This is the second in our series of guest posts for Math Week on Afterthoughts. You may want to read the preceding posts first. Here is the series Table of Contents:

- Series Introduction
- Teaching Maths the CM Way {by Jeanne}
- Five Strategies for Teaching Mathphobics {by Tammy} <–you are here
- Teaching Euclid in the Homeschool, Part I {by Willa}
- Teaching Euclid in the Homeschool, Part II {by Willa}

**Tammy Glaser**has spent the past ten years building a life, discipline, and atmosphere in which she, her husband Steve, and two adult children get in touch with the universe of things and thoughts. She majored in her first love, mathematics, and earned a master’s degree in operations research {high-level statistics}. She still teaches her daughter with autism, mentors a Charlotte Mason study group, and tutors students of all ages at church.

When students struggle in mathematics, homeschoolers are tempted to abandon ship and jump to another curriculum. Year after year, we switch programs, hoping to find the elusive perfect approach. After years of pondering this phenomenon, I have come to the conclusion that curriculum hopping might be in vain. Unlike most areas of knowledge in a Mason philosophy, success in this subject depends upon the teacher and the pupil’s habit of attention rather than upon the books {p. 7}. Thus, we might fair better by sticking with something that is working, although not perfectly, because I do not believe there is a perfect mathematics curriculum.

Every lesson ought to be an exercise in clear thinking and rapid, yet careful execution. We ought to unfold ideas gradually beginning with demonstrating them in such a way that the path ahead is obvious. Having students narrate back by telling, drawing, and showing what they know often reveals immediate gaps. Addressing misunderstandings before working on problems prevents the frustration of doing a slew of them incorrectly. If a student does not grasp how the book teaches a concept, we should research alternative strategies online and try another route.

Once a student can narrate back with solid understanding, they are ready to take on some short problems and gradually work up to something challenging later. We ought to assign enough to practice fluency and accuracy without the lesson causing frustration. If math is taking too long but the child is putting forth real, attentive effort, then we may consider cutting back on the number of problems or assigning more later in the day. Mistake-prone children might benefit from doing just a few problems and getting them checked before going through the whole page.

One way to cure dawdling and careless execution is to assign only half of the problems. The child who does them perfectly and quickly needs no more practice for the day. We mark what is wrong as wrong and not as almost right. The child who makes a few errors will have a few more problems to do from the remaining half. The mistake-prone child requires more effort.

When we ban students from righting mathematical wrongs, they avoid the frustration of untangling a tangled web of errors. Our business as teachers is to assess what went wrong and why it went wrong. Once we work on those issues, the student starts anew with fresh hope of getting it right the next time. {p. 260-261}

In tutoring students from elementary school through college, I have found five strategies that put a smile on the face of most mathphobics.

- I
**assess whether the issue is habit training.**Does the person mix up four and nine or six and zero because of sloppy handwriting? Does the student rush through their work? Is the pupil so dependent upon the calculator that number sense falls apart? One student showed her work in such a way that her paper looked more like math doodles than a problem solution. Interruptions caused her to lose her train of thought, and she could not find her way in the scribbles on her paper. Folding her paper in half and working down a column reduced careless errors and she passed her final exam in Algebra II. - Sometimes,
**the way a book covers a concept does not click with a student.**My daughter Pamela preferred starting on the right column for long division because all operations begin in the right column. I found a method of division called partial quotients which permitted starting on the right. This method was the bridge Pamela needed to shift to the more traditional algorithm eventually. **When a textbook moves students too quickly into abstract thinking, they forget why they are doing what they are doing and make too many mistakes.**One high schooler asked me to explain adding and subtracting negative numbers. She said, “I never know if the answer is going to be positive or negative.” So, we backed up to fundamentals and concrete thinking. We explored adding and subtracting integers on a number line. As she demonstrated simple problems, she began to see a pattern. Suddenly, everything clicked and she smiled, “This is genius!” Then, we explored an alternative method offered in Jacob’s Elementary Algebra. Her understanding was solid at the end of the tutoring session.- My son had trouble memorizing math facts for the four basic operations.
**I had him build tables**as described in Volume 1 by proving each fact to himself. After that, I let him consult the table when he did arithmetic problems. I prefer tables over a calculator because patterns are more obvious in a table. My son eventually learned his math facts. - When a student shares a mathematical howler, I
**ask questions to assess if it was a careless comment or a deep lack of understanding**. One college student crossed out the numerator and denominator of x squared divided by x squared and got zero. When I asked why, she said, “Because you get zero when you cancel out.” I began to probe her knowledge. “What does x squared divided by x squared mean?” She gave me a blank look. “What does x divided by x mean?” produced another empty stare. She couldn’t explain five divided by five either. So, we went back to the basics and I explained, “Division is when you take some objects and share them equally. If you had five cookies and five people and you shared them equally, how many cookies would each person get?” She answered, “One.” “What about ten cookies and ten people?” Again, “One.” Then I asked about a thousand cookies and a thousand people and a million cookies and a million people. Each time, she answered, “One.” Then, we took the next leap of logic, “So, what if you had an unknown number of cookies and the exact same number of people. How many cookies would each person get?” She answered, “One. So, when you cancel out by dividing a number by itself, you get one.”

