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Computational Thinking and Math Maturity: Improving Math Education in K-8 Schools (June 2006)

Improving Math Education in Elementary Schools: A Short Book for Teachers ( 2005)

From this Website you can download at no charge the following two books:

Moursund, D.G. (June 2006). Computational Thinking and Math Maturity: Improving Math Education in K-8 Schools.

Moursund, D.G. (January 2005). Improving Math Education in Elementary Schools: A Short Book for Teachers.

The more recent of the two books is a substantial revision of the January 2005 book. The amount of revision was so large that I changed the title of the book to better represent its content and focus.

The books are designed to be used as a supplementary text in two types of preservice and inservice courses for elementary school and middle school teachers.

  • Math Methods courses for elementary and middle school teachers.
  • Math Content courses for elementary and middle school teachers.

PDF file of the 109 page June 2006 book.

Microsoft Word file of the 109 June 2006 book. Note that this may download the document to your desktop so that you need to open it from there. The file name is K8-Math.doc.

Preface of the June 2006 book.


PDF file of the 89 page January 2005 book.

Microsoft Word file of the 89 page January 2005 book. Note that this may download the document to your desktop so that you need to open it from there. The file name is ElMath.doc.

Preface of the January 2005 book.


Additional free books and other materials written by Moursund.

Related information and references that I have found interesting. This is the spot where I collect materials for future revisions.

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Preface to the June 2006 Edition

The saddest aspect of life right now is that science
gathers knowledge faster than society gathers
wisdom. (Isaac Asimov, 1988)

An educated mind is, as it were, composed of all
the minds of preceding ages. (Bernard Le Bovier
Fontenelle, mathematical historian, 1657-1757)

This book is motivated by the problem that our K-8 school math education system is not as successful as many people would like it to be, and it is not as successful as it could be. It is designed as supplementary material for use in a Math Methods course for preservice K-8 teachers. However, it can also be used by inservice K-8 teachers and for students enrolled in Math for Elementary and Middle School teachers’ courses.

Many people and organizations have put forth ideas on how to improve our math education system. However, in spite of decades of well-meaning reform effort, national assessments in mathematics at the precollege level in the United States do not indicate significant progress. Rather, scores on these national assessments have essentially flat lined during the past 40 years.

The results of the past 40 years of attempts to improve math education suggest that doing more of the same is not likely to improve the situation. We can continue to argue about whether back to basics or a stronger focus on new math is the better approach. From time to time, both such approaches have produced small pockets of excellence. In general, however, our overall math education system is struggling to achieve even modest gains.

This book draws upon and explores four Big Ideas that, taken together, have the potential to significantly improve out math education. The Big Ideas are:

  1. Thinking of learning math as a process of both learning math content and a process of gaining in math maturity. Our current math education system is does a poor job of building math maturity.
  2. Thinking of a student’s math cognitive development in terms of the roles of both nature and nurture. Research in cognitive acceleration in mathematics and other disciplines indicates we can do much better in fostering math cognitive development.
  3. Understanding the power of computer systems and computational thinking as an aid to representing and solving math problems and as an aid to effectively using math in all other disciplines.
  4. Placing increased emphasis on learning to learn math, making effective use of use computer-based aids to learning, and information retrieval.

Math Maturity

Math maturity is a relatively commonly used term, especially in higher education. In higher education, the dominant components of math maturity are “proof” and the logical, critical, creative reasoning and thinking involved in understanding and doing proofs. The focus is on mathematical thinking, on being able to read and write math, and on being able to learn math using a wide range of resources such as print materials, courses, colloquium talks, and so on.

Many of the same ideas are applicable to defining math maturity at the precollege level. However, the cognitive development work of Piaget and others provides another quite useful approach. Piaget’s four-stage cognitive development scale is useful in tracking and facilitating cognitive development through the levels: sensory motor, preoperational, concrete operations, and formal operations. Piaget and many more recent researcher have recognized that one can look at the formal operation end of this scale both in general, and also in specific disciplines. Thus, we can explore the math education curriculum in terms of how well it helps students gain in math cognitive development.

