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Goals in Math Education

Math is a very large, vertically structured field. There are many possible goals in Math Education, and various stakeholders hold strong (and, often differing) opinions on what these goals should be.

Mathematics and Math Education are large and complex fields. Math Education changes over time. This Website explores two current powerful change agents: Brain Science and ICT. This section of the Website provides information on several important areas that we need to understand as we explore possible changes in our Math Education system.

Weaknesses in Our Current Math Education System

Six General Principles Specified by the NCTM

Ten General Standards Specified by the NCTM

A Mathematics Expertise Scale Approach

Other Topics That Might be Added to This Page

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Weaknesses in Our Current Math Education System

Much of the weakness in our current Math Education system is historical in nature and can be discerned by carefully thinking about the following diagram. It is a simplified 4-step model of using mathematics to solve a problem.

Standard estimates are that about 80-percent of Math Education at the K-12 level is focused on part 2 of the diagram.

Historically, Math Education systems focused on helping students to learn to carry out a number of different types of "step 2" using some combination of mental and written knowledge and skills. It takes a typical students hundreds of hours of study and practice to develop a reasonable level of speed and accuracy in performing addition, subtraction, multiplication, and division on integers, decimal fractions, and fractions. Even this amount of instructional time and practice -- spread out over years of schooling -- tends to produce modest results. Speed and accuracy decline relatively rapidly without continued practice of the skills.

During the past 5,000 years there has been a steady increasing body of knowledge in mathematics, science, and engineering. The industrial age and our more recent information age has lead to a steady increase in the use of "higher" math in many different disciplines and on the job. Our Education System has moved steadily toward the idea that the basic computational numeracy described above is insufficient. Students also need to know basic algebra, geometry, statistics, probability, and other higher math topics.

As these topics began to be introduced into the general curriculum, a gap developed between the math that students were learning in school and the math that most people used in their everyday lives. More and more, Math Education focused on learning math topics in a self-contained environment where what was being learned had little immediate use in the lives of the students and little use in the lives of their parents.

A pattern of Math Education curriculum developed in which one of the main reasons for learning the material in a particular course was to be prepared to take the next course. Students developed little skill at transferring their math knowledge and skills into non-math disciplines or into problems that they encountered outside of school. Only a modest number of adults maintain the math knowledge and skills that they initially developed while studying algebra, geometry, and other topics beyond basic arithmetic.

That brings us up to current times. Many high schools require students to take three years of math (during their four years of high school) in order to graduate. There is considerable pressure to have all students take an algebra course. The nature of the instruction and the learning in many of these math courses follows the "80-percent on step 2" that has been noted above. Students are now learning the underlying concepts, or how to make use of the math in other courses or outside of a formal school setting.

As a final note in this subsection, the 4-step diagram represents only part of the field of math. For example, it does not include math as a human endeavor with its long and rich history.

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Here, and perhaps elsewhere in this Website, we want to emphasize that computers can carry out step 2 both accurately and rapidly. Thus, our current Math Education system is spending about 80-percent of its time teaching students to compete with machines in a domain where the machine is far far superior to even the best of humans.

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Note to self: The February 1999 issue of the Phi Delta Kappan is a theme issue focussing on The Mathematical Miseducation of America's Youth. It is an excellent resource.

Phi Delta Kappan. Select articles are available online. Accessed 8/15/01: http://www.pdkintl.org/kappan/karticle.htm. See specifically the following three articles from the February 1999 issue:

The Mathematical Miseducation of America's Youth: Ignoring Research and Scientific Study in Education, by Michael T. Battista

Parrot Math, by Thomas C. O'Brien

Technology, Children, and the Power of the Heart, by Matthew M. Maurer and George Davidson

Reference

Glazer, Evan (Date?). Technology Enhanced Learning Environments that are Conducive to Critical Thinking in Mathematics: Implications for Research about Critical Thinking on the World Wide Web [Online]. Accessed 1/14/02: http://www.arches.uga.edu/~eglazer/EDIT6400.html. Quoting the Abstract from this draft paper:

Mathematics education reform has emphasized a need for increased experience with technology and critical thinking skills in order to better prepare students for a modern society that is dependent on access to and use of information. However, the complexity and diversity of these skills leaves some uncertainty about their successful implementation in the mathematics classroom. The goal of this paper is to reveal a model that promotes critical thinking in a technology-enhanced mathematics classroom. Moreover, this paper will also examine critical thinking opportunities that are unique to technology-enhanced environments by examining discrepancies from critical thinking in non-technology enhanced environments.

