Math Maturity

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Information Age Education (IAE) is an Oregon non-profit corporation created by David Moursund in July, 2007. It works to improve the informal and formal education of people of all ages throughout the world. A number of people have contributed their time and expertise in developing the materials that are made available free in the various IAE publications. Click here to learn how you can help develop new IAE materials.

Contents


Before We Start

Before delving into the topic of math maturity, here is some general information that may be useful to you.


Readers interested in this Math Maturity IAE-pedia document will likely also be interested in the following free book:

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.
This book was created for preservice and inservice teachers with the goal of improving the informal and formal math education of preK-8 students. The authors emphasize using simple, inexpensive games to provide students with learning environments that help to increase their levels of math maturity.
The book is a mixture of theory and practice. It contains a careful analysis of a small number of games, and links to a large number of games and related resources.
A strong emphasis in the book is on students learning to develop and test strategies for themselves. This is in marked contrast to a common approach to learning about strategies, an approach in which students are expected to memorize (and accept without question) strategies to solve or help them to solve a variety of problems.


If you want a self-study course on the topic of math maturity, here is a free book containing a detailed syllabus for such a course:

Moursund, David (August, 2010). Syllabus: Increasing the Math Maturity of K-8 Students and Their Teachers. This is a course developed by David Moursund, and it is suitable for self-study or in working with a group of your colleagues. Retrieved 9/25/2013 from http://i-a-e.org/downloads/doc_download/201-extended-syllabusfor-prism-course.html.
Brief Course Description
This course was developed as part of a math and science education project in Oregon. It was field tested as a distance education course and was successful The grant has ended and the course is no longer being taught.
This is a course for teachers of math and science focusing on increasing K-8 student levels of math maturity. It includes an emphasis on computational thinking and problem solving across the curriculum. Course participants must have Email and Web access as well as access to elementary or middle school students in order to carry out a number of the course assignments.

Some readers of this current version of the Math Maturity IAE-pedia document may want to compare/contrast it with an earlier version. This current version of the Math Maturity page is a highly revised and edited version of the original document. The original document has been renamed Math Maturity v1 and is available at http://iae-pedia.org/Math_Maturity_v1.

Part 1: Introduction and Goal

Math has long been a required part of the school curriculum. This is because some math knowledge, skills, and ways of thinking are deemed important for all students.

We know that math is quite useful in helping to represent and solve problems in many different academic situations as well as in many situations people encounter at home, at work, and at play. We know that the overall field of mathematics is very large and it still growing.

We also know that students taking math courses vary widely in how well they learn and understand the math, and how well they can apply their knowledge and skills in a variety of math-related problem-solving situations. They also vary widely in their long-term retention of the math they learn. These three topics all relate to math maturity.

How can teachers teach and students learn for increased math maturity? These are the two unifying questions addressed in this document. The intended audience is parents, teachers, teachers of teachers, students, and all others who are concerned about and involved in our informal and formal educational systems.

Math Maturity

It is well recognized that some rote memory learning is quite important in math education. However, most of this rote learning suffers from a lack of long-term retention, and from the learner’s inability to transfer this learning to new, challenging problem situations both within the discipline of math as well as to math-related problem situations outside the discipline of math.

Thus, math education (as well as education in other disciplines) has moved in the direction of placing much more emphasis on learning for understanding. There is substantial emphasis on learning some “big ideas” that will last a lifetime.

For example, the number line is one of the big ideas in math. Perhaps you can still remember being in an elementary school classroom with a segment of the number line on a poster that extended across the front of the room. Perhaps in your mind’s eye you see larger and larger positive numbers on one end of the number line, and corresponding negative numbers on the other end. Perhaps your mind’s eye visualizes fractions, decimal fractions, and irrational numbers such as the positive square root of 2.

You know that there are positive and negative numbers. You know that, for any pair of numbers, either they are the same or one is larger than the other.

Math maturity food for thought. Think about the assertion “that for any pair of numbers, either they are the same or one is larger than the other.” How do you know this? Is it a “fact” that your teacher told you, you memorized, and you now accept without question or doubt? Can you explain to yourself why this is a “true fact"? Can you give arguments that would convince someone else?

You know that any pair of numbers can be added, subtracted, or multiplied. You know that any pair of numbers can be divided—except that it is “impossible” to divide by zero. These aspects of numbers on a number line are all big ideas. However, your mind might fixate on why the number zero has been singled out for special mention. You might ask: What is so special about zero? Why is it impossible to divide by zero? You might begin to explore what it means to say that one cannot divide by zero. You might begin to explore various aspects of the number zero that make this number distinctly different from other numbers.

Math maturity food for thought. One indicator of increasing math maturity is a student’s movement from rote and unquestioning memorization to learning with and for understanding. As you were reading the previous paragraph, you encountered, "It is 'impossible’ to divide by zero.” What does this statement mean to you? Can you provide arguments that convince you and that might convince others that this is a correct statement? A more mathematically mature mind questions assertions such as, "It is impossible to divide by zero." The mind works to develop a level of understanding beyond rote learning and unquestioning acceptance of such assertions. This approach to learning with questioning and understanding is applied both in math and in other disciplines of study.

At any grade level, a teacher might encounter a student who raises such questions and who does such explorations. In the same class, there will be students who “don’t have a clue” or who “couldn’t care less” about why such questions are being asked or explored. If one student raises and explores such questions while another student “hasn’t a clue,” we are likely to take this as an indication that one student is more mathematically mature than the other. Math inquisitiveness is one aspect of math maturity.

However, this is a tricky situation. Suppose that the “powers that be” decide that all students should learn to pose such questions about zero. We can readily have students memorize the statement that it is impossible to divide by zero. We can have students memorize that zero is the only number that one cannot divide by. We can help each student to develop a mental picture of the answers that one gets by dividing a number such as 12 by smaller and smaller positive numbers. This provides the student with an explanation that it is impossible to divide 12 by zero because “the answer is larger than any positive number.” This example shows some of the difficulties inherent to developing specific test items to be used in a “test” for math maturity. It also begins to provide some insight into "infinity" and the mathematics of infinite series.

Of course, the more mathematically mature student will not be fazed by this “additional rote memory” approach to increasing the math maturity test scores of students. The more mathematically mature student will likely ask further questions, such as what does it mean to say the number line extends “forever” in each direction and that there is no largest positive number? Why is it that zero divided by 8 is the same as zero divided by 12? Is there any other number that has this same peculiarity as zero? And, suppose that it were possible to divide by zero. Would eight divided by zero be the same as 12 divided by zero?

At the earliest levels of learning math, students can encounter or pose very challenging and perplexing math questions. There are many opportunities for the more mathematically intelligent, the more mathematically cognitively developed, the more mathematically inquisitive or creative, and so on, to demonstrate an above average level of math maturity relative to his or her peers.

Math maturity food for thought. A variety of organizations have created math standards. The NCTM standards have a considerable focus on the ideas of Math Maturity, but not on this particular vocabulary.
Recently a proposed set of national standards (the Core Standards, http://www.corestandards.org/) have been published. The following is quoted from this document:
These Standards define what students should understand and be able to do in their study of mathematics. Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. [Bold added for emphasis.]
Quoting from another CCSS Math Initiative document http://www.corestandards.org/Math/Practice/MP8:
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.[Bold added for emphasis.]

The point being made is that "math maturity" is an important component of the Common Core State Standards (CCSS) Math Initiative.

Math Maturity of Teachers of Mathematics

At every grade level in elementary school and on up, teachers of math will encounter students with widely varying knowledge, skills, and interests in math, and with varying levels of math maturity. This situation suggests an important question: What do teachers themselves need to know to be effective math teachers? The following paper provides a discussion of some research on this question:

Ball, D.L., Hill, H.C., & Bass, H. (Fall 2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator. Retrieved 7/13/2010 from http://www.aft.org/pdfs/americaneducator/fall2005/BallF05.pdf.

