The history of formal math education goes back to the initial development of reading and writing, about 5,200 years ago. The various written languages that have been developed include symbols for numbers. In essence, written math notation facilitates the "paper and pencil" algorithms that are an important component of the mathematics curriculum.
More generally, written language is a powerful aid to one's brain. It helps to overcome limitations of short-term and long-term memory.
Arithmetic is so important that it is considered one of the "basics" of education. The wide spread, low cost availability of calculators has proven to be a major challenge to our math education system. Scientific calculators, graphing and equation-solving calculators, and computers have all contributed to the challenge of developing a good math education system for the Information Age.
A Little Bit of History of Math
Quoting from the Wikipedia:
- Long before the earliest written records, there are drawings that do indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back to c. 70,000 BC. Also prehistoric artifacts discovered in Africa and France, dated between 35,000 BC and 20,000 BC, indicate early attempts to quantify time.
Throughout their history, humans have faced the need to measure and communicate about time, quantity, and distance. The Ishango Bone artifact shown below is perhaps 25,000 years old.
- At one end of the Ishango Bone is a piece of quartz for writing, and the bone has a series of notches carved in groups. It was first thought these notches were some kind of tally marks as found to record counts all over the world. However, the Ishango bone appears to be much more than a simple tally. The markings on rows (a) and (b) each add to 60. Row (b) contains the prime numbers between 10 and 20. Row (a) is quite consistent with a numeration system based on 10, since the notches are grouped as 20 + 1, 20 - 1, 10 + 1, and 10 - 1. Finally, row (c) seems to illustrate for the method of duplication (multiplication by 2) used more recently in Egyptian multiplication. Recent studies with microscopes illustrate more markings and it is now understood the bone is also a lunar phase counter. Who but a woman keeping track of her cycles would need a lunar calendar? Were women our first mathematicians?
The picture given below shows Sumerian clay tokens whose use began about 11,000 years ago. Such clay tokens were a predecessor to reading, writing, and mathematics.
Quoting from the document History of Mathematics:
- Every culture on earth has developed some mathematics. In some cases, this mathematics has spread from one culture to another. Now there is one predominant international mathematics, and this mathematics has quite a history. It has roots in ancient Egypt and Babylonia, then grew rapidly in ancient Greece. Mathematics written in ancient Greek was translated into Arabic. About the same time some mathematics of India was translated into Arabic. Later some of this mathematics was translated into Latin and became the mathematics of Western Europe. Over a period of several hundred years, it became the mathematics of the world
Mathematics as a formal area of teaching and learning was developed about 5,200 years ago by the Sumerians. They did this at the same time as they developed reading and writing. The development of reading, writing, and formal mathematics allowed the codification of math knowledge, formal instruction in mathematics. It was the start of a steady accumulation of mathematical knowledge.
What is Mathematics?
As indicated above, math has a very long history. Initially, math focussed on simple counting and record keeping. The development of written languages facilitated the growth of math into a broad and deep discipline. This discipline is considered so important that is is a required area of study in schools throughout the world.
Mathematics as a Discipline
A discipline (a organized, formal field of study) such as mathematics tends to be defined by the types of problems it addresses, the methods it uses to address these problems, and the results it has achieved. One way to organize this set of information is to divide it into the following three categories (of course, they overlap each other):
- Mathematics as a human endeavor. For example, consider the math of measurement of time such as years, seasons, months, weeks, days, and so on. Or, consider the measurement of distance, and the different systems of distance measurement that developed throughout the world. Or, think about math in art, dance, and music. There is a rich history of human development of mathematics and mathematical uses in our modern society.
- Mathematics as a discipline. You are familiar with lots of academic disciplines such as archeology, biology, chemistry, economics, history, psychology, sociology, and so on. Mathematics is a broad and deep discipline that is continuing to grow in breadth and depth. Nowadays, a Ph.D. research dissertation in mathematics is typically narrowly focused on definitions, theorems, and proofs related to a single problem in a narrow subfield in mathematics.
- Mathematics as an interdisciplinary language and tool. Like reading and writing, math is an important component of learning and "doing" (using one's knowledge) in each academic discipline. Mathematics is such a useful language and tool that it is considered one of the "basics" in our formal educational system.
To a large extent, students and many of their teachers tend to define mathematics in terms of what they learn in math courses, and these courses tend to focus on #3. The instructional and assessment focus tends to be on basic skills and on solving relatively simple problems using these basic skills. As the three-component discussion given above indicates, this is only part of mathematics.
Even within the third component, it is not clear what should be emphasized in curriculum, instruction, and assessment. The issue of basic skills versus higher-order skills is particularly important in math education. How much of the math education time should be spent in helping students gain a high level of accuracy and automaticity in basic computational and procedural skills? How much time should be spent on higher-order skills such as problem posing, problem representation, solving complex problems, and transferring math knowledge and skills to problems in non-math disciplines?
