Issue Number 250 January 31, 2019

This free Information Age Education Newsletter is edited by Dave Moursund and produced by Ken Loge. The newsletter is one component of the Information Age Education (IAE) and Advancement of Globally Appropriate Technology and Education (AGATE) publications.

All back issues of the newsletter and subscription information are available online. In addition, seven free books based on the newsletters are available: Joy of Learning; Validity and Credibility of Information; Education for Students’ Futures; Understanding and Mastering Complexity; Consciousness and Morality: Recent Research Developments; Creating an Appropriate 21st Century Education; and Common Core State Standards for Education in America.

Dave Moursund’s newly revised and updated book, The Fourth R (Second Edition), is now available in both English and Spanish (Moursund, 2018c). The unifying theme of the book is that the 4th R of Reasoning/Computational Thinking is fundamental to empowering today’s students and their teachers throughout the K-12 curriculum. The first edition was published in December, 2016, the second edition in August, 2018, and the Spanish translation of the second edition in September, 2018. The three books have now had a combined total of more than 34,000 page-views and downloads.

ICT Tools and the Future of Education
Part 4a: Using Human and Computer Brains as Tools
in Solving Math Problems (4a)

David Moursund
Professor Emeritus, College of Education
University of Oregon

“Each problem that I solved became a rule which served afterwards to solve other problems.” (René Descartes; French philosopher, mathematician, scientist, and writer; 1596-1650.)
"The most important single element in problem solving is the individual working on the problem. The secret of real success is the confidence and desire to succeed. One must try and try again, vary the methods and procedures, have brains and good luck. There are no infallible rules for solving problems." (John N. Fujii; American mathematician and math educator.)

“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.)

Introduction

René Descartes is a very famous mathematician and philosopher. Perhaps you have heard of cartesian geometry (analytic geometry) and/or his statement, “I think, therefore I am.”

The first of the quotes given at the start of this newsletter captures part of what I consider to be one of the most important aspects of problem solving. Reading, writing, and computers are tools that help problem solvers to build on the previous work of themselves and others (Moursund, 1/19/2019, link.)

The two other quotes capture key elements of the discipline of problem solving.

Problem Solving Using Mathematics

Why do our schools pay so much attention to math and math problem solving? Perhaps the simplest answer is that so many of the problems faced by people living in today’s world are math related. We want to prepare students to adequately deal with those aspects of everyday life.

Because math is such an ancient and rigorous discipline, a huge number of different types of math problems have been studied and solved over the years. When someone solves a certain category of math problems and the results are published in a reputable, refereed journal, others can build on this work with a high degree of confidence. Today’s students can access, learn to use, and build on many hundreds of years of accumulated math knowledge.

In my early days of teaching math problem solving, I encountered George Pólya’s book, How to Solve It, A New Aspect of Mathematical Method (Pólya, 1957). His book contains a general six-step strategy that one can follow in attempting to solve almost any problem. The six steps of Polya’s strategy and applications of this strategy to solving math problems are the focus of this newsletter and the next one.

Note that there is no guarantee that use of this procedure will always solve the problem a person is attempting to solve. For example, the first cases of HIV/AIDS were discovered in 1981. Since then, a large number of researchers have been attempting to solve various aspects of this multifaceted problem. While significant progress has occurred, the problem is by no means solved. Similar statements can be made about other medical problems, such as cancer, stroke, and heart attack.
Here are three simple math problems to ponder:

  1. Find an integer that is greater than 5 and less than 7. (This problem has exactly one solution.)
  2. Find an integer that is greater than five and less than 8. (This problem has two solutions.)
  3. Find an integer that is greater than 5 and less than 6. (This problem has no solution.)

These three math problems illustrate two very important ideas. First, a math problem may have more than one correct solution. Second, a math problem may have no solution. I must admit that it really pains me when I hear a student saying, “My goal is to find the answer.”

Even if the problem you are working on is solvable, you may lack the knowledge, skills, time, and other resources needed to solve it. However, Pólya ’s six-step strategy might get you started on a pathway to success.

A Six-step Strategy for Attempting to Solve Many Different Types of Problems

This section contains a slightly modified version of Pólya six-step strategy It is applicable to problem solving in many other disciplines in addition to mathematics.