Here is an example from the article. Answer the following questions. Mark YES, NO, or I’M NOT SURE for each item below.

a) 0 is an even number. YES, NO, I’M NOT SURE
b) 0 is not really a number. It is a placeholder in writing big numbers. YES, NO, I’M NOT SURE
c) The number 8 can be written as 008. YES, NO, I’M NOT SURE

Now, think about why the authors used these examples and your insights into whether these questions are relevant to a third grade math teacher. Take the question about even numbers. What is an even number? Why do we want students to learn about odd and even numbers? Can you think of any example from outside a school setting where it might be useful for a student to know whether an integer is odd or even?

Try to put yourself into the mind of a young student. You know how to count and how to add integers. Both activities seem relevant to your life. Then the teacher starts explaining that there are two kinds of integers—odd and even. Both "odd" and "even" are part of your working vocabulary. Perhaps you laugh as you think about what it might mean for an integer to be labeled as "odd." … Now, back to thinking as a teacher. Are the math words odd and even just words to be memorized and a bunch of drill and practice activities to carry out? Or, do they have deeper math meaning to you? Why would anyone care whether 0 is odd or even?

Suppose that a student asks you the question: ""Why would anyone care whether 0 is odd or even?" How would you respond?

At any grade level, it is possible for students to ask math-related questions that the teacher is unlikely to be able to answer. Here is an example that some of you will find interesting and challenging. Quite likely you have memorized “to divide by a fraction, invert and multiply.” Thus:

(1/2) / (1/4) = (1/2) x (4/1) = 2

Now, discuss and explain the following:

(1/2) / (1/0) = (1/2) x (0/1) = 0

Here are two different approaches used in math teacher education to deal with this situation:

  1. Require prospective math teachers to take a “lot” of math content coursework. Don’t let people teach math who have only a modest level of competence in math content.
  2. Require prospective math teachers to take a “lot” of math methods coursework with a strong emphasis on math pedagogical knowledge. Specifically, teach preservice teachers how to deal with questions that are disruptive to the specific focus of a lesson plan and/or that the teacher cannot answer.

Math Department faculty and Teacher Education faculty have long argued the merits of these two approaches. In the preparation of elementary school teachers, the standard compromise that has been worked out consists of a certain amount of college coursework in each of the two areas listed. The nature and amount of the coursework varies in different teacher education programs throughout the country.

One of the main themes in the Math Maturity document you are currently reading is that both approaches are important, but that both need to place considerably more emphasis on math maturity. The teacher needs a reasonably high level of both math content maturity and math pedagogical knowledge maturity. Determining what constitutes an appropriate balance between these two areas will vary with the teacher and with the students that the teacher is teaching.

Math maturity provides a useful framework for addressing some of the math education challenges. Very roughly speaking, math maturity focuses on the long-term understanding, retention, and ability to make use of the math that one has studied. What lasts as one forgets the finer details of what one has studied? What lasts as rote memories fade?

Math maturity food for thought. In every discipline, teachers will encounter situations in which their content knowledge and/or their content pedagogical knowledge is not up to adequately resolving the situation. This might occur when preparing a lesson, when presenting and facilitating a lesson, when grading papers, when talking with a parent, and so on. What do you do when this occurs for you?

The more mature teachers are able to deal with such situations by saying, “I don’t know. Let’s explore this together.” If the combined resources of the teacher and the students and/or people the teacher is working with are not able to resolve the situation, the teacher’s next response might well be, “Let me do some research on this. I’ll get back to you.” And… the teacher does the needed research and personal learning, and does get back to the students, parents, or others who are involved.

Part 2: What is Math Maturity?

“Math maturity" has such a tantalizing ring to it, but what does it mean? This document offers thoughts toward what it means and how educators can use the concept to better meet their teaching and learning goals. Mathematicians use “mathematical maturity” to mean, loosely, a mixture of mathematical experience and insight that is not taught directly, but which grows and ripens from substantial exposure to complex mathematical concepts and processes.

Math has a High Inherent Level of Abstractness

Much of the power of math lies in its relatively high level of abstractness. Think about a young child learning the number words one, two, three, etc. The child eventually learns that by saying the words and making a one-to-one correspondence with a set of objects, the final number said is the quantity of objects in the set. That is a major math-learning step.

Later the child encounters the symbols 1, 2, 3, etc. (A couple of thousand years ago, many children would have learned the numeral symbols I, II, III, IV, V.) These are shorthand symbols for the words one, two, three and likely they are learned (memorized) before the child encounters and learns the alphabetic representations one, two, three, etc. Do you find it interesting that we have children learn the abstract shorthand representations for the natural language words one, two, three, etc., before we have them learn to read and spell the written forms of these words? That is, very early on in a child’s education, we move toward the abstractness and power of the language of mathematics.

Quoting David Tall (: http://www.tallfamily.co.uk/david/papers/biological-brain-math-mind.pdf):

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 [sic] 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.

One thing implied here is that, as the symbols and the manipulations become more and more abstract, and more difficult for the student to relate to what is known, then the student “learns” with less and less understanding. In many cases, a student would face a daunting task trying to work out a referent that has meaning to the student.

Some Components of Math Maturity

Slightly expanding on and modifying a Wikipedia list, mathematical maturity includes the ability and/or capacity to:

  • Communicate mathematics and math ideas orally and in writing using standard notation, vocabulary, and acceptable style.
  • Learn to learn math; complete the significant shift from learning by memorization to learning through understanding.
  • Generalize from a specific example to broad concept.
  • Transfer one’s math knowledge and skills into math-related areas and problems in disciplines outside of mathematics.
  • Handle abstract ideas without requiring concrete referents.
  • Link a geometrical representation with an analytic representation or vice versa.
  • Manifest mathematical intuition by abandoning naive assumptions and by readily drawing on one’s accumulated subconscious math knowledge and insights.
  • Move back and forth between the visual (e.g., graphs, geometric representations) and the analytical (e.g., equations, functions).
  • Recognize a valid mathematical or logical proof, and detect 'sloppy' thinking. Provide solid evidence (informal and formal arguments and proofs) of the correctness of one’s efforts in solving math problems and making proofs.
  • Recognize mathematical patterns.
  • Separate key ideas from the less significant ideas.
  • Represent (model) verbal and written problem situations as mathematical problems.
  • Effectively draw upon one’s math knowledge and skills to address novel (not previously encountered) math-related problems.

Notice that the list does not contain any specific math content. Rather, it is a list of what mathematicians do. A person might have a quite high level of math maturity and not be skilled at paper and pencil arithmetic. An increasing level of math maturity represents progress in moving in the direction of learning to think about, understand, represent, and solve math problems and math-related problems.

Also note that the list does not mention calculators and computers. Some mathematicians are highly skilled in the use of these tools and, indeed, may use them in their research and math problem solving. The disciplines of Mathematics and of Computer and Information Science strongly overlap. Many other mathematicians have a more modest but still quite personally useful level of knowledge of the overall field of computers and information science and its underlying mathematics.

Larry Denenberg (Ph.D. in applied mathematics, systems analyst, entrepreneur, and business executive) comes at “Mathematical Maturity” from a different angle:

“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!” See http://www.larry.denenberg.com/math22/LectureA.pdf

One can evaluate a math lesson plan or unit of study in terms of how it contributes to a student gaining math maturity. For example, a set of math lesson plans can be designed to help students understand why a times table works as well as aid them in their memorization of it. See http://iae-pedia.org/Good_Math_Lesson_Plans.

Maturity in Different Disciplines

The general notion of "maturity" in a discipline applies to every discipline—indeed to every voluntary adult activity. Earlier in this document it was mentioned that Mozart composed music when he was quite young. However, it is obvious to music critics that his early compositions were quite immature. Similar statements are often made about the work of “young” artists and writers.

One way to describe increasing math maturity is to talk about a person making progress toward “being” a mathematician. That is, increasing math maturity is progress toward learning to think like a mathematician and to function effectively in the culture of mathematicians.

This mathematical thinking is applied both over a wide range of components of the discipline of mathematics and also in areas outside of mathematics. Howard Gardner’s theory of Multiple Intelligences lists Logical/Mathematical as one category of intelligence. Being logical and thinking logically are applicable in many disciplines. Think of law, for example.

