eb-good math lesson plans
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Good Math Lesson Plans
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People interested in the Good Math Lesson Plans document are also apt to be interested
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• Math Project-based Learning.
• Communicating in the Language of Mathematics.
• Improving Math Education.
• Two Brains are Better than One.
• Math Education Digital Filing Cabinet.
Contents
[hide]
• 1 Introduction
• 2 A General-Purpose Lesson Plan
• 3 Discipline Specificity
• 4 What is Math?
o 4.1 Some Often-Quoted Answers
o 4.2 Patterns and Language of a Discipline
o 4.3 Some Important Math Concepts
o 4.4 Long-Enduring Results
• 5 Some Math-Specific Lesson Plan Topics
o 5.1 Increasing Math Expertise
5.1.1 Problem Solving
5.1.2 The Concept of Proof
5.1.3 Include a Focus on Important Problems
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o 5.2 Prerequisite, Review, and Remediation
5.2.1 Slower and Faster Learners 5.2.2 Student and Teacher Responsibilities
o 5.3 Teaching Self-Assessment and Self-Responsibility
o 5.4 Teaching for Transfer of Learning
o 5.5 Math Cognitive Developmental Level and Maturity Level 5.5.1 Cognitive Development
5.5.2 Math Cognitive Development 5.5.3 Math Maturity
o 5.6 Communication in Math
5.6.1 Communication and Math Content 5.6.2 Communication and Math Word Problems
o 5.7 Math Modeling
o 5.8 Computational Math
o 5.9 Lesson Plan as Self-Inservice Education
• 6 Some Roles of ICT
o 6.1 Content 6.1.1 Calculators
6.1.2 Computers
6.1.3 Information retrieval
o 6.2 Teaching and Learning
• 7 A "Full Blown" Math Lesson Plan Template
• 8 References
• 9 Authors
“Education is a human right with immense power to transform. On its foundation
rest the cornerstones of freedom, democracy and sustainable human
development.” (Kofi Annan; Ghanaian diplomat, seventh secretary-general of theUnited Nations, winner of 2001 Nobel Peace Prize; 1938-.)
"In a completely rational society, the best of us would be teachers and the rest of
us would have to settle for something less, because passing civilization alongfrom one generation to the next ought to be the highest honor and the highest
responsibility anyone could have." (Lee Iacocca, American industrialist; 1924-.)
Introduction
Lesson plans are a core theme of most preservice teacher education programs. Preserviceteachers learn how to create them, how to critique lessons others create, how to teach
working from a plan, and how to judge the results. By definition, a lesson plan is good to
the degree it helps teachers teach well and students learn well.
“Lesson plan” usually refers to a single lesson, designed for one class period. However, itcan also refer to a sequence of such plans designed for a unit of study. (Such a sequence
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may be called a unit plan.) In this document, “lesson plan” mean a plan to facilitate one
more times of organized teaching and learning.
The figure below shows that the need for written detail depends on lesson plan’saudience.
1. A personal lesson plan is an aid to memory that takes into
consideration one's expertise (teaching and subject area knowledge, skills,
and experience). It’s often quite short—sometimes just a brief list of topics
to be covered or ideas to be discussed. (For example: “Show how to derivequadratic formula by completing the square; then use spreadsheet to show
how to plug in values.” “Use Taxman software to introduce factoring.”)
2. A collegial lesson plan is designed for a limited, special audiencesuch as your colleagues, a substitute teacher, or a supervisor such as a
principal. It contains more detail than the first category. It is designed to
communicate with people who are familiar with the school and curriculumof the lesson plan writer.
3. A (high quality) publishable lesson plan is designed for publication
and for use by a wide, diverse audience. It contain still more detail than
the second category. It is designed to communicate with people who have
no specific knowledge of the lesson plan writer's school, school district,and state. It is especially useful to preservice teachers, to substitute
teachers in unfamiliar situations, and to workshop presenters seeking toelicit in-depth discussion.
This document is primarily intended for people who create and/or make use of the third
category of lesson plans. It can aid development of a preservice or inservice instructional
unit or serve as a guide during a course or workshop concerned with lesson plan creation.In addition, one can use it for self-instruction to strengthen one’s ability to organize and
conduct a course of study.
A General-Purpose Lesson Plan
Early on, preservice teachers are apt to encounter a general-purpose template or outlinefor lesson planning. The template is usually general enough so that it can be used over a
wide range of grade levels and disciplines. Preservice teachers often do assignments in
which they create specific lesson plans that, in the main, follow the template pattern.
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An example of a general-purpose lesson planning template: 1. Title and short
summary. You may find it helpful to think of a title of a lesson plan as being like a
section title in a book chapter, while the title of a unit plan is like a chapter title. Theshort summary can include information about how students will be empowered by
learning the material in the lesson.
2. Intended audience and alignment with standards. Categorization by: subject or
course area; grade level; general topic(s) within the discipline(s) being taught; length; andso on. A listing of the standards (state, or national) being addressed. Categorization
schemes are especially useful in a computer database of lessons, as it allows users
quickly to find lesson plans to fit their specific needs.
3. Prerequisites. It is difficult to state clearly the prerequisites for a particular lesson, andit is difficult to determine if students meet the prerequisites. A common approach consists
of two parts:
State (or assume) the general knowledge and skills of average students who willnormally encounter such a lesson—for example, second graders near the end of
the year or first year high school algebra.
State any special prerequisites—for example, perhaps key ideas that it is sort of
assumed that students should have covered, but that many will have not learnedvery well or will have forgotten. This type of prerequisite is often used in a
focused review at the beginning of a lesson or a unit of study. This often occurs at
the start of a school year. An often overlooked prerequisite is attitudinal. For example, those who help people learn to use a spreadsheet or word processing
program often encounter hidden anxieties about mathematical, reading, or writing
skills. Until these are remediated, the learner will have scant success with the
computer programs.
4. Accommodations. Special provisions needed for students with documented, relevant,
significant differences from "the average" learners. The differences may be attitudinal,
mental, or physical. The difference may so great that the lesson is beyond the student’scapacity or merely time-wasting busywork. Examples are a lesson in naming colors if the
learner is color-blind or teaching the C major scale when the learner has been taking
music lessons for years.
5. Learning objectives. Teachers of teachers often stress the need for very carefulstatement of the learning objectives. They may argue among themselves whether it is all
right to use the word understand, as in "Students will understand how do multidigit
subtractions with borrowing." The argument is over what it means to understand, andwhether more precise, measurable objectives need to be given. The expression
measurable behavioral objectives is sometimes used. It can be helpful to distinguish
between lower-order goals and higher-order goals, perhaps by using Bloom's taxonomy.
6. Materials and resources. These include written material for students to read,assignment sheets, worksheets, tools, equipment, CDs, DVDs, video tapes, physical
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environment, and so on. (It may be necessary to begin the acquisition process well in
advance of teaching a lesson, and it may be that some of the resources are available
online.
7. Instructional plan. This is usually considered to be the heart of a lesson plan. It tells
how to conduct the lesson. It may include a schedule, details on questions to be askedduring a presentation to learners and actions to handle contingencies. (Suppose the topic
is the U. S. Constitution and a student raises the question of whether private citizensshould be able to buy assault rifles.) If the lesson plan includes dividing students into
discussion groups or work groups, the lesson plan may include details for the grouping
process and instructions to be given to the groups.
8. Assessment options. A teacher needs to deal with three general categories of assessment: formative, summative, and long term residual impact. In order to become
efficient self-directed learners, students need to learn to do self assessment and to provide
formative assessment and perhaps summative assessment feedback to each other. A
rubric, perhaps jointly developed by the teacher and students, can help students take anincreasing responsibility for their own learning.
9. Extensions. These may be designed to create a longer or more intense lesson. For
example, if the class is able to cover the material in a lesson much faster than expected,extensions may prove helpful. Extensions may also be useful in various parts of a lesson
where the teacher (and class) decide they should spend more time on a skill or topic.
10. Teacher reflection and lesson plan revision. This is to be done after teaching the
lesson. Items in this section related to content, pedagogy, resources, and skills will makefor greater readiness “next time.” Also, such notes shared with colleagues will improve
the general level of teaching.
11. References. The reference list might include other materials of possible interest to people reading the lesson plan or to students who are being taught using the lesson plan.
Many variations on templates for a general-purpose lesson plan exist. Madeline Hunter's
work in this area is well known and widely used.
Discipline Specificity
A generalized lesson plan template is quite useful. Among other things, it helps unify the
overall processes and profession of teaching, giving all teachers some common ground.
However, each discipline has its own content and its own pedagogical content knowledge
(PCK). A good discipline-specific lesson plan reflects the uniqueness of the content and
teaching of the discipline. A good teacher in a discipline draws heavily on thatdiscipline’s proven PCK repertoire. A good lesson plan may well include a discussion of
PCK to employ when conducting the lesson.
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The remainder of this document focuses on possible components of a math lesson plan
template. Of course, such a template will include the components of the general-purpose
lesson plan. However, math has differences from any other discipline. A good math
lesson plan or unit plan reflects these differences.
Math was an informal area of study long before the development of reading and writing.
