Good Math Lesson Plans
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- 1 Introduction
- 2 A General-purpose Lesson Plan
- 3 Discipline Specificity
- 4 What Is Math?
- 5 Some Math-Specific Lesson Plan Topics
- 5.1 Increasing Math Expertise
- 5.2 Prerequisite, Review, and Remediation
- 5.3 Teaching Self-Assessment and Self-Responsibility
- 5.4 Teaching for Transfer of Learning
- 5.5 Math Cognitive Developmental Level and Maturity Level
- 5.6 Communication in Math
- 5.7 Math Modeling
- 5.8 Computational Math
- 5.9 Lesson Plan as Self-inservice Education
- 6 Some Roles of ICT
- 7 A "Full-blown" Math Lesson Plan Template
- 8 References
- 9 Links to Other IAE Resources
- 10 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 the United 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 along from one generation to the next ought to be the highest honor and the highest responsibility anyone could have." (Lee Iacocca; American industrialist; 1924-.)
Lesson plans are a core topic in most preservice teacher education programs. Preservice teachers 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 to teach well and students to learn well.
“Lesson plan” usually refers to a single lesson, designed for one class period. However, it can also refer to a sequence of such plans designed for a unit of study. (Such a sequence may be called a unit plan.) In this document, “lesson plan” means a plan to facilitate one or more times of organized teaching and learning sessions. We use the term to include one-period and multi-period lessons.
The figure below shows that the need for written detail depends on the lesson plan’s audience.
- 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 derive quadratic formula by completing the square; then use spreadsheet to show how to do the calculations.” “Use Taxman software to introduce factoring.”
- A collegial lesson plan is designed for a limited, special audience such as your colleagues, a substitute teacher, or a supervisor such as a department head or principal. It contains more detail than the first category. It is designed to communicate with people who are familiar with the lesson plan writer's school, curriculum, and standards.
- A high-quality, publishable lesson plan is designed for publication and for use by a wide, diverse audience. It contains still more detail than a plan in 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 to elicit in-depth discussion.
This present 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, a preservice or inservice teacher can use it for self-instruction.
A General-purpose Lesson Plan
Early on in their teacher education program of study, preservice teachers are apt to encounter a general-purpose template or outline for 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.
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 1-period lesson plan as being like a section title in a book chapter, while the title of a multi-period plan is like a chapter title. The short 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; and so on. A listing of the standards (state or national) being addressed. Categorization schemes are especially useful in a computer database of lesson planss, 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, and it 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 will normally encounter such a lesson—for example, second graders near the end of the school year or students entering an Algebra 1 course.
- State any special prerequisites—for example, perhaps list key ideas that it is sort of assumed that students should have covered, but that many will have not learned very well or will have forgotten. This type of prerequisite is often used in a focused review at the beginning of a lesson. 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’s capacity or is 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 a very careful statement 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, and whether 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 or objectives, 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, physical environment, and so on. It may be necessary to begin the materials and resources acquisition process well in advance of teaching a lesson, and it may be that some of these 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 asked during a presentation to learners, and actions to handle contingencies. (For example, suppose the topic is the U.S. Constitution and a student raises the question of whether private citizens should 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 an increasing 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 specific skill or topic.
10. Teacher reflection and lesson plan revision. This is to be done after teaching the lesson. Items in this section that are related to content, pedagogy, resources, and skills will make for 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.
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). pedagogical content knowledge 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 that discipline’s proven PCK repertoire. A good lesson plan may well include a discussion of PCK to employ when conducting the lesson.
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, the teaching of math has many differences from teaching any other discipline. A good math lesson 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 believe math to be second only to 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 specialized vocabulary
- Methods of teaching, learning, and assessment; its lower-order and higher-order knowledge and skills; and its critical thinking and understanding—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 sharing accumulated results
- Criteria that separate and distinguish among a:
- a) novice,
- b) person who has a personally useful level of competence,
- c) reasonably competent person, employable in the discipline,
- d) local or regional expert,
- e) national or world-class expert.
