Math is hard. This is half lie, half confession, and half true. This is no mathematical statement. Everything new is hard. Math for children, for young students, and for students approaching adulthood should come organically with life. Math for adults should be the necessary skill for the task at hand. Math for the homeschool teacher will be hard too. Parents should not just hand the math off to some online guru. The teacher should grasp anything the homeschooler masters.
Math is hard in itself.
Math is a group of languages. Math languages are logical, artificial languages and not natural languages. Artificial and unnatural, math describes only some things in the world. Different cultures needed to describe different things. Romans did not need a zero. Astronomers in Mesopotamia did. “Zero not only represents nothing but it also represents the starting point of anything” said Brahmagupta.
The most confusing part of math is the incoherent relationship among various approaches that describe the same phenomenon, or solve the same problem. Take addition and multiplication. Romans added with cross-hatches. It served the empire well for centuries. Multiplication enhanced addition in Babylon where the merchants had large and complex calculations pre-solved on clay tablets. Just because each method solves the same problem does not require them to remain logically coherent. Common arithmetic is incoherent across the logical boundaries. This is why each type of math starts by learning rules. Addition and multiplication, for example, have logical gaps. After all, is not 3+3+3 the same as 3×3 ?
Many math jokes come from the illogical boundaries. Most people have heard the old riddle: “Three people check into a hotel. The clerk tells them that the bill for their stay is $30, so each person pays the clerk $10. The clerk puts the money in the cash register.”
“Later that night, the clerk realizes that she made a mistake and should have only charged the three guests $25. She takes five one dollar bills from the register and tells the assistant to return the money to the guests.”
“On the way to the rooms, the assistant realizes that she cannot split the money evenly among the three people. As the guests don’t know that they were charged the incorrect amounts for their rooms, she decides to simply give them each $1 and pocket the extra $2 as a tip. Each guest gets $1 back, so each paid $9 for their room, ($9 X 3 equals $27). The bellhop kept $2, and $27 + $2 = $29.”
“But the guests originally handed over $30. What happened to the missing dollar?”
The answer is that there is no “missing dollar.” It is an informal fallacy caused by jumping from addition to multiplication at the wrong time and breaking the logical boundary. The different math languages, addition and multiplication,were not developed at the same time by the same people. They have different rules for consistency.
Addition was first used extensively by the Chinese almost 6,000 years ago, so it is safe to say the Chinese invented addition.1 The ancient Babylonians were probably the first culture to create multiplication tables, more than 4,000 years ago.2
Multiplication was a way for merchants to do complex additions quickly by looking at the multiplication calculations on clay tablets, multiplication tables. Today most people get to the same result either by addition or multiplication; however, each method has firm rules. These rules were violated in the joke, so the result seems to reveal a magical dollar. The method of calculating the total is simply flawed. In the end there is still $25 in the register, $3 returned by the desk clerk, and $2 appropriated by the assistant. 25+3+2=30. Right? Addition. Begin there, end there.
Math is not all just fun. Math has natural uses as a tool. Most tasks can use math. This is just common sense. All jobs involve math. Any person that handles money should instantly see through the problem above and solve it by saying in the end that there is still $25 in the register, $3 with the guests, and $2 with the assistant. $30.
The target is math fluency.
How, then, do you teach math in the homeschool. What sequence should math be taught? What if the homeschool teacher is not a math nerd? Schools usually teach math from simple calculation to increasing complexity. The assumption that the ability to solve more complex problems comes with each math is pretty good, but it need not be the rule at your school. Nothing says that geometry should not come when the student is ready for carpentry, or precision drawing. The homeschool should teach appropriate math when an immediate utility can be seen by the student. Math does not have to be a bleak and dreary time of the day. In the homeschool math can arrive as it becomes organically necessary. Walk, talk, read, write, learn numbers and then counting. Gradually. Then adding things together. Then taking things away. Then dividing things and multiplying them. Eventually, the Mandelbrot sets.
Older students can master household math. Cooking is great to begin the task of learning fractions. At this time various units of measure can also be introduced.
Household accounting with budgets and banking introduces decimals and set theory as soon as the child can learn about money. Mortgages and interest can introduce more complex math through ideas like compound interest.
Geometry is good for crafts and projects, carpentry and landscaping. Any good carpenter must understand angles and lengths. Every math involves memorizing rules. Euclid is difficult for some because all the rules are learned at once and an understanding of each application is not introduced as an example. Memorization is nowhere more important than geometry. Once the rules are mastered, all future woodworking projects are understood, can be planned, designed and completed.
Every science class is an opportunity for another set of advanced math tools. Each of the sciences requires understanding specific advanced mathematical languages. Trigonometry is vital in the earth sciences and in video games. Something as simple as planning the drainage in the back yard is a trigonometric opportunity.