However, the reasoning and logical arguments used by most people in most disciplines are different from the precise mathematical logic that is inherent to the discipline of mathematics and to the education of mathematicians. Richard Feynman captures part of this idea in his statement:

"I believe that a scientist looking at nonscientific problems is just as dumb as the next guy." (Richard Feynman; Nobel Prize winning American physicist; 1918–1988.)

Part 3: Math Education and Math Maturity

“Mathematics is the queen of the sciences.” (Carl Friedrich Gauss, German mathematician, physicist, and prodigy; 1777-1855.)
"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner, The Disciplined Mind: What All Students Should Understand, Simon & Schuster, 1999.)
"If we desire to form individuals capable of inventive thought and of helping the society of tomorrow to achieve progress, then it is clear that an education which is an active discovery of reality is superior to one that consists merely in providing the young with ready-made wills to will with and ready-made truths to know with…" (Jean Piaget; Swiss philosopher and natural scientist, well known for his work studying children and his theory of cognitive development; 1896-1980.)

These three quotes help to convey 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. Loosely speaking, we use quality and quantity to describe the universe—and all quantity is mathematics. See the Web document, “What is mathematics?” at http://iae-pedia.org/What_is_Mathematics%3F.

Improving Math Education

As indicated earlier in this document, a math teacher needs maturity in both math content knowledge and in math pedagogical content knowledge. Quite a bit of this comes through on-the-job learning. Most teachers become considerably more effective as they proceed through their first half-dozen years of teaching.

Many people believe that the math education system in the U.S. is far less effective than 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 coursework.

Dissatisfaction persists, with reason. Current efforts to improve our math education system tend to be mostly focused on the same approaches that have not proven very successful in the past. The general opinion seems to be that if we can just do more and do it better with these approaches, our math education system will improve. Perhaps it would be well to recall one aspect of math maturity: That we manifest mathematical intuition by abandoning naive assumptions and developing a more critical attitude. If one is walking but the goal seems always equally far away, one is walking in a circle.

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

  • Substantial and mounting evidence indicates that the math education curriculum in the United States is neither designed nor taught in a manner congruent with what is known about students’ math cognitive development and brain development. Brain science research 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 attack and solve novel, complex, challenging problems; and in their ability to transfer their math knowledge and skills to problems outside a strictly bounded discipline of mathematics. Furthermore, many students—particularly teenagers—regard spending time in a math class as only marginally better than time in a dentist’s chair. Nor do many adults look back fondly on their math classes.
  • 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 educational system with a focus on helping human brains and computer brains to work together at posing and solving problems. We need to prepare students to work effectively in environments where the computer capabilities increase significantly year by year. See “Two Brains are Better than One at” http://iae-pedia.org/Two_Brains_Are_Better_Than_One.
  • 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 either never had or have forgotten.

The nature and extent of forgetting what one has learned (or supposedly learned) varies from discipline to discipline. Think of your math education as consisting of a combination of some big ideas with a large number of smaller (little) ideas. For example, think about the idea of square root versus various paper and pencil methods for calculating square root. The idea of square root is bigger than that of a specific paper and pencil computational procedure. Or, to expand on this example, the idea of an n-th root of a number is bigger than the idea of square root. The idea of an n-th degree polynomial having n solutions is bigger than the idea of n-th root.

In a child’s early grades in school, math education tends to focus on content that may well be of immediate use to the child. Reading numbers, counting, telling time, gaining insight into the measurement of distance, and various other aspects of quantity are all useful to a child.

Eventually, however, the math curriculum draws away from or ahead of the typical child’s immediate needs. For example, relatively few adults find the need to divide one fraction by another. Thus, students are taught a number of math topics that many do not use outside of the math classroom. This is an invitation to forgetting. Moreover, it is one reason that many students do not have the prerequisite knowledge and skills for successful constructivist learning of new topics that assume knowledge of previous topics. This leads many students to a “memorize, regurgitate on a test, and forget” learning style in math courses.

An examination of the various types of math-related problems that ordinary adults are apt to encounter in real life can give us added insight into challenges faced by our math education system. Perhaps you have heard the quote, “Money is the source of all evil.” Certainly, money is the source of a great many problems, and math can be used to help people to address some of these problems.

For example, consider buying on credit. This may be through use of a credit card, in the purchase of a car or other vehicle with monthly payments, or in the purchase of a home with a mortgage. In all cases, the buyer faces the challenge of paying back the loan and its compounded interest. Moreover, the buyer may be faced by a complex combination of changing interest rates and balloon payments.

A great many adults lack the math knowledge and skills to deal with such situations. Sure, they may well have at one time studied interest and how to compute it. They may well have studied compound interest and how to compute it. But such coursework did not lead to a level of math maturity and math understanding that enables them to deal with the real adult life problems of borrowing and repaying.

Similar example can be found in many other math-related problems that adults face, such as:

  • Balancing a checkbook and not writing checks that will bounce.
  • Developing and living within a balanced budget.
  • Doing personal income taxes.
  • Effectively dealing with and having an understanding of probability.
  • Not being mislead by simple statistical arguments, charts, and graphs related to crime, money, income, spending, inflation, and so on.
  • Saving and investing money for the future.
  • Detecting significant math-related errors made by oneself and others.
  • Striving to understand the large numbers involved in discussions about state and federal government budgets and spending, or the numbers involved in talking about the age or size of our universe. For most people, such large numbers are merely words representing quantities beyond comprehension.

In all such example, there are general concepts that one can learn and understand (a type of math-related maturity) and there are details of mathematical manipulation and computation that one does not use very often. This infrequent use tends to create situations in which relearning or learning for the first time is essential. It also creates situations in which it is very easy to make major errors.

Learning Reading & Writing Versus Learning Math

Reading, writing, and arithmetic (math) are considered to be core components of a formal education.

The average person has considerable innate ability to learn oral communication. It is easy to see a clear parallel between oral communication—speaking and listening—and the reading and writing combination.

A young child’s level of skill in oral communication grows through being immersed in an oral communication environment and through years of practice. A child growing up in a linguistically rich home environment will grow in linguistic maturity faster than one growing up in a linguistically impoverished home environment. This increasing linguistic maturity is evidenced by increasing correctness and effectiveness of oral communication—for example, being able to understand and carry out a complex set of instructions or to communicate a complex idea. Note also that if the child’s home environment is bilingual or trilingual, the average child is quite capable of learning to communicate orally in two or three languages.

Our natural languages provide some words that facilitate counting and dealing with quantity. Thus, in societies that make substantial use of numbers, children “naturally” develop some counting and simple arithmetic skills, some “number sense,” and some understanding of the number line. A child’s emerging and growing number sense and understanding of the number line are part of the child’s growing level of math maturity.

“Telling” time using analog and digital clocks provides useful examples. A relatively young child can learn to read and say the numbers that represent a particular time on a digital clock. This is a far cry from understanding what the numbers mean. An analog clock is more challenging in learning to say the numbers that represent the time, but more user-friendly in helping a child understand lengths of time and the passage of time. Through years of practice and steady exposure to time and telling time, most people develop the level of “time maturity” that is expected of adults in our society.

Math maturity food for thought. I wonder what this “time maturity” idea says about people who are habitually late to meetings and appointments? This time-telling topic helps me to understand some of the complexity of math maturity. We know how to help students learn to tell time. But telling time is far more than just learning to say the numbers that represent the time. We are not so good at helping students learn punctuality or to make effective and responsible use of their time.

People often refer to math as a language. (See “Communicating in the Language of Mathematics” at http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics.) Certainly math includes an extensive vocabulary and symbol set for oral and written communication. The language of mathematics facilitates very precise communication within the discipline of math. However, math is much more than that. Math is a discipline of study with a huge accumulation of information and knowledge. Math is representing and solving problems, and developing proofs that make use of the language of math and the math discipline's accumulated math information and knowledge.

Thus, it is easy to see why children vary considerably in the levels of linguistic maturity and math maturity they have achieved before they enter kindergarten or the first grade. Researchers and teachers have made considerable progress in dealing with the challenges of helping a classroom full of young children learn to read. Our schools place a very strong emphasis on this teaching and learning of the reading process. Moreover, there are relatively good measures of a child’s level of reading knowledge and skills.

