The Dos and Don’ts of Teaching Mathematics

The Quest for the Royal Road to Learning
by Dr. David and Karen Shormann

Math & Science teacher. President of Digital Interactive Education. DIVE courses.

B.S. Aerospace Engineering from UT. Worked on F-16 engines.

Master’s in Marine Chemistry from UT Marine Science Institute. Researched “brown tide” algae. “Dead zone” off of Mississipi River Delta. Conducted research in AK.

Ph.D in Aquatic Science from Texas A&M. Effects of different fish species on water quality.

Some research included homeschool students. Students presented at Houston Science Fair & did well.

Just released, Journey to Novarupta audio adventure.
Teaches online live math classes.

[Life history way too long and not obviously connected to topic of talk.]

PDF copy of presentation. Email.

About Mathematics – What We Believe

  • Mathematics is a God-given tool for measuring and classifying pattern and shape.
  • Mathematics is the language of science, and a tool for studying God’s creation.
  • One of God’s first commands to Adam was to do science (Genesis 1:26-28)
    ** Science is about measuring in a repeating way
  • The Lord abhors dishonest weights and dishonest measurements (Proverbs 20:23).
    ** We are responsible for using mathematics properly.

What we will cover

  • Discuss dos and don’t of teaching mathematics in a home school.
    ** Share our 20+ years of experience teaching science and math (15+ teaching homeschoolers.)
  • Q & A

This is not about a dogmatic presentation. These are experienced based guidelines.

[Brief prayer at this point.]


  • Incomplete list…

1. Start with the end in mind

  • Set a high standard.
    ** Failure is an option! [In fact desirable]
  • A man’s heart plans his way, but the Lord directs his steps. (Proverbs 16:9)
  • Make a K-12 plan, then re-evaluate yearly.
  • Up to 80% of community college freshmen need remedial math or English.

One of the great things about home school is we can cater to our children. Tune it over time. Teach to the skill level of the child. Make a plan, but treat it as a possibility.

Defined a “liberal education” as freeing your child so they can consider as many options as possible.

Focus on math facts in early years

  • +, -, *, /
  • Exposure to other areas
    ** Other areas: measuring, volume, reading from a table, graphing, statistics, etc.
  • Memorize and understand
    ** Mathematics is primarily about deductive reasoning

Mathematics is deductive reasoning which is about applying rules. If you don’t know the rules you won’t be able to advance. You have to know where to find the rules if you don’t know them. Example you teach your child the alphabet first so they can learn to read.

Teach them what it means: 6*6 is adding 6 groups of 6. They should know that, but they shouldn’t have to think about it.

Be careful about a curriculum which says that memorization is not important. Both memorization and understanding are important. There are historical precedents for this organization structure and priority.

Realize the best preparation for algebra is…

  • Fluency with fractions, which is really just /
    ** 1/4x + 1/3 = 7
  • Fraction/decimal/percent important, too…

If you don’t know how to add two fractions together, the sample problem become quite difficult, because you don’t know what least common multiple is.

Life applications for things like percent are critical.

Using Math to build godly character

  • Integrity (Proverbs 20:23)
  • Patience (Galatians 5:22-23)
  • Perseverance with joy (James 1:2-3)
  • Responsibility (Luke 12:48)

Of all courses that you teach, math may be that you have to preserve more, both the teacher and the student.

If they have this knowledge, they can think about what else they can do with it. How can they share it with other people.

The more math you can get them to learn, the more different they’re going to be eduction-wise that anyone else in the world.

Incorporate Mathematics History

  • Did you know
    ** The Greeks almost discovered calculus, but their cyclical view of time directed them away from a study of infinity?
    ** The greatest mathematician ever was a Christian? Euler. (His method of teaching Algebra is used today.) Book: “Letters to a German Princess.” (Christian worldview incorporated into the study of nature)

Greeks shied away from infinity because of their cyclical perspective on time. Calculus is about the study of infinity.

Teach math as a language

  • Fluency comes through practice and repetition [Interesting conceptually.]
    ** Long-term retention takes 21 days, on average.
    ** Saxon curriculum repeats the same concept in multiple homework assignments over time.
  • Fluency comes through immersion
    ** In science course & projects, small business, life.
  • Fluency comes through passion
    ** SAT/ACT college prep, AP and CLEP exam prep.

Expect the same process to work for foreign languages as math facts.

“Hope and pray for your child that they develop a love of learning.”

Do take at least one CLEP or AP Exam

  • CLEP for College Algera, PreCalculus, Calculus I
  • AP for Calculus I & II
  • Consider taking at least one of these.
    ** Proof of fluency

This looks good on your homeschool transcript. Adds accountability to what you’ve done. [Blah blah. Third time we’ve heard this. Reasonable point, but no new information added.]

