Math Methods for Preservice Elementary Teachers

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Audience and Purpose

The main audience for this page is faculty who teach a Math Methods course for preservice elementary education teachers. However, most faculty who teach the Math for Elementary Teachers will also benefit from this document. The document has three main goals:

  1. To share some of the insights that researchers and experienced teachers of teachers have gained through designing and teaching the Math Methods course for preservice elementary teachers.
  2. To present some Information and Communication Technology (ICT) ideas that are important to teaching math at the elementary school level.
  3. To encourage and facilitate more sharing among Math Methods faculty.

If you are a teacher of Math Methods for preservice and/or inservice elementary teachers, you might want to share this article with your students. It will give them some insights into the challenges you face in teaching the course.

In addition, remember that some of your students will eventually become teachers of teachers. Many will do this informally in their conversations with fellow teachers. Some will eventually become workshop leaders, make conference presentations, and/or teach a Math Methods course.

Introduction

In my opinion, the preservice elementary education Math Methods course is near the top of the list of most important and most challenging education courses to teach. In some sense, our whole math education system rests in the hands of the few thousand faculty members who regularly teach this Math Methods course.

Preservice elementary school teachers enter a Math Methods course with widely varying math backgrounds and interest in math. Many profess to hate math and claim, "I can't do math." Some have taken the minimum math requirements to graduate from high school, and the minimum math content courses in college to meet requirements for entry into a teacher education program.

On the other hand, many of your students are relatively strong in math and have a good self image in their ability to learn and use math. They may have taken a strong math program in high school, and they may have taken a year sequence in calculus or still more math in college.

The students in the Math Methods course have typically completed some version of Mathematics for Elementary Teachers coursework from a math department. The math prerequisite and the number of credits in this course or course sequence varies in different programs of study throughout the country. A relatively strong course from a math department point of view might have college algebra or equivalent as a prerequisite and be a year sequence, three or four credits per term.

Many teacher education programs of study require their students to demonstrate some Information and Communication Technology (ICT) knowledge and skills, and/or take an ICT course. This course might be mainly ICT content or a mixture of ICT content and ICT Methods.

Computer Science and Mathematics are closely linked. Indeed, many Computer Science Departments were formed by faculty who split off from a Math Department. In quite a few cases, Computer Science is still part of a Math Department. This suggests that math teachers at every grade level may (could, should) have some responsibilities to help their students learn some math related ICT and Computer Science.

In brief summary, the math content, math interests, ICT content, and ICT interests of students in a Math Methods course vary tremendously across the nation. The teachers of Math Methods courses sometimes complain about the modest math and ICT content knowledge of their students. They want to spend the valuable and limited course time teaching Math Methods, not math and computer content.

This ongoing problem creates tension between teachers of Math Methods courses and teachers of math and computing content courses. In some cases, there is close cooperation among the departments and faculty who teach the content and methods courses. The same faculty members may teache both courses, or team teaching may occur. This can be especially effective in a collaboration between faculty who teach Math for Elementary Teachers and Math Methods. Both courses need to stress math content, math pedagogy, and math pedagogical content knowledge.

The Past and the Future

Here are two important ideas:

  • Our educational system looks to the past. It helps us understand, preserve, and build upon the collected knowledge, wisdom, and cultures of the past.
  • Our educational system looks to the future. It helps to prepare students for life in the future.

An educational system that overemphasizes the past tends to be out of tune with younger people. An educational system that overemphasizes the future tends to be out of tune with older people. Thus, our educational system faces a continuing challenge of seeking an appropriate balance between the past and the future.

The challenge becomes greater in times of rapid change. Today we live at a time of rapidly increasing technologically-based change. The world is growing "smaller." The world as a whole faces the challenge of sustainability.

Here are a few rapidly changing areas to think about:

  • Information and Communication Technology (ICT), including computers, robots, artificial intelligence, telecommunications, and the Internet.
  • Genetics and genomes, including cognitive neuroscience (brain science).
  • Nanotechnology—the technology of the very very small.

Math is quite important in each of these areas! Math is a discipline with a very long history. The discipline has both great breadth and great depth. The discipline is constantly growing through the work of many thousands of math researchers throughout the world.

The discipline of math is also growing because of its use and importance in many other disciplines—and these are rapidly growing and changing.

Thus, our math education system faces the ongoing challenge of helping to prepare students for life in a world where math is both an important discipline in its own right and is also an important part of our changing world. This challenge becoming greater as the pace of technological change in our world continues to increase.

I think you can see the challenges the above ideas bring to you, as a Math Methods teacher. Of course, these same challenges will face your students as they become teachers. Thus, you should give careful thought to how you will help prepare your students for their futures as teachers of math and other disciplines.

You might want to use this challenge as a topic for a writing assignment, small group discussions, and whole class discussion. Exploration of this topic might also include discussion on the Math Education Wars.

Improving Math Education

The underlying, unifying theme in this document is that our math education system can be substantially improved. There tend to be two general approaches to trying to improve our math education system:

  1. Do what we have been doing in the past, but do it better. This includes imposing more required math coursework, higher standards, and high-stakes tests. Back to basics, "Algebra for all," and high-stakes algebra testing as a requirement for high school graduation are some examples of this movement.
  2. Draw upon past and ongoing math educational research, and on progress in areas such as ICT and brain science, to implement significant research-based changes in the content, pedagogy, and assessment in math education.

Part of the Math Wars controversy is between those who favor the first approach and those who favor the second approach.

