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What is Mathematics?

Major Unifying Themes in This Document

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Foundational Information

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Computational Mathematics

The Future

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 Website Author
"Dr. Dave" Moursund

The Future

The next two decades will see major changes in our educational system due to continued rapid progress in Brain Science and ICT.

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We have a 5,000 year of history of development in the curriculum, instruction, and assessment of our formal math education system. During this time, a huge amount of new mathematical discovers occurred, and many of them gradually made their way into the K-12 curriculum. But, most of the math curriculum content at the K-12 level is is math that was developed hundreds of years ago.

Thus, as we look toward the future of K-12 math education, some of the things we will likely see include:

  1. While there will continue to be changes in the K-12 content, these will be based almost entirely on math that was developed hundreds of yeas ago. The math content changes will mainly be increases and decreases on the emphasis places on various topics. There will be a continuing emphasis on understanding of concepts and on problem solving.
  2. The major changes in math curriculum content will come about because of the steadily increasing capabilities and availability of ICT and the direct relevance of ICT in math curriculum content. More specifically, Computational Mathematics is of growing importance. [Eventually this document will contain a major section on Computational Mathematics. There is a strong parallel between Computational Mathematics and Computational Science. Computational Science is now one of the three main ways of knowing and doing science.]
  3. We will see increased use of ICT-based math applications in non-math disciplines, because ICT is an interdisciplinary tool that empowers users of math,
  4. Significant progress will occur in instructional methods (pedagogy). Our steadily increasing knowledge of Brain Science, Learning Theory, and Teaching Theory will all contribute to this progress.
  5. There will be continuing controversy on assessment. Math assessment has a long history of occurring in an environment in which students are allowed to use "traditional" math tools such as paper, pencil, and perhaps straight edge and compass. The past 20 years have see significant progress toward allowing students to use calculators while being assessed. The future will gradually bring an emphasis on hands on ICT while being assessed, and portfolio assessment.
  6. We will see continuing improvement in Computer-Assisted Learning (which, of course, includes the build-in "Help" features of all modern software tools. This will facilitate "Just in time." learning and review.
  7. Distance Learning and Computer-Assisted Learning will become more closely intertwined and will become important components of the instructional delivery system.

Each of these is affected by the continued rapid change of ICT capabilities and availability. (Such projections are a standard component of discussions about the future of ITC in education. A brief discussion of this topic is given at the end of this page.)

Each of these "predictions" is discussed in more detail in the sections that follow.

Math Curriculum Content

We do not have a Federal Government-mandated national curriculum in mathematics. However, the work of the NCTM, major book publishers, and testing agencies has led to a relatively uniform curriculum across the country. The curriculum content that we have is based on input from thousands of math education leaders and other people over a long period of time. Over the years, there has been a steady trend to:

  1. Move certain secondary school topics to lower grades. (For example, we are currently into an "Algebra for all." movement, with algebraic concepts and then formal algebra being introduced at earlier grades.
  2. Integrate some math topics of growing applicability and importance into the general curriculum. Probability provides a good example.

The NCTM curriculum content recommendations have a strong emphasis on students understanding of concepts, problem posing, and problem solving. There is a decreasing emphasis on rote memorization and on by-hand computation.

Activity 1: Many students get through certain math topics by rote memorization and by developing skill at plugging numbers in formulas and accurately carrying out the needed computations. One sees examples of this as students encounter topics such fractions, probability, and algebra. Working individually and then sharing in small groups, select a math topic that you know well. Think carefully about rote memorization of algorithms or formulas and developing skill in their use, versus understanding of the underlying concepts. What relationships can you see between the concepts and the computational processes? (For example, what does a student learn about the concepts of division by developing skill in paper and pencil long division of multi digit numbers? For another example, probably you have memorized the rule: To divide a fraction by a fraction, "invert and multiply." Think about the concept of division versus this computational algorithm. Also, think about whether you can explain why this algorithm works.)

Activity 2: "Algebra for all," along with an increased emphasis on standards and accountability, is a controversial movement in math education. Working in small groups, discuss possible roles of Brain Science and ICT in this controversy.

Roles of ICT in Math Curriculum Content

By the late 1970s, handheld calculators had become inexpensive enough so that many people begin to recommend their inclusion in the K-12 curriculum. In 1979, the National Council of Supervisors of Mathematics made such a recommendation. In 1980, the National Council of Teachers of Mathematics made this recommendation.