With the development of reading and writing somewhat over 5,000 years ago, math
became part of the core curriculum in schools. Many people feel math to be second onlyto language arts in importance in the curriculum.
What distinguishes math from other disciplines? Perhaps a good starting point in
answering this question is to delve into an exploration of what constitutes a discipline.
The various academic disciplines in our formal educational system have considerable
differences. Each academic discipline or area of study is delineated by such things as its:
• Typical problems, tasks, and activities it addresses
• Accumulated accomplishments (results, achievements, products,
performances, scope, power, uses, impact on the societies of the world,
and so on)
• History, culture, and language, including notation and specializedvocabulary
• Methods of teaching, learning, and assessment; its lower-order and
higher-order knowledge and skills; and its critical thinking and understand
—what its practitioners do to further their work and pass on their ethics,knowledge, products, and skills
• Tools, methodologies, and types of evidence and arguments used
in solving problems, accomplishing tasks, and recording and sharingaccumulated results
• Criteria that separate and distinguish among a:
a) novice,
b) person who has a personally useful level of competence,
c) reasonable competent person, employable in the discipline,
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d) local or regional expert,
e) national or world-class expert.
When you teach within a discipline, you represent that discipline. Part of your teachingtask is appropriate and adequate representation of the discipline. This means that you, as
a math teacher, can identify and explain similarities and differences between math andthe other disciplines your students have studied or are studying. This is especially
important for the various other disciplines in which math is standard component. For example, you know that students use math in business and science. What distinguishes
math from business or science?
Practitioners, teachers, and students—all face the challenge that a well established
discipline has substantial breadth and depth. A single discipline-specific unit of studyaddresses a minute fraction of the discipline. Thus, considerable thought ought to be
given as to what aspects of the discipline should be stressed and how this material
contributes to a student's overall progress toward gaining expertise within the discipline
that is or will be relevant to the student’s life.
Typical disciplines included in PreK-12 education are so vast that even if the entire PreK-
12 curriculum were devoted to the study of just one such discipline, students would learn
only a small fraction of that discipline. Indeed, students continuing their studies through a bachelor's, master's, and doctorate degree in a discipline still master only a modest
fraction of that discipline.
This observation helps us to understand the relative ease of creating a single lesson plan
in a discipline versus the challenge of creating a unit of study, a course, or an extendedcurriculum leading to a relatively high level of expertise in a discipline. It is quite
difficult to develop an extensive curriculum that fits the needs of a broad range of students who are working over a period of many years to gain a particular level of expertise in the various disciplines. This is further complicated by synergies among
disciplines.
Often, large teams of "experts" in a discipline address this challenge by working to
develop appropriate scope, sequence, and benchmarks. Most professional societies havesuch ongoing efforts. For example, the National Council of Teachers of Mathematics
plays a leadership role in developing math education standards in the United States. From
time to time populous state, such as California, will publish benchmarks in a disciplinesuch as math, and these benchmarks influence textbook companies and many other states
throughout the country.
What is Math?
Precollege math curricula in the United States are sometimes described as (and criticizedas) being "a mile wide and an inch deep." So many different topics can be taught that it is
hard to decide which to emphasize. Time is limited; curriculum developers continually
face the challenge of balancing depth and variety.
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A possible lodestone is to attend to the essence of the discipline. Thus, math educators
think carefully about the math-related aspects of attitude, content, and process. Their
answers to "What is math?" will then guide development of curriculum content, teacher attitudes, instructional processes, and assessment in our math education system. Those
who teach teachers are expected to be mathematically competent and able to
communicate a defensible answer.
Some Often-Quoted Answers
Here are three quotations that are fun and interesting, but not particularly helpful in mathcurriculum planning and development:
"Mathematics is a more powerful instrument of knowledge than any other that has
been bequeathed to us by human agency." (René Descartes, French philosopher,
mathematician, scientist, and writer; 1596–1650.)“Mathematics is the queen of the sciences.” (Carl Friedrich Gauss, German
mathematician, physicist, and prodigy; 1777–1855.)“God created the natural numbers. All the rest is the work of man.” (LeopoldKronecker, German mathematician and logician; 1823–1891.)
Patterns and Language of a Discipline
Many people attempting to answer the "What is math?" question give answers that fit thediscipline they are talking about, but that also apply to many other disciplines.
For example, is common to say that math is the study of patterns and then go on to give
examples of the types of patterns mathematicians study. A major shortcoming of this
answer is that "the study of patterns" description fits every discipline. It is only thediffering examples of discipline-specific patterns being studied and methodologies of
studying the patterns that distinguish one discipline from another.
Indeed, information is stored in a human brain as a pattern of stronger or weaker
neural connections. Science fiction stories have included machines that could quicklyimpose such patterns in nervous systems; researchers are making progress understanding
what is actually happening in a brain as it learns and as it uses its learning to solve
problems and accomplish tasks.
Another widely use answer to the "What is math?" question is that math is a language.
Indeed, it is sometimes said that algebra is the language of mathematics. Quoting LynnArthur Steen, a leading math educator:
… algebra is the language of mathematics, which itself is the language of the
information age. The language of algebra is the Rosetta Stone of nature and the passport to advanced mathematics (Usiskin, 1995). It is the logical structure of
algebra, not the solutions of its equations, that made algebra a central component
of classical education. (Steen, 1999)
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The combined assertion math is a language & algebra is the language of mathematics is
useful. However, each discipline can be considered from the point of view of
communication within the discipline. Each discipline has its own special vocabulary,notation, gesturing and movement, and other ways to represent and communicate with
others who know the discipline.
Thus, it seems more appropriate to say that math has its own special modes of
communication, rather than to say that math is a language. Math is known as being alanguage that facilitates very precise communication—perhaps more so than any other
widely used human language.
Some Important Math Concepts
Another approach to answering the "what is math" question is to name some of the really
important ideas or concepts in math that help to distinguish it from other disciplines. Hereare four examples:
"One of the most important concepts in all of mathematics is that of function."
(T.P. Dick and C.M. Patton.)
"The most powerful single idea in mathematics is the notion of a variable."(Alexander Keewatin Dewdney, 1941–, Canadian, computer scientist,
mathematician, and philosopher.)
"No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof ." (Leonardo da Vinci, quoted in Concepts of Mathematical
Modeling by Walter J. Meyer.)
"The usual approach of science of constructing a mathematical model cannotanswer the questions of why there should be a universe for the model to describe.
Why does the universe go to all the bother of existing?" (Steven Hawking)
In mathematics, the words function, variable, proof , and modeling have special
definitions that are different from the "natural language" definitions that peoplecommonly use. Thus, to appreciate the four quotes, one has to know some mathematics.
Indeed, one possible measure of a good mathematics curriculum is in terms of student
growth in understanding these four important ideas.
Long-Enduring Results
Many other people have written answers to the "What is math?" question. Math education
experts tend to agree on the need for a good answer to include a discussion of math patterns, problem posing and problem solving, communication in the language and
notation of mathematics, and the types of careful, rigorous arguments used in developing
and presenting mathematical proofs. It takes a reasonably good understanding of math in
order to understand possible meanings to the "What is math?" question.
Still another way to look at math is the longevity of some of its results. Theorems and
other mathematical results developed several thousand years ago are still true today. They
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are true throughout the world, and they will continue to be true in the future. This aspect
of math means that one can build upon and have confidence in the accumulated
mathematical knowledge.
Perhaps more so than for any other discipline, math is a discipline of broad and long-
lasting results. Of course, many other disciplines have some long-lasting results or accomplishments. For example, great music and art can endure over the ages. The design
of a tool such as a fork or a paper clip might be so good that it becomes a standard againstwhich possible new versions are measured. An invention and many of the results in
science can have very long lives.
In any case, math results have a permanency that facilitate the accumulation of results
over millennia, with the results being such that new researchers and users of math cansafely build upon these accumulated results. The following two quotations help capture
the essence of the permanency of accumulated math content knowledge.
"A mathematician, like a painter or poet, is a maker of patterns. If his patterns aremore permanent than theirs, it is because they are made with ideas." (G. H. Hardy,
English mathematician; 1877–1947.)
"In most sciences one generation tears down what another has built and what one
has established another undoes. In mathematics alone each generation adds a newstory to the old structure." (Hermann Hankel, German mathematician; 1839–
1873.)
As curriculum developers and teachers help students learn math, it is important to help
students understand and be able to make use of the steadily increasing accumulatedknowledge in this discipline. This does not mean packing more and more math results
into the students’ brains. It does mean enabling students, within their capacities, to haveconfidence in their mathematics and to be able to learn more as their needs and interestsdictate.
This accumulation of math results is very important in facilitating one of the most
important ideas in problem solving. That idea is the building on the previous results of
oneself and others. It is sometimes simplified to the statement, "Don't reinvent thewheel."
This quotation should be used with some care. In educational settings, a problem is
studied and solved for the purpose of increasing one’s expertise as a problem solver.
Looking up an answer in a library will produce an answer, but it will not contribute muchto gaining an increasing level of expertise in solving novel problems.
