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  1. Astract
  2. Introduction and Goals
  3. Craft and Science of Teaching and Learning Mathematics
  4. Curriculum Content
  5. Instructional Processes (Pedagogy)
  6. Assessment
  7. Closure

Craft & Science of Teaching & Learning


3.1. NCTM Standards. The National Council of Teachers of Mathematics published its Standards in 1989 and an updated, revised version Principles and Standards for School Mathematics in 2000. This document can be viewed as an overview of the Craft and Science of Teaching and Learning Mathematics. A large number of people worked on this revision effort for a number of years. One specific group, the Electronic Format Group, was charged to:

  • think of alternative ways to present and distribute the document that would result;
  • envision ways in which technology-based materials could be incorporated in the Standards;
  • keep the Standards 2000 Writing Group up-to-date on uses of technology;
  • assist in the work of the Standards 2000 Writing Group by finding examples of appropriate uses of technology.

The NCTM first recommended use of calculators and computers in school mathematics in 1980, a year after the National Council of Supervisors of Mathematics made such a recommendation. Notice the 2nd and 4th of the four bulleted items given above.

While the NCTM Standards are widely recognized and adopted, many states have developed their own versions of standards. Thus, there are state to state variations in math standards in the US.

In some sense, global standards for school mathematics have developed over the years. There have been several international studies of student achievements in mathematics. In recent years, the Third International Mathematics and Science Study (TIMSS) has received a lot of attention. Many educators, political leaders, and others in the US are not happy with the outcomes.

Activity 3.1: Clayton Christensen wrote a book titled The Innovators Dilemma in which he explores what happens when powerful new ideas and technology collide with existing companies and institutions. Quoting Dr. Christensen:

The dilemma is that the criteria that managers use to make the decisions that keep their present businesses healthy make it impossible for them to do the right thing for their future. What's best for your current business could ruin you for the long term. They've been trained to listen closely to customers and do things that improve their margins. Those things are mandatory to keep your present business healthy and stock price high. But they can prevent you from addressing the Internet.

In small groups, discuss your thoughts on ICT being a disruptive technology in education or in math education. We will debrief in the whole workshop group.

3.2. Mathematical expertise (expertise in being a mathematician). A simple way to think about goals in math education is to talk about a student developing an increasing level of mathematics expertise through this program of study. A very young infant has some mathematical expertise as well as the potential to gain a great deal more expertise. This expertise includes:

  • learning mathematics and learning to learn mathematics;
  • posing mathematical problems and representing problems as mathematical problems; and
  • solving math problems.

We can then talk about increasing a student's level of mathematical expertise. An important component of this expertise is being able to "use" one's mathematical knowledge when it is applicable to addressing problems in whatever environment or situation one encounters. One way to think about this is we want each student to be a mathematician at a level appropriate to the mathematical expertise the student has gained.

The following diagram illustrates a general-purpose expertise scale.

One might use such a scale in thinking about one's expertise in various general aspects of mathematics such as:

Mathematics as a human endeavor. For example, consider the math of measurement of time, distance, etc. There is a rich history of human development of these powerful aids to our modern society. Or, think about math in art, music, and other non-math fields.

Mathematics as a discipline. This is the "traditional" content of school math.

Mathematics as an interdisciplinary language and tool. To a large extent, students learn to think of what they learn in math courses as math--and they learn to use it only in math courses. Transfer of math learning is difficult for most students. But, math is an important component of each discipline. In recent years. "computational" (as in Computational Sciences) has grown to be an important component of each field of science.

The point is, mathematics is a broad and deep field. There are a variety of aspects of mathematics, as well as subfields of mathematics, in which one might gain an increasing level of expertise through formal and informal study and practice.

Activity 3.2: In small groups, make estimates on the amount of math education time and effort at the K-12 level that focusses on each of the three areas of expertise listed above. Discuss whether there are any good arguments for changing the current time and effort allocations to each of these three areas.

3.3 Mathematical Intelligence. Howard Gardner lists Logical/Mathematical as one of the eight human intelligences in his theory of Multiple Intelligences (MI). It is clear that there are huge variations in the innate levels of mathematical intelligence that people have. [See, for example: Center for Talented Youth at the Johns Hopkins University [Online]. Accessed 2/26/02: ]

The following definition appears in Moursund (1996, 2002) and is synthesized from the work of Howard Gardner, David Perkins, and Robert Sternberg.

Intelligence is a combination of the abilities to:

  • learn. This includes all kinds of informal and formal learning via any combination of experience, education, and training.
  • pose problems. This includes recognizing problem situations. and transforming them into more clearly defined problems.
  • solve problems. This includes solving problems, accomplishing tasks, answering questions, making decision, fashioning products, and so on.