Beauty in Mathematics
Relatively few K-12 teachers study enough mathematics so that they understand and appreciate the breadth, depth, complexity, and beauty of the discipline. Mathematicians often talk about the beauty of a particular proof or mathematical result. Do you remember any of your K-12 math teachers ever talking about the beauty of mathematics?
G. H. Hardy was one of the world's leading mathematicians in the first half of the 20th century. In his book "A Mathematician's Apology" he elaborates at length on differences between pure and applied mathematics. He discusses two examples of (beautiful) pure math problems. These are problems that some middle school and high school students might well solve, but are quite different than the types of mathematics addressed in our current K-12 curriculum. Both of these problems were solved more than 2,000 years ago and are representative of what mathematicians do.
- 1. A rational number is one that can be expressed as a fraction of two integers. Prove that the square root of 2 is not a rational number. Note that the square root of 2 arises in a natural manner in land-surveying, carpentering, and navigation problems.
- 2. A prime number is a positive integer greater than 1 whose only positive integer divisors are itself and 1. Prove that there are an infinite number of prime numbers.
We all know that prime numbers are important in the simplification of fractions. In recent years, very large prime numbers have emerged as being quite useful in encryption of electronic messages.
Quoting from Early June 1999 issue of Early Childhood: Where Learning Begins:
- The National Council of Teachers of Mathematics (NCTM), the world's largest organization devoted to improving mathematics education, is developing a set of mathematics concepts, or standards, that are important for teaching and learning mathematics. There are two categories of standards: thinking math standards and content math standards. The thinking standards focus on the nature of mathematical reasoning, while the content standards are specific math topics. Each of the activities in this booklet touches one or more content areas and may touch all four thinking math areas.
- The four thinking math standards are problem solving, communication, reasoning, and connections. The content math standards are estimation, number sense, geometry and spatial sense, measurement, statistics and probability, fractions and decimals, and patterns and relationships. We have described them and then provided general strategies for how you as a parent can create your own activities that build skills in each of these area
North Central Regional Educational Laboratory
Quoting from NCREL: What is Mathematics?
- Often, people equate mathematics with arithmetic. Arithmetic is concerned with numbers. When considering the mathematics curriculum, many people focus on computational skills and believe that they constitute the full set of competencies that students must have in mathematics. Traditionally, the major emphasis of the K-8 mathematics curriculum has been to teach children arithmetic - how to add, subtract, multiply, and divide whole numbers, fractions, decimals, and percentages. Mathematics involves more than computation. Mathematics is a study of patterns and relationships; a science and a way of thinking; an art, characterized by order and internal consistency; a language, using carefully defined terms and symbols; and a tool. Teachers and other educators working together to improve mathematics education must explore a broader scope of mathematics. Mathematics should include experiences that help students to shift their thinking about mathematics and define mathematics as a study of patterns and relationships; a science and a way of thinking; an art, characterized by order and internal consistency; a language, using carefully defined terms and symbols; and a tool.
Courant and Robbins Book
- Courant, Richard and Robbins, Herbert (1947). What is Mathematics? London: Oxford University Press.
This is a relatively famous book first published in 1941. It is a college-level math textbook. Quoting from the book:
- Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitutes the life, usefulness, and supreme value of mathematical science.
The preface to the 1941 edition notes that:
- For more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today the traditional place of mathematics in education is in grave danger. … The teaching of mathematics has sometimes degenerated into empty drill in problem solving, which may develop formal ability but that does not lead to real understanding or to greater intellectual independence.
This second quote is closely related to the Math Education Wars about math education reform. The reformers are pushing for a curriculum that enhances understanding and intellectual independence.They view their "opponents" as people who favor continued emphasis on what Courant and Robbins call "empty drill in problem solving."
From the Wikipedia
Quoting from the Wikipedia:
- Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions". Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.
- Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in ancient Egypt, Mesopotamia, ancient India, ancient China, and ancient Greece. Rigorous arguments appear in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.
The second paragraph emphasizes "the use of abstraction and logical reasoning." This is both a key to the nature of mathematics and a major challenge in math education. Think of the "simple" idea of using a variable x to represent an (unknown) number in a linear equation such as 2x + 5 = 9. This is a linear equation that a student might encounter in an algebra course or in a much earlier course that contains some ideas leading to algebra. A variable is a quite abstract concept. Indeed, the math symbols +, ( ), +, -. the ten digits, decimal point, and so on are all examples of math abstraction.
