Math Maturity v1

From IAE-Pedia

Note to Readers: This is the original version of the Math Maturity page. On 10/6/09 it was renamed to Math Maturity v1 and a new Math Maturity page was created. While the two pages overlap considerably, the newer page has been heavily redesigned and edited. Most readers will likely decide the new version is superior to the older one.

Note to Readers

Math maturity is a complex topic. This article represents my explorations and understanding of the topic. It is a relatively long and convoluted article, drawing on a number of reference sources.

After reading the Introduction, you might want to jump to the Math Maturity section, thereby skipping much of the discussion of the research literature and more quickly arriving at some conclusions and recommendations.

Introduction

The three quotes given above help to give the flavor of this document. Math is a very important discipline both in its own right and because of its widespread applications in other disciplines.

Many people believe that the math education system in the United States is not nearly as effective as it should be. Over the years, there have been considerable efforts to improve the effectiveness of our math education system. Many of these efforts have focused on developing better curriculum and books, providing better preservice and inservice education for math teachers, requiring more years of math courses for precollege students, and setting more rigorous standards. There has also been a strong emphasis on encouraging women and minorities to take more math.

Current efforts to improve our math education system tend to be mostly focused on the same approaches. The general feeling seems to be that if we can just do more and better in these approaches, our math education system will improve.

Here are four important areas that have received much less attention.

There is substantial and mounting evidence that the math education curriculum in the United States is not designed and taught in a manner consistent with what is known about math cognitive development of students. Research in brain science is progressing more rapidly than our implementation of the results in our educational systems.

Our current math education system is not nearly as successful as we would like in helping students gain in their math creativity knowledge and skills, in their ability to attach and solve novel, complex, challenging problems, and in their ability to transfer their math knowledge and skills to problems outside the discipline of mathematics.

Our current math education system is still rather weak in teaching and learning in a manner that appropriately deals with forgetting. We know that students in math classes eventually (or, quite quickly) forget much of what they supposedly have learned. Although we spend quite a bit of time on review, we still face the constructivist problem that we are expecting students to build (construct) new knowledge on top of knowledge that they do not have.

In recent years, math educators have also had to deal with the steadily increasing capability and available of calculators and computers. In essence, we now need an education system that deals with both human brains and computer brains, and how to prepare humans to work effectively in environments where the computer capabilities increase significantly year by year.

There are many different approaches to the study of math education and in exploration of ways to improve our math eduction system. This document explores math intelligence, math cognitive development, and math maturity. It includes a focus on how to help students increase their level of math maturity.

In very brief summary, math maturity consists of an appropriate combination of math knowledge and skills, and the ability to think, understand, and solve problems using the math knowledge and skills. With disuse over time, one forgets much of their "learned" math knowledge and skills. However, one's level of math maturity—one's level of math-oriented thinking, understanding, and problem-solving—tends to have long term retention. Much more detail about math maturity is provided later in this document.

As with other documents in this IAE-pedia, the goal is to help improve education at all levels and throughout the world. Readers need to keep in mind that there are many different approaches to improving math education.

Marshmallows and Delayed Gratification

Quite a bit of formal education involves delayed gratification. This is certainly true in math education. When a student asks, "Why do I need to learn this?" a frequent response is, "You are going to need it next year." Of course, an common response nowadays is also, "It is going to be on the test." Personally, I find such a response rather unsatisfactory.

There has been some amusing and interesting research on a type of delayed gratification of young children. You can read a New Yorker magazine article on this, or view the short video on the test.

Youngsters are tested on whether they can delay eating a marshmallow (or some other "treat") in order to get two of the treats 15 minutes later. Only about 1/3 of the four-year old US children in the original research and 1/3 of the 4–6 year old Colombian children in research on children in that country were able to delay for 15 minutes.

Follow-up research on the US children 15 years later indicated that all who were able to delay their gratification for 15 minutes had been quite successful as students and in other parts of their lives.

Here is a math education quote from the New Yorker article:

Angela Lee Duckworth, an assistant professor of psychology at the University of Pennsylvania, is leading the program. She first grew interested in the subject after working as a high-school math teacher. “For the most part, it was an incredibly frustrating experience,” she says. “I gradually became convinced that trying to teach a teen-ager algebra when they don’t have self-control is a pretty futile exercise.” And so, at the age of thirty-two, Duckworth decided to become a psychologist. One of her main research projects looked at the relationship between self-control and grade-point average. She found that the ability to delay gratification—eighth graders were given a choice between a dollar right away or two dollars the following week—was a far better predictor of academic performance than I.Q. She said that her study shows that “intelligence is really important, but it’s still not as important as self-control.” [Bold added for emphasis.]

Background on Innate Human Math Capabilities

There is research backing the idea that several month old human babies have innate ability to recognize small quantities, such as noticing that there is a difference between two of something and three of that thing. A variety of other animals have somewhat similar innate sense of quantity.

Recent research supports the idea that a human brain also has some innate ability to deal with fractions. Quoting from the article:

Although fractions are thought to be a difficult mathematical concept to learn, the adult brain encodes them automatically without conscious thought, according to new research in the April 8, 2009 issue of The Journal of Neuroscience. The study shows that cells in the intraparietal sulcus (IPS) and the prefrontal cortex - brain regions important for processing whole numbers - are tuned to respond to particular fractions. The findings suggest that adults have an intuitive understanding of fractions and may aid in the development of new teaching techniques.

"Fractions are often considered a major stumbling block in math education," said Daniel Ansari, PhD, at the University of Western Ontario in Canada, an expert on numerical cognition in children and adults who was not affiliated with the study. "This new study challenges the notion that children must undergo a qualitative shift in order to understand fractions and use them in calculations. The findings instead suggest that fractions are built upon the system that is employed to represent basic numerical magnitude in the brain," Ansari said.

The book The Math Gene (Devlin, 1999) presents an argument that the ability to learn to speak and understand a natural language such as English is a very strong indication that one can learn math.

In essence, Devin argues that a student's development of math knowledge and skills is mostly dependent on informal and formal education coming from parents, teachers, television, games, and so on. See his opening keynote presentation at the 2004 NCTM Annual Conference.

This insight helps us to understand one of the major challenges in our current math education system. A great many parents were not particularly successful in learning math, and typically they do not provide a "rich" math environment for their children. A great many elementary school teachers are not particularly strong in math. The math environments they provide in their classrooms tends to consist of "covering" the math book and its related curriculum. Their level of math maturity is modest, as is their interest in and enthusiasm for math.

AS a consequence of this, many young students do not gain nearly as high a level of math maturity as they might. This is not a consequence of their innate math abilities. Rather, it is a consequence of the informal and formal math education that they receive at home, in their community, and in their early years of schooling.

Intelligence and Intelligence Quotient (IQ)

“Did you mean to say that one man may acquire a thing easily, another with difficulty; a little learning will lead the one to discover a great deal; whereas the other, after much study and application, no sooner learns than he forgets?” (Plato, 4th century B.C.)

As the quote from Plato indicates, people have long been interested in intelligence. It has long been known thatpeople vary considerably in their rate and quality of their learning.

There is substantial research to support the contention that students of higher IQ learn faster and better than students of lower IQ. A teacher in a typical elementary school classroom may have one or two students who can learn twice as fast (and better) than the average students in the class, and one or two who learn half as fast (and not as well) as compared to the average students in the class.

Here is a little more recent history on measuring IQ. Quoting from the Wikipedia:

The Stanford-Binet test started with [the 1904 work of] the French psychologist Alfred Binet, whom the French government commissioned with developing a method of identifying intellectually deficient children for their placement in special education programs. As Binet indicated, case studies might be more detailed and helpful, but the time required to test many people would be excessive.

