Issue Number 251 February
15, 2019

This free I

All back issues of the newsletter and subscription information are available online. In addition, seven free books based on the newsletters are available:

Dave Moursund’s newly revised and updated book,

ICT Tools and
the Future of Education

Part 4b: Using Human and Computer Brains as Tools

in Solving Math Problems

Part 4b: Using Human and Computer Brains as Tools

in Solving Math Problems

David
Moursund

Professor Emeritus, College of Education

University of Oregon

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.)

This is the second of two* IAE Newsletters* exploring a
version of George Pólya’s six-step strategy for attempting to solve a
wide range of math problems (Pólya, 1957). The previous newsletter
discussed the first two steps in the diagram given in Figure 1 below,
and provided background information about* What is Mathematics?*
and *What is a Math Problem?*. In brief summary, math and math
problem solving are routine parts of our everyday lives and an
integral component of PreK-12 education.

Some of the most important ideas from the previous newsletter are captured in the quote above. Descartes’ reference to “each problem that I solved” can be interpreted to mean all of his previous learning experiences. And, of course, all mathematicians realize that they learn from their failures as they try to solve a particular problem. When we are working to solve a problem, we are building on our previous successful and unsuccessful work, and that of others who came before us. Reading and writing were a very important aid to achieving such collaboration over the ages. Now we have computers. They are another great leap forward in terms of building on one’s own previous work and that of those who came before us.

This current newsletter discusses steps 3
through 6 in the diagram given in Figure 1.

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

Background Information
for Step 3

I suspect that most people do not realize how deeply math and aids to solving math problems are imbedded into their everyday lives. For example, suppose I ask you what time it is. You glance at your watch or other time-keeping device and read the analog or digital answer in a 12-hour or 24-hour format. You are using a tool to solve a math-related problem. My brain receives the numerical answer you say to me, and translates it into personal meaning for me.

Telling time involves the use of math, but it is not a *pure *math
problem. It refers to measurement of time. Quantification is a routine
part of our everyday lives. The development of accurate clocks and
watches was a very major achievement, and it certainly changed our
lives.

Think of some more types of machine or other aids to quantification. I step on the scales and read my weight. I look at a calendar and read the date. I look at items in a store and read the prices. In all of these example, my informal and formal education has helped me to attach personal meaning to the problem being solved.

However, sometimes quantification is not that simple. Suppose I am thinking about buying a product “with no money down, and easy monthly payments.” Hmm. What is it costing me and what does this method of payment mean to me?

Here is a more complex example. Suppose that I am thinking about buying a house or condo. I need to think about closing costs, down payment, interest rates, taxes, utilities, insurance, cost of upkeep, monthly payments, and so on. The math involved in dealing with this problem is relatively complex—and beyond the math education of most high school graduates.

As an amusing aside, think about hunter-gathers before the time of the invention of agriculture. Money, buying on time, paying interest, taxes, and other aspects of “home” ownership had not yet been invented. I find it fun to think about how complex life has become, and the essential roles of informal and formal education in helping us learn to deal with the problems we face today and in our futures.

Continuing the house-buying example, I can make use of the Web. I can find an interactive site that gathers the necessary information from me in a step-by-step manner and provides me with answers for varying lengths of loans, such as 15-year, 30-year, or whatever I pick. I am still left with issues such as whether I can afford to buy a house, and whether it would be a wise decision to buy a house.

In summary, these examples help to illustrate several vital aspects of math education. First, there is understanding—making sense of a human-posed problem situation in which math may be a useful aid to resolving the problem situation. Next, there is posing a problem to be solved. Third, there is creating a pure math problem whose solution we believe will be helpful to us in solving the problem. Fourth, there is the issue of whether we can solve the math problem. Fifth, there is the issue of whether having solved the math problem proves to be useful in resolving the original problem situation.

For a huge range of math problems that people encounter in their
everyday lives, we now have tools that can help greatly in solving or
can actually solve the math problem. This brings us back to step 3 in
the six-step diagram.

Step 3: Solve the Pure
Math Problem

For thousands of years, humans have been accumulating various types of math problems that are solvable and that have known methods of solution. It may take a person years of study to learn how to solve a particular type of math problem.

Just think of how long it takes a typical student to develop speed and accuracy in doing arithmetic on whole numbers, fractions, and decimal numbers. Teachers of first year algebra at the 8th or 9th grade level find that many of their students have not yet mastered doing arithmetic on fractions.