Many math teachers often stress that "the goal is to get the answer." That is a poor
approach to math education. First of all, a math problem may have no answer, oneanswer, or many answers. Second, the goal is to learn math to fit one's current and
possible future needs.
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We now have computers and Information and Communication Technology (ICT) to
facilitate the storage, retrieval, use, and communication of humanity’s accumulated math
knowledge. Computer technology has made it possible to:
1. Store ever-more accumulated math knowledge in a form accessible
by a steadily increasing percentage of the world's population.2. Store parts of this accumulated math knowledge in a form so that
the computer system can actually carry out the procedures to solve or helpsolve a wide range of problems. Even a "lowly" handheld scientific,
graphing, equation-solving calculator stores much math knowledge in a
form that automates the solving of a wide range of problems.
You can think about computerized storage and automation as an auxiliary brain, a brainaugmentation. Progress in computer technology is aiding in the development of such
brain augmentation in each academic discipline. The potential may be greater in math
than in any other discipline because of the fact that a proven math theorem remains
proven over time and throughout the universe.
In summary, it is not easy to provide short answers to the "What is math?" question that
can serve to provide a unifying foundation for math curriculum developers, textbook
writers, and teachers. It is not surprising that considerable areas of disagreement exist.Sometimes such disagreements are classified as being part of the Math Education Wars.
Some Math-Specific Lesson Plan Topics
This section explores some possible topics that need special attention in a math lesson
plan. It also explores some themes that are especially important for math success and that
math lesson plans should specifically emphasize.
Increasing Math Expertise
Students should increase their levels of math expertise during every math unit of study.
Thus, in preparing to teach a math lesson or unit of study, begin by thinking how a
student's level of math expertise will be maintained and improved by the time and effortthe student spends on the lesson or unit of study. Keep in mind that math is a broad and
deep discipline. The various components of math are thoroughly intertwined.
Problem Solving
The absolute heart—the unifying mission—of math education is students getting better atmath problem solving. Here is a brief summary of what problem solving includes:
• Question situations: Recognizing, posing, clarifying, and
answering questions.
• Problem situations: Recognizing, posing, clarifying, and solving problems.
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• Task situations: Recognizing, posing, clarifying, and
accomplishing tasks.
• Decision situation: Recognizing, posing, clarifying, and makingdecisions.
Note that solving problems usually requires higher-order, critical, creative, and wisethinking. Further, to successfully share or demonstrate the resulting “result”—product,
performance, or presentation—usually requires communication and social skills. Finally, problem solving requires sense making. The successful problem solver makes sense of
the problem and the results of solving or attempting to solve the problem. This sense
making provides feedback to the problem solver and helps to ensure correctness of thesolution(s) produced by the problem-solving processes.
An analogy with thinking about learning to write may be helpful. You can think of
learning to write as mastering spelling, punctuation, grammar, penmanship, etc. Or you
can think of learning to write as learning to express oneself clearly in written language.
The goal is to produce written documents that are understandable to yourself and others.
Similarly, one can think of lower-order math knowledge and skills such as multidigit
paper and pencil algorithms for addition, subtraction, multiplication, and division.
Alternatively, one can think of representing real-world (or interesting, theoretical) problems using the language of math, and then solving them using one's level of expertise
in problem solving in math. In both writing and in math problem solving, some basic
skills are important. In teaching writing, however, there is significant emphasis on thehigher-order goals even as students practice some of the basics. Often, this emphasis on
higher-order skills and sense making is missing in the way teachers teach math. (It helps
to remember that what you have learned to do so well that the doing seems a lower-level
skill is not so for students.)
The Concept of Proof
The concept of proof lies at the very heart of mathematics. Thus, every math lesson or
unit of study can be analyzed in terms of its contribution to students gaining increased
understanding of proof and how to make mathematical arguments that are proofs or are proof-like.
The "proof-like" idea is a key part of problem solving. The explanation and arguments
supporting the steps used in solving a problem are proof-like. Problem solvers can do a
mental check of the steps, testing to see if they make sense and if they could readilyconvince other people that they make sense.
A math proof can be thought of as a sequence of arguments so carefully done that they
can convince well-qualified mathematicians. Students get better at constructing proofs or proof-like arguments through being instructed in these endeavors, by practicing and
receiving high-quality feedback, and by studying proofs and proof-like arguments done
by others. The concepts of proof and proof-like are closely tied in with giving partial
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credit when grading math tests or math homework. Consider a teacher grading a problem
that a student has solved or attempted to solve. It is possible to attempt to solve a
problem, use methods or steps that are incorrect and make little or no sense, and still get acorrect answer. Thus, the grader looks both at the answer(s) produced and the steps used
to obtain the answer(s). If the steps and their underlying logic are correct, but one or more
are implemented incorrectly, a student may well be deserving of considerable partialcredit.
The idea of partial credit certainly carries over to other disciplines. For example, consider
a group of teachers, each grading a student's written essay. It takes a substantial amount
of instruction and practice to teach the group of teachers to have a high level of consistency and agreement in essay grading. While reaching agreement on how to deal
with errors in spelling and grammar is relevant and fairly easy to achieve, this is a far cry
from dealing with the higher-order thinking involved in written communication.
However, there may be a lodestone for the individual teacher: How does this grade,
comment, this correction contribute toward increasing the student’s expertise?
Include a Focus on Important Problems
We want students to learn math for a variety of reasons. For example, math is a human
endeavor and an important part of our history and of many different cultures.
There are certain problems we humans face that cut across many disciplines, that are too big for any one person or small team of people to solve, and that are important to all of
us. Sustainability provides a good example.Thus, in creating and delivering a math
lesson, the teacher might hold in mind that the math students are being taught might be
useful in helping to address various aspects of the overall problem of sustainability.
Thus, as a math topic is being taught, and transfer of learning of that topic is being taught,
students might well be led to considering uses of the math topic in exploring various
issues of sustainability in the other courses they are studying.
Prerequisite, Review, and Remediation
A typical lesson builds upon and expands the current knowledge and skills of students.Students construct new knowledge and skills by building on their current knowledge and
skills. This theory is called constructivism and it is a very important educational theory.
"Forgetting" has attracted much educational research, and the results are in: In every
course area, students forget a significant amount of course content relatively soon after completing the course. The amount forgotten varies with the student. However, in many
courses the amount forgotten in a year or so is in the range of 75% to 90%.
What students usually remember is a combination of some big ideas and material that the
students use rather frequently in courses and in other parts of their lives. Teachers do well
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to assume that many students in a class which has covered the same course or courses
will have forgotten much of the material covered from those courses. That retention
varies tremendously among students makes the teacher’s task harder; that previouslyexposed students relearn easier makes it easier—provided the students’ attitudes are
good.
General and quite variable long-term residual knowledge, skills, and understandings
usually serve as an adequate foundation for future learning in some areas, but often it isnot adequate in math (except for problem-solving skills and attitudes). Thus, in preparing
to teach a math lesson, the teacher needs to think carefully about, or have ascertained,
what aspects of the math prerequisite knowledge and skills the students actually have.
As students progress through math instruction, year after year, this math prerequisitesituation gets to be a bigger and bigger challenge both to students and to teachers. For a
number of the students, the entire instructional time in a math lesson or unit could be
used up in math review, and still the students would not have the proficiency that the
teacher would like in order to deal with the mew material. For other, the review time is awaste of time.
No simple, sure fire solutions to this problem situation exist. What typically happens is
that some class time is spent in review, which bores students who have the necessary prerequisite proficiency. For some of the students, the review process is adequate, but for
many others it is inadequate. Thus, many of the students face the challenge of trying to
learn new material (construct new knowledge) by building upon an inadequatefoundation. The result does not provide them with the prerequisite knowledge and skills
for the subsequent lessons, units of study, and courses. Such students continue in a
downward spiral where they fall further and further behind.
When the "falling behind" situation gets bad enough, our educational system tends to tryto do something about it. We know, for example, that students learn faster with one-on-
one tutoring or in very small-group instruction. We know that such intense instruction,
with a longer period of time being devoted to a subject area, will help students catch up.In some cases, quality computer-assisted learning materials have some of the needed
characteristics of an individual tutor: Accuracy, cost-effectiveness, diagnosis, feedback,
interesting presentation, patience, and relevance.
Having a student devote extra time to learning math entails the question: What part of theother curriculum should receive less time?" Is this math skill or topic so important to
these slower-learning math students that they should learn less art, history, music,
physical education, or other standard components of the curriculum?
Slower and Faster Learners
The issue of prerequisite knowledge and skills is a major challenge to any math education
system. Humans vary considerably in the nature and nurture aspects of math. A typical
first grade teacher will have some students at a kindergarten or lower level in a math area,
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and some students who are at second grade or higher level. Very roughly speaking, some
students in the class will learn math at 1/2 (or less) to 3/4 the rate of average students in
the class, while others will learn math at the rate of 1.5 to 2 times (or more) of the classaverage. That the same student may find some areas of math easy and others difficult
complicates the problem.
The students will vary widely in their depth of understanding of previous and new
materials, and how well they retain (how rapidly they forget) the math they have studiedor are currently learning.
Teachers’ usual heuristic is to conduct reviews so the "average" students meet the
prerequisites, more or less.
