Math Education






 * The saddest aspect of life right now is that science gathers knowledge faster than society gathers wisdom. (Isaac Asimov; Russian-born American science fiction author and biochemist; 1920–1992.)


 * An educated mind is, as it were, composed of all the minds of preceding ages. (Bernard Le Bovier de Fontenelle; French scientist, philosopher, mathematician, and writer; 1657-1757.)

Introduction
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. 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.

Starting in the 1990, conflicts about what to include in the math curriculum and how to teach it erupted into he Math Education Wars. Computer technology certainly contributes to the complexity of what to teach and how to teach it. If a computer can solve a particular type of math problem,what should students learn about how to solve this type of mahth problem? If a computer program is quite good at teaching how to solve a certain type of math problem, what is the role of teachers in teaching this content?

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.[2] Also prehistoric artifacts discovered in Africa and France, dated between 35,000 BC and 20,000 BC,[3] indicate early attempts to quantify time.[4]

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.



Figure 1. Ishango Bone

Quoting from http://www.math.buffalo.edu/mad/Ancient-Africa/ishango.html:


 * 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.



 Figure 2. Clay tokens.

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 many 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 sub-field 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 above. 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 to be able to 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. Download a free copy of the book from https://archive.org/details/AMathematiciansApology. Hardy 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.



 Figure 3. Right triangle with hypotenuse an irrational number.


 * 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.

NCTM
Quoting from Where Learning Begins (Early Childhood, June, 1999): http://www.ed.gov/pubs/EarlyMath/whatis.html 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 areas.

Quoting from [http://www.nctm.org/ccssmposition/ Supporting the Common Core Standards for mathematics:


 * The widespread adoption of the Common Core State Standards for Mathematics (CCSSM) presents an unprecedented opportunity for systemic improvement in mathematics education in the United States. The Common Core State Standards offer a foundation for the development of more rigorous, focused, and coherent mathematics curricula, instruction, and assessments that promote conceptual understanding and reasoning as well as skill fluency. This foundation will help to ensure that all students are ready for college and careers when they graduate from high school and that they are prepared to take their place as productive, full participants in society.]




 * The Common Core State Standards are a significant component of systemic improvement in mathematics learning, but on their own they are not sufficient to produce the mathematics achievement that our country needs to be competitive in the global economy of the 21st century. Other factors are critical to realizing the potential of the Common Core:


 * Substantial opportunities for ongoing professional development to ensure that all teachers understand and are prepared to implement the Common Core State Standards for Mathematics and that all administrators and policymakers understand teachers’ needs.


 * Accommodations in teacher evaluation systems to allow time for the profession and institutions to adjust and adapt to the Common Core State Standards before evaluation systems include accountability for student achievement as one element of a valid, multifaceted teacher evaluation.

North Central Regional Educational Laboratory</Center>
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</Center>
Courant, R., & Robbins, H. (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</Center>
Quoting from the Wikipedia:


 * Mathematics is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8]


 * Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

The second paragraph emphasizes "the use of abstraction and logic." 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 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.

Math Lesson Plans</Center>
One way to answer the "What is math?" question is to provide information about what is taught in the math curriculum and how it is taught. This information is embodied it the "Teacher's Edition" of the math books students use at the precollege level.

The information is also embodied in the lesson plans that teachers create for their own use or draw from sources such as the Teacher's Edition. See Good Math Lesson Plans for an extensive discussion of Good math Lesson Plans and also for a free book on the topic.]

Problem Solving
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.



Figure 4. Six steps in solving a variety of math problems.

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.

Mathematical Modeling</Center>
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 @ Drexel (2014). Retrieved 7/23/2014 from http://mathforum.org.

Quoting from a philosophical question sent to the [http://mathforum.org/dr.math/ Math Forum: Ask Dr. Math: that included the following observation:


 * Question. 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 contain 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.

Mathematical Proofs</Center>
Many mathematicians agree that the very heart 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</Center>
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 of the number line is 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 be 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 do others along the higher-order knowledge and skills pathway.

How Well Is the U.S. Math Education System Doing?</Center>
The math education system in the United States does not do as well as the systems 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.

This situation has persisted over subsequent years. For information about TIMMS from 1995 through 2011, and the upcoming 2015 assessments, see http://nces.ed.gov/TIMSS/.

However, people in the U.S. 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 Deborah Perkins-Gough 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.