Later, Alfred Binet and physician Theodore Simon collaborated in studying mental retardation in French school children. Theodore Simon was a student of Binet's. Between 1905 and 1908, their research at a boys school, in Grange-aux-Belles, led to their developing the Binet-Simon tests; via increasingly difficult questions, the tests measured attention, memory, and verbal skill. Binet warned that such test scores should not be interpreted literally, because intelligence is plastic and that there was a margin of error inherent to the test (Fancher, 1985). The test consisted of 30 items ranging from the ability to touch one's nose or ear when asked to the ability to draw designs from memory and to define abstract concepts. Binet proposed that a child's intellectual ability increases with age. Therefore, he tested potential items and determined that age at which a typical child could answer them correctly. Thus, Binet developed the concept of mental age (MA), which is an individual's level of mental development relative to others.

A driving force in Binet's work and the work of others in the field of IQ is the goal of developing a measurement that is reasonably accurate in predicting future success in school, work, and other cognitive-related activities. For example, with accurate information one can better align formal schooling with the cognitive abilities of a student, and one can better advise a student about informal and formal academic and career choices.

Notice the initial measures of intelligence measured attention, memory, and verbal skill. An ADHD student might do poorly on such a test. A child growing up in a "rich" verbal environment will tend to score much better on such a test than a child growing up in a poorer oral communication environment.

Here are four important ideas that have been developed and/or more fully explored since the initial work of Binet:

Intelligence comes from both nature (one's inherited genetic makeup) and nurture (life experiences, informal education, formal education). A starvation-level diet or a variety of drugs and poisons (lead, mercury, PCBs) reduce cognitive ability. Researchers talk about two components of intelligence: Gf (fluid intelligence; genetically-based intelligence) and Gc (crystallized intelligence; the "nurture" component of intelligence.) See Chapter 2 of Moursund, 2006a.
A typical person's brain reaches full maturity at approximately 25 to 27 years of age. One's brain continues to change significantly over the years, showing a marked level of plasticity. Thus, continuing active use and education of one's brain can maintain and continue to improve its level of performance for a great many years after its full physical maturity is reached.
As Binet pointed out, a child's intellectual ability increases with age. This led researcher to "norm" the scores on intelligence tests, in an attempt to produce a number (called IQ) that remains relatively stable over time.
Many researchers have explored the idea of a single general intelligence factor called "g" versus multiple intelligences. There is a quite high level of correlation between "g" and the various multiple intelligences identified by Howard Gardner, Robert Sternberg, and other researchers who have focused on the general area of multiple intelligences.

In terms of the fourth point given above, logical/mathematical is one of the eight multiple intelligences identified by Gardner. Creativity is one of the three multiple intelligences identified by Sternberg. From the point of view of these two theories of multiple intelligences, learning to make effective use of one's innate math abilities and learning to make creative mathematical use of one's brain are ways to increase one's level of math maturity.

Measuring Intelligence

There are many different ways to attempt to measure intelligence. It turns out that this is a very challenging task. This is further complicated by the fact that intelligence is strongly influenced by both nature (one's genetic makeup) and nurture (informal and formal education and life experiences). The "nurture" component of intelligence is also affected by things like the quality of food one eats (starvation is bad for the brain), poisons (mercury and lead are bad for the brain), brain injuries (brain damage can severely disrupt a brain's capabilities), and so on.

Have you ever wondered why an average person has an IQ of about 100, and that this often does not change much over the years? Surely an average person develops quite a bit mentally as he or she grows from infancy to adulthood and learns a great deal during this time through informal and formal education and through life experiences.

The explanation to this situation lies in the way that intelligence is measured and reported. Measures of intelligence are usually normed in a manner that makes one's IQ a relatively stable number over the years. Historically this was done by dividing one's intelligence test score by one's chronological age. Researchers developed the idea of scaling intelligence test scores to produce a mean of 100 and a standard deviation of 15 (or 17, or 14, or …, depending on the people developing the test.)

Nowadays the scaling process is handled somewhat differently. An intelligence test is developed for a certain age range and group of people. The scores are then normed to produce a mean of 100 and a specified standard deviation such as 15 or some other number. Now, when a person in this age range takes this test, his or her IQ score is determined by looking up the test score in a table of values that converts test score to IQ by comparison with test scores of those used in the norming process.

Thus, for example, a person whose test score is close to the mean of the test scores used in the norming process will be assigned an IQ of approximately 100. Suppose that this person takes another IQ test ten years later. It may well be a different test with different questions, and designed to be suited to the person's current age. The score that he or she receives on the test will be compared to the scores achieved by people who were approximately this age when the test was created and normed. If his or her test score is near the mean of this norming group, the result will be an IQ of approximately 100.

Even with this norming process, IQ can change over time. As an example, consider a four year old who has grown up in extreme poverty and in a home and neighborhood environment that includes lead paint and other toxins, and a single parent who is holding down two jobs to make ends meet. Then the child's home environment changes markedly. Perhaps the single parent marries into greater wealth and the child now experiences a much better home and neighborhood environment. Moreover, the child goes to a high quality kindergarten and then on into high quality elementary school. Such changes can produce a marked increase in IQ.

Nature and Nurture

It is important to understand that intelligence depends on a combination of nature and nurture. On average, intelligence increases considerably as a person grows up, and it decreases as one grows old. It is the norming process in IQ that (artificially) makes it appear that one's intelligence is not changing over the years.

How much of one's intelligence is due to nature and how much is due to nurture? This is a difficult question and researchers have produced varying answers.

Here is a somewhat different way of looking at this question. A newborn with a healthy brain has a tremendous capacity to learn. The child's brain grows rapidly and learns rapidly. Just imagine the challenge of gaining oral fluency in one language. If the child happens to live in a bilingual or trilingual home and extended environment, the typical child will become bilingual or trilingual. Amazing! This represents a huge capacity to learn and to make use of one's learning.

My point is that the "average" person is very intelligent. Good informal and formal learning opportunities can greatly increase the Gc component of one's intelligence.

Studies of nature versus nurture are typically done making use of identical twins that were separated at birth. One can find varying results in the literature—with IQ being determined about 50 percent by nature all the way up to about 80 percent by nature, depending on the study.

Current research suggests that nature and nurture work together in a very complex manner, and that we have a long way to go in this area of research.

Multiple Intelligences

A human brain is a very complex organ. Many different parts and characteristics of a human brain contribute to its ability to learn and to deal with complex, challenging problems. Perhaps you have observed that some people seem to have more linguistic ability, or musical ability, or math ability that other people. Might there be significant discipline-oriented differences in intelligence? That is, might the differing characteristics of healthy brains have significant built-in inherent abilities to learn some disciplines better than others?

This question has led to a variety of multiple intelligences models and ongoing disagreements among proponents of these various models and proponents of a single factor model of intelligence. These are importatn disagreements. Suppose that nature can endow one person with the brain and hearing system to gain/have perfect pitch, another person to have much better than average propensity to develop very good spatial sense, and a third to have a propensity for developing much above average mathematical logic sense? If so, then perhaps educators would want to identify these varying characteristics in their students and better individualize programs of study to meet there varying students.

Howard Gardner is well known for his theory of Multiple Intelligences. His current theory includes eight different components, including logical/mathematical, and spatial. (Spatial intelligence is quite helpful in math as well as in other disciplines, such as art.)

Suppose that Howard Gardner's theory of multiple intelligences is essentially correct. Moreover, suppose that there are huge differences in the inherent and potential logical/mathematical abilities of students. If this is the case, then perhaps we are expecting many students in school to attempt to learn far more math than their brains are well suited to learn. Our school system's heavy emphasis on math puts many students at risk of not graduating from high school.

Personal comment: I am not tone deaf. However, my children like to label me as "tune" deaf. I tend to believe that I have somewhat below average inherent ability in music. It is not clear to me that I would have made it through high school if I had been required to take a stringent sequence of musical performance, composition, and theory courses in high school.

