The above math “problem” caused a sensation on Twitter recently with strong arguments about the “solution.” Depending on how you go about solving it, you could get either 1 or 16 as the answer. This is a perfect illustration of the insanity of trying to mix algebra with arithmetic. They are two different mathematical languages with two different sets of rules for solving problems. When you mix the two, all you get is confusion as illustrated above. The mathematical symbol for division ÷ is only used in arithmetic and the parentheses ( ) are only used in algebra. In arithmetic, the rule is to work the problem from left to right, but in algebra, you work the problem according to the rules of algebra which tell you to combine the terms inside the parentheses first whenever possible. If you solve this problem from left to right, you get an answer of 16. If you solve it algebraically, you get an answer of 1. This is why Pre-Algebra, which mixes the notation of arithmetic with the notation of algebra causes so much confusion.
Since the late 1960s Americans have been unwitting subjects in a vast experiment in math education called Pre-Algebra. In truth, there is no such mathematical discipline as Pre-Algebra. There is Arithmetic and there is Algebra. However, beginning around 1970, California math textbooks began mixing the two and calling it the “New Math.” It was already a disaster in 1970 when I started teaching, but that did not deter the continued implementation of this nonsensical program. Those who learned the two separate disciplines (Arithmetic & Algebra) in school are now grandparents. No one younger than 55 is likely to even know that algebra and arithmetic are distinct mathematical languages and can only be clearly understood if learned as separate disciplines. The New Math has gone through several versions in the last 60 years, but it always moved toward more confusion. The label “pre-algebra” was introduced about 30 years ago as more and more algebraic notation was introduced to younger and younger children.
The rationale for introducing elements of algebra in first grade (and now Kindergarten) was that children would gradually become used to algebraic notation and therefore would not be intimidated when they later encountered full-on algebra, which at that time was seldom introduced before 9th grade. There are two problems with this thinking. First, young children are concrete thinkers. They do not move toward abstract mathematical thinking until sometime between ages 12 and 15. This difficulty can be partially overcome by using concrete objects like Cuisenaire Rods or Math-U-See blocks to solve the math problems. However, the second problem, that of trying to integrate two different mathematical languages, has no real solution. In fact, it is the basis for nearly all the difficulty students today are having with math. The solution is simple: teach arithmetic first (addition, subtraction, multiplication, division, fractions, decimals, percents) and then proceed to real algebra.
For a more detailed examination of this topic, including recommendations, see http://excellenceineducationhomeschooling.com/math-madness/.