Now, more than 20 years later, we can look at what has transpired. From a hardware point of view, there has been considerable progress. A 5-function with memory handheld, solar cell powered calculator retails for $5 or less. Multifunction scientific calculators retail for under $15. Graphing and equation-solving calculators are available for under $100. Classroom sets of such calculators are available in most schools. And, of course, many students own such calculators or have ready access to them at home.

One can find a number of examples of calculator-related changes in curriculum content, instructional processes, and assessment. Here are four general examples:

  1. Certain curriculum topics have greatly changes. The by-hand calculation of square roots, and the use of math tables with interpolation to computer trig functions, have nearly disappeared from the curriculum. Computation using slide rules and using logarithms has nearly disappeared from the curriculum. The use of graphing calculators has led to significant changes both studying functions and solving equations.
  2. The use of calculators is allowed in certain types of national and state assessments.
  3. Most math textbooks at the K-8 level make mention of calculators. Some include a significant emphasis, while others downplay the topic.
  4. Many elementary teachers, especially in the primary grades, feel that student use of calculators is inappropriate. A commonly heard statement is, "I think it is okay for students to use calculators after they have mastered the computational algorithms [of multi digit addition, subtraction, multiplication, and division]."

What is important about the types of changes listed above is that the calculator is being used as a tool to do mathematics. There is a growing realization that memorizing algorithms and formulas, ad developing speed and accuracy in their use, has been over emphasized . The calculator makes it possible to place less emphasis on these topics, and shift some of this time to learning concepts, problem posing, and problem solving.

One might think that the same statements would hold for use of computers in K-12 math education. For the most part, however, this has not been the case. The computer has not been viewed as a tool to help represent and solve math problems. Rather, it has been viewed as a tool for instruction. The attitude of many teachers is expressed by the statement, "The computer is a tireless drill master, helping students master the basics of [computational] mathematics."

We are making steady progress toward providing every student with a computer to routinely use as they study and do math. Thus, it is becoming more and more feasible to provide students with good ICT as they study math, as they apply their math knowledge throughout the curriculum and their everyday lives, and as they are assessed in math. Here are a few important things to think about as this occurs:

  1. A computer is an information storage device (and, or course, can provide access to the Global Library that we call the Web.) Thus, we need to think in terms of math curriculum, instruction, and assessment in an environment in which the student has access to books and materials studied in the past and currently being studied, as well as a mathematical dictionary, a mathematical encyclopedia, lots of math reference books, and so on. This is a hugely changed environment from the current situation in which many teachers are loth to even allow students access to their current textbook while taking a test, and most students lack ready access to an adequate math reference library while they are studying and using math.
  2. The human-machine interface in a calculator is very poor relatively to what can be provided by a computer with a full sized keyboard and a decent sized screen. Moreover, the "Help" features in a computer can provide detailed, step by step, just in time instruction on use of software tools. Just in time learning and review of topics is a powerful aid to learning and using math.
  3. We now have available very powerful software that can solve a huge range of the types of problems that we currently teach students to solve by hand. (Indeed, some versions of this software are available on calculators. But, better user interface that computer can provide makes such tools much more user friendly on a computer.) This situation helps to make clear the issue of concepts versus processes. Computers can carry out mathematical processes. If such use of computers becomes routine in curriculum instruction, and assessment, then there will be much more time available to help students learn concepts, problem posing, and problem solving.

Activity 1: Working in groups of three, assign each person a different one of the three ideas listed above. Each person is to think about their topic, and then develop their own opinions of the opportunities and threats that the idea presents for the student and the teacher. Then each person is to do a brief presentation to the other members of the group, and facilitate a discussion on the topic.

Activity 2: Suppose that all students had ready access to ICT, math software applications, math library, and math Computer-Assisted Learning materials on a routine basis. In small groups, discuss how this would change math curriculum, instruction, assessment, and the general overall job of being a math teacher.

Use of Math in Non-math Disciplines

We all routinely make use of math. For example, think about the measurement of time (including day of the week, month of the year, the year), quantity (money, your salary, your age, your weight), location (how to go some place and get back home), and so on. We routinely make use of maps, graphical representations of data, and time schedules. Math has such general usefulness that it is one of the basics of education and is a routine part of the K-12 curriculum.

