As for the “How?” of mathematics—its calculations and mechanics—this aspect of teaching is probably the strongest existing suite. Most teachers I know do a good job in this area. But, once again, try to humanize this effort. Calculation and problem solving have a long history containing many curiosity-arousing facts and episodes of experimentation. For example, the Chinese wrote their common fractions “upside down” with our denominator, called “the mother”, on top; below was our numerator, “the son”. In medieval Europe, at least five algorithms for the multiplication of two multi-digit numbers were employed. The Zuni people of America had seven directions for relating location (What were they?), and the mathematically admired “Golden Rule” was a technique involving a simple proportion (Smith 1958, II:484). Problems from other cultures and distant historical periods can be assigned, affirming that our ancestors did much of the same mathematics we do today. Worked calculations performed centuries ago can be resurrected and reexamined to affirm their correctness. (See Figure 4.) Even Pharaoh’s scribes in ancient Egypt made mistakes!

Figure 4. *Synopsis Algebraica* (1693) was written for the teaching of mathematics at Christ’s Hospital, London. The interesting feature of this book is that it is a mathematical workbook; opposite each page of text is a blank page allowing for annotations and computations. Little is known about the original Swiss author, Johann Alexander. Above, an 18^{th}-century student attempts to solve problem 35 by allowing EG=x, FG=y, EF=c, DE=b and DF=a; setting up an equation to represent the mathematical situation; and solving for the required unknown, “x”. By the customs of the period, a squared quantity is designated, in print, by repeating it twice. Is the computation shown correct? An English-language version of the book was published in 1709.

Now, for the “Why?” "Because it’s going to be on the test!" is the usual response. Prodding and threatening, yes, but not cognitively satisfying. Can we convey an appreciated value, from the past, present or future? Historically, every mathematical concept has had a motivation, a purpose. While the concept may be produced by the mathematics itself, sufficient real world examples are available for reference. The right triangle determines “vertical and horizontal.” Geometry and trigonometry gave rise to efficient surveying and navigation; instructors might ask, "What was the Great Trigonometric Survey?" [1] "How much was the prize for finding a method to determine longitude at sea?" Decimal fractions were needed to make precise measurements, so "why does a clock face mark only twelve hours?" In answering or referring to such questions, the query “Why?” often guides students to a deeper understanding than alternately-directed inquiries.

While historical relevance can motivate learning, allusions to contemporary relevance should have a teaching/learning rationale. In the 1950s, I had an algebra class where I was introduced to matrix computation. The teacher was a “name dropper”. In the process of instruction, he would casually include such names and terms as “IBM”, the “Rand Corporation”, “Think Tanks”, “Game Theory”, and “Markov Chains”. Some words we recognized with respect; others, for example ”Markov Chains”, were completely foreign but fascinating in their implications. I remember researching the concept of a Markov Chain, learning something about its development and, mathematically, playing with one. Of course, I also did my assigned work with matrices. Here the teacher was knowledgeable about the applications of mathematics, contemporary and potential, and shared his knowledge with the class, encouraging and motivating us as to the uses of mathematics.

More than an appeal to authority, I believe the best way to secure a student’s respect for problem solving is to give good, relevant problems. Observing an Algebra II class that just completed a problem in mathematically modeling an irrigation situation, I heard a young participant exclaim, “Now, that’s real mathematics.” I hope we teach “real” mathematics (Swetz and Hartzler 1991, p. 52). Interpreting data from the current news media on economic predictions or political polls is a fertile field for culling mathematical problems. Modeling problems involving contemporary environmental situations—global temperature rise, species extinction, ocean and air pollution, population growth, disease prevention—can assert the value of mathematics and, further, even promote meaningful discussion: "Is the ban on elephant poaching in Africa having an effect on ivory production?" "In which country is the human population increasing the fastest? Why?" Such questions focus both on contemporary issues and on issues our young people will continue to confront throughout their lives. This is the “New Math” of the millennium together with STEM priorities. [2] Our students must have the experience and knowledge to recognize the problems and the capability to resolve them.

##### Notes:

[1] Several questions have been left open for the reader to consider. Using information-supplying electronic devices, answers are easy to secure; however, I will assist with these immediate inquires. The Great Trigonometrical Survey was a project instituted by Great Britain intended to measure and map the entire Indian subcontinent with scientific precision. It was begun in 1802 by the infantry officer William Lambton under the auspices of the East India Company. His successor was George Everest, for whom the highest mountain in the world would be named. The project was completed in 1871. What mathematical techniques and concepts were employed?

For the seeking of a convenient and accurate method of determining a measure of longitude at sea, the British Navy established a Board of Longitude. This agency offered a prize/reward of £20,000 for the solution. What classroom exercises could be developed from this incident?

For the efficient operation of a clock face, a system of modulus 12 numeration and arithmetic is employed. What is modular arithmetic?

[2] STEM is a curriculum based on the idea of educating students in four specific disciplines: science, technology, engineering and mathematics, in an interdisciplinary and applied approach. See https://www.livescience.com/43296-what-is-stem-education.html.