There is substantial evidence that teachers teach the way they were taught. Let's use PreK-8 school teachers who teach math as an example. These teachers first learned about how to teach math when they were at the preschool level, learning from their parents, caregivers, and perhaps from older siblings. The math pedagogy education of these teachers continued as they progressed through an elementary school and middle school math program of study while they were in school. Still further continuation was provided by what they observed as they studied math in high school.

Thus, the typical preservice elementary school teacher knows a great deal about how math is taught before he or she takes any education courses, a Math for Elementary Teachers sequence, or Math Methods course. Teachers of preservice and inservice teachers know that it is difficult to change this situation.

If you want your Math Methods students to learn to teach math in a "non traditional" way, you will need to stress this strongly throughout the course. Even then, your success may be quite limited.

The use of calculators and computers in math classes provides an excellent example of resistance to change. The National Council of Teaches of Mathematics has supported use of calculators in elementary school math education since 1980. Even now, more than a quarter century later, a great many preservice and inservice math teachers make their own decision as to whether they think calculators are appropriate or not. Thus, they do teach about and allow use of calculators, or they do not teach about and allow use of calculators, based mainly on their personal opinions.

Math Content Pedagogical Knowledge

The importance of pedagogical content knowledge (PCK) in teaching is a relatively new idea pioneered by Lee Shulman. The basic concept is that good teaching involves knowledge of the content area being taught, general knowledge of how to manage and teach students, and specific knowledge about how to teach the content of that discipline.

As a "far out" example, consider the challenge a research mathematician would face if asked to teach elementary school children about the number line. The research mathematician has a huge amount of math content knowledge in this area. This includes ideas about different sizes of infinity, rational and irrational numbers, transcendental numbers, repeating decimals, numbers to different bases, number theory, and so on. The math researcher can likely give a rigorous proof that (-3) x (-5) = (+15). But, how will this go over with a fourth grader? What should be taught, how should it be taught, and how does one assess the results?

How does one explain the number line to a learning disabled second grader, an average second grader, or a talented and gifted second grader? What will each already know, and how does one help these varying students build new knowledge and understanding onto what they already know?

The students are building and building upon their mental models of the number line. Many will eventually develop a useful visual model of the number line. How might a bead frame (or abacus), graph paper, or a variety of math manipulatives help in each case? What virtual manipulatives available on the Web might help? What does one do for students who are inherently poor in spatial visualization and have difficulty "seeing" things in their mind's eye?

The importance of math PCK was researched in Liping Ma's doctoral dissertation and widely disseminated via her book:

Ma, Liping (1999). Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum.

Liping's doctoral research compared elementary school math education in the United States with that in China where she grew up. She makes it clear how math content knowledge and math pedagogical content knowledge of teachers differ between the U.S. and China. Her short book is an excellent resource for preservice and inservice teachers. Also, there are many good articles about her work available on the Web.

Math Manipulatives

The previous section mentioned math manipulatives. Some people seem to view math manipulatives as a sort of toy for use by young children. Others see math manipulatives as a really valuable part of math education for children who are at the Concrete Operations and Pre-Operational stages of Piaget's 4-stage cognitive development theory.

Marilyn Burns is a well known math educator. Quoting from her article, How to Make the Most of math Manipulatives:

You find them in classrooms across the nation—buckets of pattern blocks; trays of tiles and cubes; and collections of geoboards, tangrams, counters, and spinners. They've been touted as a way to help students learn math more easily. But many teachers still ask: Are manipulatives a fad? How do I fit them into my instruction? How often should I use them? How do I make sure students see them as learning tools, not toys? How can I communicate their value to parents? Are they useful for upper-grade students, too?
I've used manipulative materials at all levels for 30 years, and I'm convinced I can't — and shouldn't — teach without them.

The article then continues with sections about a number of important math manipulative topics.

However, appropriate uses of physical math manipulative and computerized (virtual) math manipulatives is not a cut and dried issue. Doug Clements is an outstanding math education researcher. Here is an article that I consider to be a "must read" for preservice elementary school teachers:

Clements, D.H. (1999). 'Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood. 1(1), 45-60. Retrieved 5/18/08. http://www.didax.com/articles/concrete-manipulatives-concrete-ideas.cfm. Quoting from this article:
Students who use manipulatives in their mathematics classes usually outperform those who do not, although the benefits may be slight. This benefit holds across grade level, ability level, and topic, given that use of a manipulative "makes sense" for that topic. Manipulative use also increases scores on retention and problem solving tests. Attitudes toward mathematics are improved when students have instruction with concrete materials provided by teachers knowledgeable about their use.
However, manipulatives do not guarantee success. One study showed that classes not using manipulatives outperformed classes using manipulatives on a test of transfer [of learning]. In this study, all teachers emphasized learning with understanding. In contrast, students sometimes learn to use manipulatives only in a rote manner. They perform the correct steps, but have learned little more. For example, a student working on place value with beans and beansticks used the (one) bean as ten and the beanstick (with ten beans on it) as one.

For a number of years in the United States, there has been funding for a national project on virtual manipulatives. See:

Virtual Manipulatives (n.d.). National library of virtual manipulatives for interactive mathematics. Retrieved 5/18/08: http://matti.usu.edu/nlvm/nav/vlibrary.html.

Communicating in the Language of Mathematics

Communication lies at the very heart of education. This includes communication with oneself and communication with others. It includes communication over time and distance. For example, you know about Euclid and Pythagoras; they both are communicating with you over time and distance.

Teachers of math are helping their students gain increased expertise in communicating in the language of mathematics. Thus, throughout your course you will want to role model communication in math and help your students gain an increased level both in such communication and in understanding why this is important to their future.