Since whole-class review will bore the students who have mastered the material and oftenwill continue to bewilder or dishearten students who didn’t “get it” the first (or second!)
time around, teachers may try to cope by dividing the class into groups that progress at
different rates. Elementary schools often divide a class into two or three groups for mathinstruction—based on current math preparation and rate of learning. One way to do this is
to have two or three teachers’ classes work together, one teacher taking all of the slower
group, one taking all of a second group, and so on. Educational institutions beyond the
elementary grades normally offer a menu of courses.
As with the reading curriculum, it is possible to increase the amount of math learning in
the lower group by teaching them in smaller classes and extending the amount of math
instructional time per day. It is possible to meet some of the needs of the faster group by
giving them instruction on how to learn math by reading math books and by interactingwith each other, and how to do self-assessment and peer assessment.
Student and Teacher Responsibilities
The problem of math prerequisites increases as students move to high level grades. Both
teachers and students have ownership of the problem. Thus, one way for a teacher toapproach this to educate students about the problem and get students actively engaged in
addressing the problem.
This raises the issue of the extent to which a students can learn to take habitual
responsibility for their own learning, lack of learning, need for review, and need for remediation. For educators and parents, two possible aspects of this are:
1. Help students learn to understand the level of knowledge, skills,
understanding, and performance expected of them. Typically, expectations
may well vary from student to student in a class. All expectations willinclude level of performance upon completion and amount of progress in a
given period of time. Using these expectations effectively requires that
students have a firm grasp of what they mean. It requires that they get
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good assessment and feedback from themselves and others (such as the
teacher) so they know they’re on track and on schedule.
2. Provide a variety of aids to students to help them meet theexpectations. This includes helping students learn to help themselves.
Students learning to take responsibility for their own learning is one of the mostimportant tasks of educators. It is an issue in all components of the curriculum (and in
many other areas such as managing or parenting). One way to see we are doing a very poor job in this is to look at students entering post-secondary education. Colleges and
universities routinely give students a math placement test. In many institutions, fully half
of the students "discover" that they are not prepared to take any math course that carriescredit toward graduation—sometimes because they’ve regarded there math courses as
things to get through and be done with. Their test scores indicate which of a variety of
"pre-college, remedial" courses they need to begin in as they work their way throughmaterial that they have already “studied” in middle school and high school.
We now have the technology, via online tests that one can take over and over, becausethese tests can be designed to give different questions each time. There is no reason why
such tests are not readily available to all students starting at the middle school level or above. (Probably it is appropriate to make such feedback available at still younger ages.
The general idea is to help students learn to depend on themselves and on readily
available feedback systems such as computers when they want an answer to “How am Idoing?”) The report given to a student can be completely confidential, if that is what is
needed or wanted. It can contain an analysis of areas needing remediation and ways to get
needed help.
Teaching Self-Assessment and Self-Responsibility
Here is a penetrating quotation:
"In the book of life, the answers aren't in the back." (Charlie Brown, as written byCharles Schulz)
Learning requires feedback. The feedback may come from a teacher, from one's peers,
from parents , from the learner, and so on. One of the major weaknesses in our math
education system is that many students do not develop effective skills in providingfeedback to themselves—that is, often what they are doing makes no sense to themselves,
and they seldom reflect. Thus, when they make errors, they have few internal resources to
detect and then correct the errors, perhaps in part because they have little or no ownershipin the task.
One reason that this situation persists is that a lot of math education learning effort goes
on in a context where it is difficult for a student to provide self-feedback. The instruction
is presented in the form of computations to be carried out using algorithms to bememorized. A student does not learn to attach meaning to the numbers being manipulated
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on the answers being produced. The student does not gain the knowledge and skills to
check if a result "makes sense."
Some math books and s teachers suggest that students do paper and pencil calculationsand then use a calculator to check their results. However, it is very easy to make a
mistake when using a calculator. Indeed, an important aspect of learning to use acalculator is learning to detect one's errors. One way to do this is to check if an answer
makes sense.
Thus, sense making is a fundamental idea in calculation, whether it is done mentally,
using paper and pencil, or using a calculator or computer. Modern math education
programs of study include a strong emphasis on students learning to check whether an
answer makes sense. A student’s skill in sense making should be increasing year after year.
As students begin to encounter word (story) problems, we can readily see those who are
reasonably good at checking to see if an answer makes sense. Many students will solvesuch problems and produce answers that make no sense whatsoever—and be quite unable
to detect that their answers make no sense. Word problems generally admit sense-
checking. Thus, one important reason for including word problems in the math
curriculum is that they are a good vehicle to help students increase their sense-checkingskills. Other important reasons are that they entail higher-order skills and that students’
reactions to them tells much about their attitudes toward math.
Teaching for Transfer of Learning
Math is very useful in many different academic disciplines. Math is a general-purpose aid
to problem solving—indispensable if the problem situation involves quantities. Thus, it ishighly desirable to teach math in a manner that facilitates transfer of learning to other
disciplines and to actual and probable problem-solving situations students will encounter.
The 1992 article by Perkins and Salomon provides an excellent summary of this field.
Over the past two decades, educational researchers have learned a great deal about thetheories of low-road and high-road transfer of learning, and how to teach for transfer.
Low-road transfer of learning is based on automaticity. For example, various number
facts can be learned to such a high level of automaticity that they seem as if they are
instinctive when one needs them in addressing problems both in and outside of school.
High-road transfer is based on learning general-purpose strategies and learning how to
apply these strategies over a wide range of problem situations. For example, many hard
problems can be broken into sets of less difficult problems. Solve the less difficult
problems, put all the results together in an appropriate manner, and the harder problem issolved. The teaching approach is to recognize when it is appropriate to generalize a
strategy being taught in a specific discipline (such as math), give the strategy a name, and
explicitly help students to learn to apply the strategy in a variety of disciplines. Divide
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and conquer (break a big problem into a coherent set of smaller problems) is commonly
taught in math, and it is quite useful in problem solving in other disciplines. Two
examples: Learning to drive a car, and comparing/contrasting the foreign policy/ military policy of Germany and Japan from 1932-1945.
In summary, every math lesson plan should include a statement of how the new materialis transferable to problem solving in other settings, including in non-math disciplines.
This transfer of learning should occur from math learning to other disciplines, and fromlearning other disciplines to math. Suppose that you teach both math and other disciplines
(This especially applies to elementary school teachers.) When developing a math lesson
plan in which transfer from math to other disciplines is important, at the same time think
about revising your lesson plans in the other disciplines you teach. When appropriate,integrate some math into these disciplines and stress ideas that transfer from math. For
example, stating a story problem intelligibly and explaining the solution process require
language arts skills. Another example: Compare popularity ratings of U. S. Presidents
when they left office with their relative rankings by historians now, and discuss therelationship between popularity and enduring worth.
Math Cognitive Developmental Level and Maturity Level
As a student's brain matures and as a student studies math over a period of years, two
important results are:
1. The student moves up the Piagetian (math) cognitivedevelopmental scale, moving toward (math) formal operations.
2. The student grows in math maturity—getting better at thinking
mathematically, learning to learn math, and creatively use math to solvecomplex and challenging problems.
The next two sub sections provide short introductions to math cognitive development and
math maturity. For a deeper discussion, see Chapter 7 of:
Moursund, D.G. (2006). Computational Thinking and Math Maturity: ImprovingMath Education in K-8 Schools. Eugene, OR: Information Age Education.
Retrieved 12/7/07: http://i-a-e.org/eBooks/cat_view/37-free-ebooks-by-dave-
moursund.html.
Cognitive Development
Piaget is well known for his work in cognitive developmental theory. Moderninterpretations of his work emphasis both the nature and nurture aspects of a student
gaining in cognitive development. Moreover, both observation and research attest that a
student's cognitive development may proceed more rapidly in some areas than in others —”better” at language arts than in math, or vice versa.
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Many college students have become capable of formal operations in many areas, but are
not at formal operations in math. That is, they deal well with the overall complexity of
rational thought in their everyday lives and in areas not requiring use of math, but they donot deal well with the level of abstraction that is common in math at the level of first year
high school algebra and above. This is true even though they have taken three or four
years of high school math. Their seeming competence is akin to that of a well-trainedanimal, as can be discovered if they are presented with a novel problem that draws on the
math knowledge and skills they have supposedly “learned.”
Here is a “thought experiment.” Consider identical twins separated at birth. One goes to
live in a middle class household in the US, where both parents are high school teachers.One parent teaches language arts and social studies, while the other teaches math and
science.
The other child goes to live with "middle class parents" in a hunter-gatherer tribe in the
jungles of South America.
[Note to readers: It is not clear what would constitute being "middle class" in such
a tribe. The tribe might have a chief and a Shaman or priest, and that they would
be above middle class. Perhaps those who take good care of their children are able
to take a nap in the heat of the day, and have a regular turn speaking around thecampfire. In any event, there are no formal schools and no written language.]
By the time he or she finishes high school, the first child has a good chance of being well
along toward achieving general Piagetian formal operations and math formal operations.
