If Only I Could Remember! (and how Coherence can help)
by Donna Curry
From my elementary school history class, I remember, “In 1493, Columbus sailed the big blue sea.” From my science class, I memorized the colors of the spectrum because I remember ROY B GIV (Red, Orange, Yellow, Blue, Green, Indigo, Violet). And, from my math classes, I can still recall PEMDAS – Please Excuse My Dear Aunt Sally… or was that PFMNS (Please Forgive My Niece Sally)? Let’s see: (6 + 4)/5 + 3(2) – 1. Parentheses [6 + 4], then Fractions [10/5], then Multiplication [3(2)], then I do my Negatives [6 – 1] and last I do my Sums [2 + 5] for an answer of 7.
OK, maybe it isn’t PFMNS, but that makes just as much sense as PEMDAS (also known as Pink Elephants Destroy Mice And Snails!). Students who come from other countries may have been exposed to similar math mnemonics like BEDMAS or BIDMAS or even BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction). Heck, creative types have even come up with rap songs to remember the order of operations!
These strategies are ways to remember the order in which to do calculations: First, tackle what’s in parentheses, then any exponents, then multiplication and division, and lastly, addition and subtraction. But, if I use the mnemonic, and especially if I’ve only learned what the letters stand for and not the understanding behind those words, I’m likely to solve a problem like 7 – 3 + 4 incorrectly.
Why? Well, with PEMDAS I’d probably get an answer of 0, because I see that addition has to come before subtraction. After all, that’s the rule—Aunt comes before Sally! But that’s all I know about it. As you can see, PEMDAS may have its place, but not when used as an isolated memory tool without any conceptual understanding behind it.
Coherence Across the Levels
I started thinking about all the tricks and mnemonics we teach our students when I read about coherence, one of the key shifts in the College and Career Readiness Standards for Adult Education (CCRSAE). Coherence is about making math make sense. Put another way: “Mathematics is not a list of disconnected tricks or mnemonics. It is an elegant subject in which powerful knowledge results from reasoning with a small number of principles such as place value and properties of operations. The standards define progressions of learning that leverage these principles as they build knowledge over the grades.” [See www.achievethecore.org]
Wow, “a small number of principles”? That’s not what most of our students think about math. . . and definitely not what I thought about it as I was learning all my rules and procedures, tricks and mnemonics.
The CCRSAE are actually quite coherent (although this is sometimes a bit challenging to visualize). However, you can get a better sense of what coherence looks like if you download our handy overview which was specifically designed to show that coherence. And since we’ve been talking about order of operations, which involves operations and properties of numbers, let’s see what coherence looks like across levels.
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.) (1.OA.3)
Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. (3.NBT.2)
What if we started showing our students early on that they could use parentheses to organize what they are doing when they decompose? Take for example the problem:
31 – 15 = (30 + 1) – 15
Using the associative property of addition, and we can show 30 + 1 = (20 + 10) + 1 = 20 + 11 to explain how borrowing works. Another approach could be to create an easy-to-subtract number in place of the 15. For example, by adding 5 to the 15 we get 20, but to keep the subtraction situation the same (the same difference between the two amounts), we have to also add 5 to 31 and get 36, and see that 36 – 20 = 16. But how does this work? We are using the additive inverse property (a – a = 0) this way: 31 + (5 – 5) – 15 = (31 + 5) – (5 + 15) or 36 – 20, an easy subtraction problem. While the addition procedure of (-5) + (-15) = (-20) may recall algebra, it is intuitive to explain that subtracting 5 and then subtracting 15 is the same as subtracting 20 all at once.
Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30. (Associative property of multiplication.) 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. (Distributive property.) (3.OA.5)
The distributive property is a way to solidly begin to teach the order of operations. Discovering which operations work with the commutative and the associative property is also part of the initial understanding of the order of operations. If students were exposed early on to the distributive property, maybe they wouldn’t have to learn about Dear Aunt Sally since they would realize that they could either do what’s in the parentheses first, or not: 7(3 + 4) = 7(7) or 7(3) + 7(4)
If either of these processes is correct, and if we taught our students this important principle, why would we tell our students that they have to do what’s in the parentheses first?
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 x (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. (6.EE.3; p. 66)
If we want our students to be successful at Level C (much less Levels D and E!) we have to build on what they should have been exposed to in Levels A and B.
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. (4.NF.3b; p. 62)
If we built on the properties with whole numbers, students could more readily deal with fractions. For example, this is similar to the earlier whole number example where numbers were decomposed and then regrouped based on the associative property:
Example: 3 1/3 – 1 2/3
Let’s look at alternative equivalent representations of the 3 1/3 in the problem:
3 1/3 = (3/3 + 3/3 + 3/3) + 1/3 or (3/3 + 3/3) + (3/3 + 1/3) or (3/3 + 3/3) + 4/3
If we choose to represent the 3 1/3 as 2 4/3, the new problem (2 4/3 – 1 2/3) is now much easier to solve, and I did it using number properties rather than a seemingly new strategy for borrowing with fractions. Here is another way to use equivalent fractions to add or subtract mixed numbers. Since 3 1/3 = (3/3 + 3/3 + 3/3) + 1/3, we know an equivalent representation is 10/3. Similarly, 1 2/3 = (3/3 + 2/3) = 5/3. So, 10/3 – 5/3 = 5/3 or 1 2/3. In this method, there is no borrowing at all.
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. (7.NS.2a; p. 71)
As you can see, each progressive level builds on previous understanding of some core properties of number and operations. The CCRSAE help guide us in what to teach when and even provide some examples of how to teach those standards. I challenge you to closely review the CCRSAE yourself. Create your own ‘progressions’ to help you see what grounding students should have before you try to add on a new skill to their often weak foundation. Think of ways to begin to infuse your typical teaching with core basic principles. Ask students to explain why an algorithm works so maybe they will actually remember the rule or procedure later.
A version of this article first appeared in The Math Practitioner (V18.3. Fall 2013).
Editor’s Note: You can read more on one teacher’s view of all the rules we teach – and what happens when they get to develop here.
Donna Curry is the Director of the SABES PD Center for Mathematics and Adult Numeracy, a project managed by the staff of the Adult Numeracy Center at TERC. She has trained teachers nationally, and has taught and administered ABE classes for over 30 years.