The past 20 years have brought quite rapid progress in cognitive neural science and other aspects of brain science. Researchers have gained considerable insight into how the brain functions in math learning and math problem solving. This research is beginning to contribute to the design of more effective aids to learning math and to increasing math maturity.

Nature and Nurture

People are born with a certain “amount” of innate mathematical ability. In dealing with quantity, for example, this innate ability roughly corresponds to dealing with 1, 2, 3, and many. Howard Gardner has identified logical/mathematical as one of the eight multiple intelligences that in his theory of intelligence.

However, most of what we call mathematics has been invented by people. It is part of the accumulated knowledge of the human race, and it is passed on from generation to generation by informal and formal education. Children who grow up in a hunter-gather society do not learn the types of math that we expect children to learn in our information age society.

In recent years, use of brain imaging equipment and brain modeling using computers have been added to earlier tools used to study cognitive development. Research in cognitive development and cognitive acceleration suggests that our informal and formal educational system could be doing much better.

Computational Thinking

Many people now divide the discipline of mathematics into three major sub disciplines: pure math, applied math, and computational math. The term computational denotes the study and use of computer modeling and simulation. The table in figure P.1 contains data from Google searches on the three sub disciplines.

Search Expression

Google Hits 5/10/06

"applied math” OR “applied mathematics"


"pure math” OR “pure mathematics"


"computational math” OR “computational mathematics"


Figure P.1 Google searches on some math sub disciplines

Of the three sub disciplines listed, the most recent to emerge is computational. The addition of computational as a subdivision of math and various other disciplines has occurred because of the steadily increasing role of computers as an integral component of the content of many different disciplines. For example, in 1998 one of the two winners of the Nobel Prize in Chemistry received the prize for his previous 15 years of work in computational chemistry. He had developed computer models of chemical processes that significantly advanced the discipline of chemistry.

Similarly, physics is now divided into the three components: theoretical, experimental, and computational. Here is a brief quote from page 2 of the April 22, 2006 issue of Science News:

When black holes collide, they cause surrounding space-time to wiggle, generating a torrent of radiation known as gravitational waves. That’s what Einstein’s general theory of relativity predicts, but computer models [modelers] have struggled for more than 30 years to reproduce those waves. Because of the relativity theory’s mathematical complexity and the extreme gravity of black holes, modelers haven’t succeeded in getting black holes to crash.

Now, two teams independently reported that they have successfully simulated the merger of two black holes and the event’s production of gradational waves. [Bold added for emphasis.]

As you can see, computational means far more than just doing arithmetic calculations. Indeed, it has emerged as a way of thinking.

An excellent, brief introduction to computational thinking is provided in Jeannette Wing (2006). She is the Head of the Computer Science Department at Carnegie Mellon University. Quoting from her article:

Computational thinking builds on the power and limits of computing processes, whether they are executed by a human or by a machine. Computational methods and models give us the courage to solve problems and design systems that no one of us would be capable of tackling alone. Computational thinking confronts the riddle of machine intelligence: What can humans do better than computers, and What can computers do better than humans? Most fundamentally it addresses the question: What is computable? Today, we know only parts of the answer to such questions.

Computational thinking is a fundamental skill for everybody, not just for computer scientists. To reading, writing, and arithmetic, we should add computational thinking to every child’s analytical ability.

Learning to Learn Math

All teachers recognize that to be effective, they need to know the content they are teaching and they need to know how to teach the content. Much teaching knowledge and skill cuts across the school disciplines. However, there is considerable discipline-specific pedagogical knowledge and skill for each discipline. To be an effective teacher of math, one needs to know math and one needs to have significant math pedagogical knowledge and skill.

A somewhat similar idea holds for learning math. The human brain is naturally curious and has an innate ability to learn. A child is born with a modest amount of math capability, such as being able to distinguish among quantities such as one, two, and three. As a person’s brain grows and matures, one’s innate mathematical ability grows.

However, mathematical development depends heavily upon the informal and formal math learning environments that are available to the learner. In addition, math development is highly dependent on learning to learn math—in making progress toward being a more effective and efficient learner of math.