All preservice teachers learn about the idea of reading (or reading and writing) across the curriculum. However, most learn very little about reading and writing of math as being an important component of learning math. As you teach your Math Methods course, you may well want to make use of such strategies as having your students do journaling and regularly participate in small group and whole class discussions. You may well have them develop and/or modify some math lesson plans. You may want to insist that they read the textbook and other written material. All of these types of activities will give your students practice in communicating in math. (Many faculty find that this turns out to be a struggle. Think about giving a one or two question short quiz over the reading at the start of each class period as a lead-in to this skill.)

Here is an activity that you might want to try out with your students. Have them individually answer the following question. Then have them discuss their answers in small groups and in a whole class debriefing.

Question: Think back over your own math content education. Try to remember a time in your precollege education when you were expected to learn some math by reading a math book. Assess yourself in your current level of skill in learning math by reading a math book.

The point to this activity is that, for the most part, precollege math tends to be taught by "oral tradition." That is, students are not expected to learn new math topics by reading about them in their math book, on a computer, or in written handouts.

There is a lot of good material on reading and writing in math education available on the Web. Marilyn Burns is a world leader in this area. Here in one example of her "writing in math" resources available on the Web:

Options for Writing in Math. (Adapted from Marilyn Burns, Writing in Math Class. Math Solutions Publications, 1995) Retrieved 5/22/08: http://www.springfield.k12.il.us/resources/math/Assessment/optionsforwritinginmath.pdf.

Other good examples are listed in the Alexandria Jones article below:

Jones, Alexandria (8/21/08). Writing to Learn Math: Let's Play Math. Retrieved 5/19/08: http://letsplaymath.wordpress.com/2007/08/21/writing-to-learn-math/. Contains a nice assortment of links to writing to learn math materials.

Good Math Lesson Plans

In a Math Methods course, students often study lesson plans written by others and/or create their own sample math lesson plans. They may be provided with a template for general lesson plan writing or one specifically designed for math lesson plans. A good example of such a math-specific lesson plan template is available in this IAE-pedia. It stresses many of the ideas included in the document you are currently reading.

Project-based and Problem-Based Learning

Problem-based learning is often used in teaching math. Project-based learning is less often used, but is a valuable part of the math education repertoire of many teachers. Information and discussion about these topics is available in separate documents. See:

Prerequisites: A Delicate Issue

This section examines both the prerequisites one might expect of students in a Math Methods course, and also the prerequisites needed to be an effective and successful faculty member in such a course.

Prerequisites for Math Methods Students

Here are some "delicate" questions. In the Math Methods course that you teach, what is the math content knowledge and skill prerequisite? How do you determine if each of your students meets these prerequisites? How do you advise and help remediate students who do not meet your prerequisites? How much of the very limited and very valuable time in your Math Methods course is used in math content remediation?

Many Math Methods teachers pass lightly over this delicate situation. And yet, every math teacher at every grade level must face these issues in their day-to-day teaching.

You might want to raise this prerequisite issue (for your students, and for their future students) near the beginning of your course. Bring the problem out into the open, and then role model an effective way of dealing with it. Help your students understand that forgetting the math content knowledge one has studied is common for almost all students. Your students can introspect (do metacognition) on how much of the math they studied in the past they have forgotten. You and your students can explore ideas on what is remembered over time, what is forgotten because it was never understood and/or is not used, and how past learning helps in relearning.

Another approach, or a parallel approach, is through use of a mathography assignment. In this, students write about and reflect on their math backgrounds, experiences, and attitudes.

A 5/18/08 Google search on mathography produced more than 5,000 hits. See, for example:

http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm
http://www.aug.edu/~lcrawford/Web_3262/FA_2005/Mathography.htm

Taking personal responsibility is one of the things to stress in these prerequisite activities. Every student needs to learn to take personal responsibility for meeting prerequisites in a course. This type of learning and responsibility-taking needs to be fostered, beginning at the earliest grades in school. Among other things this means that every teacher needs to have knowledge and skill in helping students learn to self-assess and self-remediate. By the time students finish high school, they should be quite skilled in this area of their education.

These same types of issues apply to your students' knowledge and skills in the area of calculators and computers. Of course, you and your students know how to use a calculator. However, do you and they know how to use and how to explain the use of the M+, M-, MR, and MC keys on an inexpensive calculator? Do they know how to explain and deal with limited precision decimal arithmetic? How does the calculator's number line compare with the real number line? How does one learn to detect and correct calculator keyboarding errors? These types of questions are important to your students and to the students they will teach during their career.

Next, consider computers and the more expensive calculators. Software now exists that can solve many of the types of problems students study in math courses up through the first two years of college. If a computer can solve a particular type of math problem, what do we want students to be learning about solving that type of problem using by-hand methods? This is a difficult question. If you are not comfortable and convinced in your own personal responses to this question, how can you expect your students (and their parents) to learn to deal with it?

And, speaking of problem solving, what do your students really know about problem solving? Do they realize that problem solving is not only a key idea in math, it is also a key idea in every academic discipline? Do they understand how math can be used as an aid to helping solve the problems in non-math disciplines? Do they know how to integrate math problem solving throughout the non-math curriculum?

Here is an activity that you might want to use with your students near the beginning of the term. It is a short quiz that can be done by students working individually and then debriefing in small groups. It is not an activity to be turned in and graded. By listening to the small group discussions, and later through a whole class debriefing, you can gain some insights into the math backgrounds and understandings of your students.

  1. Give an example of a math problem that has exactly one correct answer.
  2. Give an example of a math problem that has exactly two correct answers.
  3. Give an example of a math problem that has an infinite number of correct answers.
  4. Give an example of a math problem that has no correct answer.