Students who are interested in studying math at a deeper level may well get through a year of calculus coursework while still in high school. But, all of that math is a very small part of the totality of known math that is covered thoroughly in the math literature. That is, even a doctorate in mathematics comes nowhere near preparing a person to solve the full range of math problems that occur in the various disciplines of study.

You know some of the capabilities of handheld calculators. Even an inexpensive calculator can add, subtract, multiply, and divide integers and decimal numbers. But, such an inexpensive calculator may also have keys for M+, M-, and CM (Clear Memory). To add 2/7 to 5/13 one first does 2 / 7 = and stores the result in memory by use of the M+ key. Then one calculates 5 / 13 = and adds the result to the number in memory by use of the keys + and MR. If you have such a calculator, have you learned to use these three keys? Many calculator owners have not.

For another example of the complexity of a simple calculator, consider the calculation (1/3) x 3. The result is 0.9999999 on my eight-digit calculator. Yet we expect the answer to be 1. Hmm. At what grade level do students learn to deal with this difficulty? My point is that even a simple calculator is a rather complex tool.

And, what about scientific and graphing calculators? Calculators now
are allowed to be used on a variety of state and national exams. For
example, quoting from* SAT Suite of Assessments: Calculator Policy*
(SAT, 2019, link):

If you’re taking a Subject Test in Mathematics, bring an approved calculator on test day. Test centers will not provide one.

The only Subjects Tests for which
calculators are allowed are Mathematics Level 1 and Mathematics Level
2. You must put it away when not taking a mathematics test. **A
scientific or graphing calculator is necessary for these tests. We
recommend using a graphing calculator rather than a scientific
calculator.** [Bold added for emphasis.]

It is now common that use of such calculators is a routine part of the high school math curriculum in the United States.

We now have computer systems that can solve the full range of the
types of pure math problems that students usually study in grades K-12
and in the first couple of years of a typical college math curricula.
These are called *Computer Algebra Systems* or *CAS *(Wikipedia,
2019, link):

A computer algebra system is any
mathematical software with the ability to manipulate mathematical
expressions** in a way similar to the traditional manual
computations of mathematicians and scientists.** The
development of the computer algebra systems in the second half of the
20th century is part of the discipline of "computer algebra" or
"symbolic computation", which has spurred work in algorithms over
mathematical objects such as polynomials. [Bold added for emphasis.]

Notice the bolded section. To me, this sounds very much like the first mechanization of factories that is then later followed by their automation. Before we had mass production in factories, we had manual methods of producing the products. Now, we have automated such factories by use of computers.

Some aspects of a CAS are built into higher end calculators. Some CASs are free, and others can be purchased. The Mathematica CAS developed by Stephen Wolfram comes in both a commercially available system (Mathematica, 2019, link) and a less extensive free version (Wolfram Alpha, 2019, link). Quoting from Stephen Wolfram’s blog (Wolfram, 6/21/2018, link):

On June 23 we celebrate the 30th anniversary of the launch of Mathematica. Most software from 30 years ago is now long gone. But not Mathematica. In fact, it feels in many ways like even after 30 years, we’re really just getting started. Our mission has always been a big one: to make the world as computable as possible, and to add a layer of computational intelligence to everything.

Our first big application area was math (hence the name “Mathematica”). And we’ve kept pushing the frontiers of what’s possible with math. But over the past 30 years, we’ve been able to build on the framework that we defined in Mathematica 1.0 to create the whole edifice of computational capabilities that we now call the Wolfram Language—and that corresponds to Mathematica as it is today.

In brief summary, CAS and other aids to solving a huge range of math
problems are continuing to improve through a combination of better
(smarter) programs and faster computers. I particularly like the title
of Stephen Wolfram’s blog, “We’ve come a long way in 30 years (**but
you haven’t seen anything yet!**)”. I bolded the second part
of the title that suggests such computer capabilities are going to get
better and better. It seems obvious to me that a modern math education
would include a substantial introduction of the currently available
computer aids to solving math problems. Some progress has occurred.
But, my prediction is that **you haven’t seen anything yet! **

Steps 4, 5, and 6: Move
Backwards Through Steps 2 and 1.

Remember, we started with an ill-defined problem situation that seemed to be mathematical in nature. By the end of Step 3, we have produced a mathematical answer to a pure math problem. This math result may or may not be useful to us in resolving the original ill-defined problem situation.

Perhaps the first question to ask is whether not we correctly solved the math problem. If we used mental or paper-and-pencil math, there is a reasonable chance that we made an error. If we used a calculator, we may have keyed in a number incorrectly, accidently pushed an incorrect operation key, or read the calculator results incorrectly.

