A Tidbit of HistoryMathematics as a formal area of teaching and learning was developed about 5,000 years ago by the Sumerians. They did this at the same time as they developed reading and writing. However, the roots of mathematics go back much more than 5,000 years.

Throughout their history, humans have faced the need to measure and communicate about time, quantity, and distance. The Ishango Bone (see ahttp://www.math.buffalo.edu/mad/

Ancient-Africa/ishango.html andhttp://www.naturalsciences.be/expo/ishango/

en/ishango/riddle.html) is a bone tool handle approximately 20,000 years old.

Throughout their history, humans have faced the need to measure and communicate about time, quantity, and distance. The Ishango Bone (see ahttp://www.math.buffalo.edu/mad/

Ancient-Africa/ishango.html andhttp://www.naturalsciences.be/expo/ishango/

en/ishango/riddle.html) is a bone tool handle approximately 20,000 years old.

The picture given below shows Sumerian clay tokens whose use began about 11,000 years ago (seehttp://www.sumerian.org/tokens.htm). Such clay tokens were a predecessor to reading, writing, and mathematics.

The development of reading, writing, and formal mathematics 5,000 years ago allowed the codification of math knowledge, formal instruction in mathematics, and began a steady accumulation of mathematical knowledge.

Mathematics as a DisciplineA discipline (a organized, formal field of study) such as mathematics tends to be defined by the types of problems it addresses, the methods it uses to address these problems, and the results it has achieved. One way to organize this set of information is to divide it into the following three categories (of course, they overlap each other):

Even within the third component, it is not clear what should be emphasized in curriculum, instruction, and assessment. The issue of basic skills versus higher-order skills is particularly important in math education. How much of the math education time should be spent in helping students gain a high level of accuracy and automaticity in basic computational and procedural skills? How much time should be spent on higher-order skills such as problem posing, problem representation, solving complex problems, and transferring math knowledge and skills to problems in non-math disciplines?

Beauty in MathematicsRelatively few K-12 teachers study enough mathematics so that they understand and appreciate the breadth, depth, complexity, and

G. H. Hardy was one of the world's leading mathematicians in the first half of the 20th century. In his book "A Mathematician's Apology" he elaborates at length on differences between pure and applied mathematics. He discusses two examples of (beautiful) pure math problems. These are problems that some middle school and high school students might well solve, but are quite different than the types of mathematics addressed in our current K-12 curriculum. Both of these problems were solved more than 2,000 years ago and are representative of what mathematicians do.

Mathematics as a DisciplineA discipline (a organized, formal field of study) such as mathematics tends to be defined by the types of problems it addresses, the methods it uses to address these problems, and the results it has achieved. One way to organize this set of information is to divide it into the following three categories (of course, they overlap each other):

- Mathematics as a human endeavor. For example, consider the math of measurement of time such as years, seasons, months, weeks, days, and so on. Or, consider the measurement of distance, and the different systems of distance measurement that developed throughout the world. Or, think about math in art, dance, and music. There is a rich history of human development of mathematics and mathematical uses in our modern society.
- Mathematics as a discipline. You are familiar with lots of academic disciplines such as archeology, biology, chemistry, economics, history, psychology, sociology, and so on. Mathematics is a broad and deep discipline that is continuing to grow in breadth and depth. Nowadays, a Ph.D. research dissertation in mathematics is typically narrowly focused on definitions, theorems, and proofs related to a single problem in a narrow subfield in mathematics.
- Mathematics as an interdisciplinary language and tool. Like reading and writing, math is an important component of learning and "doing" (using one's knowledge) in each academic discipline. Mathematics is such a useful language and tool that it is considered one of the "basics" in our formal educational system.

Even within the third component, it is not clear what should be emphasized in curriculum, instruction, and assessment. The issue of basic skills versus higher-order skills is particularly important in math education. How much of the math education time should be spent in helping students gain a high level of accuracy and automaticity in basic computational and procedural skills? How much time should be spent on higher-order skills such as problem posing, problem representation, solving complex problems, and transferring math knowledge and skills to problems in non-math disciplines?

Beauty in MathematicsRelatively few K-12 teachers study enough mathematics so that they understand and appreciate the breadth, depth, complexity, and

**beauty**of the discipline. Mathematicians often talk about the beauty of a particular proof or mathematical result. Do you remember any of your K-12 math teachers ever talking about the beauty of mathematics?G. H. Hardy was one of the world's leading mathematicians in the first half of the 20th century. In his book "A Mathematician's Apology" he elaborates at length on differences between pure and applied mathematics. He discusses two examples of (beautiful) pure math problems. These are problems that some middle school and high school students might well solve, but are quite different than the types of mathematics addressed in our current K-12 curriculum. Both of these problems were solved more than 2,000 years ago and are representative of what mathematicians do.

- A rational number is one that can be expressed as a fraction of two integers. Prove that the square root of 2 is not a rational number. Note that the square root of 2 arises in a natural manner as one uses land-surveying and carpentering techniques.

- A prime number is a positive integer greater than 1 whose only positive integer divisors are itself and 1. Prove that there are an infinite number of prime numbers. In recent years, very large prime numbers have emerged as being quite useful in encryption of electronic messages.

The following diagram can be used to discuss representing and solving applied math problems at the K-12 level. This diagram is especially useful in discussions of the current K-12 mathematics curriculum.

The six steps illustrated are 1) Problem posing; 2) Mathematical modeling; 3) Using a computational or algorithmic procedure to solve a computational or algorithmic math problem; 4) Mathematical "unmodeling"; 5) Thinking about the results to see if the Clearly-defined Problem has been solved,; and 6) Thinking about whether the original Problem Situation has been resolved. Steps 5 and 6 also involve thinking about related problems and problem situations that one might want to address or that are created by the process or attempting to solve the original Clearly-defined Problem or resolve the original Problem Situation. Click here for more information about problem solving.

Final RemarksHere are four very important points that emerge from consideration of the diagram in Figure 3 and earlier material presented in this section:

Final RemarksHere are four very important points that emerge from consideration of the diagram in Figure 3 and earlier material presented in this section:

- Mathematics is an aid to representing and attempting to resolve problem situations in all disciplines. It is an interdisciplinary tool and language.
- Computers and calculators are exceedingly fast, accurate, and capable at doing Step 3.
- Our current K-12 math curriculum spends the majority of its time teaching students to do Step 3 using the mental and physical tools (such as pencil and paper) that have been used for hundreds of year. We can think of this as teaching students to compete with machines, rather than to work with machines.
- Our current mathematics education system at the PreK-12 levels is unbalanced between lower-order knowledge and skills (with way to much emphasis on Step #3 in the diagram) and higher-order knowledge and skills (all of the other steps in the diagram). It is weak in mathematics as a human endeavor and as a discipline of study.

- Brain Science, which is being greatly aided by brain scanning equipment and computer mapping and modeling of brain activities, is adding significantly to our understanding of how the brain learns math and uses its mathematical knowledge and skills.
- Computer and Information Technology is providing powerful aids to many different research areas (such as Brain Science), to the teaching of math (for example, through the use of highly Interactive Intelligent Computer-Assisted Learning, perhaps delivered over the Internet), to the content of math (for example, Computational Mathematics), and to representing and automating the "procedures" part of doing math.
- The steady growth of the totality of mathematical knowledge and its applications to representing and helping to solving problems in all academic disciplines.