The second child may well never achieve general Piagetian formal operations, andcertainly is quite unlikely to achieve math formal operations.
Moving back to the math education curriculum in our school system, the households and
extended families that children grow up in vary considerably in how much they helpchildren in their general cognitive development and in their discipline-specific cognitive
development. This poses two questions for math curriculum developers:
1. How to provide appropriate math curriculum for students of
considerable different levels of math cognitive development levels?2. How to help all students to move upward in their math cognitive
developmental levels?
Any math lesson plan or unit of study can be examined from the point of view of how itcontributes to students efficiently continuing their math cognitive development. Inaddition, each math lesson can be examined from the point of view of the math cognitive
developmental level or the general cognitive developmental level needed to learn and
understand the material. There are some serious flaws in our current math scope andsequence.
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One example lies in the area of ratio and proportion. This is typically taught at a time
when students are just starting to move into formal operations. Most such students do not
yet have the cognitive maturity to deal with the level of mathematical abstraction neededto understand ratio and proportion. Thus, they are forced into a "learn by memorizing and
demonstrate knowledge by regurgitating" approach.
A second major flaw is the level of abstraction that easily can be put into a plane
geometry or first algebra course. Only a minority of high school freshmen or sophomore brains have developed enough to be ready for this level of abstraction and mathematical
rigor. Many pass the courses (indeed, perhaps even get good grades) but do not gain the
kind of understanding prerequisite for success in further math courses.
High school geometry provides a good example. About 50 years ago, the Dutch educatorsand Pierre van Hiele focused some of their research efforts on defining a Piagetian-type
developmental scale for Geometry. Their five-level scale is shown below.
Level Name Description
0 Visualization
Students recognize figures as total entities (triangles, squares), but
do not recognize properties of these figures (right angles in asquare).
1 Analysis
Students analyze component parts of the figures (opposite angles of
parallelograms are congruent), but interrelationships between
figures and properties cannot be explained.
2 InformalDeduction
Students can establish interrelationships of properties within figures(in a quadrilateral, opposite sides being parallel necessitates
opposite angles being congruent) and among figures (a square is a
rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not see how the logical order
could be altered nor do they see how to construct a proof starting
from different or unfamiliar premises.
3 Deduction
At this level the significance of deduction as a way of establishing
geometric theory within an axiom system is understood. Theinterrelationship and role of undefined terms, axioms, definitions,
theorems, and formal proof is seen. The possibility of developing a
proof in more than one way is seen. (Roughly corresponds toFormal Operations on the Piagetian Scale.)
4 Rigor
Students at this level can compare different axiom systems (non-Euclidean geometry can be studied). Geometry is seen in the
abstract with a high degree of rigor, even without concreteexamples.
Notice that the van Hieles, being mathematicians, labeled their first stage Level 0. This is
a common practice that mathematicians use when labeling the terms of a sequence.
Piaget's cognitive development scale has four levels, numbers 1 to 4. The highest level in
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the van Hiele geometry cognitive development scale is one level above the highest level
of the Piaget cognitive development scale.
A third area is probability. A number of math education researchers have explored theissue of cognitive development and learning probability. For example:
Garfield, J. and Ahlgren, A. (1988). Difficulties in Learning Basic Concepts in
Probability and Statistics: Implications for Research. Journal for Research in
Mathematics Education. 19, 1, 44--63.
Quoting the abstract of this article:
There is a growing movement to introduce elements of statistics and probability
into the secondary and even the elementary school curriculum, as part of basic
literacy in mathematics. Although many articles in the education literature
recommend how to teach statistics better, there is little published research on how
students actually learn statistics concepts. The experience of psychologists,educators, and statisticians alike is that a large proportion of students, even in
college, do not understand many of the basic statistical concepts they havestudied. Inadequacies in prerequisite mathematics skills and abstract reasoning are
part of the problem. In addition, research in cognitive science demonstrates the
prevalence of some "intuitive" ways of thinking that interfere with the learning of correct statistical reasoning. The literature reviewed in this paper indicates a need
for collaborative, cross-disciplinary research on how students come to think
correctly about probability and statistics.
The research relating the learning of probability and a student’s level of cognitive
development suggests that learning for understanding requires students to be at a formaloperations level. Remember, even though age 11 or 12 is a biological time for beginning
to move into formal operations, only about a third of students have achieved formal
operations by the time they finish high school. Thus, research in this area tells us that
K-8 students are not ready to develop a formal understanding of probability.
Math Cognitive Development
The following scale was created (sort of from whole fabric) by David Moursund. It
represents his current insights into a six-level, Piagetian-type, math cognitivedevelopment scale.
Stage & Name Math Cognitive Developments
Level 1. Piagetian
and Math
sensorimotor.
Infants use sensory and motor capabilities to explore and gain
increasing understanding of their environments. Research on very
young infants suggests some innate ability to deal with smallquantities such as 1, 2, and 3. As infants gain crawling or walking
mobility, they can display innate spatial sense. For example, they
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can move to a target along a path requiring moving around
obstacles, and can find their way back to a parent after having taken
a turn into a room where they can no longer see the parent.
Level 2. Piagetian
and Math
preoperational.
During the preoperational stage, children begin to use symbols, suchas speech. They respond to objects and events according to how they
appear to be. The children are making rapid progress in receptive
and generative oral language. They accommodate to the language
environments (including math as a language) they spend a lot of time in, so can easily become bilingual or trilingual in such
environments.
During the preoperational stage, children learn some folk math and
begin to develop an understanding of number line. They learnnumber words and to name the number of objects in a collection and
how to count them, with the answer being the last number used in
this counting process.
A majority of children discover or learn “counting on” and counting
on from the larger quantity as a way to speed up counting of two or
more sets of objects. Children gain increasing proficiency (speed,
correctness, and understanding) in such counting activities.
In terms of nature and nurture in mathematical development, both
are of considerable importance during the preoperational stage.
Level 3. Piagetian
and Math concrete
operations.
During the concrete operations stage, children begin to think
logically. In this stage, which is 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).
While concrete objects are an important aspect of learning during
this stage, children also begin to learn from words, language, and
pictures/video, learning about objects that are not concretely
available to them.
For the average child, the time span of concrete operations is
approximately the time span of elementary school (grades 1-5 or 1-6). During this time, learning math is somewhat linked to having previously developed some knowledge of math words (such as
counting numbers) and concepts.
However, the level of abstraction in the written and oral math
language quickly surpasses a student’s previous math experience.That is, math learning tends to proceed in an environment in which
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the new content materials and ideas are not strongly rooted inverbal, concrete, mental images and understanding of somewhat
similar ideas that have already been acquired.
There is a substantial difference between developing general ideasand understanding of conservation of number, length, liquid, mass,weight, area, and volume, and learning the mathematics that
corresponds to this. These tend to be relatively deep and abstract
topics, although they can be taught in very concrete manners.
Level 4. Piagetian
and Math formal
operations.
Thought begins to be systematic and abstract. In this stage,intelligence is demonstrated through the logical use of symbols
related to abstract concepts, problem solving, and gaining and using
higher-order knowledge and skills.
Math maturity supports the understanding of and proficiency inmath at the level of a high school math curriculum. Beginnings of
understanding of math-type arguments and proof.
Piagetian and Math formal operations includes being able to
recognize math aspects of problem situations in both math and non-math disciplines, convert these aspects into math problems (math
modeling), and solve the resulting math problems if they are within
the range of the math that one has studied. Such transfer of learningis a core aspect of Level 4.
Level 5. Abstract
mathematicaloperations.
Mathematical content proficiency and maturity at the level of
contemporary math texts used at the senior undergraduate level instrong programs, or first year graduate level in less strong programs.Good ability to learn math through some combination of reading
required texts and other math literature, listening to lectures,
participating in class discussions, studying on your own, studying ingroups, and so on. Solve relatively high level math problems posed
by others (such as in the text books and course assignments). Pose
and solve problems at the level of one’s math reading skills and
knowledge. Follow the logic and arguments in mathematical proofs.Fill in details of proofs when steps are left out in textbooks and other
representations of such proofs.
Level 6.
Mathematician.
A very high level of mathematical proficiency and maturity. Thisincludes speed, accuracy, and understanding in reading the researchliterature, writing research literature, and in oral communication
(speak, listen) of research-level mathematics. Pose and solve
original math problems at the level of contemporary researchfrontiers.
Math Maturity
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Mathematicians tend to prefer the concept of math maturity over the idea of math
cognitive development. A Google search (10/6/08) of the expression: "math maturity"
OR "mathematical maturity" OR "mathematics maturity" produced over 24,000 hits.Wikipedia states:
Mathematical maturity is a loose term used by mathematicians that refers to amixture of mathematical experience and insight that cannot be directly taught, but
instead comes from repeated exposure to complex mathematical concepts.
