This is the second of a two-part newsletter addressing the question, What is mathematics?. The first newsletter covered the topics:
“The laws of nature are written in the language of mathematics.” (Galileo Galilei; Italian astronomer, physicist, and engineer; 1564-1442.)
Many people consider math to be a language. (Palisco, 12/5/2014, link to 8:54 TEDx video.) It is not a general purpose language, such as English or Spanish. Rather, it is a discipline-specific language. Each discipline has developed specialized vocabulary and its own ways of communication that are specific to the discipline. Consider, for example, music notation and a person learning to read music or to write music.
The written and spoken language of mathematics makes use of a very extensive collection of vocabulary and symbols. The Concise Oxford Dictionary of Mathematics contains more than 3,000 entries.
You are used to the idea that, in a language such as English, the words are spelled out as a sequence of letters. Sentences make use of various other symbols that we call punctuation marks.
Our ordinary, everyday language makes use of many mathematical words and terms. In our base 10 number system, we have the symbols 0, 1, 2, … 9. One can think of these digits as concise mathematical abbreviations for the words zero, one, two, … nine. This mathematical notation has a very interesting characteristic. The written notation or representation of a number (that is, a written display of its digits) defines the number. This is somewhat overly simplified. We also make use of the decimal point in writing some numbers, and we use a comma to make it easier to read and understand multidigit numbers. Contrast this with the alphabetical spelling of other words in a language such as English.
Here are some additional aspects of the subject and language of mathematics.
The disciplines of business data processing, electrical engineering, and mathematics were all well-established before the first electronic digital computers were developed. The majority of the people involved in this development of the first computers had strong backgrounds in engineering and mathematics. In higher education, during the early days of such computers, the use of this technology blossomed in Business, Engineering, and Mathematics departments. In a number of institutions, each of these three areas showed an interest in creating an independent Computer Science Department. My own institution, the University of Oregon, did not have an Engineering School. The Mathematics Department fostered the creation of the university’s Computer Science Department in 1969. (Full disclosure: I was a Mathematics faculty member at that time, and I became the first Chair of the new department.)
Math is such an important discipline of study that it is a substantial part of the required education of children throughout the world. The world’s educational systems have had thousands of years of experience in deciding what math to teach, when and how to teach it, and how to assess the results. During all of this time, the discipline of and applications of math have been growing. Our knowledge about the human brain, teaching theory, and learning theory also have been growing. Moreover, substantial research and development has been done in developing aids to using (doing) math.
Visual math is an example of a math education theory. (Maier, 2003, link.) Sometimes also called math in the mind’s eye, the theory is that it is very helpful in math education and math sense-making to create and use visual representations of math content and its applications. Some people argue that all thinking is visual. A number of interesting math examples are given in the Maier article. For many years I have been on the Board of Directors of the non-profit company the Math Learning Center (MLC). Much of the curriculum they have developed for Pre-5 math education is based on visual thinking (MLC, 2020, link.)
Today, the content, pedagogy, and assessment of math are all moving targets. What should students be learning in their required math classes? What should students be learning about applications of math in each of the other disciplines they study in school? As computers and computer connectivity become better, and also become more readily available to students and to all others who have a need for such technology, what can and should math education and the rest of education do to benefit from these changes?
I do not have simple answers to such questions. However, I have written extensively about the topic for precollege math educators. (Moursund, 2020, link.)
Math modeling is a process of developing a mathematical representation of some (or all) aspects of a particular type of problem.
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science). (Wikipedia, 2020, link.)
First, consider a simple example. If A and B are both the same type of things or objects, then A + B is a mathematical model for their sum. Suppose Suzy has 7 apples and Tommy has 5 apples, and they want to know how many apples they have together. The math model says an answer is 7 + 5, which is 12.
If a calculator or computer is doing the calculation, it has no knowledge or understanding that it is adding apples to apples. It is merely adding the numbers 7 and 5. But, a human can understand what it means to add apples to apples. The math model is a combination of human understanding of the problem, together with capabilities of a machine that lacks this human understanding but can perform the necessary calculations. With this model, the human and machine can work together to (1) perform the calculations prescribed by the model, and (2) interpret and take actions based on the calculations.
This is a very important idea in math education. In traditional math education, considerable time is spent on rote memorization of math facts and procedures, with goals of accuracy and speed. Paper-and-pencil calculations of multidigit numbers provide a good example. Many students learn to do these in a machine-like manner, losing sight of the underlying possible meanings of the original problem and/or the results. Students are not developing number sense and math sense; rather they are learning to do something that calculators and computers can do both faster and more accurately.
Even this simple example about apples can be used to illustrate some difficulties and challenges in math modeling. Suppose that Suzy has 7 apples and Tommy has 5 oranges. Are apples and oranges the same type of thing or object? Hmm. What does it mean to add apples and oranges?