Learn by doing

  • Mathematics fluency comes primarily through an active, pencil and paper pursuit.
  • Make the majority of math time active, not passive.

Have the student put pencil to paper. Helps keep the kid’s mind engaged. Reading math is dry.

Be encouraging

  • Blessings for proper attitude, effort, and obedience.
    ** Material blessing, activity, etc.
  • Penalties for not, while remembering that, while we were still sinners, Christ died for us (Romans 5:8).

Moral aspect of it is important. Be careful how you give those penalties. Be mindful of you relationship with your children.

Encourage Independence

  • At some point, make your role accountability, not instruction.
    ** A student should be able to complete and grade their own work with integrity.

[Heard a variant of the independence point in the Self Propelled lecture.] Good moral aspects to doing this. There’s a lot of learning that goes on with the correction of homework mistakes.

Expect your children to have different learning styles

  • One of the greatest things about homeschooling is building strong relationships.
    ** In so doing, you will learn each child’s strengths and weaknesses, and when/where to push them harder or show mercy.

These are hard things to know and do without a good relationship with your children. “Good” defined as close and in-depth.

Show Work

  • Show enough work so that you can tell where a mistake was made.

This is part of the active vs. the passive aspect. You’re not shooting for a 50 page essay, but especially as they get to Algebra it’s hard for them to do in their head.

Geometry proofs. All of math is deductive reasoning, so it’s all like that. You’re proving yourself. In classical education, people overemphasize geometry, but it’s not just geometry which is deductive reasoning. Most of the world runs off of Calculus now.

Use a calculator sparingly

  • Try to use memory as much as possible.
  • Use a calculator for
    ** very large (>1000) or ver small (<0.01) numbers.
    ** Perimeter, area and volume calculations.
    ** Other rather tedious problems where the goal is not to practice arithmetic skills.
    ** Graphing functions, trigonometry and logarithms (mainly Algebra 2 and up.)

Typically you can do math faster than a calculator if you have the math facts memorized. 4th grade math you're not using a calculator at all.

Let them fail and learn from mistakes

  • A little "death and resurrection" during 12+ years of home education should be expected.
    ** Dying to self and obeying parents.
    ** Dying to bad attitudes and taking on trials with joy.
    ** Dying to bad habits and building good ones.
    *** Laziness, cheating.

Remind students to have a good attitude about math.

Remind them each lecture time to help them turn it around.

Cheating is a big problem that he's had. When the homework looks like a xerox of the answer key… They fail the quiz. That is a human condition which affects everybody, including homeschoolers. Don't just trust your child 100%.


Incomplete list of don'ts for teaching mathematics.

Don't list math pile up to do in large chunks

  • Ideally, math should be complete 3-5 days per week
    ** Also consider a slower pace, homeschooling through the summer
    ** Slow down, speed up, as needed.
    ** Sometimes you may need to stop and rewind – that's OK

…expect every child to be equally skilled in math

  • The won't be
    ** Effort, attitude and integrity are important.

This is character building as much as it is learning.

…be Impatient

  • Which can lead to anger
    ** You may have to review some facts or concepts a lot, depending on the student.

Realize that you've been doing this for a very long time. Give them some time to learn it because it's the first time they've done it.

…neglect mathematics, because it's difficult for you

  • The fear of the Lord is the beginning of knowledge (Proverbs 1:7).
    ** Make sure your children know to fear God, not math!
  • You most certainly can teach them math facts, while developing their love for learning
  • When they get to higher maths, your role is accountability, not instruction.

If you say that you're afraid of math to your child – they're likely to fear it as well. Don't share that with your child, even if you are/did. Especially Calculus. [LMAO]

…overempahisize the high school transcript

  • A good SAT/ACT score
  • A letter of recommendations
  • Prepare for those test by taking practice examples

…make everything a test.

  • Most curricula have daily homework sets and weekly/monthly tests.
    ** Treat homework as practice like a sport.
    ** Treat tests like "game day" where errors are emphasized.
    ** Practice hard, just not too hard. (Ephesians 6:4)

…unnecessarily repeative

Don't bore your children. Many curriculum have lots of repetition, more than might be necessary for your child.

…be surprised if your child struggles in college because they receive a poor math education

  • The #1 indicator of student success in college is the level of mathematics they complete in high school.
    ** Break point is pre-calculus.
  • Right now 40-50% of 4 year college graduates are working in jobs that did not require a college degree.

Poor education isn't the goal. But it does happen.

…surprised by the limited number of degree options if your child cannot pass calculus

  • Example: 80% of degrees at Texas A&M require at least 1 semester of calculus.
  • Calculus is simply the study of unknown rates of change…

…expect to find the royal road to learning!

  • But it doesn't hurt to try.

"There is no royal road to learning." – Euclid.

Blog: http://drshormann.dom