may remove the need for overlearning routine procedures since they can perform the task of computing the procedures. It is still debatable whether overlearning of facts helps or hinders deeper understanding and use of mathematics. Technology tools can also be designed to help students link critical steps in procedures with abstract symbols to representations that give them meaning.
Social tools. Social tools are a fairly recent consideration. In the 1990s, small-group work where students share strategies for solving problems began to be used as a powerful learning tool. This tool facilitates students’ ability to solve word problems and to understand arithmetic. Group collaboration while learning with technology can help students develop the perspectives and practices of mathematics, such as what constitutes acceptable mathematical evidence. Peers and computers can provide feedback that makes students aware of contradictions in their thinking. In this way, social tools can assist learning and transform understanding.
Mathematical Tools
A graphing calculator
These devices let students quickly generate graphs, do calculations on lists of numbers, and a huge amount of other mathematical operations. I tend to focus on the things we can do with the mathematics we know, and treat mathematical calculations as tools for solving problems. As far as I know, so far no single computer program is as easy to use and does as much as the modern graphing calculator.
Decks of cards
You can use these in probability simulations, learning mathematical card games and tricks, or just choosing which group is going to present next.
Dice
We use dice for more probability simulations, choosing partners for groups (occasionally, I usually let students choose their own group members), and playing board games. Yes, we play games in math class.
String
Whether you are using the string as a tool for more interesting mathematical phenomena (like pendulum motion), using it to tie parts of a project together, or studying the pattern of knots tied in the string, this is an invaluable tool in mathematics class.
Scissors and rulers
I have a class set of these very useful devices. Being able to construct models from shapes cut out of paper helps bring mathematics alive for students.
Compass and protractor
I also have a class set of a compass and protractor. I ask students to buy one of each of these things as part of their supplies for class, but provide them for students as well. I don’t want my lesson floundering because some 13 year old forgot his supplies for class…
Golf (and other similarly bouncy) balls
You can bounce them, roll them down inclines, and throw them through the air. All of these result in interesting mathematical models for students to inspect and analyze.
Beads
These are great for counting, using as markers, keeping track of positions of frogs jumping past each other on lily pads, and loads of uses I haven’t even thought of yet.
Long measuring tape
I have 8 of these. At one point I had 10, but they are such an incredibly useful device, they sometimes go missing. I use these to test the Pythagorean theorem out on the soccer field or apply trigonometry to finding heights of things, to test that our measures of distances using parallax are accurate. I can’t imagine not having these measuring tapes, they’ve been to 4 different countries with me. When my wife (while we were packing for Thailand) asked me if I really needed these, I just gave her the most incredulous look. Of course!
Paper clips
It’s amazing what you can do with paper clips if you see them as something other than a paper clip. Sure, I use them to hold pieces of paper together, but I have also used them to pick locks on filing cabinets when the key went missing with the previous teacher, construct 3d models, and twist them into interesting mathematical shapes.
iPhone (video camera)
I love the fact I always have a regular camera and a video camera on me at all times. It also doubles as a way to retrieve information from the Internet which is handy when having a discussion where you really want to be right. Being able to capture moments from my classroom as they happen is awesome.
My laptop
I’ve had access to a computer in my teaching since the very beginning, and I wouldn’t have it any other way. I used it at first for research for lesson ideas, and for collecting resources for class, but am now using it to connect to other mathematics teachers from our global community. The fact it also includes Google Apps, Geogebra, video and image editing, and a host of other applications is just a bonus.
Conclusion
Every maths teacher needs to know quite a lot of gadgets and tools for teaching mathematics. And the more you understand the technology and tools for teaching math, the easier it is going to be for the students who have a hard time with that subject. It’s just a way to have an enjoyable learning experience with no stress.