A domain specific definition of critical thinking in mathematics will be formed in order to identify research and literature that supports this model. A review of literature suggests that environmental factors, such as the nature of a critical thinking task, the students' learning responsibilities, and the teacher's role to foster learning, influence and enhance critical thinking opportunities. The nature of each of these elements in technology and non-technology-enhanced environments will be elaborated through the model. The paper will conclude with implications for research about critical thinking using resources from the World Wide Web.

This Evan Glazer article includes a lengthy discussion of what is meant by critical thinking in general, and in mathematics. Suppose one decides that one of the goals of mathematics education is to develop critical thinking. Then, what does progress in Brain Science, ICT, and SoTL do to help us here?

Six General Principles Specified by the NCTM

The National Council of Teachers of Mathematics (NCTM) it the leading professional society for Pre K-12 Mathematics Education in the United States. Much of the contents of this section are direct quotes from NCTM [Online] Accessed 7/28/01: http://www.nctm.org/standards/overview.htm.

NCTM's Math Standards are based on six general principles:

  1. The Equity Principle

    Excellence in mathematics education requires equity--high expectations and strong support for all students.

  2. The Curriculum Principle

    A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.

  3. The Teaching Principle

    Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.

  4. The Learning Principle

    Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

  5. The Assessment Principle

    Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.

  6. The Technology Principle

    Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.

Comments on the Six General Principles

The Six General Principles are carefully thought out, powerful statements that NCTM feels should underlie mathematics in curriculum, instruction, and assessment. Here are a few observations that relate some of these principles to this workshop and Website.

  • Note that #3 is supportive of higher-order knowledge and skills -- learning for understanding, rather than rote learning.
  • Note that #4 is strongly supportive of Constructivism.
  • Note that #6 is supportive of a careful examination of roles of ICT in all aspects of mathematics curriculum, instruction, and assessment.

When people read a list such as the Six General Principles, many tend to accept what is listed as being complete and comprehensive. It does not immediately occur to them that the list may have major flaws, and that the list is a compromise developed by a large number of people working together over a long period of time.

For example look more carefully at #6. The statement "it [technology] influences the mathematics that is taught" does not provide us with much help in deciding what to add to the curriculum or what to drop from the curriculum. For example, should we drop paper and pencil long division of multi digit numbers from the curriculum because of calculators and computers? How about paper and pencil calculation of square roots?

Or, look more carefully at #4. Constructivism is an important learning theory. But, there are other learnng theories that might also be important to math education. Situated Learning is a good example. And, teaching for transfer and learning for transfer (that is, taking into consideratoin what we know about transfer of learning) would seem to be an important principle.

Activities

Activity: Working in small groups or individually, select one of the Six General Principles. Analyze it from varying points of view such as:

  • Is it consistent with and supported by your current understanding of Brain Science?
  • Is it consistent with and supported by your current understanding of ICT?
  • Is it consistent with and supported by your personal understanding and philosophy of mathematics and mathematics education?

Share your analysis. Does your analysis suggest changes that you might want to consider making in your teaching? If "yes," then suggest a specific change that you intend to try.

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Ten General Standards Specified by the NCTM

NCTM's Standards for school mathematics describe the mathematical understanding, knowledge, and skills that students should acquire from prekindergarten through grade 12. Each Standard consists of two to four specific goals that apply across all the grades. For the five Content Standards, each goal encompasses as many as seven specific expectations for the four grade bands considered in Principles and Standards: prekindergarten through grade 2, grades 3&endash;5, grades 6&endash;8, and grades 9&endash;12. For each of the five Process Standards, the goals are described through examples that demonstrate what the Standard should look like in a grade band and what the teacher's role should be in achieving the Standard. Although each of these Standards applies to all grades, the relative emphasis on particular Standards will vary across the grade bands.