Math is useful across all of the disciplines that students study in K-12 education. Thus, curriculum designers face a significant transfer of learning problem. How can math curriculum, instruction, and assessment be designed to facilitate students being able to use their math throughout their current and future curriculum, jobs, adult lives, hobbies, and so on? Most math educators agree that we do not do very well in teaching math for such transfer of teaching, and there is considerable room for improvement in this area.

ICT is an interdisciplinary tool. This means that many different disciplines find it valuable to incorporate instruction about and use of ICT. Gradually, teachers in various disciplines are role modeling effective use of ICT in teaching, learning, and using the disciplines they teach.

This opens up opportunities for increasing the use of (ICT-enabled) math within the various disciplines. For example, consider a spreadsheet. The study and use of spreadsheets is a perfectly appropriate topic in math instruction at a variety of grade levels. But, a spreadsheet is a valuable tool in all of the sciences and social sciences, as well as in business courses. Thus, a math teacher who is teaching spreadsheet will want to make a determined effort to have students develop and use applications in a variety of disciplines. And, of course, similar statements hold for the use of the data graphing features that are built into modern spreadsheet software.

Another example is provided by statistical computations, mean, standard deviation, and such correlation., From a math teacher point of view, one measure of success would be seeing students taking the initiative to use various math-oriented ICT tools in non-math disciplines -- and even helping their teachers in these disciplines to learn how to make such use of ICT.

Activity: Select a math topic that has useful applications in non-math courses, and in which ICT is a valuable aid. Think about how you can help your students to learn the ICT aspects of this math topic in a manner that will facilitate students making this transfer of learning to other disciplines. Share in small groups.

Instructional Methods

Here are a few things that we know:

  1. The human brain has some innate abilities in counting and spatial relationships. Formal education can greatly enhance the usability and power of such innate abilities. A number of important ideas about this are covered in the book:
    1. Gardner, Howard (1991) The Unschooled Mind: How Children Think and How Schools Should Teach. Basic Books.

    Perhaps the main point that Gardner makes is that people can learn a lot without formal education. However, the contemporary standards in many disciplines can only be reached by concerted and extensive study and practice. Thus, our educational system should develop efficient and effective instructional methods to help students gain contemporary levels of expertise within the various disciplines covered in school.

  2. Each student brings their own knowledge, skills, attitudes, and other "baggage" to any particular learning task. New knowledge and skills are "constructed" by building on previous knowledge and skills. There are huge individual differences among students. Individualization of instruction makes a significant difference in the speed and quality of learning that occurs. (One way of looking at this is to compare the results of conventional instruction in classes of 20 to 30 students, with the results of individual tutoring by skilled tutors. Students learn significantly faster and better with individual tutors.)
  3. Learning can be substantially improved by actively engaging students in applying what they are learning in problem posing and problem solving environments, metacognition, project-based learning activities, and other ways that increase student engagement.
  4. It is important for teachers to role model being active, engaged learners and users of what they are teaching. For example, students need to see science teachers who do and use science in their everyday lives, writing teachers who write, art teachers who are artists, and so on.
  5. Cooperative learning and cooperative problem solving are effective instructional and learning methodologies.

Note to self: This section needs more work. Eventually we are looking for quite specific recommendations for things that a teacher can do to enhance the effectiveness of the instructional process. Here are a few thoughts:

  1. Teachers are being encouraged to make use of some form of Presentation Graphics (such as Microsoft's PowerPoint, or the Slide Show in AppleWorks) in their everyday teaching. I am not aware of solid research that indicates that this leads to students learning faster and better.)
  2. The research supporting Project-Based Learning and Problem-Based Learning is fairly strong. (Note that each is called PBL.)
  3. The Low-Road, High-Road model of Transfer of Learning is useful in teaching for transfer.
  4. Constructivism is an important Learning Theory for use in math education and many other disciplines.
  5. It is helpful to view mathematics as a language. It is effective to have students do journaling, peer tutoring, cooperative problem solving, and in other ways communicate mathematics in written and oral form.

Activity: In small groups, discuss uses of ICT-Assisted PBL in math education. What are strengths and weaknesses (opportunities and threats) of this form of instruction?

Assessment

Some Topics:

Authentic Assessment

Portfolio Assessment

Assessment in a Hands-on Environment

Computerized Adaptive Testing

Computer-Assisted Learning

Topics:

General information about CAL

Specific results in math education

Change in effective class size

 

 

Distance Learning

Topics:

Learning to learn in this environment

General research

Specific applications in math education

 

Gordon Moore's "Law" and Exponential Change

The density of transistors on a chip will double every 18 months, thus increasing the price performance of compute power by a factor of two every 1 1/2 years.