One of the interesting things in this definition is that a person's intelligence (or, intelligence in a particular MI) can be increased through appropriate education, training, and experience. Note also that providing a person with appropriate tools (such as calculators and computers) and instruction in their use will improve their ability to solve certain math problems. Does this mean that it increases their Mathematical Intelligence?

Research evidence strongly supports the idea that a person's intelligence comes from a combination of nature and nurture. We can improve the nature's contribution by appropriate health care of pregnant women and young children. Removing lead from the environment makes a significant contribution to preventing a decline in intelligence, as does appropriate food for the growing child. Rich intelligence environments contribute significantly to improving intelligence. A number of researchers have noted a worldwide increase in IQ over the past few decades.

Activity 3.3A: In small groups, discuss how you deal with the tremendous variations in student math "IQ" as you teach math to students at a particular grade level or in a particular math course.

Activity 3.3B: In small groups, share your thoughts on IQ being something that is essentially fixed versus it being something that can be significantly changed through appropriate education and in other ways.

3.4. Developmental Theory. Piaget, as well as many others, did research on stages of development. Piaget, for example, talks about a child beginning at the level of Sensory Motor, moving to Preoperational, then Concrete Operations and eventually reaching Formal Operations. Click here to read about Piaget's theory of Cognitive Constructivism.

Activity 3.4A: In small groups, share your knowledge of evidence that the math being taught in the various grades and math courses in K-12 education is at a level that is consistent with the stages of development (stages of mathematical development) of students.

Here is another Website that ediscusses the work of Piaget.

Huitt, W. and Hummel, J. (January 1998). Cognitive Development [Online]. Accessed 2/28/02:
col/cogsys/piaget.html. Quoting from the Website:

Sensorimotor stage (Infancy). In this period (which has 6 stages), intelligence is demonstrated through motor activity without the use of symbols. Knowledge of the world is limited (but developing) because its based on physical interactions / experiences. Children acquire object permanence at about 7 months of age (memory). Physical development (mobility) allows the child to begin developing new intellectual abilities. Some symbolic (language) abilities are developed at the end of this stage.

Pre-operational stage (Toddler and Early Childhood). In this period (which has two substages), 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

Concrete operational stage (Elementary and early adolescence). In this stage (characterized by 7 types of conservation: number, length, liquid, mass, weight, area, volume), 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.

Formal operational stage (Adolescence and adulthood). In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts [emphasis added by Moursund]. Early in the period there is a return to egocentric thought. Only 35% of high school graduates in industrialized countries obtain formal operations; many people do not think formally during adulthood.

Developmental Theory is certainly applicable to learning mathematics. If we attempt to teach a math topic to a student who is far from being developmentally ready for it, the child tends to have little recourse but to attempt to "get by" by memorizing and regurgitating. Secondary school math teachers see this all the time, perhaps most especially in geometry courses and more advanced courses. The "Algebra for all." movement is suspect partly because it appears to be pushing many students into classes for which they are not mathematically developmentally ready. Algebra and Formal Operations seem to be closely related.

The table given below is from Huitt and Hummel (January 1998).

Notice that only 35% of students reach formal operations by the time they finish high school. This is suggestive that there is a substantial mismatch between our secondary school math curriculum and the developmental level of students.

Arnold Arons wrote extensively about the developmental level of college students in his science courses. He noted that a significant percentage (and, an even higher percentage of education majors) were not at the Formal Operations level. Thus, they learned science by rote memorization. He noted that about 10% of students in the General Physics course at the University of Washington (this is the course for students who are seeking a serious course in Physics and who have a strong high school math background) were not at the Formal Operations level. He developed an intervention that could help bring students to Formal Operations. It was many weeks in length.

Activity 3.4B: In small groups, continue the discussion begun in Activity 1.4A. Do you have an increasing or decreasing level of confidence in the scope and sequence of our math education system?

3.5. Intelligent Computer-Assisted Learning and Brain Science, ICT, and SoTL. Significant and rapid progress is occurring in Brain Science, the Science of Teaching and Learning, and ICT. This progress impinges on every academic discipline.

In this workshop, we are exploring how this progress might translate into improvements in Curriculum Content, Instructional Processes, and Assessment in math education.

The translation of theory into practice is a major challenge to education in each discipline. A standard approach is via staff development. But, there are other ways. One way is through the development of interactive, multimedia-based, intelligent computer-assisted learning (ICAL) materials that are based on the research findings. Progress is occurring in this area. It is important to note that once a high quality ICAL unit of instruction has been developed, it can be mass distributed. (We can discuss the meaning of "high quality." We can have high quality materials based on behavioral learning theory and that are nothing more than types of drill and practice designed to help a student more quickly gain speed, accuracy, and long term retention of procedural knowledge. We can also talk about ICAL designed to help students gain higher-order, problem-solving knowledge and skills.

Activity 3.5: In the whole workshop group, share examples of CAL or ICAL software that effectively incorporated modern ideas and research from SoTL in math.

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