Our son struggled with math all his life due to the mistakes I made in instructing him because I never thought through Mason’s ideas carefully. He finally found joy in math last year when we faced his fundamental issues. He started asking for help when he realized that doing so is not a sign of ignorance or weakness. Working through Jacob’s Elementary Algebra showed him that thinking logically is the only way to understand mathematics; memorizing procedures only gets you so far. He turned to practice problems at Khan Academy to work on the habits of fluency and accuracy.

Because he spent last spring and summer overcoming his lifelong issues, he passed the math placement exam for his college, which meant he could take a 100 level class instead of remedial math. Moreover, when I asked him what his favorite classes were, he said English Composition and added, “Actually, for some reason I’m preferring the math problems over reading…. Pretty odd, huh?” and “I might take some extra math classes as soon as I get all my required math courses out of the way.” He’s earning a solid B in College Mathematics.

One day, I was helping a third-grader with multiplication problems. He made a few mistakes with his nines facts, so I told him that he will never miss his nines again after I show him a few, neat tricks. I showed him how to multiply by nines quickly on his fingers as illustrated here. As we tried different times nine facts, two women older than me were intrigued for they had never seen anything like it. They watched as I had the student write the digits 0 through 9 in a column down the page. To the right of each digit, I had him write 9 through 0 to look as follows:

09

18

27

36

45

54

63

72

81

90

To the right of that, I had him write nines multiplication problems. By the time he hit the third one, he started to giggle. My friends were amazed for they had never noticed this pattern before:

09 = 1 x 9

18 = 2 x 9

27 = 3 x 9

36

45

54

63

72

81

90

Finally, I had him add the tens and ones digit in all the products. Again, he started to giggle and my friends were awestruck by seeing something new in something so familiar.

0 + 9 = 9

1 + 8 = 9

2 + 7 = 9

3 + 6 = 9

4 + 5 = 9

5 + 4 = 9

6 + 3 = 9

7 + 2 = 9

8 + 1 = 9

9 + 0 = 9

So, I said to him, “If someone said to you that 9 times 4 is 56, would you agree or disagree?” He said, “Disagree.” When I asked why, he explained, “5 plus 6 is 11, not 9.”

Whenever I tutor a student, my ultimate aim is to help them find awe and wonder in math. That day, three people experienced the joy of multiplication.

Tammy Glaser says

I also like Greg Tang’s Best of Times for multiplication facts. Get out some objects and set this up for yourself. Suppose you don’t know 7 times 6. Set up the array for that (7 rows of 6). Now think of something you know. “Well, I know 7 times 3 is 21). You can divide that array into 7 rows of 3 and 7 rows of 3. So, 7 times 6 is the same as 7 times 3 plus 7 times 3, or 21 plus 21, or 42. That sounds like it might give you a headache trying to read it. But, use objects. The principle of taking the known (7 times 3) to find the unknown (7 times 6) is a powerful tip for solving many math problems.

Jeanne says

MEP teaches all the tricks. It is one of the great things about the programme.

Brandy @ Afterthoughts says

This website had the trick to multiply by nines on your fingers. Perhaps it would have more tricks?

Melissa Wild says

My 9 year old enjoyed the 9’s tricks as did I. Are there any other good tricks out there for an old mom looking to learn new ones? 😉 I remember testing for divisibility by 3 by adding up the digits of the number to see if it is a multiple of 3.

Melissa Wild says

My 10 year old was excited to see the 9’s tricks and had it worked out pretty quickly. I know there is a trick for testing divisibility by 3 — the digits need to add up to a multiple of 3. Any other handy tricks out there for an old mom looking to learn new ones? 😉

amy in peru says

yay! i love this series! SO glad you put it together! great job!