A good example of what is entailed by this is inherent to the idea of reading across the curriculum. We know that there is a difference between general reading skills and discipline-specific reading skills. We also know that students first learn to read and eventually can read to learn. From a math education point of view, students need to learn to read math. Progress in this endeavor is important to learning math by reading. Our current math education system is weak in helping students learn to read math and they learn math through reading.

Nowadays, most students have relatively easy access to the world’s largest library—the Web. Thus, as they learn to read math and to learn math by reading, they can take advantage of the math components of this huge and steadily growing library. Because math is an important component of many disciplines, learning to read math is an important part of learning to read across the curriculum

Computers have also brought us computer-assisted learning (CAL). In recent years, some of the best CAL falls into the category highly interactive intelligent computer-assisted learning (HIICAL). Such materials are a powerful aid to learning. Research on HIICAL in math suggest that some of the available materials are considerably more effective aids to student learning than are the traditional aids.

Contents of this Book

The 10 chapters of this book weave together various approaches to the four Big Ideas discussed above. Each chapter includes a set of activities for preservice and inservice teachers who are studying this book, and a set of activities useful in working with K-8 students. The latter activities are quite general, and they certainly do not constitute a curriculum that can be picked up and implemented at the K-8 level. Rather, they suggest some ideas to explore with young students and to try out in the K-8 curriculum.

Final Comment

As with most of my current writing efforts, this book is a “work in progress.” It is regularly being added to and revised. Your input and suggestions are welcome.

The change in title from the previous edition represents my growing insights into the problems faced by our math education system.

Dave Moursund

June 2006

Preface to the January 2005 Edition

This short book addresses the problem that our elementary school math education system is not as successful as many people would like it to be, and it is not as successful as it could be. It is designed as supplementary material for use in a Math Methods course for preservice elementary school teachers. However, it can also be used by inservice elementary school teachers and for students enrolled in Math for Elementary Teachers courses.

Procedures and Procedural Thinking

One of the big and unifying ideas in this book is procedures and procedural thinking. From the point of view of the elementary school math teachers, a major goal is to help students learn some math procedures and learn how to think in terms of using these procedures to solve problems. The same idea lies at the core of the field of computer and information science. However, there is a difference between how math people and computer people approach the big idea of procedure. They both think about two kinds of procedures:

  1. Algorithms. These are step-by-step procedures that can be proved to solve a certain type of problem or accomplish a certain type of task in a finite number of steps. You know algorithms for addition, subtraction, multiplication, and division of integers. You know many other algorithms, such as an algorithm for alphabetizing a list of words and an algorithm for looking up a word in a dictionary, and determining whether it in or is not in the dictionary you are using).
  2. Heuristics. These are step-by-step procedures that are designed to solve or help solve a certain type of problem or accomplish a certain type of task, but are not guaranteed to actually do so. As you work to solve a challenging math problem, you likely use heuristic procedures such as draw a picture, look up information in a book, ask a friend, attempt to break the problem into a set of smaller problems, and guess and check.

Computer people think specifically about creating and using procedures that can be carried out by a computer, while math people tend to focus their attention on procedures that can be carried out by people. Of course, many people are both math and computer oriented, and the disciplines of math and computers strongly overlap. People and computers working together can outdo people alone or computers alone in a very wide range of problem-solving situations.

Some Big Ideas

The math content level of this book is very modest, and the math prerequisite is also very modest. However, there are a number of ideas that require use of higher-order thinking and the ability to quest deeply for meaning and understanding.

For example, think about problem solving. Because this is a math-oriented book for elementary teachers, your first thoughts might be about the types of math problems students learn to solve while in elementary school. But, expand your thinking. Problem solving is an important aspect of every academic discipline. Of course, the nature of problems in the language arts, science, and social sciences is different than the types of problems students learn about in math.

You know that math provides tools useful in problem solving in every academic discipline. This may raise some Big Ideas questions in your mind, such as:

  1. How do I teach math in a manner that will help my students learn to make use of math as an aid to solving problems in math and in all the other academic disciplines, as well as in other parts of their everyday lives?
  2. What is the same and what is different in solving math problems versus solving problems in other academic disciplines?

