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Introduction and Goals

Craft and Science of Teaching and Learning Mathematics

Curriculum Content

Instructional Processes (Pedagogy)



Curriculum Content

4.1. Math is a cumulative, vertically structured discipline. We have many thousands of years of accumulation of mathematical research. The totality of accumulated mathematical knowledge continues to grow at a significant pace. This presents major difficulties to teachers and learners of math.

First, there is a great deal that could be learned, so careful decisions have to be made about what to have students learn. These decisions need to be revisited from time to time as the totality of mathematical knowledge continues to grow.

Activity 4.1A: In the whole workshop group, share examples of topics that have been dropped from the K-12 math curriculum in the past few decades.

Second, the teaching of a math topic at any particular grade level tends to be based on an assumption that students have certain math knowledge and skills taught in previous courses and previous years. The reality of the math classroom is that students tend to lose the knowledge and skills that they have previously demonstrated (over time, and especially over the summer). "Use it or lose it" tends to describe the situation.

Third, we have the issue of math procedural knowledge that has been practiced sufficiently so that it is well embedded (sort of "permanently" embedded) in procedural storage in the brain. Like physical skill procedural knowledge (riding a bicycle) such math knowledge tends to last for a long time, even if seldom used. Moreover, a modest amount of practice tends to help a person regain a significant fraction of declining skills. The issue is: what mathematical procedural knowledge should be practiced to the level of "permanency" in the brain? Keep in mind that it takes a substantial amount of time and effort to achieve such permanency.

Activity 4.1B: In small groups, discuss the general topic of concepts versus procedures in terms of learning and forgetting. Is it harder or easier to learn a concept, versus learning a procedure? Which tends to be better remembered over the long run? In our math education system functioning in a manner consistent with your findings?

4.2 The "Genius Principle." Ordinary students can learn and use the results developed by mathematical geniuses.

Newton and Liebnitz invented calculus about 300 years ago. At that time, calculus was at the frontiers of math research. Now, many high school students are expected to learn calculus. Of course, the totality of mathematical knowledge has double many times since then. We must select wisely as we make decisions on what topics should "trickle down" into the curriculum in undergraduate courses and K-12 courses. This trickling down process can consist of individual topics being introduced at various grade levels, and it can also consist of moving an entire course to a lower grade level.

For example, during the past 50 years, algebra as a ninth grade topic for the more mathematically able students has moved toward being "Algebra for all." at the eighth grade level.

Activity 4.2: In small groups, share your thoughts on whether it is reasonable to expect that ordinary K-12 students will come to understand some of the math ideas that were at the frontiers of math research less than 500 years ago.

4.3. Problem solving. There has been a substantial amount of research on problem solving, both in mathematics and in other domains. Some important ideas include: A) domain specificity and domain independence; B) higher-order thinking; and C) transfer of learning.

Math is an interdisciplinary tool--a useful aid to problem solving in many different disciplines. However, many students have considerable difficulty transferring their mathematical knowledge and skills from the math classroom into other problem-solving environments/subject areas. This suggests that both mathematics and the other subject areas that are not being appropriately taught.

A standard response to this assertion is finger pointing. Consider the situation of students have significant difficulty in using math in their science classes. The science teacher says that the math teacher is at fault, while the math teacher says the science teacher is at fault.

This finger pointing situation suggests the difficulty of teaching and learning for transfer. It also suggests that more attention should be paid to Situated Learning Theory. Over the past 20 years or so, a learning theory called Situated Learning has been developed. The focus is on learning by doing, and on addressing real problems. IT is a powerful aid to "doing" and to "addressing real problems." Thus, Situated Learning and IT work well together. Situated Learning and Constructivism are compatible and appear to be mutually supportive.

Activity 4.3: In small groups, discuss the meaning of "problem solving" in math. What relationship exists (if any) between problem solving and higher-order thinking skills? What do you think that students should be learning about roles of ICT in solving math problems?

4.4. Data, information, knowledge, and wisdom. We accept without question that data and information can be stored in a computer and retrieved through the use of a computer network. The Global Library that we call the Web is an important addition to our education system.

It is arguable whether knowledge and wisdom can be stored in a computer. To the extent that we agree that knowledge can be stored in a computer, we open up the possibility of major changes in our educational system. In math education, for example, we would then think about what mathematical knowledge to store in a student's head, and what to store in the computer system the student is learning to use. We need to think carefully about our goals in math education as access to calculators and computers continues to grow, and the capabilities of these machines continue to increase.

Our current math education system tends to assume that a student will have a math book to use during the school year. This book is not available the next year. Thus, information retrieval in math education tends to consist of retrieving information from one's currently available book. This means that students gain little skill in looking up math information in books that they have previously used and in the books that contain a substantial portion of the collected mathematical knowledge of the human race. The Web is making such collected knowledge more readily available--but our math education system has been slow to adjust to this situation. Our current math education system focuses on helping a student store mathematical knowledge in his/her head--where much of it is quickly forgotten if it is not being regularly used.

Activity 4.4A: In small groups, discuss what students learn about the storage and retrieval of math information while they are engaged in the K-12 math education curriculum. For example, perhaps students learn to retrieve information about historical people or events in math. Do they learn to look up math content topics?

2.5. ICT as part of the content knowledge of each academic discipline. This idea is already entrenched in math and the sciences, and in areas such as business, engineering, and graphic arts. By 1982., Computational Biology was producing important research results in Biology. One of the two winners of the 1998 Nobel Prize in Chemistry was a Computational Chemists who had made outstanding contributions to the field during the previous 15 years.

Computer modeling typically requires extensive use of mathematics as well as logical/mathematical intelligence.