Five Content Standards

  1. Number and Operations

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • understand numbers, ways of representing numbers, relationships among numbers, and number systems;
    • understand meanings of operations and how they relate to one another;
    • compute fluently and make reasonable estimates.

    Number pervades all areas of mathematics. The other four Content Standards as well as all five Process Standards are grounded in number.

  2. Algebra

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • understand patterns, relations, and functions;
    • represent and analyze mathematical situations and structures using algebraic symbols;
    • use mathematical models to represent and understand quantitative relationships;
    • analyze change in various contexts.

    Algebra encompasses the relationships among quantities, the use of symbols, the modeling of phenomena, and the mathematical study of change.

  3. Geometry

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
    • specify locations and describe spatial relationships using coordinate geometry and other representational systems;
    • apply transformations and use symmetry to analyze mathematical situations;
    • use visualization, spatial reasoning, and geometric modeling to solve problems.

    Geometry and spatial sense are fundamental components of mathematics learning. They offer ways to interpret and reflect on our physical environment and can serve as tools for the study of other topics in mathematics and science.

  4. Measurement

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • understand measurable attributes of objects and the units, systems, and processes of measurement;
    • apply appropriate techniques, tools, and formulas to determine measurements.

    The study of measurement is crucial in the preK&endash;12 mathematics curriculum because of its practicality and pervasiveness in so many aspects of everyday life. The study of measurement also provides an opportunity for learning about other areas of mathematics, such as number operations, geometric ideas, statistical concepts, and notions of function.

  5. Data Analysis and Probability

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them;
    • select and use appropriate statistical methods to analyze data; develop and evaluate inferences and predictions that are based on data; understand and apply basic concepts of probability.

    To reason statistically--which is essential to be an informed citizen, employee, and consumer--students need to learn about data analysis and related aspects of probability.

Five Process Standards

  1. Problem Solving

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • build new mathematical knowledge through problem solving;
    • solve problems that arise in mathematics and in other contexts;
    • apply and adapt a variety of appropriate strategies to solve problems;
    • monitor and reflect on the process of mathematical problem solving.

    Problem solving is an integral part of all mathematics learning. In everyday life and in the workplace, being able to solve problems can lead to great advantages. However, solving problems is not only a goal of learning mathematics but also a major means of doing so. Problem solving should not be an isolated part of the curriculum but should involve all Content Standards.

  2. Reasoning and Proof

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • recognize reasoning and proof as fundamental aspects of mathematics;
    • make and investigate mathematical conjectures;
    • develop and evaluate mathematical arguments and proofs;
    • select and use various types of reasoning and methods of proof.

    Systematic reasoning is a defining feature of mathematics. Exploring, justifying, and using mathematical conjectures are common to all content areas and, with different levels of rigor, all grade levels. Through the use of reasoning, students learn that mathematics makes sense. Reasoning and proof must be a consistent part of student's mathematical experiences in prekindergarten through grade 12.

  3. Communication

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • organize and consolidate their mathematical thinking though communication;
    • communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
    • analyze and evaluate the mathematical thinking and strategies of others;
    • use the language of mathematics to express mathematical ideas precisely.

    As students are asked to communicate about the mathematics they are studying--to justify their reasoning to a classmate or to formulate a question about something that is puzzling--they gain insights into their thinking. In order to communicate their thinking to others, students naturally reflect on their learning and organize and consolidate their thinking about mathematics.

  4. Connections

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • recognize and use connections among mathematical ideas;
    • understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
    • recognize and apply mathematics in contexts outside of mathematics.