When solving math problems—or problems in any other discipline—a person makes use of their brain as well as a wide range of tools. Pencil and paper can be thought of as technology-based tools designed to help in representing and solving math problems as well as problems in other disciplines. Here are two Big Ideas related to tools and brains:

  1. Computers—more generally, Information and Communication Technology—provide a wide variety of aids to representing and solving problems in all academic disciplines.
  2. There is a huge and steadily growing collection of information about the human brain and mind. Research in the areas of brain and mind is making a significant contribution to the science of teaching and learning. [Some good, short articles on brain science and education are available at Sylwester (2004).]

This book provides a brief introduction to math education aspects of the craft and science of teaching and learning. The four Big Ideas help to unify this book. Each chapter ends with a few applications that can be used in teaching math at the elementary school, and then a set of activities targeted at readers of this book.

As with most of my current writing efforts, this book is a “work in progress.” It is regularly being added to and revised. Your input and suggestions are welcome.

One of My Pet Peeves

Finally, I close this Preface with one of my pet peeves. Elementary school teachers often talk to students about getting “the” right answer to a problem. Students grow up with the idea that each math problem has exactly one and only one right answer. This is a wrong concept. Read the math problem examples given below. In the future, I hope you will no longer talk about getting “the” right answer in math.

  1. Find two integers that are greater than 1 and less than 10. [There are lots of correct answers.]
  2. Find two odd integers that add up to an even integer. [There are lots of correct answers.]
  3. Find two even integers that add up to an odd integer. [There are no such integers.]
  4. Find an integer that lies between 0 and 1. [There is no such integer.]
  5. Find a fraction that lies between 0 and 1. [There are lots of correct answers.]

Dave Moursund
January 2005

Related Information I Have Found Interesting

AquaMOOSE 3D. Accessed 4/4/06: Quoting fro the Website:

AquaMOOSE 3D is a desktop 3D environment that lets you play with the behavior of parametric equations in three dimensions. You can create things that are beautiful both mathematically and artistically.


Jason Elliott, Lori Adams, and Amy Bruckman. "No Magic Bullet: 3D Video Games in Education." Proceedings of ICLS 2002, International Conference of the Learning Sciences, Seattle, WA, October 23-26, 2002. (PDF, ~3MB) [This is an intersting article reporting on a failed experiment in using the software in a high school course.]

Rodriguez, Nancy C. (February 2, 2006). State to require more math for diploma: High school rules take effect in 2012. The Courier-Journal. Accessed 2/3/06:
Quoting from the Website:

FRANKFORT, Ky. — Students in Kentucky's public schools will have to take more math to earn a high school diploma under a plan unanimously approved yesterday by the state Department of Education.

Beginning with the graduating class of 2012, students will be required to take a math class each year they are in high school and complete algebra I, geometry and algebra II.

Los Angles Times (January 30, 2006). A Formula for Failure in L.A. Schools. Accessed 1/31/06:
. The newspaper article discussses the impact of requiring California students to pass first year algebra to graduate from high school. In some sense, this is part of the "Algebra for all" movement. Such a movement does not take into consideration that a significant percentage of studetns may lack the mathematicla talent (math IQ) to progress that far in math. The failure rate also reflects poor teaching and learning at earlier grades. All in all, an interesting problem.

Benbow, C. & Stanley, J. (1983). An eight-year evaluation of SMPY: What was learned? Academic precocity: Aspects of its development. The Johns Hopkins University Press. Accessed 3/6/05:

Brooks, F.P. Jr. (n.d.). No Silver Bullet: Essence and Accidents of Software Engineering. Retrieved 6/16/06:

The inherent complexity of compute programming. Does some comparison with math and the physical sciences. Quoting from the Website:

The familiar software project, at least as seen by the nontechnical manager, has something of this character; it is usually innocent and straightforward, but is capable of becoming a monster of missed schedules, blown budgets, and flawed products. So we hear desperate cries for a silver bullet--something to make software costs drop as rapidly as computer hardware costs do.

But, as we look to the horizon of a decade hence, we see no silver bullet. There is no single development, in either technology or in management technique, that by itself promises even one orderof- magnitude improvement in productivity, in reliability, in simplicity. In this article, I shall try to show why, by examining both the nature of the software problem and the properties of the bullets proposed.