Howard Gardner based his theory of Multiple Intelligences partly on studies he did on brain damaged people. If ability in a particular area—such as spatial sense—is wiped out by damage to a particular part of one's brain, then this is taken as evidence that spatial is a distinct type of intelligence.

However, many people disagree with the work of Howard Gardner. One of the difficulties is that the various parts of one's brain work together, and that there is considerable plasticity.

To take a personal example, I (Dave Moursund) was quite good at math from my early childhood on, and I had little trouble in getting a doctorate in this discipline. It certainly helped that both my mother and father were high IQ people, and both were mathematicians. However, my spatial sense is terrible—well below average. At one time it was believe that a high spatial "IQ" was an essential part of being successful in math. More recent research indicates that many successful mathematicians have poor spatial sense. There is more than one way to look at a math problem!

Cognitive Development

Cognitive development is measured and studied in terms of a stage theory. Piaget is well known for the initial four-level stage theory that he developed. According to Piaget, a child moves from the Sensory Motor Stage to the Pre Operational Stage to the Concrete Operations Stage to the Formal Operations Stage.

The following chart is from the Piaget reference given above:

Quoting from the same reference:

Data from adolescent populations indicates only 30 to 35% of high school seniors attain 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.

More modern versions of this stage theory have a much larger number of stages. See:

Commons, M.L. and Richards, F.a. (2002). Organizing components into combinations: How stage transition works. Journal of Adult Development. 9(3), 159-177. Retrieved 6/18/09: http://www.tiac.net/~commons/Commons&Richards04282004.htm.

This article provides a 15-stage Piagetian-type model of cognitive development. Quoting from the article:

The acquisition of a new-stage behavior has been an important aspect of Piaget’s theory of stage and stage change. Because of his controversial notions of stage and stage change, however, little research on these issues has taken place in the late twentieth century, at least among psychologists in the United States. The research that has taken place is being done by Neo-Piagetians. The neo-Piagetians more precisely defined stage, taking each of Piaget’s substages and showing that they were in fact stages. In addition, three postformal stages have been added. Similar changes were made with Kohlberg’s stages and substages. Commons, Richards, and Armon (1984) created a stage comparison table, comparing stage sequences from a number of different traditions, that stands today as the standard. This table shows that there is, essentially, only one stage sequence. Commons and Richards (1984a, b) presented their first General Stage Model at that time. Commons and colleagues (Commons, Trudeau, et al., 1998; Commons & Miller, 1998) later revised that model and expanded it downward, changing the name of the model to the Model of Hierarchical Complexity. Table 1a shows a complete list of the Orders of Hierarchical Complexity described in that model.

Researchers in cognitive development are faced by many of the same issues as researchers in IQ. Two of these issues are:

What (relative) roles do nature and nurture play in cognitive development?
Is cognitive development essentially domain independent or is does a theory of "multiple" cognitive developments better describe the field?

IQ and Cognitive Development are relatively closely related areas. (An IQ test and a cognitive development test may well make use of some of the same questions or activities.) A brain undergoes cognitive development from the time it first begins to form in a fetus. This cognitive development continues throughout life. (However, there can be cognitive decline due to age-related and other damage to the brain.)

IQ testing is one way to measure cognitive development. A test based on a stage theory of cognitive development is another approach to such measurement. Each of these measurement processes can be used to produce a number, such as an IQ number or a stage number.

The norming process in IQ measurement tends to produce a number that remains relatively stable over time. The stage measurement approach produces a stage level (number or designation) that increases over time as a person moves up a cognitive development scale.Relatively few people reach the top level of this 15-point scale.

Math Cognitive Development

There are a large number of hits on a Google search of math cognitive development. As an example, Stages of Math Development is a very short article that includes a math cognitive development stage theory model for children up to six years old.

Piaget did a lot of research in developing his 4-stage model of cognitive development. Besides his general interests in cognitive development, he also has a particular interest in math cognitive development. The following Dina and Pierre van Hiele geometry cognitive development scale was certainly inspired by Piaget's work. See also http://www.math.uiuc.edu/~castelln/VanHiele.pdf.

Level	Name	Description
0	Visualization	Students recognize figures as total entities (triangles, squares), but do not recognize properties of these figures (right angles in a square).
1	Analysis	Students analyze component parts of the figures (opposite angles of parallelograms are congruent), but interrelationships between figures and properties cannot be explained.
2	Informal Deduction	Students can establish interrelationships of properties within figures (in a quadrilateral, opposite sides being parallel necessitates opposite angles being congruent) and among figures (a square is a rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.
3	Deduction	At this level the significance of deduction as a way of establishing geometric theory within an axiom system is understood. The interrelationship and role of undefined terms, axioms, definitions, theorems, and formal proof is seen. The possibility of developing a proof in more than one way is seen. (Roughly corresponds to Formal Operations on the Piagetian Scale.)
4	Rigor	Students at this level can compare different axiom systems (non-Euclidean geometry can be studied). Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.

Notice that the van Hieles, being mathematicians, labeled their first stage Level 0. This is a common practice that mathematicians use when labeling the terms of a sequence. Piaget's cognitive development scale has four levels, numbers 1 to 4. The highest level in the van Hiele geometry cognitive development scale is one level above the highest level of the Piaget cognitive development scale.

The following scale was created (sort of from whole fabric) by David Moursund. It represents his current insights into a six-level, Piagetian-type, math cognitive development scale.

Stage & Name	Math Cognitive Developments
Level 1. Piagetian and Math sensorimotor. Birth to age 2.	Infants use sensory and motor capabilities to explore and gain increasing understanding of their environments. Research on very young infants suggests some innate ability to deal with small quantities such as 1, 2, and 3. As infants gain crawling or walking mobility, they can display innate spatial sense. For example, they can move to a target along a path requiring moving around obstacles, and can find their way back to a parent after having taken a turn into a room where they can no longer see the parent.
Level 2. Piagetian and Math preoperational. Age 2 to 7.	During the preoperational stage, children begin to use symbols, such as speech. They respond to objects and events according to how they appear to be. The children are making rapid progress in receptive and generative oral language. They accommodate to the language environments (including math as a language) they spend a lot of time in, so can easily become bilingual or trilingual in such environments. During the preoperational stage, children learn some folk math and begin to develop an understanding of number line. They learn number words and to name the number of objects in a collection and how to count them, with the answer being the last number used in this counting process. A majority of children discover or learn “counting on” and counting on from the larger quantity as a way to speed up counting of two or more sets of objects. Children gain increasing proficiency (speed, correctness, and understanding) in such counting activities. In terms of nature and nurture in mathematical development, both are of considerable importance during the preoperational stage.
Level 3. Piagetian and Math concrete operations. Age 7 to 11.	During the concrete operations stage, children begin to think logically. In this stage, which is characterized by 7 types of conservation: number, length, liquid, mass, weight, area, volume, intelligence is demonstrated through logical and systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions that are reversible). While concrete objects are an important aspect of learning during this stage, children also begin to learn from words, language, and pictures/video, learning about objects that are not concretely available to them. For the average child, the time span of concrete operations is approximately the time span of elementary school (grades 1-5 or 1-6). During this time, learning math is somewhat linked to having previously developed some knowledge of math words (such as counting numbers) and concepts. However, the level of abstraction in the written and oral math language quickly surpasses a student’s previous math experience. That is, math learning tends to proceed in an environment in which the new content materials and ideas are not strongly rooted in verbal, concrete, mental images and understanding of somewhat similar ideas that have already been acquired. There is a substantial difference between developing general ideas and understanding of conservation of number, length, liquid, mass, weight, area, and volume, and learning the mathematics that corresponds to this. These tend to be relatively deep and abstract topics, although they can be taught in very concrete manners.
Level 4. Piagetian and Math formal operations. After age 11.	Starting at age 11 or 12, or so, thought begins to be systematic and abstract. In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts, problem solving, and gaining and using higher-order knowledge and skills. Math maturity supports the understanding of and proficiency in math at the level of a high school math curriculum. Beginnings of understanding of math-type arguments and proof. Piagetian and Math formal operations includes being able to recognize math aspects of problem situations in both math and non-math disciplines, convert these aspects into math problems (math modeling), and solve the resulting math problems if they are within the range of the math that one has studied. Such transfer of learning is a core aspect of Level 4. Level 4 cognitive development can continue well into college, and most students never fully achieve Level 4 math cognitive development. (This is because of some combination of innate math ability and not pursuing cognitively demanding higher level math courses or equivalent levels on their own.)
Level 5. Abstract mathematical operations. Moving far beyond math formal operations.	Mathematical content proficiency and maturity at the level of contemporary math texts used at the upper division undergraduate level in strong programs, or first year graduate level in less strong programs. Good ability to learn math through some combination of reading required texts and other math literature, listening to lectures, participating in class discussions, studying on your own, studying in groups, and so on. Solve relatively high level math problems posed by others (such as in the text books and course assignments). Pose and solve problems at the level of one’s math reading skills and knowledge. Follow the logic and arguments in mathematical proofs. Fill in details of proofs when steps are left out in textbooks and other representations of such proofs.
Level 6. Mathematician.	A very high level of mathematical proficiency and maturity. This includes speed, accuracy, and understanding in reading the research literature, writing research literature, and in oral communication (speak, listen) of research-level mathematics. Pose and solve original math problems at the level of contemporary research frontiers.