Part of the goal of this activity is to break students of the habit of talking about "finding the answer" to a math problem or to a problem in any other discipline.

Variations of this activity can be used at other times during the course. For example, what kinds of problems do historians (or name any other non-math discipline) try to solve? How do they make use of math in such endeavors?

Prerequisites for Math Methods Faculty

This is another delicate issue. What constitutes a good preparation to be a teacher of a Math Methods course?

Here is my opinion. A good teacher of a Math Methods course needs to greatly exceed the prerequisites expected of students in the course. Moreover, the faculty member needs to know a great deal about elementary school students, their capabilities and limitations, how to help them learn, and so on. For example, the Math Methods teacher needs to be familiar with the major math textbook series used at the elementary school level. The Math Methods teacher needs to have knowledge of where and how math is used or could be used in the various other elementary school disciplines such as science. In addition, the Math Methods teacher needs to be skilled in working with preservice and inservice teachers.

If you are not overwhelmed by all of the above, then add to the list the math-related aspects of topics such as:

Here are examples of the challenges.

  1. Math is a broad and deep discipline. Its core is identifying and mathematically representing the math aspects of problems in all disciplines, and then solving these math problems. Calculators and computers are an aid to representing and solving a wide range of problems. See, for example, David Moursund's free book on problem solving: Introduction to Problem Solving in the Information Age. (Eugene, OR: Information Age Education(. Access at no cost in PDF and Microsoft word formats at: http://i-a-e.org/ebooks.html.
  1. Math education needs to help move students up the math component of the Piagetian cognitive developmental scale. It needs to help students increase their level of math maturity. Thus, teachers of math teachers (as well as their students) need to understand math-related Piagetian cognitive development and math maturity. See, for example, Good Math Lesson Plans.
  1. Consider the challenges of math education for elementary school Talented and Gifted students and for Learning Disabled students. Consider what you know about how dyslexia and dyscalculia affect a student's learning of math. Brain research is giving us considerable new insights into the math education needs and capabilities of these students.
  1. The quality of computer-assisted learning materials is gradually improving. Increasingly, there are teaching/learning situations in which highly interactive intelligent computer-assisted learning [[1]] can accomplish more than will a teacher working with the whole class.
  1. Finally, consider computer games. When I was a child, I grew up playing a number of different board games and card games. I learned a lot of math through playing Monopoly and other board games involving dice, spinners, money, and decision making. I learning a lot of math through playing cribbage and other card games that involved planning and strategies. These types of games still exist. However, together with many new types of games, these traditional games are now available on computers. You can think of a computerized version of a traditional game such as Monopoly as being a virtual reality. It uses virtual dice, virtual money, virtual player pieces, and so on. Indeed, the computer can even play the role of one or more of the human players. Here is the crux of the game example. How well do you know what today's children are learning about math and math related topics through the computer/video games they are currently playing and the electronic toys they play with? What do you know about possible uses of such games in a school setting to aid students in learning math and learning to use math to solve problems in other disciplines?

It is easy to extend the lists given in this section of faculty prerequisites. Perhaps now you understand why I believe that it is such a challenge to be a good teacher of the Math Methods course!

Learning Theories in Math Education

Research in education, and specifically in math education, has led to the development of many different learning theories. Thus, you are probably familiar with several such theories.

Here is an idea to try out with your Math Methods students. In a whole class or small group discussion mode, ask your students to name some pf the learning theories they are familiar with. Then have them name one or two that they believe are especially important in math education at the elementary school level, and explain why these are especially important.

My personal response includes:

  • Constructivism.
  • Transfer of learning: High-road and Low-road transfer of learning.

There are, of course, many more learning theories. The next two sub-sections discuss the two I have listed.

Constructivism in Math Education

Constructivism is a learning theory that provides a good example of the difficulty faced by teacher education programs. In brief summary, the theory says that students learn by building, or "constructinkg," upon their previous knowledge and understanding.

Catherine Fosnot is the director of Math in the City, a research and development project that provides teacher training in constructivist practice, aligned with the NCTM Standard. She is an international leader in constructivism in math education.

Constructivism is applicable in any discipline. It is especially important in math education. Quoting from http://mathforum.org/mathed/constructivism.html

What is Constructivism? "Students need to construct their own understanding of each mathematical concept, so that the primary role of teaching is not to lecture, explain, or otherwise attempt to 'transfer' mathematical knowledge, but to create situations for students that will foster their making the necessary mental constructions. A critical aspect of the approach is a decomposition of each mathematical concept into developmental steps following a Piagetian theory of knowledge based on observation of, and interviews with, students as they attempt to learn a concept."

Piaget, for example, was a strong supporter of constructivism. Quoting from this reference:

A central component of Piaget's developmental theory of learning and thinking is that both involve the participation of the learner. Knowledge is not merely transmitted verbally but must be constructed and reconstructed by the learner. Piaget asserted that for a child to know and construct knowledge of the world, the child must act on objects and it is this action which provides knowledge of those objects (Sigel, 1977); the mind organizes reality and acts upon it. The learner must be active; he is not a vessel to be filled with facts. Piaget's approach to learning is a readiness approach. Readiness approaches in developmental psychology emphasize that children cannot learn something until maturation gives them certain prerequisites (Brainerd, 1978). The ability to learn any cognitive content is always related to their stage of intellectual development. Children who are at a certain stage cannot be taught the concepts of a higher stage.