When you teach within a discipline, you represent that discipline. Part of your teaching task is an appropriate and adequate representation of the discipline. This means that you, as a math teacher, need to identify and explain similarities and differences between math and the other disciplines your students have studied or are studying. This is especially important for the various other disciplines in which math is a 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 lesson addresses 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.
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 doctoral 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 one-period lesson plan in a discipline versus the challenge of creating a multi-lesson unit of study, a course, or an extended curriculum 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 have such ongoing efforts. For example, the National Council of Teachers of Mathematics (NCTM) plays a leadership role in developing math education standards in the United States. From time to time a populous state, such as California, will publish benchmarks in a discipline such as math, and these benchmarks influence textbook companies and many other states throughout the country.
Currently, the Common Core State Standards (CCSS) project is a dominant force in the United State. See http://www.corestandards.org/.
What Is Math?
Precollege math curricula in the United States are sometimes described as (and criticized as) 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, and curriculum developers continually face the challenge of balancing depth and variety.
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 educational system. Those who teach teachers are expected to be mathematically competent and able to communicate a defensible answer to the "What is math?" question.
Some Often-Quoted Answers
Here are three quotations that are fun and interesting, but not particularly helpful in math curriculum 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.” (Leopold Kronecker; German mathematician and logician; 1823–1891.)
More math education quotations are available at http://iae-pedia.org/Math_Education_Quotations.
Patterns and Language of a Discipline
Many people attempting to answer the "What is math?" question give answers that fit the discipline they are talking about, but that also apply to many other disciplines.
For example, it 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. A human brain works by processing patterns. It is only the differing 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 quickly impose such patterns in nervous systems; researchers are making progress in 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 used 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 Lynn Arthur 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).
The combined assertion that math is a language and 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. Math is known as being a language that facilitates very precise communication—perhaps more so than any other widely used discipline-specific 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 math from other disciplines. Here are 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 cannot answer the question 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 mathematical modeling have special definitions that are different from the "natural language" definitions that people commonly 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.
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 answers 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 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 against which possible new versions are measured. Inventions and many of the results in science can have very long lives.
In any case, math results have a permanency that facilitates the accumulation of results over millennia, with the results being such that new researchers and users of math can safely 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 are more 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 new story 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 accumulated knowledge 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 have confidence in their mathematics and to be able to learn more as their needs and interests dictate.
This accumulation of math results is very important in facilitating one of the most important ideas in problem solving. That idea is one of building on the previous learning of oneself and others. It is sometimes simplified to the statement, "Don't reinvent the wheel."
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 online or in a library will produce an answer, but it will not contribute much to 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, one answer, or many answers. Second, the goal is to learn math to fit one's current and possible future needs.
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:
- Store ever-more accumulated math knowledge in a form accessible by a steadily increasing percentage of the world's population.
- Store parts of this accumulated math knowledge in a form so that the computer system can actually carry out the procedures to solve or to help to solve 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, or a brain augmentation. 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, answers 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, topics 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 effort the 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.
The absolute heart—the unifying mission—of math education is that students will increase their expertise at math 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.
- Task situations: Recognizing, posing, clarifying, and accomplishing tasks.
- Decision situations: Recognizing, posing, clarifying, and making decisions.
Note that solving problems usually requires higher-order, critical, creative, and wise thinking. 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 the solution(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 and/or keyboaring, 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 multi-digit 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 the higher-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. Here, it can help to remember that what you have learned to do so well that the doing of it may seem to you to be a lower-level skill, but it 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 readily convince 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 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 a correct 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 steps are implemented incorrectly, a student may well be deserving of considerable partial credit.
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 evaluating the written communication of the essay.
However, there may be a lodestone for the individual teacher: How does this grade, this 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 concepts or skills the students are being taught might be useful in helping them to address various aspects of the overall problem of sustainability.
