From time to time I’ve helped out in the clinic with summer students who weren’t doing well in their math classes. I was astonished at their lack of basic arithmetic skills. They couldn’t do simple divisions, so they couldn’t reduce fractions. They couldn’t add two numbers in their head, so they couldn’t simplify equations. Junior high level math was a mystery to them because they hadn’t learned the basics. The arithmetic exercises in this section are designed for students to learn and practice elementary addition, subtraction, multiplication, and division. Once these skills are mastered they will have a foundation for understanding mathematics.
When I taught business math for college freshmen there were lots of students in the same situation as the kids in the clinic. They just wanted to learn the formula for the day’s exercises and didn’t care about the math. Math was hard for them and involved a lot of memorization because they had never learned the basics of manipulating equations. They didn’t get that if, for example, you were working with interest rates, you didn’t need to memorize a different formula for calculating interest, the payment, and the time because there is only one formula and you just needed to use a little bit of algebra to rearrange it. You don’t need to memorize a bunch of special cases—just understand one formula and you can work many different problems. For example:
I = Prt where I is interest, P is principal, r is the periodic rate and t is time in months.
If you know any three items, you can solve for the fourth. You don’t have to learn four formulas. What you do have to do is learn to reduce fractions. And you can’t do that quickly and accurately unless you know by heart the multiplication and division tables. (It does get a little more complicated with compounding and figuring out the total payments but if you understand the basic formula for interest calculation, it is an easy extension.)
Once you understand the basics you can extend your knowledge to other examples relatively easily. The physics equations F=ma and E=mc2 are exactly same problem mathematically as the interest rate problem. You can solve for any of the variables by rearranging the formula in exactly the same way you rearrange the interest rate formula.
Percentage change in your client load month-to-month is a real-world example. You can memorize the formula or know that:
If TM= this month, LM = LM, and r = rate of change (positive or negative).
TM = LM + rLM
Rearrange it a bit
- TM = LM (1 + r)
- TM/LM = 1 + r
- TM/LM - 1 = r
If you want it in percent then multiply by 100.
In order to do this you need to have memorized the distributive law of multiplication, the fact that you can divide both sides of an equation by the same (non-zero) number, and that you can add the same number to both equations. If you haven’t memorized these rules, you are reduced to memorizing formulas for each part of the equation you want to solve for. But more importantly, you can’t generalize.
A huge amount of higher level math relies on rearranging and reducing equations. If students haven’t memorized the basic multiplication and addition tables then there is no way they can do more advanced math.