Skepticism is not pessimism, however. Although we see no startling breakthroughs--and indeed, I believe such to be inconsistent with the nature of software--many encouraging innovations are under way. A disciplined, consistent effort to develop, propagate, and exploit these innovations should indeed yield an order-of-magnitude improvement. There is no royal road, but there is a road.

I believe the hard part of building software to be the specification, design, and testing of this conceptual construct, not the labor of representing it and testing the fidelity of the representation. We still make syntax errors, to be sure; but they are fuzz compared with the conceptual errors in most systems. If this is true, building software will always be hard. There is inherently no silver bullet.

NCTM (n.d.). NCTM Position Statements and Positions. Accessed 8/31/05: Quoting from the Website:

NCTM Positions and Position Statements define a particular problem, issue, or need and describe its relevance to mathematics education. Each statement defines the Council's position or answers a question central to the issue. The NCTM Board of Directors approves all positions and position statements.

Recent Updates

Highly Qualified Teachers (PDF, 25 KB) (July 2005)

Computation, Calculators, and Common Sense (PDF, 29 KB) (May 2005)

Closing the Achievement Gap (PDF, 78 KB) (April 2005)

Young, Maxine L. (May 2000). Working Memory, Language and Reading. Accessed 7/23/05:

Is working memory different from short- or long-term memory? How does it affect language and reading ability in children? In the 1980s, two English researchers named Baddeley and Hitch coined the term "working memory" for the ability to hold several facts or thoughts in memory temporarily while solving a problem or performing a task. Baddeley's research also showed that there is a "central executive" or neural system in the frontal portion of the brain responsible for processing information in the "working memory." He coined the term "articulatory loop" for the process of rapid verbal repetition of the to-be-remembered information, which greatly helps maintain it in working memory.

Working memory plays an important role in math also. When a child does a page of simple single-digit math, with alternating rows of addition and subtraction problems, it is working memory that helps the child remember to add or subtract the entire row. Children use a form of working memory, called serial memory, to count the number of cookies on a plate when figuring out how many are left for lunch the next day. Remembering not to count any cookie more than once is also a function of serial memory. Adults use working memory when keeping the total price of groceries in a cart in mind, as each new item is added, so as not to exceed a predetermined amount.

The above problems may not be so obvious. So what are other indications of problems with working memory? How would a parent or teacher even begin to suspect that such problems exist? Some of the following "red flags" could indicate the presence of working memory problems: a) trouble following lengthy directions, b) problems understanding long spoken sentences, c) difficulty staying on topic in conversations, d) difficulty with multistep math problems, e) problems with reading comprehension, or f) memory problems. If an individual is suspected of having memory problems, there are several tests that can be used to distinguish between weakness with working memory and other difficulties. It is important to determine if there are working memory limitations so that appropriate intervention can be implemented.

Ranpura, Ashish (June 2000). How We Rember, and Why We Forget. Accessed 7/23/05:

Scientists divide memory into categories based on the amount of time the memory lasts: the shortest memories lasting only milliseconds are called immediate memories, memories lasting about a minute are called working memories, and memories lasting anywhere from an hour to many years are called long-term memories.

Each type of memory is tied to a particular type of brain function. Long-term memory, the class that we are most familiar with, is used to store facts, observations, and the stories of our lives. Working memory is used to hold the same kind of information for a much shorter amount of time, often just long enough for the information to be useful; for instance, working memory might hold the page number of a magazine article just long enough for you to turn to that page. Immediate memory is typically so short-lived that we don't even think of it as memory; the brain uses immediate memory as a collecting bin, so that, for instance, when your eyes jump from point to point across a scene the individual snapshots are collected together into what seems like a smooth panorama.

Another way to categorize memory is to divide memories about what something is from memories about how something is done. Skills like catching a baseball or riding a bicycle are called nondeclarative memories because we perform those activities automatically, with no conscious recollection of how we learned the skills. Declarative memories, on the other hand, are memories of facts and events that we can consciously recall and describe verbally.