If we used a computer, there are all kinds of things that we could have done wrong. Perhaps you have heard the expression, “Garbage in, garbage out.” Using a computer, we may have processed a lot of data. Where did the data come from? How accurate is it? How does the data relate to the original problem at hand?

What computer program(s) did we use to carry out the math calculations? If we are doing a statistical analysis, for example, perhaps we used a wrong statistical program. Our data may not satisfy the mathematical requirements for the program to actually produce meaningful results. If we used a computer program in an interactive manner, how do we know we didn’t make a keyboarding error or a wrong decision in the interaction. How do we know that the programs we used did not contain errors?

**The point is that just because a computer is used to solve a
pure math problem does not mean that the result is correct. **

Steps 4-6 are a process of converting the math results produced in
step 3 into language that can be understood by others, and relating
the results to the original problem situation. What do the math
results mean relative to the original problem that was stated in
natural language? We talk about *number sense*,* math sense*,
and ** common sense**. It is easy to ask whether
the math results make common sense. But, each person has their own
common sense. You might try to explain the results to various other
people, and ask them if the results seem sensible. If the results
don’t make sense, there is a good chance that errors occurred in doing
steps 1-3.

Finally, think carefully about the meaning of the results you have
obtained. At the very beginning, in step 1, you may not have posed the
exact problem that you really wanted to solve. As noted in the
previous newsletter, *problem posing* is often the most
difficult part of *problem solving*.

Also, think carefully about whether or not you or other people really
want to use the results you have produced. Perhaps the proposed
processes that must occur in order to implement the results of your
mathematical analysis would prove to be terribly wrong in terms of
such goals at human values, human rights, and the betterment of the
world.

Final Remarks

This and the previous newsletter are about using math to help solve problems and accomplish tasks. The problems and tasks are posed by people. These people need to have and use human insight into what problem is being posed and what the effects and side effects will be if a proposed solution is implemented. Math and computers can be quite helpful, both to develop possible solutions and to analyze possible effects of using or implementing the proposed solutions.

Our analysis of problem solving by making use of Pólya’s procedure
combined with electronic aids to computation suggests a need for a
major change in math education. Sometimes I find the following analogy
useful:** Spelling is to writing** as **computation
is to mathing** (doing and using math).

In writing, it certainly is desirable that one avoid making any spelling errors. But, that is a tiny part of effective written communication. Somewhat similarly, being fast and accurate at doing paper-and-pencil calculations does not make one into a productive math user.

Math educators need to pay careful attention to how much of the math
education curriculum time and assessment is spent on step 3. This is
the step at which computers are quite good and are becoming better and
better. It seems only logical to me that our math education system
should substantially decrease the time and effort that we now spend on
helping students to develop speed and accuracy in *by hand *calculations,
and spend much more time on helping students to understand and
routinely do the other five steps. These steps require critical
thinking, knowledge of the capabilities and limitations of math, and
knowledge about uses of math across the curriculum and across their
lives.

References and Resources

Mathematica (2019). Retrieved 1/24/2019 from http://www.wolfram.com/.

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. (2018). *The fourth R* (Second Edition). Eugene,
OR: Information Age Education. Retrieved 1/25/2019 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.

SAT (2019). SAT suite of assessments: Calculator policy. Retrieved 1/25/2019 from https://collegereadiness.collegeboard.org/sat-subject-tests/taking-the-test/calculator-policy.

Wikipedia (2019). Computer algebra system. Retrieved 1/24/2019 from https://en.wikipedia.org/wiki/Computer_algebra_system.

WolframAlpha (2019). Retrieved 1/24/2019 from https://www.wolframalpha.com/?source=frontpage-immediate-access.

Wolfram, S. (6/21/2018). We’ve come a long way in 30 years (but you
haven’t seen anything yet!). *Wolfram Blog*. Retrieved
1/24/2019 from https://blog.wolfram.com/2018/06/21/weve-come-a-long-way-in-30-years-but-you-havent-seen-anything-yet/?source=frontpage-latest-news.

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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About Information Age Education, Inc.

Information Age Education is a non-profit organization dedicated to improving education for learners of all ages throughout the world. Current IAE activities and free materials include the IAE-pedia at http://iae-pedia.org, a Website containing free books and articles at http://i-a-e.org/, a Blog at http://i-a-e.org/iae-blog.html, and the free newsletter you are now reading. See all back issues of the Blog at http://iae-pedia.org/IAE_Blog and all back issues of the Newsletter at http://i-a-e.org/iae-newsletter.html.