Cognitive Acceleration In Mathematics Education

Shayer, Michael and Mundher, Adhami (June 2006). The Long-Term Effects from the Use of CAME (Cognitive Acceleration In Mathematics Education): Some Effects from the Use of the Same Principles in Y1&2, and the Maths Teaching of the Future. Proceedings of the British Society for Research into Learning Mathematics. Retrieved 8/14/09: www.bsrlm.org.uk/IPs/ip26-2/BSRLM-IP-26-2-17.pdf. Quoting from the article:

The CAME[1] project was inaugurated in 1993 as an intervention delivered in the context of mathematics with the intention of accelerating the cognitive development of students in the first two years of secondary education. This paper reports substantial post-test and long-term National examination effects of the intervention. The RCPCM project[2], an intervention for the first two years of Primary education, doubled the proportion of 7 year-olds at the mature concrete level to 40%, with a mean effect-size of 0.38 S.D. on Key Stage 1 Maths. Yet, instead of the intervention intention, it is now suggested that a better view is to regard CAME as a constructive criticism of normal instructional teaching, with implications for the role of mathematics teachers and university staff in future professional development.

BACKGROUND TO CAME AND RCPCM

In the mid-70s CSMS[3] survey 14,000 children aged 10 to 16 were given three Piagetian tests to assess the range of thinking levels at each year. Figure 2 shows the findings.

By [age] 14 only 20% were showing formal operational thinking (3A&3B). This mattered at the time because current O-level science and maths courses, designed for grammar-school children in the top 20% of the ability range, required this level from the end of Y8. In the 80s the Graded Assessment in Maths scheme for the ILEA found that by the age of 12 the children’s mathematics competence had a 12-year developmental gap between the above-average and those at what would later be National Curriculum Levels 1 and 2.

The CASE project shows the need to teach math in a manner that helps to increase cognitive development. Quoting from the article:

Enough evidence has been presented to show that the research has engendered class management skills in the teachers involved that realise Vygotsky’s insistence that teaching should foster development as well as subject knowledge: that it should always aim ahead of where students presently are.

David Tall

David Tall is a Professor of Mathematical Thinking at the University of Warwick. Quoting from the linked site:

My interests in cognitive development in mathematics have matured over the years. Initially, as a mathematics lecturer at university, seeing mathematics through the eyes of a mathematician, I used mathematical theories (such as catastrophe theory) to formulate cognitive theory. After researching concepts of limits and infinity, I designed computer software to visualize calculus ideas using the idea of 'local straightness' rather than formal limits, and expanded my interest in visualization. Working first with Michael Thomas on algebra, then Eddie Gray on arithmetic, I was able to see the links between symbolism in arithmetic, algebra and calculus. This led directly to the theory of procepts which is concerned essentially with symbols that represent both process and concept and the ability to switch flexibly between processes 'to do' and concepts 'to think about'. Eddie and I were then able to see what we termed 'the proceptual divide '- the bifurcation between those children who remain focused on more specific procedures rather than develop the flexibility of proceptual thinking.

As I worked on various aspects of mathematics through the years I began to realise that there were three very different developments that occur in mathematics based on our perceptions of objects, actions on objects which are represented by symbols that can be manipulated as mental objects, and on the properties of physical and mental objects that we reflect upon and use as a basis for deduction. This led me to categorise mathematical thinking into three different worlds: a conceptual-embodied world of objects perceived and conceived, a proceptual-symbolic world of symbols (as process and concept) that arise from symbolising actions, and an axiomatic-formal world that arises from properties that are defined and concepts that are constructed through formal proof. I found that each world develops quite differently, with different cognitive sequences of development, different uses of language and different forms of proof. For more information on advanced mathematical thinking. As I worked on various aspects of mathematics through the years I began to realise that there were three very different developments that occur in mathematics based on our perceptions of objects, actions on objects which are represented by symbols that can be manipulated as mental objects, and on the properties of physical and mental objects that we reflect upon and use as a basis for deduction. This led me to categorise mathematical thinking into three different worlds: a conceptual-embodied world of objects perceived and conceived, a proceptual-symbolic world of symbols (as process and concept) that arise from symbolising actions, and an axiomatic-formal world that arises from properties that are defined and concepts that are constructed through formal proof. I found that each world develops quite differently, with different cognitive sequences of development, different uses of language and different forms of proof. For more information on advanced mathematical thinking, click here.

Stage Theory in Math

In a 4/18/09 email message "Michael Lamport Commons" <commons@tiac.net> wrote concerning a 15 point scale with the stages numbered 0–14:

What we have found is that to pass courses such as linear algebra, multivariate calculus and the like requires systematic stage 11 reasoning, one stage beyond formal stage 10. To pass abstract mathematics classes like modern algebra, topology, etc, when they are proof based, require metaystematic stage 12 reasoning. We also find that this is the earliest domain in which such reasoning develops.

A detailed discussion of the 15-level stage theory is available in the following reference:

Commons, M. L., Miller, P. M., Goodheart, E. A., & Danaher-Gilpin, D. (2005). Hierarchical Complexity Scoring System (HCSS): How to Score Anything. Retrieved 5/5/09: http://www.tiac.net/~commons/Scoring%20Manual.htm. Unpublished Scoring Manual Available from Dare Institute, Commons@tiac.net

Abstract: The Model of Hierarchical Complexity presents a framework for scoring reasoning stages in any domain as well as in any cross cultural setting. The scoring is based not upon the content or the participant material, but instead on the mathematical complexity of hierarchical organization of information. The participant’s performance on a task of a given complexity represents the stage of developmental complexity. This paper presents an elaboration of the concepts underlying the Model of Hierarchical Complexity (MHC), the description of the stages, steps involved in universal stage transition, as well as examples of several scoring samples using the MHC as a scoring aid.