We can gain added insight into how students attain math maturity by thinking about learning to write in a natural language such as English. We want children to learn to write correct sentences, paragraphs, sections of a document, and complete documents. We expect correct spelling and grammar, correctness of small details, at a much higher level of quality than we accept in everyday spoken language.

Somewhat similar expectations hold for math. We expect a very high level of precision and correctness in communicating in and using math. Moreover, math is a vertically structured discipline. Thus, we expect students to retain the math they have learned at earlier grade levels and to use it effectively in learning and using the math they encounter in their later grade levels.

Here is one way to think about increasing math maturity. Over the years, a student studies math in steadily increasing breadth and depth. Increasing math maturity is a steady increase in mastering the knowledge and skills needed to make effective use of the math one has studied. This includes the ability to solve math-related problems, to learn additional math in a constructivist manner, and to increase one’s level of understanding of and insight into the math that one has studied.

Think about this in terms of the Algebra 1 course that is rapidly moving toward becoming a course required of all students. Teachers of Algebra 1 report that a great many of their students lack the understanding of fractions that is essential in Algebra 1. Certainly all of the students have had extensive instruction about fractions. Many have done well on tests and homework assignments involving fractions—but all too often by the "memorization with little understanding" approach.

Many Algebra 1 students lack other aspects of math maturity that are essential to success in the course. They are not good at solving problems in which they must draw on their accumulated knowledge of math. They are not good at reading math with understanding. This is evidenced in their inability to deal with math word problems and their inability to read math textbooks well enough to learn math by reading the textbook. Other signs of inadequate math maturity include lack of persistence in dealing with challenging problems and poor insight into how one learns math. Finally, a student’s ability to deal with delayed gratification often turns out to be a useful measure of a student’s likelihood of success in an Algebra 1 course.

Part 4: Instant and Delayed Gratification

You have heard about the concepts of instant gratification and delayed gratification. Indeed, you can probably do a good job of self-analysis in these areas by comparing yourself with other people you know. Are you better or worse than your friends in saving money (delayed gratification) instead of building up large credit card debts (instant gratification)?

Math is a very broad and deep discipline. Many quite important math ideas are based on and depend on other math ideas. A math curriculum is designed so that when a student is beginning the study of a new topic, the necessary prerequisites have been covered. One way to think about this is that part of the gratification from studying and learning a math topic is that one is laying the foundations for learning and understanding topics to be studied in the future. This is a type of delayed gratification situation. Students vary considerably in how well they are able to handle working to prepare for such delayed gratification.

Marshmallows and Delayed Gratification

Much of current formal education asks students to delay gratification. This is certainly true in math education. When a student asks, "Why do I need to learn this?" a frequent response is, "You’re going to need it next year." A variation on this is, “You’ll need it later on in life.” A common third reason nowadays is also, "It is going to be on the (standardized) test."

Such responses should ring hollow to the impartial observer. First of all, the question of “Why do I need to learn this” has an unspoken text: “I do not find this emotionally satisfying or interesting enough to learn it for its own sake.” The questioner lacks intrinsic motivation.

Many of the answers provided by parents and teachers are extrinsic motivation answers. “It’s going to be on the test” is cogent only if the student cares about getting good grades or high scores, parental approval, admission into college, etc.

“You’re going to need it next year” invites the obvious question, “Why will I need to learn next year’s math?” The answer in essence will be, “You’ll need it later on in life.” This statement starts becoming untrue, except for a rather small minority of students, about the time the student begins to study algebra. Relatively few people find the need to make use of Algebra 1 in their everyday adult lives.

The Marshmallow Test

This section discusses some very interesting (and amusing) research about delayed gratification. Some details of this research are available in a New Yorker magazine article at http://www.newyorker.com/reporting/2009/05/18/090518fa_fact_lehrer?currentPage=all. View short videos at http://www.youtube.com/watch?v=xNvvL9j_SIs and at http://www.youtube.com/watch?v=6EjJsPylEOY.

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 U.S. children in the original research and 1/3 of the 4-to-6 year-old Colombian children in research on children in that country were able to delay for 15 minutes.

Follow-up research on the U.S. 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.

Quoting from the New Yorker article:

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.

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

[Angela Lee Duckworth, an assistant professor of psychology at the University of Pennsylvania] 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.]

Websites and Instant Gratification

Jacob Nielsen is a world class expert on effective Web design and communication. The following is quoted from Jacob Nielsen's Alertbox, November 24, 2009:

Summary: The granularity of user decisions is much finer on the Web, which is dominated by the instant gratification of the user's needs in any given instant. Content must cater to this rapid pace.
I've been reading The Tale of Genji — the masterpiece of Japanese literature that was written a thousand years ago and is widely considered the world's first novel. I hope I don't offend my Japanese friends by saying that I find the action pretty slow moving. It's a wonderful book, but Lady Murasaki spends many pages to advance the plot even the tiniest bit.
From a historical perspective, it's clear why The Tale of Genji racked up the page count. The ladies of the Japanese court had nothing else to read. After all, this was the only novel in existence in the entire world. It's not as if Lady Murasaki had to compete with Stephen King or Tom Clancy.
Today, of course, we're in the opposite scenario: everything we write competes with trillions of Web pages, all a few clicks away. As a result, most people actually read very few words on the Web.
The velocity of media consumption has increased dramatically. Readers no longer linger over lovingly described passages detailing a lord's style of dress. They click here, they click there, they click everywhere. But they don't stay.
People's consumption of print media is different than their use of websites, leading to the many differences in designing for print versus the Web. [Bold added for emphasis.]

Teaching for Delayed Gratification

The marshmallow research indicates that gratification habits are well established before students begin the first grade. That is not surprising. A child is born with needs, such as the need for food. A child is born with means to help satisfy these needs. Thus, a child who is hungry or uncomfortable cries in a manner that tends to elicit immediate attention from a parent or other caregiver.

If you have raised children, you likely know about issues of demand feeding versus scheduled feeding. Should a child be fed whenever the child “demands” to be fed, or should the child be fed according to a specified time schedule? Suppose that a very young baby is on a demand-feeding schedule. Can this baby adjust to a time-feeding schedule? In most cases, the answer is yes. That is, the instant (on demand) gratification is replaced by the delayed gratification time schedule of feeding.

People who raise young children face an ongoing struggle with a child’s demands for instant gratification. A commonly used caregiver approach includes statements such as, “Eat your vegetables and then you can have desert.” “Pick up your toys and then you can watch television.” “Do your homework and then you can play with your new computer game.”

Researchers on delayed gratification have not yet determined the strengths of nature versus nurture in gratification. They know that delayed gratification habits can be encouraged and taught. Quoting again from the New Yorker article referenced earlier:

He knows that it's not enough just to teach kids mental tricks—the real challenge is turning those tricks into habits, and that requires years of diligent practice. "This is where your parents are important," Mischel says. "Have they established rituals that force you to delay on a daily basis? Do they encourage you to wait? And do they make waiting worthwhile?" According to Mischel, even the most mundane routines of childhood—such as not snacking before dinner, or saving up your allowance, or holding out until Christmas morning—are really sly exercises in cognitive training: we're teaching ourselves how to think so that we can outsmart our desires. But Mischel isn't satisfied with such an informal approach. "We should give marshmallows to every kindergartner," he says. "We should say, 'You see this marshmallow? You don't have to eat it. You can wait. Here's how.'

The KIPP (Knowledge is Power Program) schools are sometimes used as an example of schools teaching delayed gratification. See http://www.kipp.org/. Quoting from the Wikipedia:

Each middle school student receives a paycheck at the end of the week of KIPP dollars they have earned based on academic merit, conduct, and overall behavior. KIPP dollars may be spent on whatever the student chooses, from books to laptop computers. End-of-year trips are also earned. They vary from school to school. [Bold added for emphasis. Notice the delayed gratification aspect of this reward system.]