The Piagetian cognitive development theory posits that children begin to reach "formal operations" at approximately the age of 12. Research suggests that more than half of students are not yet fully at the formal operations level by the time they finish high school. Thus, much of the mathematics that they encounter in school is being taught at a level of abstraction and formalization that is beyond their level of cognitive development. This leads a majority of students into a "memorize and regurgitate with little understanding" approach to math learning.
Good Math Lesson Plans
One way to answer the "What is math" question is to provide information about what is taught in the math curriculum. The trouble with this approach is that often the math curriculum does not provide good insight into the broader aspects of the overall discipline of mathematics.
A good math curriculum helps students to develop an understanding of possible answers to the what is math question. As students progress through the math curriculum their understanding of math and of answers to the question should be growing. Indeed, each math lesson plan should contribute to this understanding.
The following diagram can be used to discuss representing and solving applied math problems at the K-12 level. This diagram is especially useful in discussions of the current K-12 mathematics curriculum.
The six steps illustrated are:
- Problem posing
- 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
- Thinking about whether the original Problem Situation has been resolved.
Steps 5 and 6 also involve thinking about related problems and problem situations that one might want to address or that are created by the process or attempting to solve the original Clearly-defined Problem or resolve the original Problem Situation.
The previous section mentioned math modeling. The Math Forum@Drexel is a good source of information for teachers and others interested in mathematics education.
- The Math Forum (1999). Ask Dr. Math. The Math Forum@Drexel. Retrieved 12/2607: http://mathforum.org/library/drmath/view/51940.html.
A person sent in a question philosophical question which included the following observation:
- You cannot say 2 + 3 = 5, because 2 is not three is not five. How can two things (2 and 3), neither of which is identical to 5, be identical to five if they are united? 5 in itself is also an independent "being." If it weren't, it indeed could exist as a collection of at least two other 'entities'.
Here is part of the response:
- Mathematics deals not with reality, but with an abstraction of reality: a "model" of just one aspect of the reality we use it to describe. For example, a number such as 2 doesn't represent any particular pair of things, but the idea of "two-ness." Man has found through long experience that things can be counted, and that the resulting numbers accurately describe one aspect of reality: if I counted two apples yesterday, and nothing has been done to them, then when I count them again there will still be two. The number doesn't tell us their color, or how they taste, or who owns them; but it describes something about them that is true of any pair of apples. It is as much something I perceive about them as is their color or taste; but when I talk about numbers, I am abstracting one property from the rest, thinking only of the "two-ness" and ignoring the "apple-ness."
The response provides a good way to think about the abstractness of math and the idea of math modeling. A number such as 2 is a mathematical model. The number 2 is a model of a certain property of a collection that happens to contain two apples, or a collection that happens to contain two people, or a collection that happens to two candy bars. Mathematical modeling is a very important and fundamental aspect of math, but is it a relatively abstract idea. Many people teaching and/or learning math do not have a good grasp of the concept of math modeling.
Many mathematicians agree that the very hear of mathematics lies in mathematical proof. Proofs in Mathematics provides a discussion of this idea. It also provides a number of examples of important mathematical proofs that require relatively little background in math. A number of other interesting, amusing, and challenging examples are provided.
Quoting from the Wikipedia entry Mathematical Proof:
- In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. A proof is a logical argument, not an empirical one. That is, one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture.
The Wikipedia article includes a discussion of a number of different methods of mathematical proof.
Lower-order and Higher-order
Every discipline can be analyzed in terms of lower-order knowledge and skills versus higher-order knowledge and skills. Educators in each discipline are faced by the challenge of deciding how much emphasis to place on lower-order and on higher-order at each phase of the educational process. In math education, for example, students typically learn counting numbers (words) one,two, three, and so on well before they enter kindergarten. However, their understanding on the number line if minimal.
Thus, our formal educational system is faced by the task of helping students come to know and understand the number line. Ask yourself, what constitutes lower-order knowledge and skills and what constitutes higher-order knowledge and skills in terms of learning about the number line? This is not an easy question to answer. However, you probably agree that students need to learn some things about the number line when they are in kindergarten or the first grade, and that they need to continue to learn more about the number line as they proceed through subsequent years of schooling.
However, there is more to math education than just learning about the number line. Thus, students need to me gaining lower-order and increasingly higher-order math knowledge and skills in a variety of math areas as they proceed through the math education curriculum. This situation tends to create a delicate balancing act, especially because some students can move much more rapidly than others along the higher-order knowledge and skills pathway.
The math education system in the United States does not do as well as the system in a number of other countries.
Here are some paragraphs from a 2005 American Institutes for Research report:
- Despite a widely held belief that U.S. students do well in mathematics in grade school but decline precipitously in high school, a new study comparing the math skills of students in industrialized nations finds that U.S. students in 4th and 8th grade perform consistently below most of their peers around the world and continue that trend into high school.
