International assessments show that U.S. students tend to do worse in mathematics as they get older. The Trends in International Math and Science Study (2011), shows that U.S. students scored 41 points higher than the international average in 4th grade but only 9 points higher than the international average in 8th grade. And although being higher than the international average may seem like a good thing, that overall average includes Botswana, Kazakhstan, Tunisia, and some other countries we might not want to emulate. As for the high-performing countries, like Korea and Singapore, U.S. students scored more than 100 points lower.
The situation in high school is even worse. “The Nation’s Report Card,” the National Assessment for Educational Progress (2013), shows that 12th grade math scores are stagnant. Jack Buckley, head of the National Center for Education Statistics said, "When we start looking at our older students, we see less improvement over time.” In other words, the longer our students spend in math class, the worse they do! Why is this?
In elementary school, students learn arithmetic, which mostly involves addition, subtraction, multiplication and division of whole numbers, fractions and decimals. By 8th grade, they have begun to transition into algebra. According to the National Research Council (2001), “… algebra builds on the proficiency that students have been developing in arithmetic and develops it further. In particular, the place value numeration system used for arithmetic implicitly incorporates some of the basic concepts of algebra, and the algorithms of arithmetic rely heavily on the laws of algebra.”
If you ask an algebra teacher why many students struggle, he or she will probably tell you it’s because they don’t fully understand arithmetic. It’s not that they don’t know how to do arithmetic, but that they do not understand arithmetic, i.e. the properties of arithmetic. In other words, they do not know how to take numbers apart, move them around, and do creative and effective things with them. Since algebra involves the manipulation of abstract symbols (variables, coefficients, terms, etc.), students who are comfortable with manipulating and re-composing numbers tend to do well in algebra. Students who did well in math only because they could get correct answers may struggle when they get to algebra, due to a lack of facility with arithmetic manipulation.
In high performing East Asian schools, elementary students are taught how to manipulate numbers in creative ways as early as first grade. For example, even as they are committing math facts to memory, students learn various ways to decompose numbers and move them around in order to add, subtract, multiply and divide them. Although this is sometimes called “mental math,” it is more than merely learning how to calculate in your head. Take the following example of how adding 9 + 4 is taught to young students in Singapore and Japan.
Split 4 into 1 and 3.
Add 9 and 1 to make 10.
10 and 3 make 13.
Although this “making ten” method may seem odd if you’ve never seen it before, it is based on the associative property of addition, which tells us that we can regroup the addends in any way we want.
9 + 4 = 9 + (1 + 3) = (9 + 1) + 3 = 10 + 3 = 13
Of course, such a formalistic explanation would not be given to first graders. In fact, in Singapore math, formal vocabulary and the rules of arithmetic are not taught until middle school.
Here is another first-grade example involving the subtraction of 12 – 8 by subtracting from 10 first.
Split 12 into 10 and 2.
Subtract 8 from 10 to get 2.
2 and 2 make 4.
Again, this may seem strange if you’ve never seen it before, but it is how high performing countries approach these problems and it is again based on the commutative property of addition, which says that we can add numbers in any order, a very cool idea I might add, no pun intended.
12 – 8 = 10 + 2 – 8 = 10 – 8 + 2 = 2 + 2 = 4
The reason this subtraction it is based on a property of addition is that subtracting a positive number is the same as adding a negative number.
12 – 8 = 10 + 2 + (– 8) = 10 + (– 8) + 2 = 2 + 2 = 4
Young students can learn these properties informally, and by the time they reach middle school the formal explanations will be easier to comprehend because they have been applying them throughout their elementary careers.
Another example is the multiplication table of 8 (students often struggle with their 8s). If we look at the table and the corresponding arrays of dots, we notice an interesting pattern—we can put two facts together to make another one. For example, if we put a 4 x 8 array together with a 2 x 8 array we get a 6 x 8 array.
Thus, we can find 6 x 8 by adding 32 and 16.
We can also do this with other facts. For example, we can find 7 x 8 by putting 5 x 8 and 2 x 8 together. This is based on the distributive property of multiplication.
7 x 8 = (5 + 2) x 8 = (5 x 8) + (2 x 8) = 40 + 16 = 56.
Of course, the addition and multiplication facts must become known from memory during elementary school. But right now we are talking about the time before that, when 7 x 8 is a computation problem, rather than a known fact retrieved from memory.
Unfortunately, many people (even some math teachers) seem to think that mathematics consists solely of learning facts, formulas and procedures and getting the right answer. Facts, formulas, and right answers are essential, of course; it’s just that they aren’t enough. There’s no reason to think that fluency with algorithms alone will deliver kids into algebra with a readiness for what they will face there.
Because the powerful yet accessible mathematics that is common in East Asian elementary textbooks is absent from most U.S. textbooks, American students are at a distinct disadvantage when they get to algebra. This is beginning to change, however, with the advent of the Common Core State Standards. Take the following first grade Common Core standard (1.OA.3):
Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
Here is another example from the third grade Common Core (3.OA.5):
Apply properties of operations as strategies to multiply and
divide. Examples: … Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)
Statements like these are no coincidence, because the mathematics standards of high performing East Asian countries were an important source of evidence in developing the Common Core. (See the bibliography in the Common Core beginning on page 91.)
One of the main reasons for having coherent and thoughtful mathematics standards is to prepare students for success in algebra. But having coherent standards is not enough. These ideas must be incorporated thoughtfully and systematically into U.S. curricula and textbooks and, more importantly, American classrooms.
Of course, children also need to learn facts, procedures and formulas. The Common Core sets explicit expectations for knowing addition and multiplication facts, and computing fluently with the standard algorithms of arithmetic. But if this is all students learn, even if they do “well” in it, they are likely to reach the middle grades unprepared for algebra.