Graham-Rowe, Duncan (27 June 2005). New calculator makes solving tricky sums easy. news service. Accessed 6/29/05:

The article itself does not impress me, as the study done is not particularly meaningful. However, it does suggest that I need to look for studies that have been done on error rates when students use calcualtors, and sources of these errors.

Viadero, Debra (October 13, 2004). Teaching Mathematics Requires Special Set of Skills: Researchers Are Looking for New and Better Ways of Mastering Concepts. Education Week. Accessed 2/23/05:
Quoting from the article:

Now, though, after 20 years of systematic study, Ms. Ball is beginning to get a handle on just what that special kind of knowledge is. With her university colleagues, she has defined what teachers need to know to teach math effectively and devised new ways of measuring whether they know it.

This is no mere exercise in academic hairsplitting. Ms. Ball’s work suggests that having what she calls “mathematical knowledge for teaching” matters for students’ learning. Her research shows that students whose teachers score high on the measures developed by Ms. Ball and her research partners learn more math over the school year than do students of low-scoring teachers.

David Klein, David et al. (January 2005). The State of State Math Standards 2005. Accessed 1/8/05:

Quoting from the Forward by Chester E. Finn, Jr.:

Though the rationale for changing the emphasis was not to punish states, only to hold their standards to higher expectations at a time when NCLB is itself raising the bar throughout K-12 education, the shift in criteria contributed to an overall lowering of state "grades." Indeed, as the reader will see in the following pages, the essential finding of this study is that the overwhelming majority of states today have sorely inadequate math standards. Their average grade is a "high D"—and just six earn "honors" grades of A or B, three of each. Fifteen states receive Cs, 18 receive Ds and 11 receive Fs. (The District of Columbia is included in this review but Iowa is not because it has no statewide academic standards.)

Tucked away in these bleak findings is a ray of hope. Three states—California, Indiana, and Massachusetts— have first-rate math standards, worthy of emulation. If they successfully align their other key policies (e.g., assessments, accountability, teacher preparation, textbooks, graduation requirements) with those fine standards, and if their schools and teachers succeed in instructing pupils in the skills and content specified in those standards, they can look forward to a top-notch K-12 math program and likely success in achieving the lofty goals of NCLB.

Math Scores Accessed 1/12/05


Important topics not yet adequately covered in the book

(Random thoughts)

1. Cognitive Acceleration is an important idea. I tend to think about it in two different ways. first, our schools are designed to help students learn more, better, and faster than they might if lf they were left to their devices. That is, one goal of schools is cognitive acceleration.

A second approach is to think in terms of domain independence. Cognitive development depends on nature and nurture. Create environments in which nature is given needed assistance, and nurture is designed to be broad-based.

2. Research into practice. HIICAL is key to this. What research can be translated into practice through the materials that can be mass produced and mass distributed—put directly into the hands of the learners? How do we educate students to take advantage of this approach to education? Learning to learn, learning to read to learn, learning to take individual responsibility for one's learning, learning to self assess—all are important ideas.

3. Reading in math. It seems to me there are several difficulties. First, there is the need to be able to read. Second, there is the need to understand what one is reading. This means that one must understand enough math to have a good chance to understand or to learn (while doing the reading) the needed new math. In that sense, reading math is often a learning process, building upon previous learning. Third, there is the need to read problems in each discipline where one wants to use math as an aid to solving the problem. This means that one needs to read the content area with understanding, understand the math-related aspects of the content, understand the math needed to represent the problem, and then proceed to use math to solve the problem. Then, one moves backward, interpreting the results in light of the original problem situation, coming to conclusions applicable to the non-math discipline.

Word problems (story problems) are an effort to provide practice in the above. However, the above tends to be overwhelming, as it requires expertise in reading in general, reading in the content area of the problem, developing a math representation of a problem in that content area, solving a math problem, and then interpreting the results in the original discipline area. For that reason, most story problems tend to require very little knowledge of the non-math content area. Many are written in kind of (a pidgin math??) language, so that students can be taught to translate from that language into a pure math problem.