    Mathematics is an integrated field of study, even though it is often partitioned into separate topics. Students from prekindergarten through grade 12 should see and experience the rich interplay among mathematical topics, between mathematics and other subjects, and between mathematics and their own interests. Viewing mathematics as a whole also helps students learn that mathematics is not a set of isolated skills and arbitrary rules.

  5. Representation

    Instructional programs from prekindergarten through grade 12 should enable all students to--

    • create and use representations to organize, record, and communicate mathematical ideas;
    • select, apply, and translate among mathematical representations to solve problems;
    • use representations to model and interpret physical, social, and mathematical phenomena.

    Representations are necessary to students' understanding of mathematical concepts and relationships. Representations allow students to communicate mathematical approaches, arguments, and understanding to themselves and to others. They allow students to recognize connections among related concepts and apply mathematics to realistic problems.

Comments on the Ten General Standards

Remember that formal mathematics is a 5,000 year old discipline. It is a cumulative, vertically structured discipline. Thus, the work of Pythagorus and Euclid, more than 2,000 years ago, is still important in our current mathematics curriculum. Currently there are thousands of mathematics researchers throughout the world who are building new results on the old, seeking out new mathematical ideas, and continuing to build the accumulated knowledge of the field.

The Ten General Standards can be viewed as ten strands that weave together. They cover topics that the NCTM considers to be important in our current society. Knowledge and skills in the areas covered is useful in many different aspects of learning, play, work, and being a responsible adult in our society.

As with the Six General Principles, we should recognize that the Ten General Standards were developed by a large number of people working over a long period of time. Many compromises had to be made. And, it may well be that continued rapid changes in ICT and Brain Science are not adequately reflected in the Standards.

For example, consider the "compute fluently" part of the third bullet in Standard #1. What should the specific goals be? Should we try to "train" students to have a substantial level of speed and accuracy in carrying out multi digit computations with integers, decimals, and fractions? It takes a great deal of effort over a number of years for a student do meet current paper and pencil computational standards. One might ask whether this is an appropriate use of a student's time -- especially given the fact that most students are unable to develop and maintain a high level of speed and accuracy. (A passing score on a computation test might be 70% or 80%. In multi digit computations, completing one computation per minute might be considered a high rate of speed. This is not a very useful level of speed and accuracy. We expect a computer to carry out billions of such computations in a few seconds. with no errors.)

Activities

Activity: Working in small groups or individually, select one of the Ten General standards. Analyze it from varying points of view such as:

  • Is it consistent with and supported by your current understanding of Brain Science?
  • Is it consistent with and supported by your current understanding of ICT?
  • Is it consistent with and supported by your personal understanding and philosophy of mathematics and mathematics education?

Share your analysis. Does your analysis suggest changes that you might want to consider making in your teaching? If "yes," then suggest a specific change that you intend to try.

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A Mathematics Expertise Scale Approach

The material given in this section is original, not quoted from the NCTM Standards.

For any domain of human endeavor, one can think about levels of performance along an Expertise Scale.

When a learner first begins informal and formal instruction in a given domain, the learner is a novice. With training, education, and experience over a period of time, the learner moves up the scale. The scale point "Useful Level of Competence" is not carefully defined. It varies with the learner. And, since "useful" may also be defined by people other than the learner, the definition may well change over time and be dependent on the society that is making the definition. At the current time, a medical doctor with knowledge and skills that were the standards of medical school in 1900 would not currently be considered to have a Useful Level of Competence in medicine.

Earlier on this web page we noted that the Ten NCTM Standards can be considered as interwoven strands. Let's look at this idea in a simpler setting. Here are three major types of domains that might form the basis for setting goals in mathematics education:

  1. Mathematics as a human endeavor. Mathematics has a very long history. Mathematics has beauty. Mathematics is an important aspect of aspect of past and current cultures. Mathematics is "the queen of the sciences."
  2. Mathematics as as an interdisciplinary language and tool. Mathematics can be used to help represent, communicate about, and solve problems in many different disciplines. Many jobs and other aspects of responsible adult life in our society require some mathematical knowledge and skills.
  3. Mathematics as a discipline. The formal study of and research in mathematics is at least 5,000 years old. It is a deep and wide discipline with a huge amount of accumulated knowledge.