Part 5: Math Intelligence

This part of the document talks about Math Intelligence and the following section talks about Math Cognitive Development. The basic ideas presented are that math intelligence is a component of overall intelligence and that math cognitive development is a part of overall cognitive development. These ideas provide a framework for working to improve one’s level of math intelligence and maturity in using one’s math intelligence. It also provides a framework for working to improve one’s level of math cognitive development and one's maturity in using one’s level of math cognitive development.

Taken together, these ideas provide one way to approach the field of math maturity. The levels and types of math maturity that a person can achieve depend on the nature and nurture aspects of their math intelligence and the nature and nurture of their math cognitive development.

Background on Innate Human Math Capabilities

People vary in their general intelligence. Moreover, some are far more gifted in mathematics than are others. Thus, one of the points to consider when discussing math maturity is the extent to which math intelligence plays a role in how high up on a math maturity scale a student might go.

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 (http://www.nctm.org/conferences/content.aspx?id=1270) for his opening keynote presentation at the 2004 NCTM Annual Conference.

Devlin’s work helps us to identify two major weaknesses in our current overall 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. These teachers tend to "cover" the math book and its related curriculum without any in-depth understanding. These teachers would rank low on a math maturity scale, and their interest in and enthusiasm for math is, at best, modest. As a consequence of this level of teaching, many young students grow toward math maturity with much less achievement than their innate math potential would make possible.

Research indicates that several-month-old human babies have an innate ability to recognize small quantities, such as noticing that there is a difference between two of something and three of that something. A variety of other animals have a somewhat similar innate sense of quantity. This initial number sense can be viewed as an initial (innate) level of math maturity.

Recent research supports the idea that a human brain also has some innate ability to deal with fractions. See http://www.medicalnewstoday.com/articles/145587.php. 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 development of more effective 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.

In summary, informal and formal math education and math experiences build on an initial innate level of math maturity. The initial innate levels are demonstrated well before any oral language skills are developed.

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 is a well known fact that people vary considerably in the rate and quality of their learning.

Substantial research supports the contention that students of higher IQ learn faster and better than do 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 do the “average” students, and one or two who learn half as fast (and not as well) as do the “average” students.

The “and better” and “not as well” parts of the previous paragraph are important in thinking both about math education in general, and in thinking about math maturity. Increasing levels of math maturity are highly dependent on achieving an increasing level of understanding, fluency, transferability, and so on of the math one has had a chance to learn.

Here is a little history on measuring IQ (quoting from 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).

Notice that 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 will a child who was seldom read to and has had relatively few opportunities to practice conversation. Clearly some of the measurements of an IQ test provide indicators of potential levels of achievement of math maturity.

Multiple Intelligences

The human brain’s complexity and ability to create new neural pathways means that all normal people can learn and can deal with complex, challenging problems. Still, there are significant differences in mental abilities among people who have the same overall intelligence. It’s easy to observe that some people have more linguistic ability, musical ability, or math ability than do other people. It’s easy to conclude that a given healthy brain has significant built-in inherent abilities to learn in some disciplines better than in others.

Examples demonstrate that genetics can endow one person with the brain and hearing system to have perfect pitch and another person with genius-level mathematical-logic sense.

At age three, Carl Friedrich Gauss saw a paper on which his father was doing complicated calculations. He noted an error and worked out the correction mentally, and passed on the information—all within a few seconds.

Mozart composed some relatively good music by the time he was eight years old.

Given that a multitude of such examples exist, one would think that educators would seek to identify these varying characteristics in their students and then individualize programs of study to nurture these talents—whether modest or vast—for the sake of both learners and society in general.

Howard Gardner and Robert Sternberg

The research and writings of Howard Gardner and Robert Sternberg provide us with two models of (approaches to thinking about) multiple intelligences.

Howard Gardner’s theory currently includes nine different areas or types of intelligence:

  1. linguistic,
  2. logic-mathematical,
  3. musical,
  4. spatial,
  5. bodily kinesthetic,
  6. interpersonal,
  7. intrapersonal,
  8. naturalistic, and
  9. existential

One way to think about this is in terms of a bell-shaped curve of intelligence levels for each of the eight areas. A person might be at a different point on each curve for each of the eight areas.

Robert Sternberg’s triarchic theory of intelligence contains three different areas of types of intelligence:

  1. creativity,
  2. analytic (sometimes referred to as school smarts), and
  3. practical (sometimes referred to as street smarts).

From a math maturity point of view, one person might exhibit more math-related creativity than another, and such creativity may be increased by study and practice. In terms of school smarts versus street smarts, Folk Math provides some interesting insights. (See http://iae-pedia.org/Folk_Math.) A person tends to develop an increased level of math maturity that serves the person in their life environment.

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? (See http://wilderdom.com/personality/L4-1IntelligenceNatureVsNurture.html.) 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 “g” (general factor of intelligence) component of one's intelligence and the multiple intelligences also. See http://en.wikipedia.org/wiki/G_factor_(psychometrics.

Studies of nature versus nurture typically make use of identical twins who were separated at birth. Findings vary, with indications of nature determining from about 50 percent to about 80 percent of IQ, depending on the particular 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 understanding this area of research. In practical educational terms, it matters little. To date, humans cannot change the results of the genetic lottery, but humanity certainly can make nurturing a priority.

Part 6: Math Cognitive Development

The field of cognitive development provides another approach to exploring math maturity. As a child’s brain grows and matures, the brain is increasingly able to deal with abstraction in math and in other disciplines.

Piagetian Stage Theory

Jean Piaget is well known for the initial four-level stage theory 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.

Cognitive development often is measured and studied in terms of a stage theory. A 9/20/09 Google search of math OR mathematics "cognitive development" produced about 420,000 hits. As an example, Stages of Math Development (see http://www.cem.msu.edu/%7Eleej/development-math.html) is a very short article that includes a math cognitive development stage theory model for children up to six years old.

The following Piaget reference chart from http://chiron.valdosta.edu/whuitt/col/cogsys/piaget.html provides data on students moving through the stages in his 4-level model.

Cognitive Development.jpeg

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. (Teachers in small high schools can observe that their male students seem to be boys until some time in their senior year. Then “suddenly” they speedily learn what they couldn’t seem to “get” earlier.)

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

  1. The (relative) roles that nature and nurture play in cognitive development.
  2. Whether cognitive development is essentially domain-independent or is better described by a theory of "multiple" cognitive development.

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. IQ tests produce a number that tends to remain stable over time, because of the norming process that is used. This norming process hides a person’s steadily increasing intelligence. A stage theory cognitive development test also produces a number. However, this number tends to increase over time, as a person grows in intelligence and in cognitive development. This growth is not hidden by a norming process.

Michael Commons and Stage Theory

Michael Commons is a world leader in cognitive development stage theory. See http://en.wikipedia.org/wiki/Michael_Commons. His work and the work of others has expanded on and refined Piaget’s work.

Commons and Richards (2002) provide 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.

The following is quoted from email from Michael Commons to David Moursund 5/10/09:

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[s] buy [that is, accept that strong interventions can produce] an increase of 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.

The second quoted part provides Common’s opinion about the extent to which concerted and extensive instruction and practice can raise a person's level on the 15-level scale.

In brief summary, a person moves up in both intelligence and in cognitive development as his or her brain grows and matures, and through informal and formal educational experiences. The levels that a person reaches in intelligence and in cognitive development when growing up in a somewhat “average” or “typical” environment can be somewhat increased by more extensive, demanding, high quality informal and formal education.

Stage Theory in Math

This section provides a five-level Geometry cognitive development scale based on the research of Dina and Pierre van Hiele. It also contains a six-level general math cognitive development scale that David Moursund created for use in his own work.

Geometry Cognitive Development

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.

Overall Math Cognitive Development

The following scale was created (sort of from whole fabric) by David Moursund. It represents his current insights into a six-level, Piagetian-type, math cognitive 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, and 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 (1) developing general ideas and understanding of conservation of number, length, liquid, mass, weight, area, and volume, and (2) learning the mathematics that corresponds to these concepts. 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 taking cognitively demanding higher-level math courses, or not pursuing similar equivalent studies in other courses and/or 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 textbooks 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, in writing research literature, and in oral communication (speaking, listening

) of research-level mathematics. Pose and solve original math problems at the level of contemporary research frontiers.