  1. Understand the problem. See Einstein’s quote at the beginning of this newsletter. Among other things, this includes working toward having a clearly defined problem. You need an understanding of the given initial situation, goal, resources, and ownership. This requires knowledge of the domain(s) of the problem, which could well be interdisciplinary. This reminds me of the following quote from the English biologist, Thomas Huxley:
    "Try to learn something about everything and everything about something."
    Most problems are interdisciplinary. The recommendation is to develop a high level of expertise in at least one discipline as well as a very broad general education. A person might strive toward a high level of expertise, both in a vocational area and in an avocational area. Perhaps you have heard people talking about their “day job” where they earn a living, and their “evening job” that is an avocation such as music that they do for fun.
  2. Determine a plan of action. This is a thinking activity. What strategies will you apply? What resources will you use, how will you use them, and in what order will you use them? Are the resources adequate to the task? These are challenging questions. They suggest the need for a good education that includes a strong emphasis on learning to use the resources that are available and are commonly used in many different problem-solving areas. Information retrieval is a skill useful in all areas of problem solvin
  3. Think carefully about possible consequences of carrying out your plan of action. Place major emphasis on trying to anticipate undesirable outcomes. What new problems will be created? You may decide to stop working on the problem or return to step 1 as a consequence of this thinking. As a somewhat humorous example, many years ago I decided to plant some bamboo in my yard. My older brother said this was a mistake, as the bamboo would eventually spread and would be very hard to eradicate. I went ahead with the planting, and this did eventually prove to be a serious mistake.
  4. Carry out your plan of action. Do so in a thoughtful manner. This thinking may lead you to the conclusion that you need to return to one of the earlier steps. Note that this reflective thinking leads to increased expertise. The statement to "do this in a thoughtful manner" is fundamental to developing increased expertise. I strongly believe that our education system is weak is this area.
  5. Check to see if the desired goal has been achieved by carrying out your plan of action. Then do one of the following:
    • If the problem has been solved, go to step 6.
    • If the problem has not been solved and you are willing to devote more time and energy to it, make use of the knowledge and experience you have gained as you return to step 1 or step 2.
    • Decide to stop working on the problem. This might be a temporary or a permanent decision. Keep in mind that the problem you are working on may not be solvable, or it may be beyond your current capabilities and resources.
  6. Do a careful analysis of the steps you have carried out and the results you have achieved to see if you have created new, additional problems that need to be addressed. Reflect on what you have learned by solving the problem. Think about how your increased knowledge and skills can be used in other problem-solving situations. Keep in mind the Fujii quote at the beginning of this newsletter:
“The most important single element in problem solving is the individual working on the problem. The secret of real success is the confidence and desire to succeed. One must try and try again, vary the methods and procedures, have brains and good luck. There are no infallible rules for solving problems.”


A Six-step Diagram Version of Pólya’s Strategy to Solve Math Problems

There are many different types of math problems, and there are many different ways to attempt to solve a math problem. For many years, I have used the diagram in Figure 1 as an aid to teaching teachers about roles of computers in solving math problems. It is based on Polya’s six-step strategy given in the previous section, but specifically targets math-oriented problems.

Figure 1
Figure 1. A six-step procedure for solving math problems.

Perhaps the first question to consider is, “What is a math problem?” Quoting from the Wikipedia (2019b, link):

A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics.

Unfortunately, to understand this definition one first needs to know the meaning of the methods of mathematics.

For example, one method of math is counting. A young child is given a small collection of wooden blocks and asked the math question, “How many blocks are there?” The child says the words one, two, three, and so on while touching or perhaps moving one block at a time in the collection. When the last block has been processed, the child says the last number used in counting the blocks is the answer.

This is a profound mathematical achievement for a young child. The child has learned a process for determining the number of objects in a small collection. It works equally well for wooden blocks, plastic blocks, and pieces of candy.

Such counting games are a good learning activity to help the child to develop number sense. In terms of counting blocks, the number six comes before the number ten. Six blocks are not as many as ten blocks and a collection of six blocks is smaller than the collection of all ten blocks. All of these concepts are a form of number sense.