Still quoting from the Wikipedia, other aspects of mathematical maturity include:
• the capacity to generalize from a specific example to broad concept
• the capacity to handle increasingly abstract ideas
• the ability to communicate mathematically by learning standardnotation and acceptable style
• a significant shift from learning by memorization to learning
through understanding• the capacity to separate the key ideas from the less significant
• the ability to link a geometrical representation with an analytic
representation
• the ability to translate verbal problems into mathematical problems• the ability to recognize a valid proof and detect 'sloppy' thinking
• the ability to recognize mathematical patterns
• the ability to move back and forth between the geometrical (graph)and the analytical (equation)
• improving mathematical intuition by abandoning naive
assumptions and developing a more critical attitude
Quoting Larry Denenberg:
Thirty percent of mathematical maturity is fearlessness in the face of symbols: the
ability to read and understand notation, to introduce clear and useful notation
when appropriate (and not otherwise!), and a general facility of expression in the
terse—but crisp and exact—language that mathematicians use to communicateideas. Mathematics, like English, relies on a common understanding of definitions
and meanings. But in mathematics definitions and meanings are much more often
attached to symbols, not to words, although words are used as well. Furthermore,the definitions are much more precise and unambiguous, and are not nearly as
susceptible to modification through usage. You will never see a mathematical
discussion without the use of notation!
You can evaluate a math lesson plan or unit of study in terms of how it contributes tostudents gaining in math maturity.
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The general notion of "maturity" in a discipline applies to every discipline—indeed to
every job, vocation, or pastime. However, mathematics teachers have been engaged with
the notion more often than teachers of other academic disciplines.
Communication in Math
Our overall educational system clearly acknowledges the need for students to improvetheir oral and written communication. A foundational concept in learning any discipline
is to learn to communicate with understanding, whether with others, with oneself, with
books, with computers, etc. One way of thinking is holding a conversation with oneself.Such thinking in math makes use of the vocabulary, notation, and graphical
representations of math. Fluency in a language of math greatly increases efficiency and
chances for success when addressing a problem in that area of math. Communication(read, write, speak, present and visualize, and listen) with understanding is absolutely
fundamental to learning, using, and doing math. Many people confuse the idea of reading
math with the idea of reading math word problems (math story problems). Although these
activities are somewhat related, it is important to distinguish between the two.
Communication and Math Content
Traditionally, by the time a student finishes the third grade, teachers expect that the
student can read well enough to begin to use reading as a significant aid to learning. Our
traditional curriculum increases the emphasis on learning by reading as a student progresses through the higher grades and on into college.
Traditionally, in middle school reading is the dominant aid to learning content. Moreover,
our educational system includes an emphasis on "reading and writing across the
curriculum." However, we do a very poor job of implementing this idea in matheducation. By the time most students finish high school, they have not yet learned to
learn math content by reading, and their skills in communicating math content by writing
are correspondingly poor.
Of course, there are alternatives to communicating through reading and writing. Indeed,oral communication existed long before reading and writing were invented. Our current
methods for teaching math depend heavily on oral communication backed up with written
communication. A teacher talks ("stand and deliver"), makes marks on a whole-class-viewable medium, and then has students make use of worksheets or some other form of
written assignment. Few humans will learn to read a math book through this approach.
Math does not really lend itself to oral communication. Even two math professors will
quickly move into a combination of oral and written (using a chalkboard, if one isavailable) communication when discussing a math problem. Moreover, as one might
expect, math professors have developed considerable skill in reading math books and
journals.
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Recall that one of the most important ideas in problem solving is building on the previous
work of oneself and others. In math, one builds on the previous work of others by
learning what they have done (and how), and by learning to learn what they have done.The collected human accumulation of math knowledge is so large that learning to learn
what has been done, and then using this knowledge and skill "just in time" (that is, when
one needs the specific knowledge) is essential The "just in time" idea is also important torelearning when needed, including learning how to access key information or procedures.
Remember: Over time, almost all students will forget much of the math they’ve learned
unless they use it regularly. Therefore, they should do their initial learning in a manner
that makes relearning faster and easier—even if this initial learning take somewhat longer than more “efficient” methods. (If its unlikely that the student will ever have occasion to
use a certain aspect of math, you might consider whether it should be taught at all.) One
place to practice math relearning is in dealing with prerequisites. Each math prerequisitesituation in a math lesson plan can be viewed as an opportunity to help students hone
relearning skills.
Communication and Math Word Problems
The diagram below captures the essence of many different math problem-solving
situations. The six steps shown are:
1. Problem posing and problem recognition to produce a clearly
defined problem;
2. Mathematical modeling;
3. Using a computational or algorithmic procedure to solve acomputational or algorithmic math problem;
4. Mathematical "unmodeling";5. Thinking about the results to see if the Clearly-defined Problemhas been solved; and
6. Thinking about whether the original Problem Situation has been
resolved. Steps 5 and 6 also involve thinking about related problems and problem situations that one might want to address or that are created by
the process or attempting to solve the original Clearly-Defined Problem or
resolve the original Problem Situation.
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In steps 1 and 2 a person works to understand a problem situation and makes a decision
as to whether it might be useful to attempt to solve the problem using math. A persondeciding to take a math-oriented approach to resolving the problem situation attempts to
represent or model the problem situation using the language of mathematics. This math
modeling leads to having a math problem that may of may not be solvable, and that mayor may not be solvable by the person attempting to solve the problem.
In step 6, the person who has a solution to the math problem extracted when dealing with
Step 1 checks the degree to which the results achieved are relevant to the original
problem situation and decides whether the overall process has been useful in trying toresolve the original problem situation.
The great majority of K-8 math education is focused on students learning to do step 3
using paper and pencil algorithms. Step 3 is what calculators and computers are best at.
Thus, the great majority of math education at the K-8 levels is spent helping studentslearn to compete with calculators and computers in areas that are not well suited to the
capabilities of a human mind but that are well suited to computers.
Problem posing, along with steps 1, 2, 4, 5, and 6 are all areas in which humans are better
than computers. Since inexpensive calculators have become widely available andrelatively reliable (beginning in about 1980), there has been a modest (often, heavily
fought against) trend toward reduced instructional time being spent in teaching paper and
pencil approaches to step 3. The time saved is being spent on problem posing, sensemaking, and the steps 1, 2, 4, 5, and 6.
Many thousands of articles, books, and Websites address ways students can learn to solveword problems. Often, such material treats the task as one of mechanically, a non-
thinking process of translating the words into math language. For example, in a word problem and often means +. If students are doing word problems involving percentages,
they are taught that in this situation, "the word of often means ‘times’." That is, students
are taught a number of tricks or rules of thumb that may help translate a word probleminto a “pure” math problem. They memorize and use these tricks with some success in
getting correct answers, but with little understanding.
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This approach to word problems misses the whole point or students learning to deal with
challenging "real world" problems situations in which math might be a useful aid to
resolving the situations. It misses the whole point of translating (representing, modeling)such read world problems situations into math problems.
Another difficulty exists. In the typical schoolbook word problem, students “know” thatthere is a solution. Therefore, there must be a way to get the solution, the “correct” result.
Therefore, there ought to be a mechanical way to get that result. Now, when a studentfaces a problem situation or a “story problem” that may not have a solution, the student is
apt to go to be totally frustrated.
Math Modeling
The six-step diagram given in the previous section emphasizes math modeling. Math
modeling is one of the most important aspects of math. This section provides a little moreinformation about math modeling.
Some answers to the "What is mathematics?" question focus on math being a language
that can be used to develop models (called math models) of certain aspects of objects that
people want to study. Thus, for example, suppose I observe 3 children playing withmarbles. One child has 10 marbles, one has 8 marbles, and one has 14 marbles.
The number 3 can be thought of as being a mathematical model of the children at play. It
says nothing about their age, their size, their sex, or their skin color. Similarly, the
numbers 10, 8, and 14 are mathematical models of the marbles that the various childrenare playing with. These numbers tell us nothing about the color, size, or quality of the
marbles.
Perhaps someone raises the question, what is the average number of marbles per child?
This question is not a clear, well defined question. The word "average" has a variety of meanings. In math, three of the definitions are 1) mean; 2) medium; and 3) mode.
Suppose that we decide that the question is, “What is the mean number of marbles per
child in this group?”
Since we know what “mean” signifies, we quickly come to setting up the calculation (10
+ 8 + 14)/3. At this stage of the problem-solving process, we have a pure math
calculation problem. The calculation (10 + 8 + 14)/3 is completely divorced from
children and marbles. Possible results include 10 2/3 or 10 R2 or 10.666….
We then attempt to make meaning from the results of the calculation. We might, for
example, conclude that there are 10 marbles per child, with 2 left over. We might say that
there are 10 2/3 marbles per child. That might be a little troublesome—I don't recall ever
seeing 2/3 of a marble. We might give the answer as the repeating decimal 10.666… Now we have a mathematical expression that involves an infinite number of digits! But
infinity is a very complex idea.
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In this children and marbles situation, we created a mathematical model and we solve the
pure math problem represented by the math model. We never got the chance to find out
why one might want to have an answer to the question. For example, it might have beenthat the children were squabbling with each other because some had more marbles than
others. It might have been that some had prettier marbles or larger marbles. Maybe the
purpose of the question was to use as a starting point in stopping the squabbling, perhaps by dividing the marbles more equally among the children.
The lack of meaning or purpose in the question is fairly typical in the types of problems
given in math books. The lack of non-mathematical context, meaning, or purpose makes
in much more difficult for problem solvers to detect possible errors in the modeling and problem-solving processes.