On a more general note, "Computational" is now one of the major categorizations of mathematics, along with Applied and Theoretical.

Activity 4.5: In a whole workshop group, discuss what students are currently learning about each of the three categories of math: Theoretical, Applied, and Computational. Pay particular attention to what students are learning about math modeling and its applications to problems throughout the math and non-math curriculum

4.6. Mathematical knowledge can be stored in an ICT system in a "The ICT system can do it for you" form.

The 4-function handheld calculator provided a modest example of this. Such a calculator "knows" how to add, subtract, multiply, and divide. We have scientific calculators, graphing calculators, equation-solving calculators, and calculators that can solve a huge range of algebra, calculus, and other higher-level math problems. We have computer systems that are even more powerful, and that are certainly more convenient to learn to use and to use. (A lot of computer power is used to improve the human-machine interface.)

The following diagram helps illustrate a major potential change in math education due to ICT. The five steps illustrated are 1) Problem posing; 2) Mathematical modeling; 3) Solving a math problem; 4) Mathematical "unmodeling"; and 5) Thinking about the results to see if the original problem has been solved, other problems that may need to be solved, and so on. Step 3 in this diagram can often be done by an ICT system. That is because mathematicians have developed procedures for solving a wide variety of math problems, and many of these procedures can be carried out by a computer.

One can view an ICT system as a type of library--a dynamic library that is able to carry out procedures rapidly and accurately. Clearly, this constitutes a significant alternative to having a person memorize procedures and practice them to a level of automaticity, speed, and accuracy that is required for such procedures to be a useful tool.

For example, think about the concept that a simple closed curve encloses an area. Formulas exist for computing many such areas, and a variety of procedures are available to find areas for more complex shapes. Learning this concept is a lot different than memorizing lots of area formulas, and practicing them to achieve automaticity, speed, and accuracy. An ICT device can both store formulas and can carry out the computations specified in formulas.

We need to remember that one of the most important ideas in problem solving is building on the previous work of others, and building on one's own previous work. ICT is a powerful aid in this endeavor.

Activity 4.6: In a whole workshop group, discuss implications of teaching students to use ICT in Step 3 of the diagram given above, and substantially decreasing the amount of time students are currently spending learning to do Step 3 by hand.

4.7. Strategies in math problem solving. Research suggests

  • Learning some strategies and gaining experience in using them in a variety of settings (keep in mind Transfer of Learning) is an effective way to improve one's problem solving abilities in math and other disciplines.

  • Research suggests that students do not know very many strategies of proven effectiveness. Thus, it is relatively easy to improve a student's problem solving abilities within a discipline by helping the student to learn some of the common problem-solving strategies within the discipline (that the student has not previously learned.)

Polya's strategy is useful in math and in most other disciplines. Here is a (modified by Moursund) six-step version of his strategy:

  1. Understand the problem. Among other things, this includes working toward having a clearly defined problem. You need an initial understanding of the Givens, Resources, and Goal. This requires knowledge of the domain(s) of the problem, which could well be interdisciplinary.
  2. Determine a plan of action. This is a thinking activity. What strategies will you apply? What resources will you use, how will you use them, in what order will you use them? Are the resources adequate to the task?
  3. Think carefully about possible consequences of carrying out your plan of action. Place major emphasis on trying to anticipate undesirable outcomes. What new problems will be created? You may decide to stop working on the problem or return to step 1 as a consequence of this thinking.
  4. Carry out your plan of action. Do so in a thoughtful manner. This thinking may lead you to the conclusion that you need to return to one of the earlier steps. Note that this reflective thinking leads to increased expertise.
  5. Check to see if the desired goal has been achieved by carrying out your plan of action. Then do one of the following: If the problem has been solved, go to step 6. If the problem has not been solved and you are willing to devote more time and energy to it, make use of the knowledge and experience you have gained as you return to step 1 or step 2. Make a decision to stop working on the problem. This might be a temporary or a permanent decision. Keep in mind that the problem you are working on may not be solvable, or it may be beyond your current capabilities and resources.
  6. Do a careful analysis of the steps you have carried out and the results you have achieved to see if you have created new, additional problems that need to be addressed. Reflect on what you have learned by solving the problem. Think about how your increased knowledge and skills can be used in other problem-solving situations. (Work to increase your reflective intelligence!)

The following quote from W. C. Fields seems relevant to Polya's strategy for attaching a problem:

"If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it."

Here are two computationally intense strategies that could be added to the math curriculum:

  • Strategies employing lots of compute power. Exhaustive search (trial and error) is sometimes a feasible way to solve a problem when the solution space contains only a finite number of points. Faster computers make it possible to search larger solution spaces. What types of problems can be modeled using mathematics and then solved by exhaustive search?
  • Hill climbing. Many math problems can be solved by finding relative minima and/or relative maxima for functions of one or more variables. Similar techniques are often useful in finding "pretty good solutions" to optimization problems. This topic can be discussed in terms of incremental improvements of products, processes, productions, and so on. Process writing provides a good example.

Activity 4.7A: In a whole workshop group, discuss some of the general math problem-solving strategies that students learn in our K-12 curriculum.

Activity 4.7B: In small groups, discuss examples of what math learning strategies are currently being taught in our K-12 math education system.

4.8 Computer programming. Computer programming and procedural thinking are closely related. For awhile in the history of computers in education there was a strong trend toward teaching computer programming in math classes, as part of math education. This has largely disappeared.

Activity 2.8A: In small groups discuss the plusses and minuses of teaching computer programming "to the masses" at the precollege level. For example, this discussion might include a focus on teaching Logo and/or BASIC at the elementary or middle school level.

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