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What is Mathematics?

Major Unifying Themes in This Document

Syllabus

Foundational Information

Learning Theories

Mind and Body Tools

Science of Teaching & Learning

Project-Based Learning

Computational Mathematics

The Future

Recommendations

References

Some Current Research Projects

 Website Author

"Dr. Dave" Moursund

Major Themes That Help to Unify this Website Document

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There are seven recurring and unifying themes in this Website document. Each provides opportunities for exploration of possible significant improvement in our math education system. This Webpage provides a short introduction to each of the seven themes, as well as clickable links that lead to more detail.

Theme 1: Students Competing with ICT Systems. Our math education system spends an inordinate amount of time attempting to teach students to memorize procedures and develop both speed and accuracy in carrying them out--things that machines do much better than people.

Theme 2: Students and Mathematical Developmental Theory. The content of significant parts of the current PreK-12 math curriculum is not appropriately aligned with what we know about developmental theory in general, and mathematics development theory (math maturity) in particular.

Theme 3: Teachers Competing with ICT Systems. Increasingly, ICT is producing and/or providing Intelligent Computer-Assisted Learning and Distance Learning materials that compete effectively with (indeed, are more effective than) some of the current teacher activities.

Theme 4: Teachers and Developmental Theory. Many teachers who teach math at the PreK-12 level are not at a mathematics developmental level (that is, they have not achieved an appropriate level of math maturity) suitable to the teaching task.

Theme 5: Transfer of Learning. Many teachers do not effectively teach for transfer of math learning to environments outside of their classrooms, and many students do not learn in a manner that facilitates and promotes transfer. Current progress in Brain Science, along with progress in ICT, brings some important new ideas to teaching and learning for transfer in math education.

Theme 6: Effective Procedure and Computer Programming. The fields of mathematics and of computer & information science have a strong overlap. Computer scientists define an "Effective Procedure" to be a step by step set of instructions that can be mechanically interpreted and carried out by a specified agent such as a computer. Both mathematicians and computer scientists are interested in procedural thinking--posing, representing, studying, and and solving problems that involve procedural thinking. Because of this, one can make a strong case that computer programming is an appropriate topic to include in a math curriculum.

Theme 7: Many People Disagree with Some or All of Themes 1-6. Mathematics education is at an interesting crossroad. There is major conflict between those who believe strongly in a Back to Basics point of view and those who are supporting substantial changes in the curriculum. Quoting from the Mathematically Correct Website:

"The advocates of the new, fuzzy math have practiced their rhetoric well. They speak of higher-order thinking, conceptual understanding and solving problems, but they neglect the systematic mastery of the fundamental building blocks necessary for success in any of these areas. Their focus is on things like calculators, blocks, guesswork, and group activities and they shun things like algorithms and repeated practice. The new programs are shy on fundamentals and they also lack the mathematical depth and rigor that promotes greater achievement. "

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Theme 1: Students Competing with ICT Systems

Our math education system spends an inordinate amount of time attempting to teach students to memorize procedures and develop both speed and accuracy in carrying them out--things that machines do much better than people.

Consider the general situation that people face as they attempt to apply mathematics to help solve problems. The following diagram is a highly simplified six step process that is broadly applicable. (Please be aware that this diagram does not represent the full field of mathematics. See What is Mathematics? for more detail.)

Figure. Diagram of math problem solving.

The six steps are:

  1. Understand the problem situation and translate it into a clearly defined problem. What is the given initial situation and what is the goal? What are the resources and rules that apply to solving the problem? (Moursund, 2002). Remember, the problem situation may come from any discipline or be interdisciplinary. You are going to use mathematics in attempting to represent and resolve the problem situation because you feel this may be a useful approach.
  2. Model the problem situation as a math problem. That is, translate the problem situation into a math problem. This is somewhat akin to what one does in translating a word problem into an equation or set of equations to be solved.
  3. Solve the math problem.
  4. Translate the results back into a statement about the Problem to be Solved. This can be thought of as unmodeling, sort of the opposite of step 2.
  5. Check to see if the Problem to Be Solved has actually been solved.
  6. Check to see if the original problem situation is resolved (solved). If it hasn't been resolved, reformulate the problem situation and/or problem and start over at step 1 or 2.