Computational Math
Math and some of the sciences have traditionally been divided "pure" and "applied"
components—pure and applied math, theoretical and experimental physics, theoreticaland observational astronomy, etc. Computers have changed this situation. Math and thevarious sciences have added a "computational" category to their main subdivision—see
“computational _____ in Wikipedia. "Computational" refers to developing and making
use of computer models and simulations in the discipline or intersection of disciplines.
Within math and the various sciences there are now many computational-oriented journals. For example:
• Journal of Computational Mathematics
• Journal of Computational and Applied Mathematics
• Journal of Computational Mathematics and Mathematical Physics
•
Journal of Computation and Mathematics • Journal of Computational Mathematics and Optimization
• Communications in Applied Mathematics and Computational
Science
• Computational Mathematics and Mathematical Physics
• International Journal of Computational Science and Mathematics
In 2006, Jeannette Wing summarized the "computational" idea in her seminal article oncomputational thinking. Also, note the title of David Moursund's free 2006 book, Computational Thinking and Math Maturity: Improving Math Education in K-8 Schools.
The use of computers to actualize mathematics in models and simulations is now veryimportant in math, all of the sciences, and in many other disciplines. Computational
thinking and computational math are now very important aspects of doing math. The
overall math education curriculum needs to pay far more attention to these topics than it
currently does.
Lesson Plan as Self-Inservice Education
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Many teachers feel that they learn more about teaching during their first few year on the
job than they did during their teacher education program. Moreover, recent research
suggests that years of experience is a good predictor of teacher success in helping tolearn. Here is summary of an important 2006 study by Andrew Leigh:
Using a data set covering over 10,000 Australian primary school teachers andover 90,000 pupils, I estimate how effective teachers are in raising students’ test
scores from one exam to the next. Since the exams are conducted only every twoyears, it is necessary to take account of the work of the teacher in the intervening
year. Even after adjusting for measurement error, the resulting teacher fixed
effects are widely dispersed across teachers, and there is a strong positivecorrelation between a teacher’s gains in literacy and numeracy. Teacher fixed
effects show a significant association with some, though not all, observable
teacher characteristics. Experience has the strongest effect, with a large effect inthe early years of a teacher’s career. Female teachers do better at teaching
literacy. Teachers with a masters degree or some other form of further
qualification do not appear to achieve significantly larger test score gains.
Each teaching unit is an opportunity to learn. Value this opportunity to learn. Think interms of maintaining and increasing your knowledge and skills in three areas (as well as
in the art of interacting with other staff, parents, and students):
1. General pedagogy. This is professional knowledge that cuts across
subject areas and, to a considerable extent, across grade levels. For example, consider how much instructional use you make of interactive
multimedia and Web-based video materials. If you use little or none, a
good place to begin is with interactive math manipulatives. Select a single
example that meshes with a lesson you are teaching and use thedemonstration to increase student interest in and insight into the topic.
Subject matter content. Any topic you’re teaching has far more content
than you’re teaching. You and your students will benefit as you graduallyexpand the depth and breadth of your knowledge of any specific topic
you’re teaching.
2. Pedagogical content knowledge (PCK). Any topic can be presentedmany different ways. The larger your presentation repertoire on a topic,
the more apt you are to meet the diverse needs of your students.
In math education, every unit should include a significant and well-integrated focus on
problem solving. In a problem-solving environment, students will develop or adapt avariety of methods that solve a particular problem, and a variety of methods that fail to
solve the problem. This environment is one in which you need to be actively engaged,
and it provides an opportunity to maintain and improve your own problem-solvingknowledge and skills.
Another suggestion. Build a personal library of math puzzles and math problem
challenges appropriate to the levels of students and for the courses you teach. A quick
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Web search will yield a plethora. As your collection grows, you can move from providing
students with a "Challenge Math Problem of the Month" to weekly, and then perhaps
daily challenge problems. These should be optional assignments—challenges that somestudents will enjoy exploring. You’ll learn by seeking out such problems, trying to solve
them on your own, reading student solutions, and perhaps sharing your problems with
other math teachers.
Some Roles of ICT
Information and Communication Technology plays two roles in a good math lesson plan.
ICT is part of math content, and it provides aids to teaching and learning math.
Content
A separate section of this document discussed Computational Thinking and
Computational Math. These topics include math modeling and simulation, and are a key
component of the content of a modern math curriculum.
There are a variety of other ICT-related math content areas. A few are briefly discussed
below.
Calculators
The National Council of Teachers of Mathematics supports and encourages students tolearn to use calculators in the early grades. You might ask, "What's to learn?" With a very
minimum of instruction (often provided by students showing each other) students can
learn to use add, subtract, multiply, and divide on the standard 6-function calculator.
However, even this very beginning level has significant teaching and learning challenges.Here are a few issues to consider:
1 Students can do the four basic operations before they understand the possible
meanings of these operations. The calculator readily creates a mismatch betweena student's understanding of what the calculator is doing and a student's
understanding of the number line, integers, fraction, and decimals. Say a young
student uses a calculator to divide 6 by 2, and sees 3 followed by a decimal point.How likely is it that child notices the decimal point?. So far, the child experiences
no great surprises or problems. The child then divides 6 by 4 and sees 1.5. Hmm.
What does that mean? Perhaps the child continues, dividing 6 by 5 and getting
1.2. Hmm, what does that mean? Continuing, 6 divided by 6 does not produce anysurprise, but what about 6 divided by 7 to produce a result 0.8571428?
A subtraction may produce a negative number for an answer. Multiplication of large
numbers may produce an "overflow," perhaps indicated by an E.
2 Why is the calculator called a 6-function calculator? Does the child know whata mathematical function is? (How much will the student be helped by your saying,
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A mathematical function is an abstract entity that associates an input drawn from
a fixed set to a corresponding output according to some rule?)
3 What is the meaning and use of the key that we, as adults, know is the squareroot key? A similar question applies to the key labeled %.
4 If the calculator has mc, mr, m-, and m+, etc. keys, what do they mean and what
are they used for?
Of course, scientific calculators with their large number of built-in functions and their scientific notation provide still more teaching and learning challenges. The challenge is
further increased by graphing and equation-solving calculators.
The point is, just giving students calculators unaccompanied by encouraging instruction
is poor planning and poor teaching.
Computers
Teachers often assume that those students who play games on computers or do email, or use a Social Networking Website, or… "understand" computers. And so they do—in
some ways. Those ways are unlikely to include much in the way of math. You, as a goodteacher, will think about what you want students in your math class to know about roles
of computers in representing and solving math problems, and the applicability of these
processes and results to “real-world” activities.
For example, what do you want your students to know about use of a spreadsheet? This isa huge topic. Just take the small subtopic of using a spreadsheet to graph data. A young
student can use such software to crate a colorful pie chart well before the student learns to
create one by hand. Hmm. Communicators can represent a set of data in many different
ways. How does a student learn which are apt to be most effective in a particular situation?
Modern spreadsheet software contains a huge number of built-in functions or routines, or
access a multitude of templates. Our math education system has not yet committed itself to being the curriculum area primarily responsible for teaching students about
spreadsheets and their roles in representing (modeling) and solving computationally-
oriented problems.
Information retrieval
One of the most important ideas in problem solving is building on the work that othershave done. This is particularly important in math since results developed over thousands
of years by math researchers are available when you attempt to understand and solve a
current math problem.
Ask yourself: "Where in the math curriculum do students learn to retrieve and make use
of past mathematical accomplishments? The standard math curriculum strives to store
some of this accumulated knowledge in students' heads. However, it does little to teach
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students how to read math well enough so that they can retrieve and use math results that
are available in the literature.
By and large, precollege students are not even given good access to the math books theyhave used in their previous years of schooling. They are not taught how to do Web
searches as an aid to retrieving math information. They are not taught to read math wellenough to benefit from the resources available on the Web, in math libraries, or even in
math textbooks.
Teaching and Learning
To a surprising extent, math is still taught using "oral tradition." A teacher does a "standand deliver" presentation. All students receive the same presentation. A few students may
get a chance to ask questions, but this opportunity is often severely limited. Similarly, the
teacher may ask the class a few questions in an attempt to ascertain if students understandthe new material. Time is too short for an individual response from each student.
Students then do seatwork and perhaps homework. The seatwork and homework tend to
entail repetitions of the process the teacher demonstrated.
Student exercises may be on a worksheet or come from a textbook. The worksheetapproach tends to separate students from any chance to look at a book that covers the
material, and perhaps learn to read the book and learn by reading.
Book-based seatwork and homework provide students an opportunity to look back at a
previous section in the book and perhaps review the ideas presented by the teacher. Someexercises may draw upon material from earlier chapters and sections of the book. The
student rarely has access to the previous years' books or to alternate presentations of thetopic.
ICT has brought us powerful alternatives to this approach. Computer-assisted learning(CAL) and Distance Learning are two major, proven aids to teaching and learning.
Most schools need more computer facilities if ICT is to play a major instructional role.