- The study, “Reassessing U.S. International Mathematics Performance: New Findings from the 2003 TIMSS and PISA,” focused on students in the United States and 11 other industrial countries that participated in all three assessments: Australia, Belgium, Hong Kong, Hungary, Italy, Japan, Latvia, Netherlands, New Zealand, Norway, and the Russian Federation. U.S. students consistently performed below average, ranking 8th or 9th out of twelve at all three grade levels. These findings suggest that U.S. reform proposals to strengthen mathematics instruction in the upper grades should be expanded to include improving U.S. mathematics instruction beginning in the primary grades.
- Countries that score well on items that emphasize mathematical reasoning (a higher-level skill) also score well on items that require knowledge of facts and procedures (a lower-level skill), suggesting that reasoning and computation skills are mutually reinforcing in learning mathematics well. Compared to other countries, students in the United States students do not do well on questions at either skill level. [Bold added for emphasis.]
Pay particular attention to the last of the paragraphs. It suggests that the current balance in US schools between lower-order and higher-order math knowledge and skills is not as effective as it might be.
How Well is the US Math Education System Doing?
The previous section suggests that the US math education system is not doing as well as the system in a number of other countries.
However, people in the US argue that our system is making good progress. They note the huge increase over the past hundred years in the percentage of students completing high school, and thus completing the math requirements for high school graduation. They note the significant increase in the past 50 years of the percentage of students taking three or more years of high school math.
Moreover, they point to evidence that our math education system has been improving in recent years. Quoting from the November 2007 issue of Educational Leadership:
- If state accountability tests can't tell us whether student mathematics achievement has improved, what is the most reliable indicator to shed light on this question? The National Assessment of Educational Progress has measured trends in mathematics achievement over time, and its most recent report shows a steady, long-term rise in achievement for 4th and 8th graders.
- According to The Nation's Report Card: Mathematics 2007, the average score on the NAEP assessment for 4th graders has increased 27 points and the average score for 8th graders has increased 19 points since the first assessment year in 1991. Between 2005 and 2007, the average score for 4th graders rose 2 points, and the average score for 8th graders rose 3 points.
- These gains in NAEP math scores in grades 4 and 8 have been broad and consistent. They have occurred at all levels of performance, resulting in more students scoring at or above the Basic and Proficient levels. They have occurred in each of the content areas tested—number properties and operations, measurement, geometry, data analysis and probability, and algebra—and across all 42 states participating in state-level analysis. Scores have risen regardless of students' poverty level (as determined by their eligibility for free and reduced-price lunch). And the gains have also been apparent across almost all racial/ethnic groups.
- Among all this good news, a major concern remains: Achievement gaps between white students and minority students have generally not narrowed. For 8th graders, the white-black gap remains at 32 points—smaller than it was in 2005, but not significantly different from 1990. The only significant narrowing of the white-black gap over time occurred for 4th grade students, where it fell from 32 points in 1990 to 26 points in 2007. This gap has remained static, however, for the last two years.
Here are four very important points that emerge from consideration of the diagram in Figure 1 and earlier material presented in this document:
- Mathematics is an aid to representing and attempting to resolve problem situations in all disciplines. It is an interdisciplinary tool and language.
- Computers and calculators are exceedingly fast, accurate, and capable at doing Step 3.
- Our current K-12 math curriculum spends the majority of its time teaching students to do Step 3 using the mental and physical tools (such as pencil and paper) that have been used for hundreds of year. We can think of this as teaching students to compete with machines, rather than to work with machines.
- Our current mathematics education system at the PreK-12 levels is unbalanced between lower-order knowledge and skills (with way to much emphasis on Step #3 in the diagram) and higher-order knowledge and skills (all of the other steps in the diagram). It is weak in mathematics as a human endeavor and as a discipline of study.
There are three powerful change agents that will eventually facilitate and force major changes in our math education system.
- Brain Science, which is being greatly aided by brain scanning equipment and computer mapping and modeling of brain activities, is adding significantly to our understanding of how the brain learns math and uses its mathematical knowledge and skills.
- Computer and Information Technology is providing powerful aids to many different research areas (such as Brain Science), to the teaching of math (for example, through the use of highly Interactive Intelligent Computer-Assisted Learning, perhaps delivered over the Internet), to the content of math (for example, Computational Mathematics), and to representing and automating the "procedures" part of doing math.
- The steady growth of the totality of mathematical knowledge and its applications to representing and helping to solving problems in all academic disciplines.
Math Cognitive Development and Math Maturity
Human babies are born with some innate math abilities and some innate spatial abilities. (Logical/mathematical and spatial are two of Howard Gardner's list of eight Multiple Intelligences that people have).
During the first 25 years of life, a person's brain grows and matures. Jean Piaget was a pioneer in studying cognitive development. He is well known for his four-stage theory: sensory-motor, pre operational, concrete operations, and formal operations.