As a math topic is being taught, and transfer of learning of that topic is being taught, students might well be led to consider 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 their courses and in other parts of their lives. Teachers do well to assume that many students in a class that previously has covered the same course or courses will have forgotten much of the material covered from those courses. The fact that retention varies tremendously among students makes the teacher’s task harder; that previously exposed 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 is not 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 already 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 prerequisite situation becomes 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 others, the review time is a waste 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 inadequate foundation. 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 becomes bad enough, our educational system tends to try to 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 the other 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, and some students who are at a 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 class average. That the same student may find some areas of math easy and other areas 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 studied or 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 often will 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 math instruction—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 interacting with 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 higher level grades. Both teachers and students have ownership of the problem. Thus, one way for a teacher to approach this is to educate students about the prerequisites problem and get students actively engaged in addressing the problem.
This raises the issue of the extent to which 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:
- 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 will include 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 good assessment and feedback from themselves and others (such as the teacher) so they know they’re on track and on schedule.
- Provide a variety of aids to students to help them meet the expectations. This includes helping students learn to help themselves.
Helping students learn to take responsibility for their own learning is one of the most important 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 evaluate 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 carries credit toward graduation—sometimes because they’ve regarded their earlier 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 through material that they have already “studied” in middle school and high school.
We now have the technology for students to test their own levels of competence via online tests that one can take over and over, because these 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 computer testing when they want an answer to “How am I doing?” 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 for students to get any 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 by Charles 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 providing feedback to themselves—that is, often what they are doing makes little or no sense to them, and they seldom reflect on whether their answers are correct. 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 ownership in the task.
One reason 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 be memorized. A student does not learn to attach meaning to the numbers being manipulated or to the answers being produced. The student does not gain the knowledge and skills to check whether or not a result "makes sense."
Some math books and some teachers suggest that students do paper and pencil calculations and 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 a calculator 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 identiy those who are reasonably good at checking to see if an answer makes sense. Many students will solve such problems and produce answers that make no sense whatsoever—and be quite unable to detect that their answers make no sense. Word problems generally are useful for developing 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-checking skills. Other important reasons are that they entail higher-order skills, and that students’ reactions to them tells a teacher 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 is highly desirable to teach math in a manner that facilitates transfer of learning to other disciplines, and also 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 the theories 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 out 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. Students can learn to solve the less difficult problems, put all the results together in an appropriate manner, and then the harder problem is solved. 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 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 material is 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 from learning 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 both require language arts skills. Another example: Compare popularity ratings of U.S. Presidents when they left office with their relative rankings by historians today, and discuss the relationship 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:
- The student moves up the Piagetian (math) cognitive developmental scale, moving toward (math) formal operations.
- The student grows in math maturity—getting better at thinking mathematically, learning to learn math, and to creatively use math to solve complex 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: Improving Math 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.
Piaget is well known for his work in cognitive developmental theory. Modern interpretations of his work emphasize 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.
Many college students have become capable of formal operations in many areas, but are not yet 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 the use of math, but they do not 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.
The households and extended families that children grow up in will vary considerably in how much they help children in their general cognitive development and in their discipline-specific cognitive development. This poses two questions for math curriculum developers:
- How can we provide appropriate math curriculum for students who are at considerably different math cognitive developmental levels?
- How can we 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 it contributes to students efficiently continuing their math cognitive development. In addition, 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 and sequence.
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 needed to 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 incorporated into a plane geometry or first algebra course. Only a minority of high school freshman 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 educators Dina and 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.
|0||Visualization||Students recognize figures as total entities (triangles, squares), but do not recognize properties of these figures (right angles in a square).|
|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||Informal Deduction||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. The interrelationship 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 to Formal 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 concrete examples.|
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 developmental scale has four levels, numbers 1 to 4. Thus the highest level designated "4" in the van Hiele geometry cognitive developmental scale is actually one level above the highest level of the Piaget cognitive developmental scale, also designated as "4."