I don't know whether I believe the last part of the above paragraph. But, the first part is important. In real world problem solving, there is context that helps a lot in determining if one is going in a right direction and if answers being produced make any sense. Often this context is lost, or difficult to discern, in story problems.

The above is all stated in terms of reading. Actually, it needs to be broadened. One is in an environment that includes problem situations. One uses one's senses to gain information about some of the problem situations. One then poses (math) problems corresponding to some of these problem situations. The environment may be impinging on all of one's sensory organs. Some of the problem situations may lend themselves to being translated into math problems. This depends on one's knowledge and skills in maht, and one's math problem-posing abilities.

4. One of the most important aspects of math is that it a discipline (more so than any other discipline, I think) in which knowledge can be accumulated in a manner that one can use it and build upon it. An interesting way to think about this is to consider the large education problem of translating theory into practice. Each discipline has accumulated results, knowledge, theory, or whatever. Some of the results, theory, etc. is strong enough, good enough, correct enough so that one cane safely build upon it. Thus, in every discipline, one aspect of problem solving is to translate a discipline-specific problem situation into a discipline-specific problem and then make use of accumulated results in the field to try to solve the problem.

The problem posing and translation process tends to be dependent on having an appropriate level of expertise in the discipline. The translation process (problem posing and representation process) may well result in a problem that has been solved before. Thus, accessing the accumulated results in a discipline is important. This requires information retrieval skills as well as knowledge of the discipline. I frequently have trouble retrieving information because I do not know the vocabulary of the discipline in which I need information. I cannot accurately represent my needs, because I do not speak/read the language with good understanding.

This provides another argument for learning to read math (or to read, write, speak, listen math). In math education, it would be helpful to have a large set of information retrieval tasks requiring students to retrieve information, understand the information, and use the information to help solve a problem or accomplish a task. We need these in diverse content areas (to fit in with my story problem concerns), over the full range of math that we want students to be learning, and also outside of or helping to stretch this full range.

5. What makes one math book better than another? I am sure there are lots of answers. However, I think over the years the sense of direction has been to make math books "thicker" so that more and more of what we want students to be able to do is illustrated in an manner that they can "do" essentially by copying with little understanding. That is, authors strive to reduce the level ofs struggle faced by students. How does one write to create a correct level of struggle? What is correct for one student may not be correct for another. This suggests that attempts to reduce the level of struggle may be easier than attempts to create a struggle of approximately the right level. It also suggest the need for individualization, which is not possible in traditional printed text.

This fits in well with the supported reading idea of the UO Supported Reading CATE grant, and my recent discussion with Mark Horney. Each student needs to be faced by an appropriate struggle. Since the book cannot adjust to the student, we need to approach this by computerizing things, by providing variable levels and types of help, and by educating the learner to select a path that is has an appropriate level of help/support.

There are conflicting goals in this situation. One goal is to provide a level of support that is just right so that the student learns well and efficiently. The other goal is to help the student get better at learning with a lower level of support. The sense of direction might be to wean the level of support needed so that it is approximately the normal amount provided to students and other learners by the instructional materials that are most readily available.

6. Constructivism is certainly important in math. However, it is essential to construct knowledge that is consistent with existing knowledge. This is tricky. The precision of communication that is inherent to math is a key aspect of the discipline. Words such as function and variable to mathematicians hundreds of years toe develop and to agree upon definitions. Math research is then done and written using these long agreed upon definitions. My mental models (mind's eye pictures, etc.) for various math vocabulary and ideas will not be the same as what other people have. However, my use of the terms has to be of such a high level of precision so that the communication that results can stand the scrutiny of others and the test of time. There can be more than one correct proof of a theorem, and one proof may be more beautiful or elegant than another. However, math people need to agree upon what constitutes a correct proof. As I said earlier in this paragraph, this is tricky. I suspect that most math educators who talk about constructivism need to think more carefully.

7. Standards based math education. One the one hand, I certainly support the idea of developing standards and having learning and assessment related to standards. However, what if the standards are flawed. For example, we might set standards for student learning that are above the cognitive developmental level that many students are able to achieve. Or, we might set standards that do not appropriately take into consideration computers and/or other rapidly changing aspects of our world. My feeling is that we are making both of these mistakes.