Part 7: Math Learning Disorders

An intact human brain is very complex, is naturally curious, and has a great capacity to learn. Some people have a reading disorder called dyslexia. Others have a condition that called dyscalculia, which is an arithmetic computation disorder. There are a number of other identified and well-studied significant differences (from the “average”) that a person might have. Moreover, a person can suffer from dyslexia or dyscalculia, and at the same time be talented and gifted.

Math maturity food for thought. The paragraph above contains the words dyslexia and dyscalculia. What are your insights into roles that these two cognitive disabilities may play in learning math and gaining in math maturity? Dyslexia is generally considered to be a reading disorder, but it often also relates to difficulties in learning math. Dyscalculia is a brain disorder that relates to difficulties in learning to do arithmetic calculations.

When you encountered these terms, dyslexia and dyscalculia, was a significant level of understanding of the two terms immediately retrieved from your brain? Or did you immediately use a search engine on the Web (or some other source) to find more information about these two topics?

A key aspect of increasing math maturity consists of the combination of:

  1. Developing the habits of mind as well as the knowledge and skills of trying to figure out and understand new concepts on the basis of what you have previously had a chance to learn.
  2. Developing the reading and information retrieval knowledge and skills, and the habits of mind, to access and make use of the collected knowledge of the human race.

Introduction to Math Learning Disorders

There are a number of specific math-related learning disabilities. Math disabilities tend to be significant barriers to a student's moving up on a math cognitive development or math maturity scale.

Quoting from http://en.wikipedia.org/wiki/Dyscalculia:

Mental disabilities specific to mathematics were originally identified in case studies with patients who suffered specific arithmetic disabilities as a result of damage to specific regions of the brain. More commonly, dyscalculia occurs developmentally, as a genetically linked learning disability which affects a person's ability to understand, remember, or manipulate numbers or number facts (e.g., the multiplication tables). The term is often used to refer specifically to the inability to perform arithmetic operations, but it is also defined by some educational professionals and cognitive psychologists such as Stanislas Dehaene and Brian Butterworth as a more fundamental inability to conceptualize numbers as abstract concepts of comparative quantities (a deficit in "number sense"), which these researchers consider to be a foundational skill, upon which other mathematical abilities build.

Quoting from a ScienceDaily article available at http://www.sciencedaily.com/releases/2005/10/051027082434.htm2005:

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.

Dyscalculia

Here is information quoted from a 2006 National Center for Learning Disabilities article (see http://www.ldonline.org/article/Dyscalculia) 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.

Central Auditory Processing Disorders

Here is information quoted from an article by Jerome Schultz (see http://www.familyeducation.com/experts/advice/0,1183,23-8392,00.html).

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: Math Learning Disability

David Geary is a highly prolific researcher in math cognitive development. Here is an example of his work:

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).

Part 8: 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; German-born theoretical physicist and 1921 Nobel Prize winner; 1879–1955.)
My familiarity with various software programs is part of my intelligence if I have access to those tools. (David Perkins; Professor, Harvard University.)

These two quotes capture the essence of this section of this document. 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 them 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—wonderful aids to one's brain. We have developed such machines as cars, airplanes, and bulldozers. We have developed highly automated manufacturing facilities.

Now, we have Information and Communication Technology (ICT). 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.

Some Brain & Mind Insights of David Tall

David Tall works at the Mathematics Education Research Centre, University of Warwick, UK.

Tall, David (December 2000). Biological Brain, Mathematical Mind & Computational Computers (how the computer can support mathematical thinking and learning). Retrieved 10/2/09: http://www.tallfamily.co.uk/david/papers/biological-brain-math-mind.pdf.

This article contains a number of ideas and/or examples that relate to math maturity. Here is an example quoted from the article. The Square brackets indicate material added by David Moursund.

Consider, a ‘linear relationship’ between two variables. This might be expressed in a variety of ways:
  • an equation in the form y=mx+c,
* a linear relation Ax+By+C=0,
* a line through two given points,
* a line with given slope through a given point,
* a straight-line graph,
* a table of values, etc.

Here is another study by David Tall:

Tall, David (1996). Can all children climb the same curriculum ladder? Retrieved 10/2/09: http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1996c-cur-ladder-gresham.pdf.

Here is the abstract for the article:

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. [Bold added for emphasis.]

The following quote from David Tall's article above suggests a decreasing level of math maturity of high school students:

At the age of 16 the number of children passing their General Certificate of Secondary Education at grades A, B, C is currently increasing year by year (although there is a worrying trend that the number with the lowest grades is remaining stubbornly stable). At age 18 there is an improving spectrum of passes at A-level, although mathematics is becoming a less popular subject. Despite an apparent trend for top students to get better marks in school examinations, university lecturers claim that the students arriving at university lack basic skills. In particular:
(i) They lack fluency in arithmetic and algebraic skills,
(ii) They are less able to solve problems involving several steps,
(iii) They do not perceive the need for absolute precision and proof in mathematics.
The contrast between the apparent success as seen from a school perspective yet failure from a university perspective has led to unseemly accusations flying in all directions. My own perception of the phenomenon is that the two viewpoints are focusing on different things and arguing at cross-purposes. To be able to unravel the conundrum, we need get an insight into what is happening when individuals learn mathematics and begin to “think mathematically”. By doing so it is hoped that some light can be shed on the situation. [Bold added for emphasis.]

The term “think mathematically” expresses a number of the ideas in math maturity. However, the term math maturity does not appear in Tall’s article.

Successful students [that is, more mathematically mature students] develop the idea of ‘linear relationship’ as a rich cognitive unit encompassing most of these links as a single entity.
Less successful [that is, less mathematically mature students] carry around a ‘cognitive kit-bag’ of isolated tricks to carry out specific algorithms. Short-term success perhaps, [but increased] long-term cognitive load and failure.

Two Brains (Human and Computer)

Previous sections of this document have examined math maturity in terms of the nature and nurture aspects of:

  1. One’s innate and developed math intelligence—viewed as a component of one’s overall innate and developed intelligence.
  2. One’s innate and developed math cognitive development—viewed as a component of one’s overall innate and developed cognitive development.
  3. Having some math knowledge and skills in which one can demonstrate a variety of the topics/components that together help to define math maturity.

This document views math maturity as something that is difficult to quantify but that can be increased over time. A unifying goal in math education is to help students move toward higher levels of math maturity.

The discipline of mathematics has long accepted the need for and value of various types of technology such as clay tablet and stylus, abacus, paper and pencil, slide rule, straight edge and ruler, compass and protractor, books and math tables, and so on. It has long accepted the usefulness of chalk boards (and different colors of chalk), overhead projectors, physical models of various geometric figures, dice and spinners, math manipulatives, and so on. Moreover, the discipline of math has long accepted the idea of accumulated mathematical knowledge through a combination of very precise definitions, communication, and proofs.

Thus, math maturity needs to be discussed and measured in the contemporary world—a person’s level of math maturity is maturity within a context or environment of many generally accepted contemporary assumptions about communicating math, knowing and doing math, and learning math.

The advent of electronic calculators and computers has proven to be a major challenge to math education content, pedagogy, and assessment. What content should a student be learning in math, how should it be taught, and how should students (and teachers) be assessed? Computers bring new dimensions, and often reasonably rapidly changing dimensions, to such questions.

Six-Function Calculator Maturity

Think about an inexpensive 6-function solar-powered handheld calculator. What might we mean by saying that a student has a particular level of calculator maturity or that one student has a higher level of calculator maturity than another?

Math maturity food for thought. Should our math education system even care about this sort of question? Does it make any sense to talk about a level of maturity in using various types of tools? Thus, for example, does it make sense to talk about the level of maturity a person has as a car driver? (Automobile insurance companies are certainly concerned about this.)

Most people find little difficulty in learning to use a 6-function calculator at a level that they find personally useful. Here are a few questions that you might use to help judge your own level of calculator knowledge, skills, and maturity.