A third area is probability. A number of math education researchers have explored the issue of cognitive development and learning probability. For example:
- Garfield, J., & Ahlgren, A. (1988). Difficulties in Learning Basic Concepts in Probability and Statistics: Implications for Research. Journal for Research in Mathematics Education. 19, 1.
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 have studied. 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 to a student’s own level of cognitive development suggests that learning for understanding requires students to be at a formal operations 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 cognitive developmental scale.
|Stage & Name||Math Cognitive Development|
|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 small quantities such as 1, 2, and 3. As infants gain crawling or walking mobility, they can display innate spatial sense. For example, they 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, such as 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 learn number 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 the new content materials and ideas are not strongly rooted in verbal, concrete, mental images and understanding of somewhat similar ideas that have already been acquired.
There is a substantial difference between developing general ideas and 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 in math 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 learning is a core aspect of Level 4.
|Level 5. Abstract mathematical operations.||Mathematical content proficiency and maturity at the level of contemporary math texts used at the senior undergraduate level in strong 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 in groups, 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. This includes speed, accuracy, and understanding in reading the research literature, writing research literature, and in oral communication (speak, listen) of research-level mathematics. Pose and solve original math problems at the level of contemporary research frontiers.|
Mathematicians tend to prefer the concept of math maturity rather than 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 a mixture 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 standard notation 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 communicate ideas. 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 to students gaining in math maturity.
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 improve their 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 the idea of 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.
Thus, in middle school reading is the dominant aid to learning content. In fact, 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 math education. By the time most students finish high school, they do not yet have the ability 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 developed. 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 students 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 is available) communication when discussing a math problem. Moreover, as one might expect, math professors have developed considerable skill in reading math books and journals.
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); this is accomplished by learning to how 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 to relearning 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 takes 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 prerequisite situation 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 Situations. The six steps shown are:
- Problem posing and problem recognition to produce a Clearly-defined Problem;
- Mathematical modeling;
- Using a computational or algorithmic procedure to solve a computational or algorithmic math problem;
- Mathematical "unmodeling";
- Thinking about the results to see if the Clearly-defined Problem has been solved; and
- 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 of attempting to solve the original Clearly-Ddfined Problem or to resolve the original Problem Situation.
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 person deciding 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 may or 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 to resolve 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 students learn 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 began to become widely available and relatively reliable 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, sense making, and on steps 1, 2, 4, 5, and 6.
Many thousands of articles, books, and websites address ways students can learn to solve word problems. Often, such material treats the task as one of mechanical steps, 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 problem into a “pure” math problem. They memorize and use these tricks with some success in getting correct answers, but with little understanding.
This approach to word problems misses the whole point or students learning to deal with challenging "real world" problem situations in which math might be a useful aid to resolving the situations. It misses the whole point of translating (representing, modeling) such real world problems situations into math problems.
Another difficulty exists. In the typical schoolbook word problem, students “know” that there 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. Then, when a student faces a problem situation or a “story problem” that may not have a solution, the student is apt to be totally frustrated.
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 more information 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 with marbles. 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 children are 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.
In this children and marbles situation, we created a mathematical model and we solved 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 original question posed. For example, it might have been that 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 it 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 of the types of problems in many math books. The lack of non-mathematical context, meaning, or purpose makes it much more difficult for problem solvers to detect possible errors in the modeling and problem-solving processes.
Math and some of the sciences have traditionally been divided into "pure" and "applied" components—pure and applied math, theoretical and experimental physics, theoretical and observational astronomy, etc. Computers have changed this situation. Math and the various 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 in the 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 on computational 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 very important in math, in 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
Many teachers feel that they learn more about teaching during their first few years 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 their students to learn. Here is summary of an important 2006 study by Andrew Leigh:
- Using a data set covering over 10,000 Australian primary school teachers and over 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 two years, 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 positive correlation 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 in the 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 in terms 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):
- 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 the demonstration to increase student interest in and insight into the topic.