Many people in the math community prefer to use the idea of math maturity rather than math cognitive development. Math maturity tends to have a connotation of learning to think like a mathematician. It is not dependent on having studied specific areas of math. Rather , it is dependent on increasing maturity in thinking about and using (in problem solving and proofs) the math that one has learned.
This four-stage theory posits that children begin to enter the concrete operations stage at about age 7, and children begin to enter the formal operations stage at approximately age 12.
A number of researchers have observed and argued that many people never learn to function at a formal operations level. For example, here is a quote from the Wikipedia:
- The formal operational period is the fourth and final of the periods of cognitive development in Piaget's theory. This stage, which follows the Concrete Operational stage, commences at around 11 years of age (puberty) and continues into adulthood. It is characterized by acquisition of the ability to think abstractly, reason logically and draw conclusions from the information available. During this stage the young adult is able to understand such things as love, "shades of gray", logical proofs, and values. Lucidly, biological factors may be traced to this stage as it occurs during puberty (the time at which another period of neural pruning occurs), marking the entry to adulthood in Physiology, cognition, moral judgment (Kohlberg), Psychosexual development (Freud), and psychosocial development (Erikson). Some two-thirds of people do not develop this form of reasoning fully enough that it becomes their normal mode for cognition, and so they remain, even as adults, concrete operational thinkers. [Bold added for emphasis.]
The two-thirds assertion is obviously open to debate, as we are dealing with an idea (formal operations) and a level of its acquisition (fully) that are not very well defined. The following graph and the accompanying quote are from:
- Huitt, W. and Hummel, J. (January 1998). Cognitive Development. Accessed 1/13/08: http://chiron.valdosta.edu/whuitt/col/cogsys/piaget.html.
- Formal operational stage (Adolescence and adulthood). In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts. Early in the period there is a return to egocentric thought. Only 35% of high school graduates in industrialized countries obtain formal operations; many people do not think formally during adulthood. [Bold added for emphasis.]
The stage theory can be applied to various areas of human development. Thus, for example, one can look at how a particular child is doing in math cognitive development. Math is a vertically structured discipline of study, and it has a steadily increasing emphasis on abstraction as one moves "upward" in the discipline. The math curriculum at various grade levels might be designed to fit with the "average" math cognitive development of students, and be designed to help increase the level of math cognitive development.
There appears to be quite a bit of evidence that beginning at approximately the 4th or 5th grade in the US math education system, quite a bit of the content is at a higher level of abstraction than is suitable for students at the concrete operations level. Thus, for example, ratio and proportion, divisions by fractions, and percentage may all be topics that require a majority of students to attempt to get by through a process of memorization with very little understanding. Algebra, geometry, and probability fall into this category.
If this conjecture is correct, it helps to provide an answer of why so many students grow up hating math and saying "I can't do math." In essence, much of the math the students are encounter in their math coursework is above their math cognitive developmental level. They do not understand what they are doing. It does not make sense to them. In this environment, they is constant likelihood of failure.
Math Formal Operations and Learning Computer Programming
The following article argues that College Algebra is an appropriate prerequisite for a college level computer programming course, as the College Algebra requires students to deal with the formal operations aspects of math. It is an example Piagetian-oriented research into student learning difficulties in courses that require a high level of formal operations.
White, Garry. (2003). Standardized mathematics scores as a prerequisite for a first programming course. Mathematics and Computer Education. Retrieved 1/13/08: http://findarticles.com/p/articles/mi_qa3950/is_200301/ai_n9174067.
Quoting from the paper:
- Research has shown formal operations, such as thinking in abstractions and logically, develop at different ages or not at all (Griffiths, 1973; Schwebel, 1975; Pallrand, 1979; Bastian et al., 1973; Epstein, 1980). Many high school students (and adults) fail to attain full formal operational thinking (Renner & Lawson, 1973; Renner et al, 1978). Several studies have shown that a majority of adults, including college students and professionals, fail at many formal operational tasks (Petrushka, 1984; Sund, 1976). Many college students fail to attain full formal operational thinking (Griffiths, 1973; Schwebel, 1975; Schwebel, 1972).
- Learning complex abstract concepts found in algebra courses appears to require Piaget's formal operation cognitive level (Pallrand, 1979; Niaz, 1989; Nasser, 1993). Formal operations level has been found to relate to success in algebra courses (Bloland and Michael, 1984; Bates, 1978; Forman, 1980). This may be due to mathematics problem solving and Piagetian logic using similar areas of the brain as indicated by EEG's (Kraft, 1976; Rotejnberg & Arshavsky, 1997.)