People are better than ICT systems in all but one of these six steps. Step 3 is where ICT systems excel. But, based on interactions with many hundreds of teachers, my understanding is that our PreK-12 math education system spends about 75% of its time on step 3. There, the goal is to have students memorize arithmetic, algebraic, and other mathematical procedures, and to develop speed and accuracy at carrying them out. Research in Brain Science and in Mathematics Education suggests that:

  1. It takes a lot of time and effort for a student to memorize the math procedures in the PreK-12 curriculum.
  2. Humans are inherently very slow and inaccurate relative to ICT systems in carrying out math procedures. (Today's microcomputers can carry out a billion computations per second, with no errors! We have Computer Algebra Systems that can solve a huge range of "pure" computational and symbolic math problems.)
  3. Unless a person regularly practices carrying out a particular math procedure, they tend to lose both speed and accuracy. Indeed, a person may well "forget" how to do a procedure.

For the above-stated reasons, this Website includes a substantial focus on the conjecture that a good math education system teaches students to work with ICT systems, rather than teaching students to (poorly,inadequately) compete with ICT systems.

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Theme 2: Students and Mathematical Developmental Theory

The content of significant parts of the current K-12 math curriculum is not appropriately aligned with what we know about developmental theory in general, and mathematics development theory in particular. Piaget identified four stages of human development. These are quite general, not focusing specifically on particular areas of human intelligence and potential. Thus, for example, the four stages identified by Piaget may not align well with development in the Logical/Mathematical intelligence area that Howard Gardner lists as one of the eight multiple intelligences he has identified,

In brief summary, the four stages of development identified by Piaget are:

  1. Sensorimotor stage (Infancy). In this period, intelligence is demonstrated through motor activity without the use of symbols. Knowledge of the world is limited (but developing) because it is based on physical interactions / experiences. Children acquire object permanence (memory) at about 7 months of age. Physical development (mobility) allows the child to begin developing new intellectual abilities. Some symbolic (language) abilities are developed at the end of this stage.
  2. Pre-operational stage (Toddler and Early Childhood). In this period, intelligence is demonstrated through the use of symbols, language use matures, and memory and imagination are developed, but thinking is done in a nonlogical, nonreversable manner. Egocentric thinking predominates.
  3. Concrete operational stage (Elementary and Early Adolescence). In this stage intelligence is demonstrated through logical and systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions that are reversible). Egocentric thought diminishes.
  4. Formal operational stage (Adolescence and Adulthood). In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts. Early in the period there is a return to egocentric thought (see #2).

Only 35% of high school graduates in industrialized countries have obtain formal operations by the time they graduate; many people do not think formally during adulthood. This is good data to hold in mind as we examine the implications of an "Algebra for all" curriculum to be taught in the eighth grade. Notice the bold-faced part of the discussion of formal operations (see #4) given above. Algebraic ideas can be taught at the very earliest grade levels. But, the formal study of algebra includes a strong focus on knowledge,skills, problem solving, and proof that are at the Formal Operational Stage of Piaget's scale.

A careful examination of the PreK-12 math curriculum identifies a number of potential areas in which the learning goals may be inconsistent with the mathematical developmental stage of typical students who are being expected to learn the curriculum. The curriculum content may be well below the mathematical developmental stage of students, or it may be well above it.

For example, high school geometry has a number of aspects that appear to be at Piaget's Formal Operations stage. Teachers of high school geometry indicate that this course "begins to separate the men from the boys." Their meaning is that many students attempt to learn high school geometry via the rote memorization techniques that have gotten them through earlier math courses. This does not work well. Rote memorization is increasingly ineffective as one moves into still higher math courses.

Stigler and Hiebert's 1999 book, The Teaching Gap, argues that math education in the eighth grade in the U.S. does not push very hard on problem solving and higher-order thinking skills. They compare and contrast with the eighth grade math curriculum in Germany and in Japan. It may well be that they have identified U.S. curriculum that is below the math developmental level of students. This type of analysis is used to help support the movement toward Algebra at the eighth grade.

Math education can be significantly improved by a more realistic alignment of mathematical development of students with the content they are expected to learn. ICT might play a significant role in this. Much of the PreK-12 math curriculum content can be divided into "concepts" and "procedures." It may well be both the order of topics in the curriculum, as well as topics that are in or not in the curriculum, might change significantly as ICT systems continue to gain in use as an aid to carrying out procedures.

The "algebra for all" movement is a particular challenge for learning disabled students. As a very rough estimate, one might estimate that this group of students learns math about 1/2 as fast as students in the "normal" range. If this is the case, than such a studetn who stays in school and takes math every year, grades 1-12, might be expected to learn math approximnately up through the sixth grade standards. Since algebra I is typically an eighth grade or ninth grade level, such a studetn has no chance of meeting an Algebra I requirement

With this information in mind, think about the following article from the Sacrameto Bee Newspaper. Accessed 11/28/03: http://www.sacbee.com/content/news/education/story/7775877p-8714861c.html

For seniors, no algebra, no diploma:
Class of 2004 is the first required to get a passing grade in the subject.

