When I was in high school, I enjoyed math and wanted to pursue it as a career. I also felt a lot like Alice, I almost received the same response as she did from the Cheshire Cat when I asked the guidance counselor, math teachers and others what math classes to take, what opportunities were open to me with a math degree, and what math I should know, e.g., algebra, calculus, geometry, trigonometry, probability, statistics, and accounting. I was on my own. One thing I learned once I took the SAT and ACT, and was actually in college, is that I not only needed specific math courses, but the courses should have taught specific topics, e.g., arithmetic and geometric sequences, estimating, significant digits, PEMDAS, and set theory. Some of these I had to learn on my own because my high school did not teach some of these skills and concepts.

Well I’m on my way

Me and Julio Down by the Schoolyard – Paul Simon

I don’t know where I’m goin’, I’m on my way

I’m takin’ my time, but I don’t know where

**So, what should you do?** First, determine if you want to go to a vocational school to learn a trade (blue collar), or a college to learn the basics of a profession (white collar). Second, once that decision is made, then what additional math, if any, do you need to take to reach your destination? That is, perform a gap analysis.^{7} Do you need learn a particular math skill, e.g., data analysis or interpreting graphs? Do you need to learn a particular concept, e.g., sets and set theory or probability? The Internet is teeming with websites (e.g., this one) that can help you learn the skills and concepts you need.

I invite you to read Bryan Caplan’s article^{8}, *What’s College Good For?*, to assess what level of education you need to make $$$. Here is an excerpt:

“The disconnect between college curricula and the job market has a banal explanation: Educators teach what they know—and most have as little firsthand knowledge of the modern workplace as I do. Yet this merely complicates the puzzle. If schools aim to boost students’ future income by teaching job skills, why do they entrust students’ education to people so detached from the real world? Because, despite the chasm between what students learn and what workers do, academic success is a strong signal of worker productivity.”

Bryan Caplan

### Books

What about textbooks that you use or may use in your academic career? Find other books on the same subject to help you to understand the concepts. When I took number theory in college, the book the professor chose was not the greatest. I obtained a copy of *Elementary Number Theory*, by J.V. Uspensky, & M.A. Heaslet, to supplement my study. Much easier to understand! I urge you not to rely solely on the course’s textbook. Other books, videos and websites on the subject can make your learning fun!

Se we have to ask: Why do school textbooks change? How are textbooks selected?

- Money. Having new editions makes the older editions have zero resale value. Remember the publishers only make money selling
*new*books, so having a book around for more than one year means that they make less and less money every time.^{2} - I work for an education nonprofit that does create math text books. We have changed our textbooks when (1) new standards come out as they did in 2010 when CCSS [
*Common Core State Standards*] was introduced and states had to have materials that were aligned with CCSS, both in content standards placement and the emphasis on the Standards for Mathematical Practice; (2) we have learned from research of new ways to introduce, cover, explain, or learn mathematics; or (3) a lesson or problems might be outdated or had errors.^{3}[See**More Than One Way to Do Math**below.] - Math textbooks today assert that their “scripted” way of teaching is best for all schools, all classrooms, all teachers and all topics, and that the biggest challenge is to ensure that they are used with “fidelity”. Much like “one size fits all” training is no longer relevant to the world’s workforce; this model of teaching is obsolete. The educational world has pretty much come to accept textbooks as necessary for teaching math, understanding how rigid and cumbersome these books are, but lacking alternatives. Of course, when resources are the same for everyone, they can’t be well suited to
*anyone.*We have found, encouragingly, that a number of math instructors and curriculum directors are actively looking for a better way.^{4} - As you visit classrooms, you probably notice that most, if not all, of those classrooms use a standard textbook series. The reasons for this are many, depending on the design and focus of the curriculum, the mandates of the administration, and/or the level of expertise on the part of classroom teachers.
^{5}

**Choosing Textbooks for Grade Schools**

In 1964 Feynman made the rare decision to serve on a public commission for choosing mathematics textbooks for California’s grade schools. As Gleick (author of *Genius: The Life and Science of Richard Feynman*) describes it:

“Traditionally this commissionership was a sinecure that brought various small perquisites under the table from textbook publishers. Few commissioners — as Feynman discovered — read many textbooks, but he determined to read them all, and had scores of them delivered to his house.”

This was the era of new math in children’s textbooks: introducing high-level concepts, such as set theory and non decimal number systems into grade school.

Feynman was skeptical of this approach but rather than simply let it go, he popped the balloon.