Chemistry allows an understanding of all the middle and advanced mathematical languages, and so on.
A good home study course would begin with the foundational math from the basic arithmetic, fractions, percentages to equations, functions and graphs. The full homeschool course would include relative fluency with algebra, geometry, trigonometry, through differentiation, integration and vectors. In the end it should have included complex numbers and matrices. All this can be had from the thousands of sites on the Internet or through any number of very fine homeschool math packages.
The logical calculus is not taught early enough in the US. Typically, public school requirements for math studies may not even include the courses of study above, but if they do, they usually end with precalculus. Homeschool teachers should set the goal beyond this to be competitive with schools that provide most of the technical staffing in the us. The classes usually labeled AP Calculus.
If you begin to teach math at home, a good target is functional mastery of the fundamentals tested in the college placement exams for better schools. The sequence of learning and the context for teaching do not matter. The checklist of material students need to pass the college placement tests are, according to Derek Owens:3
- The problems that Calculus solves, introduction to derivatives, finding rates of change from graphs, from equations, and from data, Numerical derivatives, Introduction to Integrals, Approximating integrals from graphs, from equations and from data, the Trapezoid Rule
- A graphical approach to limits, Describing function behavior with limits, Asymptotes, Rational Functions, Polynomial end behavior, The Limit Theorems, Evaluating limits, Continuity, The Intermediate Value Theorem
- A graphical look at derivatives, Difference Quotients, the Derived Function, Notation, Numerical calculations of derivatives, Tangents and Linear Approximation, Differentiability and Continuity, the Chain Rule, the Product Rule, the Quotient Rule, Leibniz’ Proofs, Derivatives of Trig Functions, Implicit Differentation, Derivatives of Inverse Functions, Derivatives of Inverse Trig Functions
- The Extreme Value Theorem, Rolle’s Theorem and the Mean Value Theorem, First and Second Derivatives, Concavity and Inflection Points, Graphs and Curve Sketching, The Calculus of Motion, Max-Min problems, Related Rates, Practice
- Antiderivatives, Integrals, Infinitesimals, Riemann Sums, Definite Integrals, The Fundamental Theorem of Calculus, Properties of Definite Integrals, Numerical Methods, Integration by Substitution, Average Value
- Derivatives of exponential functions, Derivatives of logarithmic functions, Derivatives and integrals of base b exponents, Integrals with variable limits, Logarithmic Differentiation, Integrals of trig functions, Intro to Differential Equations, Examples and applications, Slope Fields, Euler’s Identity
- The area of a plane region, The Calculus of Motion, Real world applications, Integrating to find volumes, Plane Slicing, Solids of Revolution, Cylindrical Shells
Cover these areas before matriculation and the periodic achievement tests required for your homeschool will not trouble you or your student. Most homeschool parents are not fluent themselves in the components of pre-calculus and calculus. Look for getting help on any bullet in the list that seems a mystery. This is essential for getting the better technical jobs or getting into the better programs at university.
Fluency in higher mathematics is not critical for learning computer science, but helps.
Computer literacy should begin early with some computer language suited for young programmers. For example, Scratch is a free programming language for kids. It was developed by MIT’s Lifelong Kindergarten Lab. Blockly is Googles’ offering. Many languages like these can begin in kindergarten. Once the hook is set, more complex projects can be developed in Python, a free programming language every adult with a computer should and probably does not know. This really opens up the project world of companies like Adafruit. This is the world of Maker science and technology. Projects like this can alleviate the relative tedium of learning abstract math.
The goal is math facility.
The little human should be able to use math tools at every juncture of life. Not everyone will find a vocation that uses discrete math or calculus, but it is not a door that the homeschool should shut on young minds. It is easy to teach. Everything can be taught online. Everything can be taught from the myriad Goodwill books on math. Remember, something invented in the 17th Century is just as accurate in an earlier edition of the textbook.
Using math should be as natural as reading and writing. One student may never master geometry but discover a passion for statistical analysis and the logical calculus. She may not grow to be a civil engineer, but a computer programmer instead.
Exposure to and a grasp of the basics in all the fundamental mathematics is critical for success in modern technical society. Math exercises memory, logical application, and problem solving. Sure, math is hard, but every new thing in life is hard. The homeschool teacher must make math as organic to the day as any other lesson. If you have a math wizard, you may find that by the time he or she graduates, you have learned enough discrete math to untangle algorithms yourself.
3 This list derives from a course developed by Derek Owens who graduated from Duke University in 1988 with a degree in mechanical engineering and physics. He taught physics, honors physics, AP Physics, and AP computer science at The Westminster Schools in Atlanta, GA from 1988-2000.