Michael Commons' work in this area is important for two reasons:

1. It emphasizes that a stage theory can be developed that cuts across domains and culture.

2. It makes use of a mathematical complexity structure.

Here is another important paper by Commons and Richards:

Commons, M. L. & Richards, F. A. (2002). Four postformal stages In J. Demick (Ed.), Handbook of adult development. New York: Plenum. Retrieved 6/18/09: http://www.tiac.net/~commons/Four%20Postformal%20Stages.html. Quoting from this document:

The term "postformal" has come to refer to various stage characterizations of behavior that are more complex than those behaviors found in Piaget's last stage—formal operations—and generally seen only in adults. Commons and Richards (1984a, 1894b) and Fischer (1980), among others, posited that such behaviors follow a single sequence, no matter the domain of the task e.g., social, interpersonal, moral, political, scientific, and so on.

Most postformal research was originally directed towards an understanding of development in one domain. The common approach to much of the work on postformal stages has been to specify a performance on tasks that develop out of those described by Piaget (1950, 1952) as formal-operational or out of tasks in related domains (e.g., moral reasoning). The assumption has been made that the predecessor task performances (formal operations), are in some way necessary to the development of their successor performances and proclivities (postformal operations). Unlike many of the other theories, the Model of Hierarchical Complexity (MHC), presented here (Commons, et al., 1998), generates one sequence that addresses all tasks in all domains and is based on a contentless, axiomatic theory.

This document summarizes the research on identifying and describing four Piagetian-type cognitive development stages that are above Piaget's Stage 4: Formal Operations. The assertion is that these four stages apply equally well in every discipline. The quote from Commons given earlier in this section provides an example of examining these higher-level stages within the discipline of mathematics.

IQ and Stage Theory

Email from Michael Commons to David Moursund 5/10/09 says:

The MHC [Model of Hierarchical Complexity] shows that stages are absolute and do not need in any way norms. Hierarchical Complexity is a major determinant of how difficult a task is. So stage and IQ should be quite correlated. My guess, is about an r of .5.

…

The evidence for stage change is a lot more clearly studied than IQ change. Most intervention buy 1 or 2 stages at the most. I know of no studies showing more.

The first quoted part reemphasizes that IQ measures are normed and Cognitive Development measures on a Piagetian-type stage scale are not. Commons suggests that IQ and Stage level are moderately correlated.

As noted earlier in this document, the norming process used to measure IQ tends to make the measure of one's IQ remain fairly stable over time. However, there are interventions that can increase IQ. It is not clear whether there are long term studies that indicate such interventions lead to long term increase in IQ.

Finally, Common's last statement above and general research on stage theory indicate that intervention can increase the rate that people move through stages, and that interventions can move the top stage level reached up one or two steps on a 15-stage measure.

Math Learning Disorders

Quoting from an article from ScienceDaily:

Mayo Clinic Researchers Find Math Learning Disorder Is Common

ScienceDaily (Oct. 27, 2005) — ROCHESTER, Minn. -- In a recently published study, Mayo Clinic researchers determined Math Learning Disorder (LD) is common among school-age children. Results show that boys are more likely to have Math LD than girls. The research also indicates that although a child can have a Math LD and a reading LD, a substantial percentage of children have Math LD alone. In fact, the cumulative incidence of Math LD through age 19 ranges from 6% to as high as 14%, depending on the Math LD definition. LD is used to describe the seemingly unexplained difficulty a person of at least average intelligence has in acquiring basic academic skills—skills that are essential for success at school, work and for coping with life in general. The results appear in the September-October issue of Ambulatory Pediatrics.

…

In the current study, Mayo Clinic researchers used different definitions of Math LD, analyzed school records of boys and girls enrolled in public and private schools in Rochester, Minn., and examined information from the students' medical records. They also looked at the extent to which Math LD occurs as an isolated learning disorder versus the extent to which it occurs simultaneously with Reading LD. This study is the first to measure the incidence—the occurrence of new cases—of Math LD by applying consistent criteria to a specific population over a long time. By considering the coexistence of Math LD and Reading LD across the students' entire educational experience (i.e., from grades K-12), the research presents a more comprehensive description of this association.

Examples

Here is some material quoted from a 2006 National Center for Learning Disabilities article on dyscalculia:

What is dyscalculia? Dyscalculia is a term referring to a wide range of life-long learning disabilities involving math. There is no single form of math disability, and difficulties vary from person to person and affect people differently in school and throughout life.

What are the effects of dyscalculia? Since disabilities involving math can be so different, the effects they have on a person's development can be just as different. For instance, a person who has trouble processing language will face different challenges in math than a person who has difficulty with visual - spatial relationships. Another person with trouble remembering facts and keeping a sequence of steps in order will have yet a different set of math-related challenges to overcome.

Here is some material quoted from an article by Jerome Schultz: The article is about central auditory processing disorders (CAPD) as it relates to learning math.

Let me take this opportunity to help our readers understand CAPD a bit better, since this condition often goes unrecognized or is misdiagnosed as ADHD. The American Speech-Language-Hearing Association (ASHA) established a task force in 1996 to gain a better understanding of central auditory processing disorders (CAPD) in children.

…

Learning multiplication tables involves auditory pattern recognition, and temporal factors (the order of the language). Differentiating 8 x 7 = 56 from 6 x 7 = 42 is very difficult, since these are abstract symbols for a particular quantity. If she just says them over and over again, she may remember one...until she hears the next one. Your daughter has to be instructed in a concrete visual, hands-on way to understand ("see" in her mind's eye) that a number represents a quantity. Otherwise, the times tables are just another jumble of numbers.

David Geary

David Geary is a highly prolific researcher in math cognitive development. Here are some David Geary papers available on the Web.

David C. Geary, Mary K. Hoard, Jennifer Byrd-Craven, Lara Nugent, and Chattavee Numtee (July/August 2007). Cognitive Mechanisms Underlying Achievement Deficits in Children With Mathematical Learning Disability. Child Development. Retrieved 4/9/09. To access this paper go to http://web.missouri.edu/~gearyd/articles_math.htm, find the paper in the list of papers, and click on its link. Quoting from this paper:

Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n=15) and low achieving (LA, n=44) were identified. These groups and a group of typically achieving (TA, n=46) children were administered a battery of mathematical cognition, working memory, and speed of processing measures (M=6 years). The children with MLD showed deficits across all math cognition tasks, many of which were partially or fully mediated by working memory or speed of processing. Compared with the TA group, the LA children were less fluent in processing numerical information and knew fewer addition facts. Implications for defining MLD and identifying underlying cognitive deficits are discussed.

Geary, David C. (2007). An Evolutionary Perspective on Learning Disability in Mathematics. Developmental Neuropsychology. Retrieved 4/9/09: http://web.missouri.edu/~gearyd/files/Geary%20et%20al.%20[2007%20Child%20Development].pdf. Quoting from this paper:

When viewed from the lens of evolution and human cultural history, it is not a coincidence that public schools are a recent phenomenon and emerge only in societies in which technological, scientific, commercial (e.g., banking, interest) and other evolutionarily-novel advances influence one’s ability to function in the society (Geary, 2002, 2007). From this perspective, one goal of academic learning is to acquire knowledge that is important for social or occupational functioning in the culture in which schools are situated, and learning disabilities (LD) represent impediments to the learning of one or several aspects of this culturally-important knowledge. It terms of understanding the brain and cognitive systems that support academic learning and contribute to learning disabilities, evolutionary and historical perspectives may not be necessary, but may nonetheless provide a means to approach these issues from different levels of analysis. I illustrate this approach for MLD. I begin in the first section with an organizing frame for approaching the task of decomposing the relation between evolved brain and cognitive systems and school-based learning and learning disability (LD). In the second section, I present a distinction between potentially evolved biologically-primary cognitive abilities and biologically-secondary abilities that emerge largely as a result of schooling (Geary, 1995), including an overview of primary mathematics. In the third section, I outline some of the cognitive and brain mechanisms that may be involved in modifying primary systems to create secondary abilities, and in the fourth section I provide examples of potential the sources of MLD based on the framework presented in the first section.