Now, think about what a preservice elementary teacher might have learned about constructivism before beginning to take teacher education courses. The whole idea of prerequisites is based on constructivism. Thus, preservice teachers have repeatedly experienced lessons in which their teacher begins by reviewing some of the prerequisite knowledge assumed in a lesson.

This is a constructivist approach—but most teachers do not make explicit that they are using a constructivist approach. Moreover, math is a vertically structured discipline of study. Thus, any new material may well draw upon (be built on) content that a student has supposedly learned in previous math instruction in previous weeks, months, and years. It is not feasible to begin each new math lesson with a review of all previous math lessons.

This may well be the root of the "I can't do math" beliefs of many adults. In a vertically-sequenced course of study, it is quite easy for a student to encounter new material that assumes prerequisite knowledge and skills that were previously covered, but not understood and/or learned very well by that student.

As a Math Methods teacher, you probably will want to role model constructivist math teaching and learning, both to help your students with the current lesson and to help them to gain an increased level of expertise in constructivism.

High-road and Low-road Transfer of Learning

Teaching for transfer is one of the seldom-specified but most important goals in education. We want students to gain knowledge and skills that they can use both in school and outside of school, both immediately and in the future.

The following free book strongly emphasizes and illustrates teaching for transfer of learning.

Moursund, D.G. (2006). Introduction to using games in education: A guide for teachers and parents. Eugene, OR: Information Age Education. Access and download at no cost in PDF and Microsoft word formats at: http://i-a-e.org/ebooks.html.

Quoting from the Moursund book:

The low-road/high-road theory of learning has proven quite useful in designing curriculum and instruction (Perkins and Solomon, 1992). In low-road transfer, one learns something to automaticity, somewhat in a stimulus/response manner. When a particular stimulus (a particular situation) is presented, the prior learning is evoked and used. The human brain is very good at this type of learning.
Low-road transfer is associated with a particular narrow situation, environment, or pattern. The human brain functions by recognizing patterns and then acting upon these patterns. Consider the situation of students learning the single digit multiplication facts. This might be done via work sheets, flash cards, computer drill and practice, a game or competition, and so on. For most students, one-trial learning does not occur. Rather, a lot of drill and practice over an extended period, along with subsequent frequent use of the memorized facts, is necessary.
Moreover, many students find that they have difficulty transferring their arithmetic fact knowledge and skills from the learning environment to the “using” environment. One of the difficulties is recognizing when to make use of the memorized number facts. In school, the computational tasks are clearly stated; outside of school, this is often not the case.
This helps to explain why rote memory is useful in problem solving, but critical thinking and understanding are essential in dealing with novel and challenging problems. It also supports the need for broad-based practice even in low-road transfer. We want students to recognize a wide range of situations in which some particular low-road transfer knowledge and skills is applicable.
Math education in schools tries to achieve an appropriate balance between rote memory and critical thinking by making extensive use of word problems or story problems. In word problems, the computations to be performed are hidden within a written description of a particular situation. The hope is that if a student gets better at reading and deciphering word problems—extracting the computations to be performed and the meaning of the results—that this will transfer to non school problem-solving situations.
It turns out that it is quite difficult to learn to read well within the discipline of mathematics. Many students have major difficulties with word problems and with learning math by reading math textbooks. Their depth of understand of math and their ability to read math for understanding stand in the way of their being able to deal with novel, challenging math problems that they encounter.
High-road transfer for improving problem solving is based on learning some general-purpose strategies and how to apply these strategies in a reflective manner. The build on previous work strategy is an excellent candidate to use to begin (or, expand) your repertoire of high-road transferable problem-solving strategies. To do this, think of a number of personal examples in which you have used this strategy as an aid to problem solving. Mentally practice what you did in each case. In the near future, each time you make use of this strategy, consciously think about its name and the fact that you are using it. Also, in the future when you encounter a challenging problem, consciously think through your repertoire of high-road transferable problem-solving strategies. Your goal is to increase your ability to draw upon this repertoire of aids to use when faced by a challenging problem.
The break it into smaller pieces strategy is another example of a high-road transferable strategy. This strategy is often called the divide and conquer strategy, and that is the name that will be used in the remainder of this book. It is helpful to have short, catchy names for strategies. A large and complex problem can often be broken into a number of smaller, more tractable problems. It is likely that many of your students do not have a name for the strategy and do not automatically contemplate its use when stumped by a challenging problem.

A Teacher's Collection of Professional Resources

Over his or her years of preparing to become a teacher and then being a teacher, every teacher builds a personal professional collection of teaching resources. Many of these resource materials are stored only in the teacher's brain. However, quite a bit can also be stored in physical filing cabinets, on shelves, and electronically in computers.

Mental and Physical Knowledge and Skill

A teacher's physical and mental knowledge and skills are used for just in time decision making and implementing the decisions. Talking to a class, responding to a question, observing and facilitating a group of students engaged in a team activity, and writing on a chalkboard/whiteboard/smartboard all require making use of one's mental and physical knowledge and skills. In essence, these are all rapid response situations.

Reading and writing allow us to supplement the knowledge and skills we store in our heads. Skill in storing, retrieving, understanding, and making use of retrieved information is now one of the basics of education. All students need to develop their own personal balance between what they store in their heads, store on paper, and store in computerized information storage and retrieval systems.

Mental and physical knowledge and skills improve through practice (including studying) and use. Both decline from one's peak performance level through disuse. This decline from peak performance is a key concept in teaching and learning. Most students forget most of what is covered in a course unless they are in a situation where it is is used and the knowledge and skills are periodically refreshed.

So, as a teacher of teachers, you should think about this very carefully. Suppose that your students will forget well over three-fourths of what you cover in your class by the time a year has gone by. How does this knowledge affect what and how you teach preservice math teachers?