We know, of course, that many students are capable of learning to program in BASIC, Logo, and other student-oriented programming languages long before they reach formal operations. In particular, we have the work of Seymour Papert. After completing two different doctorate programs in mathematics, doctorate, Seymour Papert studied under Jean Piaget for five years.This background, plus his interest in artificial intelligence, was key to his work in helping to develop the Logo programming language. His research and writings in use of Logo with relatively young children is supportive of the idea that this rich and challenging programming environment helps move students toward mathematical formal operations.
Quoting from the Wikipedia:
- At MIT, Papert went on to create the Epistemology and Learning Research Group at the MIT Media Lab. Here, he was the developer of an original and highly influential theory on learning called constructionism, built upon the work of Jean Piaget in Constructivism learning theories. Papert worked with Jean Piaget during the 1960s and is widely considered the most brilliant and successful of Piaget's proteges; Piaget once said that "no one understand my ideas as well as Papert." Papert has rethought how schools should work based on these theories of learning.
- Papert has also been widely known for focusing on the impact of new technologies on learning in general and in schools as learning organizations in particular. To this end, Papert used Piaget's work while developing the Logo programming language while at MIT. He created Logo as a tool to improve the way that children think and solve the problems. A small robot called the "Logo Turtle" was developed and children have been encouraged to solve the problem with the Logo turtle. A main purpose of the Logo Foundation research group is to strengthen the ability to learn knowledge. Papert insists a language or program that children can learn -- like Logo -- does not have to lack functionality for expert users.
Free Book: Improving Math Education in K-8 Schools
The following book is available in Microsoft Word and PDF formats under a Cretive Commons license.
- Moursund, D.G. (2006). Computational Thinking and Math Maturity: Improving Math Education in K-8 Schools. Access at http://uoregon.edu/~moursund/Books/ElMath/ElMath.html. Self published.
Preface to the Book
- The saddest aspect of life right now is that science gathers knowledge faster than society gathers wisdom. (Isaac Asimov, 1988)
- An educated mind is, as it were, composed of all the minds of preceding ages. (Bernard Le Bovier Fontenelle, mathematical historian, 1657-1757)
This book is motivated by the problem that our K-8 school math education system is not as successful as many people would like it to be, and it is not as successful as it could be. It is designed as supplementary material for use in a Math Methods course for preservice K-8 teachers. However, it can also be used by inservice K-8 teachers and for students enrolled in Math for Elementary and Middle School teachers’ courses.
Many people and organizations have put forth ideas on how to improve our math education system. However, in spite of decades of well-meaning reform effort, national assessments in mathematics at the precollege level in the United States do not indicate significant progress. Rather, scores on these national assessments have essentially flat lined during the past 40 years. The results of the past 40 years of attempts to improve math education suggest that doing more of the same is not likely to improve the situation. We can continue to argue about whether back to basics or a stronger focus on new math is the better approach. From time to time, both such approaches have produced small pockets of excellence. In general, however, our overall math education system is struggling to achieve even modest gains.
This book draws upon and explores four Big Ideas that, taken together, have the potential to significantly improve out math education. The Big Ideas are:
- Thinking of learning math as a process of both learning math content and a process of gaining in math maturity. Our current math education system is does a poor job of building math maturity.
- Thinking of a student’s math cognitive development in terms of the roles of both nature and nurture. Research in cognitive acceleration in mathematics and other disciplines indicates we can do much better in fostering math cognitive development.
- Understanding the power of computer systems and computational thinking as an aid to representing and solving math problems and as an aid to effectively using math in all other disciplines.
- Placing increased emphasis on learning to learn math, making effective use of use computer-based aids to learning, and information retrieval.
Unifying Themes in the Book
Four major themes run throughout the book. They are described in the next four subsections of this Wiki page.
Math Maturity. Math maturity is a relatively commonly used term, especially in higher education. In higher education, the dominant components of math maturity are “proof” and the logical, critical, creative reasoning and thinking involved in understanding and doing proofs. The focus is on mathematical thinking, on being able to read and write math, and on being able to learn math using a wide range of resources such as print materials, courses, colloquium talks, and so on.
Many of the same ideas are applicable to defining math maturity at the precollege level. However, the cognitive development work of Piaget and others provides another quite useful approach. Piaget’s four-stage cognitive development scale is useful in tracking and facilitating cognitive development through the levels: sensory motor, preoperational, concrete operations, and formal operations. Piaget and many more recent researcher have recognized that one can look at the formal operation end of this scale both in general, and also in specific disciplines. Thus, we can explore the math education curriculum in terms of how well it helps students gain in math cognitive development.
The past 20 years have brought quite rapid progress in cognitive neural science and other aspects of brain science. Researchers have gained considerable insight into how the brain functions in math learning and math problem solving. This research is beginning to contribute to the design of more effective aids to learning math and to increasing math maturity.