Time is short. A state law signed in October 2001 makes Algebra 1 a graduation requirement beginning with this year's graduating class.

Chavez takes a final stab at passing algebra in January but he's not enamored of the prospect.

"It's boring," said Chavez, a student at Encina High School in the San Juan Unified School District who hopes to attend a San Francisco fashion college.

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Theme 3: Teachers Competing with ICT Systems

Increasingly, ICT is producing and/or providing Intelligent Computer-Assisted Learning and Distance Learning materials that compete effectively with (indeed, are more effective than) some of the current human teacher activities. Computer-Assisted Learning (CAL) and Intelligent Computer-Assisted Learning (ICAL) include a broad range of ICT-based types of aids to learning. ICAL includes modeling of the human learner and incorporates appropriate theories of teaching and learning of the content area. ICAL includes drill and practice, tutorial, simulation, and virtual reality.

There are many things that a human teacher can do much better than an ICAL system. However, there are a number of ICAL systems that are more effective than a human teacher in a limited context. An example is provided by the car driver training, airplane pilot, and spaceship pilot simulators in which the trainee can experience a variety of dangerous emergency situations. Other examples are provided by one-on-one individualized instruction in which student responses are analyzed in real time and used to make decisions on the next piece of (highly interactive) instruction to be provided. Even a good human tutor cannot do as well as some of these ICAL systems (within quite limited domains), and certainly a teacher with 25 students cannot provide such a level of individualization of instruction and feedback.

Another approach to this topic is to analyze the issue of translating research (theory) into practice. Every year, there are thousands of research studies that identify potential ways to improve the teaching and learning processes. Staff development helps to bring a few of these ideas to some of our teachers. Clearly, however, our inservice teacher education system is overwhelmed by the nature and extent of the research. A standard approach to this difficulty is to integrate the research ideas into the materials made available to students and teachers that are used to support curriculum, instruction, and assessment. ICT provides us with some powerful new aids in this endeavor. It also provides us with aids to staff development, such as Distance Learning via the Web.

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Theme 4: Teachers and Developmental Theory

Many teachers who teach math at the PreK-12 level are not at a mathematics developmental level that is suitable to the teaching task. (That is, their math maturity is not appropriate to the teaching task.) Teaching is a complex and demanding profession. Imagine a careful analysis of the Craft and Science of Teaching and Learning. This analysis would lead to identification of a number of areas or strands in which a teacher needs to have a significant level of expertise that is related to being an effective teacher. For each of these, we can imagine a sequence of developmental stages. We can then imagine research being done on the alignment of expectations being placed on teachers and their developmental stages in these various expectation areas.

To be quite specific, consider the stages of development in the various curriculum areas that a teacher teaches. Perhaps a teacher is being asked to teach a course at a formal operations level, but the teacher is not at the formal operations level within the discipline of that course.

Or, consider stages of development in the Multiple Intelligence area that Howard Gardner calls Interpersonal Intelligence. Perhaps one must reach "Formal Operations" in this area to develop needed levels of knowledge and skills (that is, needed levels of expertise) in interacting with the variety of students and adults that a teacher encounters while on the job.

Or, consider the issue of teaching lower-order knowledge and skills versus teaching higher-order knowledge and skills. Many argue that all instruction should be a careful balance between lower-order and higher-order. Many teachers are not able to achieve an appropriate balance because they, themselves, lack the higher-order knowledge and skills and do not function at the Formal Operations level in some of the areas that they teach.

My conjecture is that this is an especially serious problem for many elementary teachers who are asked to teach math and science in a manner that prepares students to do well in state and national assessment that focuses on problem solving and higher-order thinking. I conjecture that this is also a significant problem for teachers who are teaching outside areas in which they meet state content area certification requirements (such as having at least a college minor in the content area). Nearly 1/3 of math instruction at the secondary school level is taught by such teachers.

One might also conjecture that some (many?) secondary school teachers who are especially strong in the math content area (due to a combination of being at a Formal Operations level in Logical/Mathematical intelligence and having received appropriate and extensive education in math) may have failed to develop needed levels of expertise in the Interpersonal skills area of Gardner's Multiple Intelligences.

Food for thought: In the above discussion I have not mentioned Emotional Intelligence. Clearly this is an important factor in developing the "people skills" needed to be an effective teacher. This section points to the need for a better understanding of the developmental level of teachers in a variety of "intelligences" and how this relates to teachers gaining needed expertise in the various areas necessary to be a good teacher. I know that teachers can pass courses and also can be functional in areas that do not have require coursework, even when they are not at a formal operations level in the specific intelligences underlying the learning. However, I suspect that this contributes to being less effective (indeed, ineffective) as a teacher.