“He argued to his fellow commissioners that sets, as presented in the reformers’ textbooks, were an example of the most insidious pedantry: new definitions for the sake of definition, a perfect case of introducing words without introducing ideas.”^{1}

### Supplemental Material

There are many teachers who use supplemental material to teach math. Some materials are free, e.g., Math Goodies, and some have a cost, e.g., TeachersPayTeachers. Hats off to you!

### More Than One Way to Do Math

Granted that some math changes over time. However, some math does not, e.g., fractions, adding, subtracting, multiplying, dividing and others. There are many ways to accomplish these skills and there is no one “correct” way. Teach as many as possible so the student can use the one that fits his or her learning style.

#### Integration by Parts Example

When I learned integration by parts the professor presented it a straight forward manner, e.g., shown on Paul’s Online Notes.

Later I learned that you can perform the integration by parts using a 2-by-2 box to easily solve the problem. (See **How to Do Integration by Parts** on the dummies.com website.) Q.E.D.

#### Multiplying Fractions Example

When my children were learning fractions we had some interesting and frustrating moments. My daughter told me “I will never use fractions.” Today she is teaching her children how to do fractions. Also, when she started college, she was going to be an interior designer. One of her first assignments was to draw a house layout to 1/4th scale. She eventually understood the importance of fractions.

One method for teaching fractions is the butterfly method. What a wonderful visual method to help students understand how to add and subtract, and multiply and divide fractions.

The TeachableMath website had this to say about the butterfly method:^{6}

“In our opinion, tricks like the butterfly method should be avoided when students are first introduced to fractions. There are several reasons, e.g.,

- There is no conceptual understanding in the instruction.
- It reinforces the belief that fractions is just a bunch of tricks.
- What happens if you add three or more fractions?”

I agree. It is not the method but how and when the method is introduced to the student by the teacher. We must remember:

“We need to teach children how to think rather than what to think” ~ Margaret Mead

## References

^{1} “Richard Feynman On Teaching Math To Kids And The Lessons Of Knowledge – Farnam Street”. 2016. *Farnam Street*. https://fs.blog/richard-feynman-teaching-math-kids/.

^{2} Anonymous. “Why Are College Math Textbooks Updated Almost Every Year?”. 2021. *Quora*. https://www.quora.com/Why-are-college-math-textbooks-updated-almost-every-year.

^{3} Wootton, Karen. “Why do they update math textbooks in school? Isn’t math always the same?” 2019. *Quora*. https://www.quora.com/Why-do-they-update-math-textbooks-in-school-Isnt-math-always-the-same

^{4} “Why Today’s Math Textbooks Just Don’t Add Up”. 2021. *The Learning Counsel*. https://www.thelearningcounsel.com/article/why-todays-math-textbooks-just-dont-add.

^{5} ⭐ “Textbooks: Advantages And Disadvantages”. 2021. *teachervision*. https://www.teachervision.com/curriculum-planning/textbooks-advantages-disadvantages.

^{6} “The Butterfly Method In Fractions And The Danger Of Overemphasizing Tricks – Teachablemath”. 2016. *TeachableMath*. https://teachablemath.com/butterfly-method-fractions-danger-overemphasizing-tricks/.

^{7} “What Is Gap Analysis: Definition, Method And Template With Example | QuestionPro”. 2018. *QuestionPro*. https://www.questionpro.com/blog/gap-analysis/.

Gap analysis is defined as a method of assessing the differences between actual and expected performance. The term “gap” refers to the space between “where we are” (the present state) and where “we want to be” (the target state). A gap analysis can also be referred to as need analysis, need assessment or need-gap analysis.

^{8} Caplan, Bryan. 2017. “The World Might Be Better Off Without College For Everyone”. *The Atlantic*. https://www.theatlantic.com/magazine/archive/2018/01/whats-college-good-for/546590/.

## Additional Reading

“Gap Analysis: Reaching Your Ideal Future State”. 2022. *mindtools.com*. https://www.mindtools.com/pages/article/gap-analysis.htm.

Harrison, Amy, and View Harrison. 2016. “My Math Resources – Compare Fractions: Butterfly And XD Methods”. *My Math Resources*. https://mymathresources.com/compare-fractions/.

“How To Conduct A GAP Analysis: Tips, Tools, And Examples!”. 2020. *someka*. https://www.someka.net/blog/how-to-conduct-gap-analysis/.

“How to Do Integration by Parts More than Once”. 2022. *dummies.com*. https://www.dummies.com/article/academics-the-arts/math/calculus/how-to-do-integration-by-parts-more-than-once-192233.

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.