  • One of the things that we know is that relearning can be quite a bit faster than initial learning. What do you do in your teaching to help students prepare for the relearning tasks they will encounter throughout their professional careers?
  • What parts of the content do you really really want your students to remember for a very long time? Is it the same for each student? Note how this ties in with constructivism and with individual differences. This is a challenging question.
  • When your students become teachers and begin to teach math, what do you want them do do for their students in terms of the two questions given above?

An elementary school teacher faces the challenges and responsibilities discussed above over the full ranges of courses he or she is teaching. The Math Methods teacher needs to focus special attention on the math related aspects of these challenges and responsibilities.

Physical Materials: Physical Filing Cabinets

Think of some of the physical things that an elementary school math teacher is apt to want to have readily available. There might include:

  1. A personal library of math-related books, journals, magazines, and articles.
  2. Math-related books and pamphlets for student use.
  3. Math-related teaching supplies for use by the teacher and students. This might include a classroom set of rulers, protractors, compasses, calculators, math manipulatives, colored pencils, colored chalk or white board markers, colored paper, graph paper, geoboards, and so on.
  4. A collection of frequently used black line acetate masters for use with an overhead projector. Various sizes of graph "paper" provide a good example.
  5. Grade books, samples of work done by students in the past, and perhaps a student plus teacher-created math portfolio for each student.
  6. Video tapes, CDs, and DVDs used in teaching math.
  7. Etc.

Some of these materials will be supplied by the school, and some will be in a teacher's private collection, often purchased with the teacher's own money.

Here is an aside. What happens when a teacher moves to teaching at a different grade level or moves to another school or district?. It would surely help if all of one's own personal materials were carefully labeled and/or on an inventory list. It would also help a preservice teacher to know what materials are apt to be (should be) supplied by the school.

If the total collection of physical resources is relatively small and is used frequently, then it may well be possible to remember where everything is, when it will be needed, how it will be used, and so on. The retrieval process might be completely dependent on one's own mental storage and retrieval capabilities.

However, as the collection grows, many teachers find this organization and retrieval task to be mentally overwhelming. The use of physical filing cabinets to deal with the paper parts of such a collection goes back more than a hundred years. With a little instruction or through trial and error, teachers learn how to organize paper materials into file folders and store the folders in alphabetical or subject matter order in a filing cabinet. (While this may seem like a simple task, it isn't. Where do you file a document that might well fit into several different folders? How do you find such a document years later?)

Virtual Materials: Digital Filing Cabinets (DFC)

Nowadays, most preservice and inservice teacher own a computer and also have easy access to one for routine classroom use. All of the software and documents in this computer can be considered to be virtual materials. In addition, all the materials that one can access on the Web and other networks from this computer are virtual materials.

A teacher is continually adding to, editing, and deleting from their Digital Filing Cabinet (DFC) resources. A teacher may want to share some of the contents with students and some with other teachers. Some of the content needs to be personal and private—not available to other people.

A simple approach to this would be to have your students create a Math Education file folder on their own computer, perhaps with several folders in it, to be used when they begin their teaching careers. Sub-folders might be Lesson Plans, Classroom Resources, School Resources, Personal Resources, Courses and Workshops, and so on.

Think of the Math Education folder as being a filing cabinet, and each main sub-folder as being a drawer in the filing cabinet. Each sub-folder contain electronic documents. For example, in the Courses and Workshops sub-folder one can store detailed syllabi, electronic handouts, and other electronic materials from each math course, math methods course, and math-related workshop one has attended.

Through appropriate formal instruction, self instruction, and "on the job" training (that is, practical experience), a person can gain increased expertise in this area. A teacher who fails to develop a relatively high level of expertise in organizing resources will encounter many frustrating retrieval situations.

The next section explores the idea of a Wiki as an approach to helping to solve a preservice or inservice teacher's problem of the organization of Virtual Materials.

A Wiki Math Digital Filing Cabinet

Nowadays, essentially all college students make routine use of the Wikipedia. The Information Age Education-pedia (IAE-pedia) uses the same software as the Wikipedia.

Students who are taking a Math Methods for Elementary Teachers course will certainly want to learn about the huge range of math-related materials available on the Web in general, and specifically in the math part of the IAE-pedia.

One of long-term Information Age Education (IAE) projects is to develop Digital Filing Cabinets (DFC) for teachers at various grade levels and in various curriculum areas. The document you are currently reading is specifically designed for people who teach Math Methods courses to preservice and inservice elementary school teachers. A DFC for such teachers of teachers might have one drawer specifically for teachers of math methods courses, one drawer for primary school teachers of math, one drawer for upper elementary school teachers of math, one drawer for teachers of the math content courses that preservice teachers take, and so on.

As a Math Methods teacher, you may well want to set up and maintain a "model" DFC for preservice and inservice elementary school math teachers. It can contain the detailed syllabus for your course, including all handouts, assignments, and so on. It can contain copies of readings that are in the public domain, and it can contain an extensive annotated bibliography of other required readings and useful references.

If you are generous about sharing, your Wiki can contain lecture notes, copies of videos that you have made to use in your teaching, and links to videos and other material you use in your teaching.

Set up your model DFC so that your students can contribute to it through editing, adding comments, adding annotated references, and so on. An annotated reference includes a paragraph briefly describing the content of the referenced document and why it is relevant to a preservice or inservice math teacher.