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Theme 5: Transfer of Learning

Many teachers do not effectively teach for transfer of math learning to environments outside of their classroom, and many students do not learn in a manner that facilitates and promotes transfer. Current progress in Brain Science, along with progress in ICT, brings some important new ideas to teaching and learning for transfer. Two widely used theories of transfer are "Near and Far Transfer" and "High-Road and Low-Road Transfer." Such theories provide a basis for designing curriculum, instruction, and assessment in a manner that promotes transfer of learning.

In recent years, Situated Learning Theory has helped us to understand that learning occurs in an environment, and that transfer of learning outside that environment is not a trivial task. We are all aware of students who are able to pass a math test on a particular topic, but who are unable to transfer this math knowledge and skills to other courses they are taking and to situations outside of the school environment.

Research on transfer of learning provides us with increased knowledge of how to teach for transfer to other environments and over time. Brain Science is playing an important role in this research.

In addition, ICT provides a powerful vehicle to aid in transfer. I think of this Math Website as an auxiliary brain. This Website and I are able to handle a variety of teaching situations. In some sense, my brain carries the concepts, and the auxiliary brain carries the details that help explain and justify the concepts. A somewhat different way to think about this is the concepts versus procedures ideas in Item 1 of this list. If I don't have to spend so much time mastering procedures and memorizing details (that I soon forget), I can spend more time learning concepts and practicing them in a variety of settings. In essence, the latter approach is High-Road transfer.

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Theme 6: Effective Procedure and Computer Programming

The fields of mathematics and of computer & information science have a strong overlap. Computer scientists define an "Effective Procedure" to be a step by step set of instructions that can be mechanically interpreted and carried out by a specified agent, such as a computer. Both mathematicians and computer scientists are interested in procedural thinking--posing, representing, studying, and and solving problems that involve procedural thinking. Because of this, one can make a strong case that computer programming is an appropriate topic to include in a math curriculum.

The following book focuses on Theme 6.

diSessa, Andrea (2000). Changing Minds: Computers, Learning, and Literacy. Cambridge, MA: The MIT Press.

In addition, Theme 6 is strongly supported because it is the 5th language discussed in the following book.

Logan, Robert.(2000). The Sixth Language: Learning a Living in the Internet Age. Stoddard.

In our PreK-12 educational system there is a considerable history of attempts to teach computer programming in BASIC and LOGO. The LOGO movement focussed strongly on elementary school students. There is substantial research on the successes and failures of these endeavors.

As an example, LOGO can be used to create a rich problem-solving environment in which students pose, represent, and solve complex problems. The initial expectation was that such problem-solving knowledge and skills would transfer to problem solving in other disciplines. By and large this did not occur, for two reasons:

  1. The teachers did not have very good insight into problem solving and into transfer of learning. Thus, they did not teach general principles of problem solving and they did not teach for transfer.
  2. The amount of LOGO programing that students learned was quite modest and was insufficient to facilitate problem-solving oriented transfer of learning .

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Theme 7: Many People Disagree with Some or All of Themes 1-6.

The following is quoted from the Mathematically Correct Website [Online]. Accessed 10/6/02: http://www.mathematicallycorrect.com/.

Mathematics achievement in America is far below what we would like it to be. Recent "reform" efforts only aggravate the problem. As a result, our children have less and less exposure to rigorous, content-rich mathematics.

The advocates of the new, fuzzy math have practiced their rhetoric well. They speak of higher-order thinking, conceptual understanding and solving problems, but they neglect the systematic mastery of the fundamental building blocks necessary for success in any of these areas. Their focus is on things like calculators, blocks, guesswork, and group activities and they shun things like algorithms and repeated practice. The new programs are shy on fundamentals and they also lack the mathematical depth and rigor that promotes greater achievement.

Concerned parents are in a state of dismay and have begun efforts to restore content, rigor, and genuinely high expectations to mathematics education. This site provides relevant background and information for parents, teachers, board members and the public from around the country.

In many ways, this represents a Back to Basics approach to math education. Many people (including many mathematicians) support this movement. Their point of view should not be lightly dismissed. Their goal is to improve the mathematics education that our students are gaining. They are seeking an appropriate balance between the teaching of lower-order and higher-order knowledge and skills. They tend to feel that over use of calculators, computers, constructivism, situated learning, cooperative learning, and so on has lead to a decrease in the overall effectiveness of our mathematics education system.

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