Note that bandwidth of connectivity is currently doubling approximately every nine months.

Storage has been increasing at a similar rate.

Super Computers

In 1983 there were 74 "Super Computers" in the world. Each had a "blinding speed" of 200 to 300 million operations per second.

This year (2001) many millions of people throughout the world are buying microcomputers that are several times this fast and cost about 1/5,000 to 1/10,000 as much in 1983 dollars.

My (no longer very new) laptop has approximately the same compute power as the Super Computer of 1983

The new iBook has an educational list price of $1,299. It is a 500 million operation/second machine. It is more powerful than the 1983 Super Computers.

Short Term Future

Today's Super Computers have speeds in the one to six teraflops (trillions of floating point operations per second) Thus, they are approximately a thousand times as fast as top of the line microcomputers.

Back in the early 1980s, a Super Computer had one or a modest number of Central Processing Units. Now, of course, higher overall speeds are being achieved by building a machine that incorporates a large number of relatively inexpensive microprocessors. There is a brief news item:

NSF Funds $53 Million Computing Grid. The National Science Foundation will fund development of a $53-million computing grid called the Distributed Terascale Facility (DTF), which when it is completed in 2002 will be more than 1,000 times faster than the IBM "Deep Blue" supercomputer famous for defeating chess champion Garry Kasparov in 1997. DTF will be used by four U.S. research centers: the National Center for Supercomputing Applications (NCSA) at the University of Illinois at Urbana-Champaign; the San Diego Supercomputer Center at the University of California at San Diego; Argonne National Laboratory; and the California Institute of Technology. The system will be built by IBM, using Intel Itanium McKinley chips and the Linux operating system, and more than 1,000 IBM server systems will be connected through a 40-gigabit-per-second network created by Qwest Communications. NCSA's Daniel Reed said the grid will transform "the way science and engineering research is done.'' (Reuters/New York Times 9 Aug 2001) http://partners.nytimes.com/
reuters/technology/tech-tech-supercomput.html (NewsScan Daily, 10 August 2001)

Deep Blue could analyze more than 200,000,000 chess board positions per second. It had 256 processing units. Thus, the gain in speed in the new machine is achieved by a combination of more processors and the greater speed of the processors.

The concept of a "grid" is very important. We are moving toward huge numbers of people having access to both the Web (for connectivity and the Global Library) and the grid (for access to processing power). A slightly different way of looking at this situation is that the Web will consist of connectivity, huge amounts of storage, and huge amounts of processing power. People using the Web will be able to draw upon the amounts of storage and processing that they need -- and typically will not have any idea where the storage or processing power is located.

15 Year Projection

Fifteen years is the time for 10 doublings of chip power if Moore's Law continues to hold. Ten doublings is a factor of 1,024.

This might mean that children now in the 1st grade will have routine access to computers that are a thousand times as fast as today's microcomputers -- that is, a trillion operations per second.

"Way Out" Future Ideas

Ray Kurzweil, Hans Moravec, and others indicate that another 15 years of Moore's law holding will bring us supercomputers with the capability of a human mind. Thirty years will see microcomputers with this level of capability.

The meaning of these forecasts is unclear to me. But, they seem relevant to education.

The following book takes a quite different approach to predicting the future:

Dertouzos, Michael (2001). The Unfinished Revolution: Human-Centered Computers and What They Can Do for Us. New York: HarperCollins Publishers. Quoting from the dust jacket:
If cars were as difficult to drive as our computers are to operate, they would never leave the garage. Yet every day we put up with infuriating complications and incomprehensible error messages that spew forth from out technology: software upgrades crash our machines, Web sites take forever to download. e-mail overwhelms us. We spend endless time on the phone waiting for automated assistance.

In effect, we continue to serve our machines' lowly needs, instead of insisting that they serve us -- a situation that will only get worse as millions of new mobile devices arrive on the scene.

Our world doesn't have to be this way. It shouldn't be this way. …

Activity: Continued progress in ICT hardware and software is steadily increasing the capabilities (the power, the knowledge) of ICT systems. In small groups, discuss some of the changes we should be making in our current math education system to prepare students to make effective use of current and future ICT as an aid to learning and using math.

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