You want your model DFC to contain good resources. One aspect of a good resource is that it is free or quite inexpensive, and that it is long-lasting. Examples of long-lasting resources include the National Council of Teachers of Mathematics (NCTM) and the International Society for Technology in Education (ISTE) websites. There is a reasonably good chance that both will be available for many years to come, and that they will be periodically updated.

There are a number of advantages in having a Digital Filing Cabinet. Let's use this IAE-pedia as an example. Each document in the IAE-pedia belongs to one or more Categories. It is very easy to place a document in several categories. The total set of documents in this IAE-pedia can be searched using a built-in search engine. (See the Search Box in the left column of this page.)

The IAE-pedia is stored on the Web, which means it can be accessed by anybody with access to the Web. It is possible to create a personal Wiki that is stored on the Web or that is just stored on one's home or school computer. Thus, one can have a home Wiki stored on a portable medium that can easily be moved from home to school, just like one might do with a paper document.

Another important advantage of having documents in electronic format is that they are easily updated. As an example, suppose a teacher retrieves a lesson plan and related handouts that he or she has used before. The teacher reads through the document online, correcting errors, updating the date, and incorporating some additional ideas. The teacher then prints out the handouts, or puts them into a computer file that students are expected to access. Immediately after teaching the lesson, the teacher goes through the lesson plan and other material again, making comments about what works and what needs additional work, correcting errors, adding more explanation to the students, and so on. A lesson plan and its handout materials become a living, growing document that increases in value with each use.

Materials in a Digital Filing Cabinet can be retrieved by use of a search engine. However, this misses the important idea that many documents are related to each other and you may want to retrieve a number of related documents all at the same time. The concept of Categories in a Wiki is exactly designed for that purpose. Click on the Special Pages menu item in the left column of this page. This takes you to a long list of options. Click on Categories. It tells you how many articles are in each category. Clicking on a a specific Category takes you to a page that contains the titles and links to these articles in that Category.

Look in the Toolbox in the left column menu. There you will see a What Links Here button. Click on that button to retrieve a list of all pages in the IAE-pedia that have links to the page you are currently reading. This sort of "backward" linking is useful in editing documents and in retrieving documents that relate to the one you are currently reading.

Communities of Practice

To a large extent, teachers of Math Methods courses work in relative isolation. Many teacher education programs have only one or two faculty members who teach the Math Methods course for preservice elementary school teachers. In many cases, Math Methods is taught by adjunct faculty members who are only on campus when they are teaching their course and holding their office hours. Others may be teaching the course completely online. Thus, many Math Methods teachers have relatively little regular opportunity for close and continuing professional interaction with other Math Methods teachers.

The following brief sections list some areas where every Math Methods teacher can contribute. This contribution can be by adding to and editing this Math Methods for Preservice Elementary Teachers Web Page. It can be done by Providing your ideas in the Discussion Page (see the Menu at the top of this document). It can be done by engaging your fellow teachers of Math Methods course in discussions and building a personal Community of Practice.

You can contribute to a worldwide Math Methods Community of Practice by following some of the ideas listed in the next few sections of this document.

Sharing of Course Syllabi

There are, of course, several high-quality widely used Math Methods textbooks. These help to provide a "definition" of what constitutes a Math Methods course.

However, many Math Methods teachers draw heavily on their own collection of materials and ideas. They teach from a detailed syllabus that is not widely shared with the rest of the math education community. It would be very helpful if Math Methods faculty could easily browse detailed course syllibi of their colleagues. Please put your detailed syllabus on the Web and add a link to it either in this section of the current document, or in the Discussion Page accompanying this Web page.

Sharing of Quizzes, Exams, and Other Assessment

Readers are encouraged to contribute samples of quizzes and exams they use in their Math Methods courses.

For example, do you quiz your students over required reading assignments?

Do you give open book, open notes, or open computer quizzes or exams?

How do you assess your students written assignments, participation during in-class discussion, participation in Web-based discussions, and so on?

Do you make use of portfolio assessment or require your students to do an assignment that will likely produce a product to add to their portfolio?

Sharing of Math Education Videos

Do you make use of video material in teaching your Math Methods course? For example, do you make use of video of elementary school students being taught math? Here are two good examples of free video materials:

Additional Math Education Free Videos are also available.

Teaching with the Aid of ICT

What types of Information and Communication Technology do you use in teaching your Math Methods courses? What works well, and what doesn't work well? Do you make use of "clickers?" Do you routinely use a computer and projection system as you interact with your class? Please share your experiences and insights.

Research That Supports Significant Improvement

What specific math education research do you bring to the attention of students in your Math Methods course? Are there specific papers that you have used more than once? Do you help your students to start building their own collection of such research papers? Is there any paper that you believe most preservice elementary school math teachers should read? Are there some math education researchers that you feel most preservice math teachers should be exposed to?

Additional Readings and Resources

Devaney, Laura (10/17/07). Solution aims to transform math assessment: Already revolutionizing early-literacy assessment via handheld technology, Wireless Generation seeks to boost elementary math. Retrieved 5/31/08: http://www.eschoolnews.com/resources/math-solutions/math-solutions-articles/index.cfm?rc=1&i=50120;_hbguid=06f86ec9-bd7f-47b3-8e01-540b3067fd29.

Quoting from the article:

In school systems across the country, teachers are using handheld computers and a software solution from Wireless Generation, called mCLASS, to administer the Dynamic Indicators of Basic Early Literacy Skills (DIBELS) to elementary-age students. The technique has helped boost students' reading scores dramatically. Now, the positive impact this approach has had on reading soon could be replicated in math.
Wireless Generation, along with the Teachers College at Columbia University and the University of Missouri-Columbia, recently received a four-year, $1.5 million grant from the U.S. Department of Education's Institute of Education Sciences to develop a math-related version of mCLASS.