Just for the fun of it, think about the added complexity of counting the number of blue cars that one sees going by while standing at a street corner. The child cannot touch the cars, and they are soon out of sight. Many of the cars are not blue. And, there are many shades of blue—perhaps it is not clear whether a particular car should be called blue. (And, what if the child is color blind? If a child has some form of color blindness, what and when do you want the child to learn about this?)

The remainder of this newsletter and most of the next one are based on the diagram of Figure 1. The emphasis is on the question of what aspects of this six-step strategy are best suited to human brains and what are best suited to computer brains?

The message is that learning and doing mathematics nowadays requires that students learn to effectively use a combination of their own brain power and that of computer brains to solve math problems. This observation applies to problem solving across the curriculum (Moursund, 2018b, link.)

Step 1: Converting the Problem Situation into a Clearly Defined Problem to Be Solved

The previous newsletter in this series discussed four characteristics of a well-defined problem (Moursund, 1/15/2019, link):

  1. Given initial situation.
  2. Goal.
  3. Resources, along with guidelines, rules, regulations, etc.
  4. Ownership.

This aspect of problem solving is applicable to problems in every discipline of study. Thus, as students study and practice problem solving across the curriculum and outside the curriculum, they can and should explore a wide range of problems. I like to use the phrase, “View the world through problem-solving colored glasses.” That is, think of the many kinds of situations that you encounter in life as problem situations. Practice analyzing these situations and converting them into well-defined problems.

Note that Step 1 is a human endeavor. While access to computers is often helpful, a major aspect of this step is for the problem solver to develop a human understanding of the problem to be solved and the importance of solving the problem. This human understanding is often essential in detecting errors while working to solve a problem. In math, we talk about number sense, algebra sense, spatial/geometry sense, and so on. Good math education helps students to develop these senses.

Here is a repeat of the Einstein quote at the beginning of this newsletter:

“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.”

By proper question, Einstein is referring to the need to have a clearly defined problem. In his opinion, this is the hardest and most important part of problem solving.

This assertion should make you think about the types of problems that students study in school. Problem posing and then refining the posed problem into a well-defined (clearly-defined) problem is one of the most important aspects of problem solving.

For simplicity, consider two types of problems that all students encounter in their math classes. One type presents students with a problem stated in math notation, using the language of mathematics (Moursund, 2018a, link). It might be a problem to add a collection of numbers, or it might be a problem to solve an algebraic equation. This type of problem is an example of pure math and makes no reference to any other discipline of human knowledge and understanding. Here are two examples of “pure” math problems.

  1. 3 + 8 =
  2. Solve for x in the equation: x – 6 = x/2

A word problem or a story problem is stated in a combination of non-math words and math words. A more general term is applied math problem. For example:

Today is Suzy’s birthday. She is now twice as old as she was six years ago. How old is Suzy now?

This problem talks about a girl named Suzy, her age, and her birthday. These are all non-math words and ideas. To understand the math in this problem, one must know that twice means two times and that six means 6. The name Suzy is a female name, but the specific name and that person’s gender really have nothing to do with the problem. They are extraneous information.

Birthdays come once a year unless one is born on February 29th of a leap year. Hmm. The problem is not clearly stated. Suppose Suzy was born on February 29th? Did you catch that ambiguity as you were reading and trying to solve the problem? Presumably, Einstein would have detected this ambiguity as he pondered the problem. Perhaps he would have been aware of the following information quoted from the Wikipedia (2019a, link):

… in the Gregorian calendar, each leap year has 366 days instead of 365, by extending February to 29 days rather than the common 28. These extra days occur in years which are multiples of four (with the exception of centennial years not divisible by 400). Similarly, in the lunisolar Hebrew calendar, Adar Aleph, a 13th lunar month, is added seven times every 19 years to the twelve lunar months in its common years to keep its calendar year from drifting through the seasons. In the Bahá'í Calendar, a leap day is added when needed to ensure that the following year begins on the vernal equinox.

So, what appeared to be a simple math word problem turned out to be quite complex! This example helps to explain why so many students find word problems in math to be extremely difficult. The problem draws on both math and non-math. To solve the problem, one must translate (formulate) a non-math problem into a pure math problem. To do this requires knowledge of the non-math area. In this example, the relevant non-math area is a complex area of astronomy and calendaring.