Think about a math educator's Expertise Scale for each of these three domains.

A person who teaches mathematics has some level of expertise along each of these three expertise-strand scales. Presumable the combination of these levels of expertise is adequate to meet contemporary standards for being a teacher of mathematics. Unfortunately, for many math teachers, this is not the case. Moreover, a math teacher is faced by difficulties such as:

  1. Contemporary standards increase with time (consider the medical doctor of 1900) and with the changing needs of a society.
  2. A person's knowledge and skills within a domain tend to decrease over time if they are not frequently used. (Do you remember all of the details of the math you learned in high school and college?)
  3. ICT and Brain Science are rapidly expanding domains of knowledge and skills that are important in each of the three mathematics domains that are listed above.

Activities

Activity 1: Working alone, decide on a point on each of the three scales that you feel would most appropriately represent the level of expertise that a math teacher doing your math teacher work should have at the current time. Describe each of these three points in a manner that communicates well to you and can be used to communicate to another person. Then, place yourself onto each of the three scales. Analyze your strengths and weaknesses relative to the standards that you have defined.

Activity 2: Next, add two more expertise domains to the diagram. One is labeled "ICT and Mathematics Education" and the other is named "Brain Science and Mathematics Education." Repeat Activity 1 for these two new scales. Share your insights in a small group discussion.

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Other Topics That Might be Added to This Page

  1. Incremental and continuous change versus disruptive (large jump) change. Clayton Christensen's work in the business field.
  2. Discussion of varying points of view both as to what constituters mathematics at the K-12 levels, and what it means to improve math education at these levels.
    1. Totality of accumulated math data, information, knowledge, & wisdom is huge and growing.
    2. Societal needs in this area have changed and are changing.
    3. There are diverse opinions of goals, how to reach them, & how to measure progress.
  3. Learning to learn. To what extent do we help our students learn how to learn mathematics and to be independent, self-sufficient learners, lifelong learners of mathematics? This seems like a particularly important topic.

    The human mind (at both a conscious and unconscious level) is designed to learn and is a lifelong learner. (It perceives and processes threats and opportunities, and learns from doing so.) Thus, when we talk about being a lifelong learner, we need to specify more carefully what we really want to have happen. We are defining some external things (such as the steadily accumulating totality of data, information, knowledge, and wisdom) and indicating that a person should actively, consciously engaged in learning certain parts of this. Hmm. We then need to consider:

    • New dimensions, such as distance learning, computer-assisted learning, intelligent computer-assisted instruction, learner-centered software, and brain theory.
    • "Just in time" learning.
    • Continual learning (a routine, everyday part of one's job and life).
  4. Modeling and Simulation. I assume that this topic is one of the major themes in mathematics education. Certainly ICT plays a significant role in it.
    • The 1998 Nobel Prize in chemistry was awarded to two computational chemists.
    • Computer-based modeling and simulation are now a powerful aid to knowing and doing all of the sciences as well as many other disciplines such as economics and architecture.
    • An excellent example of use of mathematical modeling in disease control is given at:

      Researchers' Mathematical Model Provides Chagas Disease Insights. NSF News Release 24 July 2001. [Online]. Accessed 7/27/01: http://www.nsf.gov/od/lpa/news/press/
      01/pr0159.htm

  5. ICT-Assisted Problem Solving.
    • One of the most useful strategies in problem solving is breaking big problems into smaller, more manageable sub problems.
    • Increasingly, IT is a tool that can solve these sub problems -- thus, greatly increasing the problem-solving capabilities of computer users. (This ties in with Effective Procedures.)
    • Trial and error -- or exhaustive search.
    • Library research (for example, using the Global; Library that we call the Web).
    • Graphing, motion graphics, and other aids to visualization
  6. ICT as Content of Various Parts of Mathematics. Examples include spreadsheet, geographic information systems, computer-aided design, and mathematics systems such as Mathematica and Maple.

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