Aha! They are both types of fruit. So, the actual problem we are now looking at is that Suzy has 7 pieces of fruit and Tommy has 5 pieces of fruit, and we want to know how much fruit the two have together. The model works okay if that is the problem we want to solve.
But, suppose that Suzy and Tommy are trying to figure out how to divide the fruit to provide equal servings of fruit to each of four people. Do we want each of the four to receive the same weight of fruit, or do we perhaps want each to receive equal volumes of the two different fruits? This is getting more and more complex! Our difficulty is that we have not defined the problem carefully enough. The challenge is to carefully define (clearly state) the problem we want to solve.
Here is a quote from Albert Einstein that is to this discussion:
“If I had an hour to solve a problem and my life depended on the solution, I would spend the first 55 minutes determining the proper question to ask, for once I know the proper question, I could solve the problem in less than five minutes.” (Albert Einstein; German-born theoretical physicist and 1921 Nobel Prize winner; 1879–1955.)
In summary, math educators want students to increase both their number sense and math sense. They also want students to increase their knowledge and skills in posing problems and in solving problems. Appropriate student use of calculators and computers can lead to a decrease in time spend on memorization, speed, and accuracy of paper-and-pencil computational skills. This can facilitate an increase in the amount of learning and practice time that students will have available for learning to pose and solve problems, and to develop number sense and other math sense.
For a far more complex example of math modeling, consider weather forecasting. A number of different groups of people have worked for years to develop models of weather that can be used in making weather forecasts. This is a hugely difficult problem. As the underlying science, data gathering, speed of computers, and the math model have improved, weather forecasting has steadily improved. The history of these efforts is both amusing and quite enlightening. Early attempts produced forecasts with very inaccurate accuracy and took days to produce a forecast of the next day’s weather. (Wikipedia. 2020, link.)
I believe that each discipline has its own humor targeted specifically to practitioners in the discipline. The humor helps to define the discipline. That certainly is the case for math.
I am reminded of a statement I have read about “A man who jumped on his horse and rode off in all directions.” How is this possible? A mathematical answer is that the man rode in a circle or ellipse.
Here is a better example, first published by Wade Clarke in 2005. It certainly tickled my funny bone. (Math Warehouse, 2020, link.) A teacher wants her students to demonstrate that they know and can use the formula relating the lengths of the sides of a right triangle. She asks them to “Find X” in the diagram given in Figure 1. One student’s response was to circle the X in the diagram!
My personal collection of math humor is available in the IAE-pedia. (Moursund, 2020, link.)
Mathematics is a very old, broad, deep, and vibrant discipline of study. It is a routine part of our everyday lives and a global endeavor. It provides an excellent example of people throughout the world and over thousands of years working together to accomplish mutually beneficial goals.
“Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.” (David Hilbert; German mathematician; 1862-1943.)
Every person who interacts with children has the responsibility of helping them to learn and understand the language of mathematics and a wide variety of its uses. Math education is a challenge to our educational systems because of its great depth, breadth, and applications, both across all areas of the curriculum and across life. It also is a challenge because of the continued rapid progress in the capabilities of computers and artificial intelligence.
Computers are a powerful aid to teaching, learning, and using math. Because of this, ICTing across the curriculum is an important component of a good (modern) educational system.
Over the years, I have taught and written extensively about math education and problem solving. Most of what I have written on these topics is available on the IAE website. The IAE-pedia currently has 291 entries. (IAE Main Page, 2020, link; IAE-pedia Most Popular Entries, 2020, link.) Nearly half of the 20 most popular entries are about math education. The IAE website also contains a more extensive list of my math education writings. (Moursund, 2020, link.)
David Moursund is an Emeritus Professor of Education at the University of Oregon, and editor of the IAE Newsletter. His professional career includes founding the International Society for Technology in Education (ISTE) in 1979, serving as ISTE’s executive officer for 19 years, and establishing ISTE’s flagship publication, Learning and Leading with Technology (now published by ISTE as Empowered Learner). He was the major professor or co-major professor for 82 doctoral students. He has presented hundreds of professional talks and workshops. He has authored or coauthored more than 60 academic books and hundreds of articles. Many of these books are available free online (IAE Books, 2020, link.)
Moursund founded Information Age Education (IAE) in 2007. IAE provides free online educational materials via its IAE-pedia, IAE Newsletter, IAE Blog, and IAE books. Information Age Education is now fully integrated into the 501(c)(3) non-profit corporation, Advancement of Globally Appropriate Technology and Education (AGATE) that was established in 2016. David Moursund is the Chief Executive Officer of IAE and AGATE (IAE, 2020, link; AGATE, 2020, link.)
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