In summary, Step 1 of Polya’s six-step procedure is a human endeavor. Access to computers is often helpful, since the problem solver often needs more information than is provided in the statement of the problem. While writing the birthday example, I used both the Google search engine and the Web to find information about leap years.

Step 2. Represent the Problem as a Math (Computational, Algorithmic) Problem

When I was teaching Information and Communication Technology (ICT) and math education to secondary school preservice and inservice math teachers, I would observe that solving quadratic equations was commonly taught in algebra courses. I would then ask them to give me some practical problems in which a person would need to solve quadratic equations. I rarely received any good examples. Most of my students who taught students how to solve quadratic equations did so because it was in the required curriculum.

I believe math is useful in almost every discipline of study. So, I ask myself, “Where and how should students be learning to use the math that is an integral component of the non-math disciplines they are studying?”

Consider the following two examples:

  1. What is 3 times 7?
  2. A person is walking at a speed of three miles an hour, and walks for seven hours. How far does the person walk?

The first is a pure math problem, while the second might be considered to be a physics problem because it involves the variables of distance, rate, and time. The process of converting the second problem into the first problem is called math modeling.

Math modeling is part of every discipline of study that makes use of math as an aid to representing and solving its problems. Thus, another name for Step 2 is math modeling.

A Simple Example of Difficulties in Learning Math Modeling

Two young children and their parents are playing together. One of the parents glances at the clock and says, “It is a quarter to eight. Time for you kids to get ready for bed.”

One of the children replies, “I know what a quarter is. It’s a round shiny piece of money that is worth 25 cents. But I don’t know what it means when you say that it is 25 cents to eight.” The slightly older child adds, “I know that there are four quarters in a dollar.”

Think about what it takes to understand that the parent is actually saying, “It is one-fourth of an hour before 8 o’clock.” At what age does a typical child understand the fraction one-fourth?

And, how long is an hour? This is a complex question. Quoting from the Wikipedia (2018b, link):

Earth rotates once in about 24 hours with respect to the Sun, but once every 23 hours, 56 minutes, and 4 seconds with respect to the stars (see below). Earth's rotation is slowing slightly with time; thus, a day was shorter in the past. This is due to the tidal effects the Moon has on Earth's rotation.

The point is that both time and measuring time are very complex areas of study.

Reading and Mathing Across the Curriculum

Reading is an important aspect of many different disciplines of study. Educators have agreed on the idea of teaching reading across the curriculum. Schools have teaching specialists who are very skilled in teaching reading and writing. But, every teacher is expected to be skilled in teaching reading and writing, and also teaching the use of reading and writing in their areas of specialization, reading across the curriculum.

It seems to me that math educators should be thinking more about mathing across the curriculum or math modeling across the curriculum.

Every teacher has a responsibility of teaching math-aspects of the disciplines they teach. I find it helpful to make an analogy between reading across the curriculum, and mathing across the curriculum. Currently, we tend to teach math as a somewhat isolated, self-contained subject. More and more problems people want to solve lend themselves to math and computer modeling, and can be solved using computers. The field of Computer and Information Science has developed the term computational thinking. In essence, computational thinking means to think about solving problems in terms of the capabilities and limitations of computers. Computer simulation has become a widely used aspect of computational thinking and an important area in mathematics.

Computer Simulation

Computer modelling consists of writing a computer program version of a mathematical model for a physical, biological, business, or other type of system. Computer simulations that are run according to such programs can produce knowledge out of reach of mathematical analysis or natural experimentation (Wikipedia, 2018a, link):

Computer simulation is the reproduction of the behavior of a system using a computer to simulate the outcomes of a mathematical model associated with said system. Since they allow to check the reliability of chosen mathematical models, computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics (computational physics), astrophysics, climatology, chemistry, biology and manufacturing, human systems in economics, psychology, social science, health care and engineering. [Bold added for emphasis.]

To be Continued

The next newsletter will contain a discussion of Steps 3-6 and make additional recommendations about how to improve our current math education system.