  1. You “know” that it is impossible to divide by zero. But, you probably also know that it does not destroy (burn out) a calculator when you have it divide by zero. Why?
  2. You “know” that (1/3) x 3 = 1. What do you suppose the answer is when you do the computation using the 6-function 8-digit calculator? Why?
  3. What do you suppose will happen if you use your inexpensive 8-digital calculator to do the multiplication: 18250000 x 3947000? How might you go about using your calculator as an aid to doing this calculation?
  4. The 6-function calculator has memory keys—they may be labeled MR, M-, M+, and MC. Do you know how these calculator features work well enough to explain them? Can you give real-life examples of where and when the memory key features of a calculator are useful?
  5. The 6-function calculator has a square root key. How can a calculator, all by itself, carry out such a complicated computational process through just a single key press on your part?
  6. How can you tell if your calculator‘s circuitry has become damaged, leading to it making errors?
  7. How can you detect keyboarding errors that you happen to make when using a calculator?
  8. How can you (or your students) tell when a calculation might best be done mentally, using paper and pencil, using a calculator, using a computer, or some combination of these four approaches?

Questions like these illustrate the difficulty of separating math content knowledge from math maturity. For each of the questions, it is possible to provide specific instruction on answers to the question. For example, consider the third question. A student can be taught that the calculator can be used to do the computation 1825 x 3947, and that the answer to the actual computational problem is produced by adding seven zeros to the end of the calculator produced answer. Students who figure this out on their own are demonstrating greater math and calculator maturity in this specific area than those who cannot figure it out on their own.

Consider question 4. The more mature math and calculator-adept student is apt to learn about calculator memory by independent investigation involving systematic trial and error. A great many adults that own a calculator with memory have never learned to use the memory capabilities and/or have forgotten how to use this calculator capability.

Part of an answer to question 7 comes from making mental estimations of an approximate answer to a calculation. Another part of an answer lies in situations when the calculation is for a “real world” problem in which one can have an understanding of when an answer makes sense or does not make sense. To do this, you need to draw on your knowledge and understanding of the “real world.” If you are trying to decide the relative weights of an average elephant and an average fly, you should conclude you have made an error (in keyboarding, or in logic) if your calculator tells you that the average elephant weighs about 145 times as much as the average fly.

Question 8 often leads to arguments among the various stakeholders in math education. A top-down approach to developing math curriculum and standards tends to ignore the individual differences and interests of students. Students vary considerably in how easily they learn various aspects of computational arithmetic and how well they retain this knowledge and skill. A “one size fits all” approach to mental, paper and pencil, and calculator-assisted or computer-assisted computation does a disservice to a great many (perhaps all) students.

Computer Maturity

Take another look at the quotation from David Perkins given earlier:

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

Perkins is saying that, for him, a computer is an intelligence booster. Obviously, merely giving a student a computer does not automatically boost the student’s intelligence or IQ. The boost in intelligence comes from a combination of things such as:

  1. Learning how to use a computer, and learning some of its capabilities and limitations. Developing a level of computer knowledge and skills that is personally useful.
  2. Integrating one’s computer knowledge with one’s other knowledge and skills. Think of this as a two-way constructivist process. One constructs the new computer knowledge and skills on top of and integrated into one’s current knowledge and skills. One also reconstructs his or her current knowledge and skills into and on top of the newly acquired computer knowledge and skills. All new learning follows this two-way interaction between old and new.
  3. Transferring one’s maturity in previously studied areas into the new area. As one’s level of computer maturity grows, one transfers this into (uses this to augment) one’s maturity in non-computer areas.

An important aspect of increasing maturity in any discipline is one’s increasing ability to do the two-way constructivist transfer of learning and the two-way transfer of and use of maturity as new learning occurs. A different way of looking at this is to consider learning maturity. An indication of increasing maturity as a learner is an increasing ability to do the transfer of learning and transfer of maturity just described.


Part 9: Education for Increasing Math Maturity

This Math Maturity document contains a number of ideas about what constitutes an increasing level of math maturity, and it presents some ideas about what teachers can be doing to help students increase their levels of math maturity. However, you will note:

  1. There are no references to or good examples of math maturity assessment instruments.
  2. There are no extensive sets of materials (lesson plans, examples for use at various grade levels, etc.) for use by teachers interested in integrating more math maturity content materials into their math teaching.

Math Education Reform

However, that is far too bleak a picture. The math reform movement has a strong emphasis on improving math maturity levels of students. It just doesn’t make much use of the term math maturity.

Here is an example. Earlier in this document there is a list that helps to define math maturity. One of the listed items is “complete the significant shift from learning by memorization to learning through understanding.” Thus, one way to think about the math education wars is that they pit people who support “back to basics, rote memory learning of math” against people who support “learning math in a manner that promotes growth in math maturity.” (See http://iae-pedia.org/Math_Education_Wars.) Of course, this is a gross over-simplification. However, it does suggest that the National Council of Teachers of Mathematics (NCTM)is supportive of teaching for increased math maturity.

For another example, consider the issue of assessment. Math teachers often do a “math maturity assessment” of their students. They do this through:

  • One-on-one conversation with students. This can be an inherent part of the interaction whenever a student asks for individual help.
  • Observing the breadth and depth of answers a student (or, the whole class) gives during class discussions.
  • Analysis of a student's homework and test answers.
  • Reading student journals, if the teacher has students doing math journaling.

Two Math Maturity Assessment Issues

Here are two issues to think about:

  1. Assessment of a student’s level of math maturity, with this being done by some combination of self-assessment and assessment by the teacher or some assessment instrument.
  2. Assessment of a teacher’s level of math content maturing and math pedagogical knowledge maturity, with this being done by some combination of self-assessment and assessment by other means.

Each of these is an area needing a substantial amount of research and development. However, a good start can be made without such an investment. Such a start can be made by:

  1. Helping teachers to develop a personal, professionally useful level of knowledge and understanding about math maturity. This can be occurring in every preservice math course and math methods course designed for preservice and inservice teachers.
  2. Having teachers explicitly introduce the idea of math maturity to their students and then helping their students learn to self-assess. There is quite a lot of research on students learning to do self-assessment and making use of self-assessment. A student can learn to ask and to think about questions such as: “Do I understand what I am doing and why I am doing it?" “Is my understanding just rote memorization, or do I ‘really understand’ what I am doing?” “What can I do to demonstrate to others that I understand and know what I am doing?”

Part 10: Summary and Final Remarks

Provided a student is not too severely physically or emotionally limited, the student’s math maturity steadily increases over time through:

  1. General overall increases in cognitive development through his or her brain growing and maturing.
  2. Learning math in a manner that facilitates higher-order creative and critical thinking, problem solving, theorem proving, communicating in and about math, and learning to learn math.
  3. Working with math teachers, folk mathematicians, and others who have a higher level of math maturity than the student, 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.

Math maturity can be increased through good teaching, good instructional materials, and by the active collaboration of learners. Student involvement is an essential component of such math maturity improvement activities. As students gain insights into what constitutes an increasing level of math maturity, they can self assess, reflect on the math they use and do both in school and outside of school, do meta cognition about their math insights, and so on. All of these activities can help a student to gain in math maturity

Challenging Students Mathematically

One complaint about our current math education system is that it is not sufficiently cognitively challenging.

Yeung, Bernice (October 2009). Arithmetic Underachievers Overcome Frustration to Succeed. Math test scores soar if students are given the chance to struggle. Edutopia: The George Lucas Foundation. Retrieved 9/18/09: http://www.edutopia.org/math-underachieving-mathnext-rutgers-newark. Quoting from the article:

New Jersey teachers have found a surprising way to keep students engaged and successful: They let underachieving youngsters get frustrated by math.
While working with minority and low-income students at low-performing schools in Newark for the past seven years, researchers at Rutgers University have found that allowing students to struggle with challenging math problems can lead to dramatically improved achievement and test scores.
"We've found there is a healthy amount of frustration that's productive; there is a satisfaction after having struggled with it," says Roberta Schorr, associate professor in Rutgers University at Newark's Urban Education Department. Her group has also found that, though conventional wisdom says certain abilities are innate, a lot of kids' talents and abilities go unnoticed unless they are effectively challenged; the key is to do it in a nurturing environment.
"Most of the literature describes student engagement and motivation as having to do with their attitudes about math -- whether they like it or not," Schorr says. "That's different from the engagement we've found. When students are working on conceptually complex problems in a supportive environment, they do better. They report feeling frustrated, but also satisfaction, pride and a willingness to work harder next time."