Geary, David (March 2006). Dyscalculia at an Early Age: Characteristics and Potential Influence on Socio-Emotional Development. Encyclopedia on Early Childhood Development. Retrieved 4/9/09: http://web.missouri.edu/~gearyd/files/GearyANGxp.pdf. Quoting from the first part of this paper:

Introduction. Dyscalculia refers to a persistent difficulty in the learning or understanding of number concepts (e.g. 4 > 5), counting principles (e.g. cardinality – that the last word tag, such as “four,” stands for the number of counted objects), or arithmetic (e.g. remembering that 2 + 3 = “5”). These difficulties are often called a mathematical disability. We cannot yet predict which preschool children will go on to have dyscalculia, but studies that will allow us to develop early screening measures are in progress. At this time and on the basis of normal development during the preschool years, it is likely that preschoolers who do not know basic number names, quantities associated with small numbers (< 4), how to count small sets of objects, or do not understand that subtraction results in less and addition results in more are at risk for dyscalculia.

Subject: How Common is Dyscalculia? Between 3 and 8% of school-aged children show persistent grade-to-grade difficulties in learning some aspects of number concepts, counting, arithmetic, or in related math areas. These and other studies indicate that these learning disabilities, or dyscalculia, are not related to intelligence, motivation or other factors that might influence learning. The finding that 3 to 8% of children have dyscalculia is misleading in some respects. This is because most of these children have specific deficits in one or a few areas, but often perform at grade level or better in other areas. About half of these children are also delayed in learning to read or have a reading disability, and many have attention deficit disorder.

Geary, David C. (1999). Mathematical Disabilities: What We Know and Don't Know. Retrieved 4/9.09: http://www.ldonline.org/article/5881. Quoting from the paper:

There have only been a few large-scale studies of children with MD [Mathematical Disability] and all of these have focused on basic number and arithmetic skills. As a result, very little is known about the frequency of learning disabilities in other areas of mathematics, such as algebra and geometry. In any case, the studies in number and arithmetic are very consistent in their findings: Between 6 and 7% of school-age children show persistent, grade-to-grade, difficulties in learning some aspects of arithmetic or related areas (described below). These and other studies indicate that these learning disabilities are not related to IQ, motivation or other factors that might influence learning.

The finding that about 7% of children have some form of MD is misleading in some respects. This is because most of these children have specific deficits in one or a few subdomains of arithmetic or related areas (e.g., counting) and perform at grade-level or better in other areas of arithmetic and mathematics. The confusion results from the fact that standardized math achievement tests include many different types of items, such as number identification, counting, arithmetic, time telling, geometry, and so fourth. Because performance is averaged over many different types of items, some of which children with MD have difficulty on and some of which they do not, many of these children have standardized achievement test scores above the 7th percentile (though often below the 20th).

Math Maturity

The previous parts of this document have explored Math IQ and Math Cognitive Development. They provide a theoretical underpinning for the discussion of math maturity given in the remainder of this document.

Components of Math Maturity

The term math maturity is widely used by mathematicians and math educators. For example, a middle school teacher may say, “I don’t think Pat has the necessary math maturity to take an algebra course right now.” It is clear that the teacher is not talking about Pat’s math content knowledge.

Probably Pat has completed the prerequisite coursework. Perhaps Pat is weak in math reasoning and thinking, tends to learn math by rote memorization, has little interest in math, and shows little persistence in working on challenging math problems. The teacher feels that with this background, Pat is apt to struggle in algebra and likely fail the course.

At the university level, the dominant component in the literature of math maturity is “proof” and the logical,critical, creative reasoning and thinking involved in understanding and doing proofs. A person with a high level of math maturity has studied math at a level that requires substantial understanding of proof and regular demonstration of the ability to do proofs.

K-12 math has only a modest emphasis on formal proofs. However, as students move up in the math curriculum, they face a growing challenge to make mathematical arguments that describe and justify the steps they take in solving challenging math problems. This is a type of logical/mathematical (proof) activity.

The following list contains some components of math maturity. An increasing level of math maturity is demonstrated by:

1. An increasing capacity in the logical, critical, creative reasoning and thinking involved in understanding and solving problems and in understanding and doing proofs.

2. An increasing capacity to move beyond rote memorization in recognizing, posing, representing, and solving math problems. This includes transfer of learning of one’s math knowledge and skills to problems in many different disciplines.

3. An increasing capability to communicate effectively in the language and ideas of mathematics. This includes:

A. Mathematical speaking and listening fluency.

B. Mathematical reading and writing fluency.

C. Thinking and reasoning in the language and images of mathematics.

4. An increasing capacity to learn mathematics—to build upon one’s current mathematical knowledge and to take increasing personal responsibility for this learning.

5. Improvements in other factors affecting math maturity such as attitude, interest, intrinsic motivation, focused attention, perseverance and delayed gratification, having math-oriented habits of mind, and acceptance of and fitting into the “culture” of the discipline of mathematics.

Email from Joseph Dalin 6/14/09

The following email message sent to David Moursund was in response to a email message about math and Talented & Gifted education sent to an National Council of Supervisors of Mathematics distribution list.

Hi,

The major question is: what is a Talented and Gifted persons or students? What is his/her special capabilities? Memorizing or the capability of understanding? Do the Talented and Gifted students learn differently? Do they understand symbolic abstract language better, or significantly better, than ordinary students?

The basic education of mathematical understanding and creative thinking is gained through solving problems of cases which deal within the child's environment, experience and conceptual system.

Algebra is a symbolic abstract nonhuman language. In order to understand such a language there is a need to:

Have a "mathematical thinking maturity” and than—to translate the symbolic representations into graphic representation and learn through self experience, exploration and discovery (which is the way human beings learn).

It can be achieved by comprehensive integration of Visual-dynamic-quantitative computer software into the teaching and learning process of school mathematics.

Such an approach should be applied for all students, not only Talented and Gifted students.

I don’t believe that, in general, students of 5th grade have the “mathematical learning maturity” for learning algebra 1.

I don’t believe that most students of 7th grade are capable to leaning algebra 1 through its symbolic representation only. Anyhow, what’s the rush? Learning is a long journey….

I don’t believe that most students, even in higher grades, are capable to really understand Algebra by learning only through its symbolic representations. That’s the main reason of poor achievement, failure and frustration of school mathematics education which is based on teaching symbolic mathematics.

I am ready to share my experience.

Dr. Joseph Dalin, Director, Israeli Institute for the Integration of Computers in Mathematics Education.

Three of the paragraphs have been bolded for extra emphasis. The message is that math maturity is a key issue in determining when a student (whether TAG or not) should begin an algebra course. The message also suggests that current widely used methods for teaching introductory algebra do not adequately address the challenge of learning to deal with a high level of abstractness and translating it into personally meaningful understanding.

Clyde Greeno

The following is quoted from an email message sent by Clyde Greeno to the National Council of Supervisors of Mathematics distribution list on 4/9/09:

The entrenched "developmental" algebra curriculum (like the HS algebra curriculum) is a direct decedent from SMSG's calculus-preparatory HS algebra—which has served more to filter students out of the mathematics curriculum than to empower them for success within it. [No wonder that educators now are concerned about the "Algebra 2 for everyone" movement among state legislatures.]

Extensive clinical research has revealed that the primary cause for students' difficulties with algebra is simply that algebra curricula within the SMSG lineage badly violate scientifically established principles of the developmental psychology of mathematical learning—i.e. of "mathematics as common sense". [The original SMSG version was created two decades before America began to understand Piaget.] Ironically, the resulting "developmental" algebra curriculum is anything but developmental. The coming reformation will be guided by psychology.

Clinical methods quickly reveal that students learn the usual essentials of algebra better, faster, and more easily through the context of functions. That is partly because the field of algebra really is all about the study of operations/functions—even the SMSG curriculum was covertly about functions—even though that context still is badly hidden by the current curriculum.