Nature and Nurture. People are born with a certain “amount” of innate mathematical ability. In dealing with quantity, for example, this innate ability roughly corresponds to dealing with 1, 2, 3, and many. Howard Gardner has identified logical/mathematical as one of the eight multiple intelligences that in his theory of intelligence.
However, most of what we call mathematics has been invented by people. It is part of the accumulated knowledge of the human race, and it is passed on from generation to generation by informal and formal education. Children who grow up in a hunter-gather society do not learn the types of math that we expect children to learn in our information age society.
In recent years, use of brain imaging equipment and brain modeling using computers have been added to earlier tools used to study cognitive development. Research in cognitive development and cognitive acceleration suggests that our informal and formal educational system could be doing much better.
Computational thinking. Many people now divide the discipline of mathematics into three major sub disciplines: pure math, applied math, and computational math. The term computational denotes the study and use of computer modeling and simulation. The table in figure P.1 contains data from Google searches on the three sub disciplines.
|Search Expression||Google Hits 5/10/06|
|"applied math” OR “applied mathematics"||50,200,000|
|"pure math” OR “pure mathematics"||5,450,000|
|"computational math” OR “computational mathematics"||3,090,000|
- Figure 1 Google searches on some math sub disciplines
Of the three sub disciplines listed, the most recent to emerge is computational. The addition of computational as a subdivision of math and various other disciplines has occurred because of the steadily increasing role of computers as an integral component of the content of many different disciplines. For example, in 1998 one of the two winners of the Nobel Prize in Chemistry received the prize for his previous 15 years of work in computational chemistry. He had developed computer models of chemical processes that significantly advanced the discipline of chemistry.
Similarly, physics is now divided into the three components: theoretical, experimental, and computational. Here is a brief quote from page 2 of the April 22, 2006 issue of Science News:
- When black holes collide, they cause surrounding space-time to wiggle, generating a torrent of radiation known as gravitational waves. That’s what Einstein’s general theory of relativity predicts, but computer models [modelers] have struggled for more than 30 years to reproduce those waves. Because of the relativity theory’s mathematical complexity and the extreme gravity of black holes, modelers haven’t succeeded in getting black holes to crash.
- Now, two teams independently reported that they have successfully simulated the merger of two black holes and the event’s production of gradational waves.
As you can see, computational means far more than just doing arithmetic calculations. Indeed, it has emerged as a way of thinking.
An excellent, brief introduction to computational thinking is provided in Jeannette Wing (2006). She is the Head of the Computer Science Department at Carnegie Mellon University. Quoting from her article:
- Computational thinking builds on the power and limits of computing processes, whether they are executed by a human or by a machine. Computational methods and models give us the courage to solve problems and design systems that no one of us would be capable of tackling alone. Computational thinking confronts the riddle of machine intelligence: What can humans do better than computers, and What can computers do better than humans? Most fundamentally it addresses the question: What is computable? Today, we know only parts of the answer to such questions.
- Computational thinking is a fundamental skill for everybody, not just for computer scientists. To reading, writing, and arithmetic, we should add computational thinking to every child’s analytical ability.
Learning to Learn Math. All teachers recognize that to be effective, they need to know the content they are teaching and they need to know how to teach the content. Much teaching knowledge and skill cuts across the school disciplines. However, there is considerable discipline-specific pedagogical knowledge and skill for each discipline. To be an effective teacher of math, one needs to know math and one needs to have significant math pedagogical knowledge and skill.
A somewhat similar idea holds for learning math. The human brain is naturally curious and has an innate ability to learn. A child is born with a modest amount of math capability, such as being able to distinguish among quantities such as one, two, and three. As a person’s brain grows and matures, one’s innate mathematical ability grows.
However, mathematical development depends heavily upon the informal and formal math learning environments that are available to the learner. In addition, math development is highly dependent on learning to learn math—in making progress toward being a more effective and efficient learner of math.
A good example of what is entailed by this is inherent to the idea of reading across the curriculum. We know that there is a difference between general reading skills and discipline-specific reading skills. We also know that students first learn to read and eventually can read to learn. From a math education point of view, students need to learn to read math. Progress in this endeavor is important to learning math by reading. Our current math education system is weak in helping students learn to read math and they learn math through reading. Nowadays, most students have relatively easy access to the world’s largest library—the Web. Thus, as they learn to read math and to learn math by reading, they can take advantage of the math components of this huge and steadily growing library. Because math is an important component of many disciplines, learning to read math is an important part of learning to read across the curriculum
Computers have also brought us computer-assisted learning (CAL). In recent years, some of the best CAL falls into the category highly interactive intelligent computer-assisted learning (HIICAL). Such materials are a powerful aid to learning. Research on HIICAL in math suggest that some of the available materials are considerably more effective aids to student learning than are the traditional aids.