Math Education Free Videos. A growing list of inks to free math education videos that are available on the Web.

Math Forum (n.d.). Constructivism. The Math Forum@Drexel. Retrieved 9/24/07: http://mathforum.org/mathed/index.html.

Moursund, D.G. (October 2007). Introduction to problem solving in the Information Age. Eugene, OR: Information Age Education. Download free book in PDF and Microsoft Word formats at: http://i-a-e.org/ebooks.html.

Moursund, D.G. (2006). Computers in Education for Talented and Gifted Students: A Book for Elementary and Middle School Teacher. Download free book in PDF and Microsoft Word formats at: http://i-a-e.org/ebooks.html.

Moursund, D.G. (2006). Introduction to using games in education: A guide for teachers and parents. Eugene, Oregon: Information Age Education. Download free book in PDF and Microsoft Word formats at: http://i-a-e.org/ebooks.html.

Moursund, D.G. (2006.Computational thinking and math maturity: Improving math education in K-8 schools. Download free book in PDF and Microsoft Word formats at: http://i-a-e.org/ebooks.html.

Open Source Materials (n.d.). Free, open source software of potential use to teachers of teachers and to teachers at all levels.

Perkins, David N., and Salomon, Gavriel (1992). Transfer of Learning: Contribution to the International Encyclopedia of Education. 2nd ed. Oxford: Pergamon Press. Retrieved 5/23/08: http://learnweb.harvard.edu/alps/thinking/docs/traencyn.htm.

Roher, Doug, and Pashler, Harold (2007). Increasing retention without increasing study time. Retrieved 9/1/07: http://repositories.cdlib.org/cgi/viewcontent.cgi?article=5990&context=postprints. Here is the abstract of this short paper:

ABSTRACT - Because people forget much of what they learn, students could benefit from learning strategies that provide long-lasting knowledge. Yet surprisingly little is known about how long-term retention is most efficiently achieved. Here we examine how retention is affected by two variables: the duration of a study session and the temporal distribution of study time across multiple sessions. Our results suggest that a single session devoted to the study of some material should continue long enough to ensure that mastery is achieved but that immediate further study of the same material is an inefficient use of time. Our data also show that the benefit of distributing a fixed amount of study time across two study sessions – the spacing effect – depends jointly on the interval between study sessions and the interval between study and test. We discuss the practical implications of both findings, especially in regard to mathematics learning. [Bold added for emphasis.]

Wiggins, Grant, and McTighe, Jay (May 2008). Put understanding first. Educational Leadership. Retrieved 5/23/08.

This is an informative article that focuses on teaching for understanding and for transfer of learning at the high school level.

Sustainability—A Work in Progress

A great many people feel that sustainability is a major problem facing the people of our world. Quoting from the Wikipedia:

One of the first and most oft-cited definitions of sustainability, and almost certainly the one that will survive for posterity, is the one created by the Brundtland Commission, led by the former Norwegian Prime Minister Gro Harlem Brundtland. The Commission defined sustainable development as development that "meets the needs of the present without compromising the ability of future generations to meet their own needs." The Brundtland definition thus implicitly argues for the rights of future generations to raw materials and vital ecosystem services to be taken into account in decision making.

It is possible to incorporate the sustainability idea into the content of many different courses. For example, language arts students might be given assignments that include reading and writing about sustainability. In science they might study the effects that dams have on efforts to sustain fish populations.

Many math textbooks used at the precollege level include examples that can be construed to being related to sustainability. For example, here is a word problem:

Mary wants to buy a cashmere sweater that costs $62. She has an allowance of $3.50 a week. She figures that she can save $2 per week from this allowance. How many weeks will it take her to save enough money to buy the sweater?

Notice that this is a consumption-oriented problem situation. Mary has income and she "wants" to buy a cashmere sweater. There is no indication of why she wants the sweater or if she "needs" the sweater. What is cashmere? Are there other types of sweater fabric that could be more sustainable? How does this tie in with Mary's carbon footprint?

Aha! At what age can students begin to understand some of the basic ideas of sustainability, carbon footprint, ways to save energy, and so on? Are there math problems that are appropriate to the math curriculum for young students that can help to teach and support sustainability?

It seems to me there are two goals here:

  1. Help preservice and inservice elementary school teachers of math realize that they can and should be teaching students about math-related aspects of sustainability.
  2. Provide some sources of resources that will help in teachers implementing such ideas.

There are a number of websites for users to calculate various carbon footprints. An IAE-pedia article that can be helpful to teachers is Math Word Problems Divorced from Reality. See http://iae-pedia.org/math_word_problems_divorced_from_reality.

Links to Other IAE Resources

This is a collection of IAE publications related to the IAE document you are currently reading. It is not updated very often, so important recent IAE documents may be missing from the list.

This component of the IAE-pedia documents is a work in progress. If there are few entries in the next four subsections, that is because the links have not yet been added.

IAE Blog

All IAE Blog Entries.

IAE Newsletter

All IAE Newsletters.

IAE-pedia (IAE's Wiki)

Home Page of the IAE Wiki.

Math Word Problems Divorced from Reality.

Popular IAE Wiki Pages.

I-A-E Books and Miscellaneous Other

David Moursund's Free Books.

David Moursund' Learning an Leading with Technology Editorials

Author or Authors

The initial version of this article was written by David Moursund.

Note to self 10/19/08. The intended audience of the current version of this document is mainly teachers of Math Methods for Elementary Teachers. It is not a document intended specifically for students in such a course. Probably this document should be split into two documents.