Education for Increasing Math Maturity

This section is a work in progress.

Math maturity increases over time through:

General overall increase in cognitive development.
Learning math in a manner that facilitates higher-order creative thinking, problem solving, theorem proving, communicating in and about math, and learning to learn math.
Working with math teachers who have a higher level of math maturity than oneself, and being taught at a level that is a little above one's current level of math maturity.

Math maturity is strongly affected by one's informal education, formal education, and life experiences. As one's brain grows and as one is engaged in informal and formal education, one's overall intelligence grows and one's level of cognitive development grows. If one's education and experiences have an appropriate math component, one's math maturity will increase.

Assessment of Math Maturity

This section is a Work in Progress and definitely needs input from a lot of people.

It is relatively easy to make use of the term math maturity and to claim it is an important concept or goal in math education. It is much more difficult to develop assessment instruments that can be used for self-assessment (by students), for assessment by people interested in measuring how well our math education system is doing in helping students to develop math maturity, and as an aid to student placement in courses.

Good math teachers are able to estimate the math maturity of their students through a one-on-one conversation, by listening to the breadth and depth of questions a student raises in class, by listening to the breadth and depth of answers a student gives to questions raised during a class, through analysis of a student's homework and test answers, and so on. There are many clues available in these information sources. However, it is a major challenge to identify them and teach less qualified teachers to learn to observe and make use of these information sources.

Good math teachers can determine if a student has the math maturity to effectively deal with the content the teacher wants to teach and whether a student is apt to be bored by the level and pace of what is to be taught.

With some practice, students can gain skill in self-assessing their level of math maturity and progress they are making in increasing their level of math maturity. Part of a useful approach is a self-assessment based on insights into learning by rote memory versus learning for understanding. Another is self-assessment on dealing with "challenging" problems that draw upon math covered a few weeks ago, much earlier in the school year, and in previous school years.

Still another approach is through self-assessment of how well one can explain to oneself and to others the thought processes and understanding used in attacking challenging problems and proofs. In this, however, one needs to be aware that people who are good at math often have intuitive or not readily explained leaps of insight. Such leaps often do not lend themselves to the "show your work" type of requirement that most teachers require of their students.

Computers and Math Maturity

“Computers are incredibly fast, accurate, and stupid. Human beings are incredibly slow, inaccurate, and brilliant. Together they are powerful beyond imagination.” (Albert Einstein)

"My familiarity with various software programs is part of my intelligence if I have access to those tools." (David Perkins,1992.)

The two quotes capture the essence of this section. An intact human mind and body has tremendous capabilities. However, it also has severe limitations. Over many thousands of years humans have been developing tools that help to overcome some of these physical and mental limitations.

Thus, for example, we have developed telescopes for "far seeing" and microscopes for "near seeing" that far exceed the capabilities of the human visual system. We have developed reading, writing, and arithmetic that are wonderful aids to one's brain. We have developed machines such as cars, airplanes, and bulldozers. We have developed highly automated manufacturing facilities.

Now, we have Information and Communication Technology. It plays a role in many of our previously developed tools, and it provides a new type of intelligence. Machine intelligence (artificial intelligence) can be thought of as a new type of brain, or as an auxiliary brain.

In terms of the document you are now reading, the major question is the nature and extent to which the computer brain adds to the capabilities of human intelligence and human cognitive development. That is, as educators we now need to think in terms of nature, nurture, and machine intelligence.

Here is a concrete example. Spatial intelligence in one of the eight Multiple Intelligences on Howard Gardner's list. We have long had maps and compasses to help people deal with certain types of spatial problems. We now have computerized maps (for example, think in terms of Google Earth) and GPS systems that can aid in solving some of the spatial problems that people face. In essence, such tools increase the intelligence of their users.

David Tall's Work

The article Tall (2000) discusses the cognitive load that is inherent to learning and using mathematics. The cognitive load is reduced through learning the language and symbolism of math so that one can use it rapidly and accurately at a subconscious level—in the same way that one uses their native language in speaking and writing.

Calculators and computers can play a significant role in math education. Quoting from Tall (2000):

The development of symbol sense throughout the curriculum

faces a number of major re-constructions causing increasing difficulties to more and more students as they are faced with successive new ideas that require new coping mechanisms. For many it leads to the satisfying immediate short-term needs of passing examinations by rote-learning procedures. The students may therefore satisfy the requirements of the current course and the teacher of the course is seen to be successful. If the long-term development of rich cognitive units is not set in motion, short-term success may only lead to increasing cognitive load and potential long-term failure.

…

Given the constraints and support in the biological brain, the concept imagery in the mathematical mind can be very different from the working of the computational computer. A professional mathematician with mathematical cognitive units may use the computer in a very different way from the student who is meeting new ideas in a computer context.‎

The article then goes on to explain some of Tall's insights into what/how the brain is learning math in a graphing calculator or computer environment versus in the traditional paper and pencil environment. One way to think of this is in terms of the automation of tasks. A mathematician's brain has automated many tasks, and this has come through a considerable amount of practice. An alternative to this mental automation is, in a number of cases, to learn to use a graphing calculator or computer.

Thus, a math student who is a heavy user of graphing calculators and computers will be developing a type of math maturity that is different than that being developed by a person who does not become proficient in the use of these math tools. Tall does not argue that one type of approach to math education is superior to the other—just that certain aspects of the final results in a student's math-brain will be different. (See also: Moursund, 1986, 1988.)

Automaticity

This section is a work in progress.

Research into how people solve problems and gain in expertise within a particular problem-solving domain have helped us to understand how study and practice lead to increased automaticity and less demands on one's brain. Thus, for example, an expert chess player can recognize and process possible desirable moves in a complex board position much more rapidly and accurately than can an less qualified chess player. This speed of recognition and analysis has come from many thousands of hours of study and practice.

A similar type of learning occurs in math. Through thousands of hours of study and practice, a mathematician's brain automates a large number of problem recognition and possible action tasks. Thus, when faced by a new and challenging math problem, the expert mathematician is able to devote more brain power to the new and challenging parts while automaticity takes care of the familiar parts.

The chess and math examples apply to all areas in which a person can achieve a high level of expertise.

"As a task to be learned is practiced, its performance becomes more and more automatic; as this occurs, it fades from consciousness, the number of brain regions involved in the task becomes smaller." (A Universe Of Consciousness How Matter Becomes Imagination. Edelman & Tononi, 2000, p.51)

That is, through repetition of a task, a brain becomes more efficient at carrying out the task. But, it can take a lot of repetitions plus occasional practice to build and maintain this efficiency. An alternative in some cases is to just turn such repetitive task over to a computer or a calculator. Quoting from David Tall's 1996 article Can All Children Climb the Same Curriculum Ladder?:

This presentation presents evidence that the way the human brain thinks about mathematics requires an ability to use symbols to represent both process and concept. The more successful use symbols in a conceptual way to be able to manipulate them mentally. The less successful attempt to learn how to do the processes but fail to develop techniques for thinking about mathematics through conceiving of the symbols as flexible mathematical objects. Hence the more successful have a system which helps them increase the power of their mathematical thought, but the less successful increasingly learn isolated techniques which do not fit together in a meaningful way and may cause the learner to reach a plateau beyond which learning in a particular context becomes difficult.

Here is another quote from a 2000 article by David Tall:

The computer is quite different from the biological brain and therefore can be of value by providing an environment that complements human activity. Whilst the brain performs many activities simultaneously and is prone to error, the computer carries out individual algorithms accurately and with great speed. Computer calculations with numbers and manipulation of symbols has some similarities with the notion of procept. Internal computer symbolism is used both to represent data and also to perform routines to manipulate that data. However, there are significant differences. The computer is simply a device which manipulates information in a way specified by a program. It has none of the cognitive richness (or baggage) of the concept image available to the human mind to guide (or confuse) problem-solving activities.