Sex Differences in Learning and Doing Math
In recent years there have been a number of studies looking for innate math-related differences and similarities between girls and boys. The following article provides a nice, readable summary of some of the findings.
Diane F. Halpern, Camilla P. Benbow, David C. Geary, Ruben C. Gur, Janet Shibley Hyde and Morton Ann Gernsbacher (December 2007). Sex, Math and Scientific Achievement. Scientific American. Retrieved 12/19/07: http://www.sciam.com/article.cfm?id=sex-math-and-scientific-achievement.
This is a complex topic. The following quoted material captures some of the findings that relate to schooling.
- Because grades and overall test scores depend on many factors, psychologists have turned to assessing better-defined cognitive skills to understand these sex differences. Preschool children seem to start out more or less even, because girls and boys, on average, perform equally well in early cognitive skills that relate to quantitative thinking and knowledge of objects in the surrounding environment.
- Around the time school begins, however, the sexes start to diverge. By the end of grade school and beyond, females perform better on most assessments of verbal abilities. In a 1995 review of the vast literature on writing skills, University of Chicago researchers Larry Hedges (now at Northwestern University) and Amy Nowell put it this way: “The large sex differences in writing … are alarming. The data imply that males are, on average, at a rather profound disadvantage in the performance of this basic skill.” There is also a female advantage in memory of faces and in episodic memory—memory for events that are personally experienced and are recalled along with information about each event’s time and place.
- There is another type of ability, however, in which boys have the upper hand, a skill set referred to as visuospatial: an ability to mentally navigate and model movement of objects in three dimensions. Between the ages of four and five, boys are measurably better at solving mazes on standardized tests. Another manifestation of visuospatial skill in which boys excel involves “mental rotation,” holding a three-dimensional object in memory while simultaneously transforming it. As might be expected, these capabilities also give boys an edge in solving math problems that rely on creating a mental image.
A 2008 research study led by Janet Hyde (University of Wisconsin) indicates that girls are now doing as well as boys on standardized math exams. A summary of the research is available in the article:
- Erickson, Amanda and Sadovi, Carlos (7/25/08). Report: Girls are equal to math test. Chicago Tribune. Retrieved 7/26/08: http://www.chicagotribune.com/news/chi-girls-math25jul25,0,7779197.story. Quoting a few tidbits from the article:
- Researchers comparing the math scores of 7 million students nationwide from 2005 to last year found that girls and boys do equally well on math tests taken from the 2nd to 11th grades.
- While previous studies have reached a similar conclusion, a new study by five professors at the University of Wisconsin and the University of California, Berkeley is by far the most sweeping, using data from 10 states.
- … by 2000, high school girls and boys were studying calculus at the same rate, the educators said.
- Though women are getting an equal number of math bachelor's degrees, fewer go on to get advanced math or engineering degrees.
AIR (11/22/07). New Study Finds U.S. Math Students Consistently Behind Their Peers Around the World. American Institutes for Research. Retrieved 11/16/07: http://www.air.org/news/documents/Release200511math.htm.
Battista, Michael T. (February 1999). Ignoring research and scientific study in education. Phi Delta Kappan. Retrieved 4/26/06 http://www.pdkintl.org/kappan/kbat9902.htm.
Bogomolny, Alexander (2007). What is What in Mathematics? Retrieved 12/20/07: http://www.cut-the-knot.org/WhatIs/index.shtml. Contains discussions about possible answers to about a dozen different "What is? questions in math, such as "What is Addition?", "What is Arithmetic?" and "What is Probability?"
Clements, D.H. (1999). 'Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood. 1(1), 45-60. Retrieved 4/25/06. http://www.gse.buffalo.edu/org/buildingblocks/NewsLetters/Concrete_Yelland.htm. The Web reference is for a slightly updated version of the original article.
Lewis, Robert H. (2000). Mathematics: The most misunderstood subject.Retrieved 12/26/07: http://www.oregon529network.com/what/faqs.html. Quoting from the website:
- The great misconception about mathematics -- and it stifles and thwarts more students than any other single thing -- is the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student's duty is to memorize all this stuff. Such students seem to feel that sometime in the future their boss will walk into the office and demand "Quick, what's the quadratic formula?" Or, "Hurry, I need to know the derivative of 3x^2 - 6x +1." There are no such employers.
Perkins-Gough (November 2007). Special Report / Are U.S. Students Getting Better in Mathematics? Educational Leadership. Volume 65 Number 3.
Rehmeyer, Jullie (February 2008). Math on Display: Visualizations of mathematics create remarkable artwork. Science News Online. Retrieved 3/13/08: http://www.sciencenews.org/articles/20080216/mathtrek.asp
Author of Authors
The initial version of this document was written by David Moursund.