Part 11: References and Resources

Ball, D.L., Hill, H.C., & Bass, H. (Fall 2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator. Retrieved 7/13/2010 from http://www.aft.org/pdfs/americaneducator/fall2005/BallF05.pdf.

Bentsen, T. (11 Apr 2009). Adult Brain Processes Fractions 'Effortlessly.' Medical News Today. Retrieved 9/19/09: http://www.medicalnewstoday.com/articles/145587.php.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 issue of The Journal of Neuroscience.
"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.

Commons, M.L., & 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 paper presents details on a 15-stage Piagetian-type cognitive developmental scale. Quoting the Abstract:

This paper investigates the nature of transition between stages. The Model of Hierarchical Complexity of tasks leads to a quantal notion of stage, and therefore delineates the nature of stage transition. Piaget’s dialectical model of stage change was extended and precisely specified. Transition behavior was shown to consist of alternations in previous-stage behavior. As transition proceeded, the alternations increased in rate until the previous stage behaviors were “smashed” together. Once the smashed-together pieces became co-ordinated, new-stage behavior could be said to have formed. Because stage transition is quantal, individuals can only change performance by whole stage. We reviewed theories of the specific means by which new-stage behavior may be acquired and the emotions and personalities associated with steps in transition. Examples of transitional performances were. Contemporary challenges in the society increasingly call for transition to post-formal and post-conventional responses on the part of both individuals and institutions as the example illustrate.

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.

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.

Crace, J. (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."

Devlin, K. (2000). The Math Gene. Basic Books. Learn more about Devlin at http://www.stanford.edu/~kdevlin/.

Dewar, G . (2008). In search of the smart preschool board game: What studies reveal about the link between games and math skills. Parenting Science. Retrieved 10/5/09: http://www.parentingscience.com/preschool-board-game-math.html. This article reports on various research projects done using board games and young children. Quoting from the article:

There is compelling evidence that certain kinds of board games boost preschool math skills. And these early skills are strongly predictive of math achievement scores later in life (Duncan et al 2008).
For instance, consider the research of Geetha Ramani and Robert Siegler (2008).
Ramani and Siegler asked preschoolers (average age: 4 years, 9 months) to name all the board games they had ever played.
The more board games that a kid named, the better his performance in four areas:
* Numeral identification
* Counting
* Number line estimation (in which a child is asked to mark the location of a number on a line)
* Numerical magnitude comparison (in which a child is asked to choose the greater of two numbers)

The four bulleted items are all important aspects of a combination of growing math knowledge and skills, and growing math maturity.

Geary, D.C. (2007). An Evolutionary Perspective on Learning Disability in Mathematics. Developmental Neuropsychology. Retrieved 9/18/09: http://web.missouri.edu/~gearyd/articles_math.htm. 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.

David C.G., Hoard, M.K., Byrd-Craven, J., Nugent, L., & Numtee, C. 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.

Glbblguy (n.d.). 1 marshmallow or 2? A study on the benefits of delayed gratification. Retrieved 9/8/09: http://www.gatherlittlebylittle.com/2008/03/1-marshmallow-or-2-a-study-on-the-benefits-of-delayed-gratification/.

This blog entry contains one person’s insights into the gratification ideas in the marshmallow research. It makes the observation that a great many of the people who have become millionaires exhibited the ability of working toward gratification that would occur far in the future. The article contains the following example quoted from Money Savvy Generation: http://www.gatherlittlebylittle.com/2008/03/1-marshmallow-or-2-a-study-on-the-benefits-of-delayed-gratification/

At age 12 you decide not to buy soda or extra snacks – either during the school week or on weekends or vacations. You save $4.00 a day. You put $4 a day in a savings vehicle such as a long-term IRA CD at five percent annual interest and leave it alone. At age 67, your savings is:
(a) $1,159
(b) $25,355
(c) $80,352
(d) $427,025
Answer: (d), or $427,025. Note that $80,352 is from the daily deposits and the remaining $346,673 is interest! [Comment by David Moursund: This is an example of very very long term delayed gratification.]

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.

This article provides an introduction to Piaget’s 4-stage theory of cognitive development and the role this theory provides in constructivist learning. In listing the four stages, Huitt indicates:

4. 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.]

Lehrer, J. (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.

This article presents a nice introduction to and overview of gratification and the marshmallow test research.

Louisiana Content Standards (n.d). Louisiana Content Standards for Programs Serving Four-Year Old Children. Retrieved 9/8/09: http://doa.louisiana.gov/osr/lac/28v77/28v77.doc.

This document includes a 4-stage Piagetian-type math developmental scale for preschool age children.

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

Moursund, D. (2006b). Computational thinking and math maturity: Improving math education in K-8 schools. Eugene, OR: Information Age Education. A free book available at http://i-a-e.org/downloads/doc_download/3-computational-thinking-and-math-maturity-improving-math-education-in-k-8-schools.html.

Stein, R. (2010). History in a Math Course for Teachers. Retrieved 1/9/2012 from http://www.hpm-americas.org/wp-content/uploads/2010/04/HistoryinaMathCourseforTeachers.pdf. Quoting from the introduction to the paper:

When I teach math courses for teachers, I try to put history into the course for the usual reasons:
  • To make mathematics more “human”
  • To make mathematics more understandable
  • To give teachers ideas and materials that can help them teach math more effectively.
However, achieving these goals within the context of a math course for teachers turns out to be :: an interesting challenge. This paper describes some of my efforts to meet that challenge.

Tall, D. (1996). Can all children climb the same curriculum ladder? Retrieved 10/2/09: http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1996c-cur-ladder-gresham.pdf. Here is the abstract for the article:

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. [Bold added for emphasis.]

Tall, D. (December 2000). Biological Brain, Mathematical Mind & Computational Computers (how the computer can support mathematical thinking and learning). Retrieved 9/18/09: http://www.tallfamily.co.uk/david/papers/biological-brain-math-mind.pdf.

This article contains a number of ideas and/or examples that relate to math maturity. Here is an example quoted from the article. The Square brackets indicate material added by David Moursund.

Consider, a ‘linear relationship’ between two variables.
Successful students [that is, more mathematically mature students] develop the idea of ‘linear relationship’ as a rich cognitive unit encompassing most of these links as a single entity.
Less successful [that is, less mathematically mature students] carry around a ‘cognitive kit-bag’ of isolated tricks to carry out specific algorithms. Short-term success perhaps, long-term cognitive load and failure.

Yeung, B. (October 2009). Arithmetic Underachievers Overcome Frustration to Succeed. Math test scores soar if students are given the chance to struggle. Edutopia: The George Lucas Foundation. Retrieved 9/18/09: http://www.edutopia.org/math-underachieving-mathnext-rutgers-newark. Quoting from the article:

New Jersey teachers have found a surprising way to keep students engaged and successful: They let underachieving youngsters get frustrated by math.
While working with minority and low-income students at low-performing schools in Newark for the past seven years, researchers at Rutgers University have found that allowing students to struggle with challenging math problems can lead to dramatically improved achievement and test scores.
"We've found there is a healthy amount of frustration that's productive; there is a satisfaction after having struggled with it," says Roberta Schorr, associate professor in Rutgers University at Newark's Urban Education Department. Her group has also found that, though conventional wisdom says certain abilities are innate, a lot of kids' talents and abilities go unnoticed unless they are effectively challenged; the key is to do it in a nurturing environment.
"Most of the literature describes student engagement and motivation as having to do with their attitudes about math -- whether they like it or not," Schorr says. "That's different from the engagement we've found. When students are working on conceptually complex problems in a supportive environment, they do better. They report feeling frustrated, but also satisfaction, pride and a willingness to work harder next time."

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Author or Authors

The initial version of this document was developed by David Moursund. Editing, along with providing many very good suggestions, was done by Dick Ricketts.