Final Remarks

As noted earlier in this newsletter, math is a large and steadily growing discipline of study. Currently math instruction at the Pre-12 levels is designed to help students gain:

  1. Some understanding of the discipline of “pure” math. We want students to learn to solve a variety of such problems using a combination of their brains and simple aids such as paper, pencil, and inexpensive calculators. We also strive to help students understand the concept of a math proof and the careful, rigorous thinking required to understand and solve a variety of math problems.
  2. Applications of math to help represent and solve a variety of problems in non-math disciplines where math can be a significant aid in representing and solving the problems.


Some understanding of the discipline of “pure” math. We want students to learn to solve a variety of such problems using a combination of their brains and simple aids such as paper, pencil, and inexpensive calculators. We also strive to help students understand the concept of a math proof and the careful, rigorous thinking required to understand and solve a variety of math problems.
 
Applications of math to help represent and solve a variety of problems in non-math disciplines where math can be a significant aid in representing and solving the problems.

Some understanding of the discipline of “pure” math. We want students to learn to solve a variety of such problems using a combination of their brains and simple aids such as paper, pencil, and inexpensive calculators. We also strive to help students understand the concept of a math proof and the careful, rigorous thinking required to understand and solve a variety of math problems.

For thousands of years, tools such as writing implements, surfaces to write on, and the abacus have been fundamental to learning and using math. Gradually, mechanical calculating devises, the slide rule, and many other aids to doing math calculations were developed. Now we have very powerful calculators and computers. We have the discipline of computer programming, and a steadily growing collection of very useful programs.

Computer programming is part of the discipline of Computer and Information Science. This discipline has many components, such as the study of artificial intelligence and the development and use of computerized machinery (including robots). We have created the Web, a huge and steadily growing library of collected human knowledge, and made it readily available to billions of people.

Mathematics plays an important role in all of this progress in Computer and Information Science. In turn, the progress in Computer and Information Science has become an important component of mathematics. Computers are powerful change agents in the math content we want our students to learn and also in the math problem solving we want them to be able to do.

References and Resources

Moursund, D. (1/19/2019). Tools help us build on the work of self and others. IAE Blog. Retrieved 1/19/2019 from https://i-a-e.org/iae-blog/entry/tools-help-us-build-on-the-work-of-self-and-others.html.

Moursund, D. (1/15/2019). Tools and the future of education, Part 3: ICT and math education. IAE Newsletter. Retrieved 1/18/2019 from https://i-a-e.org/newsletters/IAE-Newsletter-2019-249.html.

Moursund, D. (2018a). Communicating in the language of mathematics. IAE-pedia. Retrieved 1/10/2019 from http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics.

Moursund, D. (2018b). Technology and problem solving. IAE-pedia. Retrieved 1/29/2019 from http://iae-pedia.org/Technology_and_Problem_Solving.

Moursund, D. (2018c). The fourth R (Second Edition). Eugene, OR: Information Age Education. Retrieved 1/3/2019 from http://iae-pedia.org/The_Fourth_R_(Second_Edition) Download the Microsoft Word file from http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/307-the-fourth-r-second-edition.html. Download the PDF file from http://i-a-e.org/downloads/free-ebooks-by-dave-moursund/308-the-fourth-r-second-edition-1.html. Download the Spanish edition from http://iae-pedia.org/La_Cuarta_R_(Segunda_Edici%C3%B3n).

Pólya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton, NJ: Princeton.

Wikipedia (2018a). Computer simulation. Retrieved 12/28/2018 from https://en.wikipedia.org/wiki/Computer_simulation.

Wikipedia (2018b). Earth’s rotation. Retrieved 12/28/2018 from https://en.wikipedia.org/wiki/Earth%27s_rotation.

Wikipedia (2019a). Leap year. Retrieved 1/11/2019 from https://en.wikipedia.org/wiki/Leap_year.

Wikipedia (2019b). Mathematical problem. Retrieved 1/11/2019 from https://en.wikipedia.org/wiki/Mathematical_problem.


Author

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. See http://iaepedia.org/David_Moursund_Books .

In 2007, Moursund founded Information Age Education (IAE). IAE provides free online educational materials via its IAE-pedia, IAE Newsletter, IAE Blog, and IAE books. See http://iaepedia.org/Main_Page#IAE_in_a_Nutshell . 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 AGATE.

Email: moursund@uoregon.edu.

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