Important Ideas</Center>
Here are four very important points that emerge from consideration of the diagram in Figure 4 in the section on Problem Solving presented earlier 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. [See Figure 4.]
 * 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(HIICAL), 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 nine Multiple Intelligences that people have: http://iae-pedia.org/Howard_Gardner/.

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. Here is a free book on Math Maturity:


 * Moursund, D., & Albrecht, R. (2011). Using math games and word problems to increase the math maturity of K-8 students. Eugene, OR: Information Age Education. Download PDF file from http://i-a-e.org/downloads/doc_download/211-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html. Download Microsoft Word file from http://i-a-e.org/downloads/doc_download/210-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html.

Piaget's 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 two graphs and the accompanying quote are from:

Huitt, W., & Hummel, J. (January, 1998). Cognitive Development. Retrieved 7/25/2014 from http://www.edpsycinteractive.org/topics/cognition/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.]



Figure 5. Percentage of students in Piagetian stages.</Center>



Figure 6. Data on students achieving the Formal Operations level.</Center>

Note added 7/23/2014: I spent quite a bit of time browsing the Web in an effort to find more information about the 35% figure mentioned in the Huitt and Hummel book. I was not able to find any studies done in the past 20 years that verified that 35% figure. Quoting again from the Huitt and Hummel book:


 * However, data from similar cross-sectional studies of adolescents do not support the assertion that all individuals will automatically move to the next cognitive stage as they biologically mature simply through normal interaction with the environment (Jordan & Brownlee, 1981). Data from adolescent populations indicates only 30 to 35% of high school seniors attained the cognitive development stage of formal operations (Kuhn, Langer, Kohlberg & Haan, 1977). For formal operations, it appears that maturation establishes the basis, but a special environment is required for most adolescents and adults to attain this stage.

The four-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 U.S. 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 that students 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, there is a constant likelihood of failure.

Math Formal Operations and Learning Computer Programming</Center>
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 of 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, 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 the use of Logo with relatively young children is supportive of the idea that this rich and challenging programming environment helps to 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[2]. 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 on Computational Thinking and Math Maturity
Moursund, D. (2007). Computational Thinking and Math Maturity: Improving Math Education iin K-8 Schools. Eugene, OR: Information Age Education. Download the book in Microsoft Word format at http://i-a-e.org/downloads/doc_download/4-computational-thinking-and-math-maturity-improving-math-education-in-k-8-schools.html, Download the book in PDF forma at hhttp://i-a-e.org/downloads/doc_download/3-computational-thinking-and-math-maturity-improving-math-education-in-k-8-schools.html.

Quoting from the Preface:


 * 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.

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.

Halpern, D.F., Benbow, C.P., Geary, D.C., Gur, R.C., Hyde, J.S., & Gernsbacher, M.A. (December 2007). Sex, Math and Scientific Achievement. Scientific American. Retrieved 7/25/2014 from http://www.sciam.com/article.cfm?id=sex-math-and-scientific-achievement.

The following quoted material captures some of the findings that relate to sexual differences and 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.

Here is a quote from a 2010 IAE Blog entry:


 * One of the signs of improvement in our educational system is that girls are now allowed and encouraged to do as well as boys in math. Back when I was growing up, girls were discouraged from taking advanced math courses in high school and from being math or science majors.




 * Tenenbaum, David (10/11/2010). Large study shows females are equal to males in math skills. University of Wisconsin-Madison News. Retrieved 11/3/2010 from




 * The extensive report from which it came is currently one of the free articles made available by the American Psychological Association. See also the related article by Else-Quest, Hyde, and Linn at http://www.apa.org/pubs/journals/releases/bul-136-1-103.pdf.

Quoting from the Tenenbaum article referenced above:


 * The mathematical skills of boys and girls, as well as men and women, are substantially equal, according to a new examination of existing studies in the current online edition of Psychological Bulletin.


 * One portion of the new study looked systematically at 242 articles that assessed the math skills of 1,286,350 people, says chief author Janet Hyde, a professor of psychology and women's studies at the University of Wisconsin-Madison.


 * These studies, all published in English between 1990 and 2007, looked at people from grade school to college and beyond. A second portion of the new study examined the results of several large, long-term scientific studies, including the National Assessment of Educational Progress.


 * In both cases, Hyde says, the difference between the two sexes was so close as to be meaningless.

Author or Authors
The initial version of this document was written by David Moursund.