Going to a computer lab or bringing in a classroom set of laptops once a week clearly is
of limited value. If you face that situation, then think carefully about the most effectiveuse for this scarce resource. Think carefully about what you want students to be learning
about roles of computers in math content, in math teaching, and in math learning. How
can the limited computer time make a significant contribution to your overall mathcurriculum in attitude, knowledge, and skills?
A "Full Blown" Math Lesson Plan Template
Here is a Level 3 (see the diagram at the beginning of this Web page) general-purpose
template for math lesson plans. It is a template for lesson plans to be used in teaching preservice and inservice teachers. It includes all components of an interdisciplinary
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general-purpose lesson plan template, and it contains a number of components specific to
teaching math and learning to be a better teacher of math.
As you develop a lesson plan or prepare to teach from a lesson plan, think about theteacher prerequisite knowledge and skills needed to do a good job of teaching the lesson.
Before you teach a lesson, do a self-assessment to determine if you have the needed mathcontent knowledge, the general pedagogical knowledge, and the math pedagogical
knowledge. If you detect possible weaknesses, spend time better preparing yourself toteach the lesson, and spend time thinking about what you will learn as you teach the
lesson. (See item 10 in the list given below.)
1. Title and short summary —like a section title in a book chapter (lesson plan) or a
chapter title (unit plan). The title of a math lesson plan or unit should communicate purpose to the teacher and to students. It serves in part as an advance organizer. The short
summary is part of the advance organizer and should include a statement of how the
lesson or unit serves to empower students.
2. Intended audience and alignment with Standards —categorization by: subject or
course area; grade level; general math topic being taught; length; and so on. A listing of
the math standards (state, province, national, etc.) being addressed. Categorization
schemes are especially useful in a computer database of lessons, allowing users quicklyto find lesson plans to fit their specific needs.
3. Prerequisites —a critical component in math lesson planning and teaching. See the
Prerequisite, Review, and Remediation section of this document. Math teachers and their
students face the difficulty that a significant proportion of the class may not meet the prerequisites. Such students are not apt to learn the new material very well, and the lack
of success will likely add to "I can't do math" and "I hate math" attitudes.
4. Accommodations —special provisions needed for students with documentedexceptionalities and other students with math learning and math understanding
differences from "average" students. This ties in closely with how to deal with students
who clearly lack needed prerequisite math knowledge and skills, and how to deal with
students who’ll be bored by the “normal” planned lesson.
5. Learning objectives —the “there” in “getting from here to there.” Teachers of teachers
often stress the need for stating learning objectives precisely. They often use the
expression measurable behavioral objectives. Some additional important aspects of the
earning objectives section of a math lesson or unit of study are:
a. Each lesson and unit of study needs to maintain and improve each student's
overall level of math expertise. It is important that students understand the idea of
math expertise, how it grows through study, practice, and use, and how itdecreases through lack of use (forgetting). Students need to learn to take personal
responsibility for their levels of expertise. Every lesson should include an
emphasis on self assessment, self responsibility, sense-making, and problem
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solving. Problem solving and proof are closely related topics; problem solving
should be in ways that lay the foundations learning about proofs in math. Informal
(and, eventually, more formal) proof-like arguments should be part of every unitof study.
b. Keep in mind that math notation, vocabulary, and ideas have a significant level
of abstraction. Math modeling is a process of extracting a "pure" math problemfrom a problem situation. This extraction or modeling process is a very important
aspect of learning and understanding math. It is a challenge to teachers and to
students. Carefully examine the learning objectives in a lesson to see how they fitin with the Piagetian math cognitive developmental level of your students and
how they help your students to move upward in their math cognitive
development.
c. Make a clear distinction between lower-order and higher-order knowledge andskills. Both are essential to problem solving, and it is important for students to be
learning and making use of both lower-order and higher-order aspects of problem
solving in an integrated, everyday fashion. Note, of course, lower-order and
higher-order are dependent on the math cognitive developmental level and mathmaturity of your students. Higher-order pushes the envelope—it helps students to
increase their level of math development and math maturity. This ties in closelywith (a) given above.
d. Each unit of study should include specific instruction on transfer of learning. A
unit of study is long enough so that students can learn a strategy, or significantly
increase their knowledge and understanding of a strategy, and gain increased skillin high-road transfer of this learning to problem solving across the curriculum.
e. Communication in Math. Part of this is students gaining skill in communicating
with themselves—mental sense-making. Pay special attention to students learninghow to read math well enough so that they can learn math by reading math. Think
about every math lesson as including both some math content for students to read
and some math word problems in which students can practice using their mathknowledge and improve their general math problem-solving skills.
f. Keep in mind the steadily growing importance of Computational Thinking in
math and in other disciplines. Stress roles of ICT and a student's brain/mind incomputational thinking. Help students learn the capabilities and limitations of
brain/mind versus calculators and computers in representing and working to solve
math problems. Stress how math is used to develop math models of problem
situations to be explored and possibly solved in each discipline. Math is of growing importance in many disciplines because of its role in computational
thinking and in using math models to represent and help solve the problems in
these disciplines.
6. Materials and resources —These include reading material, assignment sheets,worksheets, tools, equipment, CDs, DVDs, videotapes, etc. You may need to begin the
acquisition process well in advance of teaching a lesson, and it may be that some of the
resources are available online. Keep in mind Marshall McLuhan's statement, "Themedium is the message." If you want students to learn to be mathematically proficient in
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an adult world where calculators, computers, and other ICT are ubiquitous, strive to
create such a teaching, learning, and assessment environment in your classroom.
7. Instructional plan —This is usually considered to be the heart of a lesson plan. It provides instructions to the teacher to follow during the lesson. It may include details on
questions to be asked during the presentation to students. If the lesson plan includesdividing students into discussion groups or work groups, the lesson plan may include
details for the grouping process and instructions to be given to the groups.
a. A carefully done math lesson plan includes a discussion of math content
pedagogical knowledge that has been found useful in helping students learn the
topic.
b. If students are going to be making use of math manipulative, calculators,computers, and other ICT learning aids, pay special attention to the general
pedagogy requirements and the PCK requirements of dealing with a large number
of students. The cognitive and organizational load on a teacher dealing the first
few times with a one-on-one computer situation is rather overwhelming.
8. Assessment options —A teacher needs to deal with three general categories of
assessment: Formative, summative, and long-term residual impact. Students need to learn
to do self assessment and to provide formative assessment (evaluation during the processto aid progress) and perhaps summative assessment feedback (passing judgment on the
final result) to each other. A rubric, perhaps jointly developed by the teacher and
students, can be a useful aid to helping students take increased responsibility for their own learning.
9. Extensions —These may be designed to create a longer or more intense lesson. For
example, if the class is able to cover the material in a lesson much faster than expected,extensions may prove helpful. Extensions may also be useful in various parts of a lessonwhere the teacher (and class) decide as the lesson is being taught that more time is needed
on a particular topic.
10. Teacher learning on the job —View each math lesson and unit of instruction as an
opportunity to increase your knowledge and skills in math content, math pedagogy, andgeneral pedagogy. Set specific learning goals and objectives for yourself. After teaching a
lesson or a unit of study, reflect on what you have learned. Add some notes to your lesson
plan that reflect your increased knowledge and skills, and that provide a sense of direction for focusing your learning the next time you teach the lesson or unit.
11. References —The reference list might include other materials of possible interest to
people reading the lesson plan or to students who are being taught using the lesson plan.
References
Garfield, Joan (1995). How Students Learn Statistics. International Statistics Review.
Retrieved 1/25/08: www.stat.auckland.ac.nz/~iase/publications/isr/95.Garfield.pdf.
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Handly High School (n.d.).
http://www.pen.k12.va.us/Div/Winchester/jhhs/math/mathhome.html. Quotations about
Mathematics and Education. Retrieved 11/25/07:http://www.pen.k12.va.us/Div/Winchester/jhhs/math/quotes.html.
Leigh, A. (2007) Estimating Teacher Effectiveness from Two-Year Changes in Student’sTest Scores. Retrieved 12/5/07: http://rsss.anu.edu.au/documents/TQPanel.pdf .
Math Forum (n.d.). The Math Forum at Drexel University. Retrieved 12/12/07:http://www.mathforum.org/. The Math Forum Internet Mathematics Library is a treasure
trove of links categorized by topic or educational level. The Website also offers
kindergarten to graduate-level lesson plans, software, student project ideas and
homework help.
Math Resources from the southern Oregon Education Service District. Retrieved 2/5/08:
http://www.soesd.k12.or.us/Page.asp?NavID=741. A nice collection of computer-based
resources of use to teachers and to teachers of teachers.
Steen, Lynn Arthur (1999). Algebra for all in eighth grade: What's the rush? Appeared inMiddle Matters, the newsletter of the National Association of Elementary School
Principals, Vol 8, No. 1, Fall 1999, pp. 1, 6-7. Retrieved 11/23/07:
http://www.stolaf.edu/people/steen/Papers/algebra.html.
Quotations (n.d.). Welcome to the Garden Quotes: Quotes about mathematics. Retrieved1/24/08: http://www.quotegarden.com/math.html.
Authors
David Moursund and Dick Ricketts.