This quote captures some of the idea of students learning by rote (sort of like a computer) and students learning with understanding as well with mental links to related topics that they know something about. This richer learning is a goal in math education. Here is a related quote from Tall:

The development of symbol sense throughout the curriculum therefore faces a number of major reconstructions which cause increasing difficulties to more and more students as they are faced with successive new ideas that require new coping mechanisms. For many it leads to the satisfying immediate short-term needs of passing examinations by rote-learning procedures. The students may therefore satisfy the requirements of the current course and the teacher of the course is seen to be successful. However, if the long-term development of rich cognitive units is not set in motion, short-term success may only lead to increasing cognitive load and potential long-term failure.

Declining Critical Thinking

Wolpert, Stuart (1/27/09). Is technology producing a decline in critical thinking and analysis? UCLA Newsroom. Retrieved 1/29/2009: http://newsroom.ucla.edu/portal/ucla/is-technology-producing-a-decline-79127.aspx. Quoting from the report:

As technology has played a bigger role in our lives, our skills in critical thinking and analysis have declined, while our visual skills have improved, according to research by Patricia Greenfield, UCLA distinguished professor of psychology and director of the Children's Digital Media Center, Los Angeles.

Learners have changed as a result of their exposure to technology, says Greenfield, who analyzed more than 50 studies on learning and technology, including research on multi-tasking and the use of computers, the Internet and video games. Her research was published this month in the journal Science.

Reading for pleasure, which has declined among young people in recent decades, enhances thinking and engages the imagination in a way that visual media such as video games and television do not, Greenfield said.

…

Visual intelligence has been rising globally for 50 years, Greenfield said. In 1942, people's visual performance, as measured by a visual intelligence test known as Raven's Progressive Matrices, went steadily down with age and declined substantially from age 25 to 65. By 1992, there was a much less significant age-related disparity in visual intelligence, Greenfield said.

"In a 1992 study, visual IQ stayed almost flat from age 25 to 65," she said.

References

Bentsen, Todd (11 Apr 2009). Adult Brain Processes Fractions 'Effortlessly.' Medical News Today. Retrieved 4/16/09: http://www.medicalnewstoday.com/articles/145587.php.

Quoting from long term follow-up on initial "marshmallow" research:

Once Mischel began analyzing the results, he noticed that low delayers, the children who rang the bell quickly, seemed more likely to have behavioral problems, both in school and at home. They got lower S.A.T. scores. They struggled in stressful situations, often had trouble paying attention, and found it difficult to maintain friendships. The child who could wait fifteen minutes had an S.A.T. score that was, on average, two hundred and ten points higher than that of the kid who could wait only thirty seconds.

Commons, M. L., Miller, P. M., Goodheart, E. A., & Danaher-Gilpin, D. (2005). Hierarchical Complexity Scoring System (HCSS): How to Score Anything. Retrieved 5/5/09: http://www.tiac.net/~commons/Scoring%20Manual.htm. Unpublished Scoring Manual Available from Dare Institute, Commons@tiac.net.

Crace, John (1/24/2006). Children are less able than they used to be. The Guardian. Retrieved 6/21/09: http://www.guardian.co.uk/education/2006/jan/24/schools.uk.

Quoting from the article:

New research funded by the Economic and Social Research Council (ESRC) and conducted by Michael Shayer, professor of applied psychology at King's College, University of London, concludes that 11- and 12-year-old children in year 7 are "now on average between two and three years behind where they were 15 years ago", in terms of cognitive and conceptual development.

"It's a staggering result," admits Shayer, whose findings will be published next year in the British Journal of Educational Psychology. "Before the project started, I rather expected to find that children had improved developmentally. This would have been in line with the Flynn effect on intelligence tests, which shows that children's IQ levels improve at such a steady rate that the norm of 100 has to be recalibrated every 15 years or so. But the figures just don't lie. We had a sample of over 10,000 children and the results have been checked, rechecked and peer reviewed."

Huitt, W. (n.d.). Educational Psychology Interactive: Piaget's Theory of Cognitive Development. Retrieved 6/18/09: http://chiron.valdosta.edu/whuitt/col/cogsys/piaget.html.

Lehrer, Jonah (5/18/09). Don't! The Secret of Self Control. The New Yorker. Retrieved 6/18/09: http://www.newyorker.com/reporting/2009/05/18/090518fa_fact_lehrer?currentPage=all.

MacDonald, Sharon. (1997) The Portfolio and Its Use : A Road Map for Assessment. Southern Early Childhood Association. Retrieved 6/6/09: http://www.cem.msu.edu/~leej/development-math.html.

Moursund, David (2006a). Computers in education for talented and gifted students: A book for elementary and middle school teachers. Eugene, OR: Information Age Education. Retrieved 5/4/09: http://i-a-e.org/downloads/doc_download/13-computers-in-education-for-talented-and-gifted-students.html.

Moursund, David (2006b). Computational thinking and math maturity: Improving math education in K-8 schools. Eugene, OR: Information Age Education. Retrieved 5/6/09: http://i-a-e.org/downloads/doc_download/3-computational-thinking-and-math-maturity-improving-math-education-in-k-8-schools.html.

Moursund, D.G. (1986, 1988). Computers and problem solving: A workshop for educators. Eugene, OR: Information Age Education. Access at http://i-a-e.org/submit-a-document/doc_download/188-computers-and-problem-solving-a-workshop-for-educators.html.

This book includes an emphasis on thinking about problem solving partly from the point of view of developing a repertoire of smaller problems or problem-solving activities to a high level of automaticity or making use of comptuters as an substitute for part of this learning task.

Tall, David (December 2000). Biological Brain, Mathematical Mind & Computational Computers (how the computer can support mathematical thinking and learning). Retrieved 4/11/09: http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2000h-plenary-atcm2000.pdf.

Author or Authors

The initial version of this document was developed by David Moursund

Math Maturity, Defined

This is from the from Good Math Lesson Plans Page of the IAE-pedia.

Mathematicians tend to prefer the concept of math maturity over the idea of math cognitive development. A Google search (10/6/08) of the expression: "math maturity" OR "mathematical maturity" OR "mathematics maturity" produced over 24,000 hits. Wikipedia states:

Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught, but instead comes from repeated exposure to complex mathematical concepts.

Still quoting from the Wikipedia, other aspects of mathematical maturity include:

the capacity to generalize from a specific example to broad concept
the capacity to handle increasingly abstract ideas
the ability to communicate mathematically by learning standard notation and acceptable style
a significant shift from learning by memorization to learning through understanding
the capacity to separate the key ideas from the less significant
the ability to link a geometrical representation with an analytic representation
the ability to translate verbal problems into mathematical problems
the ability to recognize a valid proof and detect 'sloppy' thinking
the ability to recognize mathematical patterns
the ability to move back and forth between the geometrical (graph) and the analytical (equation)
improving mathematical intuition by abandoning naive assumptions and developing a more critical attitude

Quoting Larry Denenberg:

Thirty percent of mathematical maturity is fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact—language that mathematicians use to communicate ideas. Mathematics, like English, relies on a common understanding of definitions and meanings. But in mathematics definitions and meanings are much more often attached to symbols, not to words, although words are used as well. Furthermore, the definitions are much more precise and unambiguous, and are not nearly as susceptible to modification through usage. You will never see a mathematical discussion without the use of notation!

You can evaluate a math lesson plan or unit of study in terms of how it contributes to students gaining in math maturity.

The general notion of "maturity" in a discipline applies to every discipline—indeed to every job, vocation, or pastime. However, mathematics teachers have been engaged with the notion more often than teachers of other academic disciplines.