- Pedagogical content knowledge (PCK). Topics in any discipline can be presented in many different ways. The larger your presentation repertoire on a topic, the more apt you are to meet the diverse needs of your students.
- Subject matter content. Any topic you are teaching has far more content than you are teaching. You and your students will benefit as you gradually expand the depth and breadth of your knowledge of any specific topic you are teaching.
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 a variety of methods that solve a particular problem, and a variety of methods that fail to solve that 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-solving knowledge 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 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 some students 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 collection of problems with other math teachers.
Some Roles of ICT
Information and Communication Technology (ICT) plays two roles in a good math lesson plan. ICT is part of math content, and it provides aids to teaching and learning math.
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.
The National Council of Teachers of Mathematics (NCTM) supports and encourages students to learn 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 between a student's understanding of what the calculator is doing and a student's understanding of the number line, integers, fractions, 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 the child to notice 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 any surprise, but what about 6 divided by 7 to produce a result of 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 what a mathematical function is? How much will the student be helped by your saying, "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 square root 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.
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 good teacher, 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 is a 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 types are apt to be most effective in a particular situation?
Modern spreadsheet software contains a huge number of built-in functions or routines, or can 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.
One of the most important ideas in problem solving is building on the work that others have 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 students how to read math well enough to be able to 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 they have 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 well enough 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 "stand and 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 whether students understand the 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 worksheet approach 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 math 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. Some exercises 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 the topic.
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 effective use 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 math curriculum in attitude, knowledge, and skills?
A "Full-blown" Math Lesson Plan Template
Here is a Level 3 general-purpose template for math lesson plans (see the diagram at the beginning of this Web page). It is a template for lesson plans to be used in teaching preservice and inservice teachers. They can learn to use it in developing their own lesson plans. It includes all components of an interdisciplinary 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 the prerequisite knowledge and skills needed by the teacher in order to do a good job of teaching the lesson. Before you teach a lesson, do a self-assessment to determine if you have the math content knowledge, the general pedagogical knowledge, and the math pedagogical knowledgethat will be needed. If you detect possible weaknesses, spend time better preparing yourself to teach 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 quickly to 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 student attitudes of "I can't do math" and "I hate math."
4. Accommodations—special provisions needed for students with documented exceptionalities 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 learning 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 it decreases 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 solving. Problem solving and proof are closely related topics; problem solving should be taught in ways that lay the foundations for learning about proofs in math. Informal (and, eventually, more formal) proof-like arguments should be part of every unit of 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 problem from 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 fit in 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 and skills. 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 that, of course, lower-order and higher-order are dependent on the math cognitive developmental level and math maturity of your students. Higher-order pushes the envelope—it helps students to increase their level of math development and math maturity. This ties in closely with (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 skill in 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 learning how 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 math knowledge and also 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 in computational 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 textbooks and related reading material, assignment sheets, worksheets, tools, equipment, CDs, DVDs, access to relevant websites, 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, "The medium is the message." If you want students to learn to be mathematically proficient in 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 for 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 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.
- 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, websites,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 who is dealing for the first few times with a one-on-one computer situation can be 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 also to provide formative assessment (evaluation during the process to 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 and self-assessment.
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 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, and general 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 and Resources—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.
Garfield, J. (1995). How students learn statistics. International Statistics Review. Retrieved 1/25/08: http://www.stat.auckland.ac.nz/~iase/publications/isr/95.Garfield.pdf.
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’s test 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.
Quotations (n.d.). Welcome to the garden quotes: Quotes about mathematics. Retrieved 1/24/08: http://www.quotegarden.com/math.html.
Steen, L.A. (1999). Algebra for all in eighth grade: What's the rush? Middle Matters, the newsletter of the National Association of Elementary School Principals. Retrieved 11/23/07: http://www.stolaf.edu/people/steen/Papers/algebra.html.
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