OK, maybe it isn\u2019t 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<\/a> 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<\/a>!<\/p>\n These strategies are ways to remember the order in which to do calculations:\u00a0 First, tackle what\u2019s in parentheses, then any exponents, then multiplication and division, and lastly, addition and subtraction. But, if I use the mnemonic, and especially if I\u2019ve only learned what the letters stand for and not the understanding <\/em>behind those words, I\u2019m likely to solve a problem like 7 \u2013 3 + 4 incorrectly.<\/p>\n Why? Well, with PEMDAS I\u2019d probably get an answer of 0, because I see that addition has to come before subtraction. After all, that\u2019s the rule\u2014A<\/strong>unt comes before S<\/strong>ally! But that\u2019s 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.<\/p>\n 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: \u201cMathematics 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.\u201d<\/em> [See www.achievethecore.org<\/a>]<\/p>\n Wow, \u201ca small number of principles\u201d? That\u2019s 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.<\/p>\n 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<\/a> which was specifically designed to show that coherence. And since we\u2019ve been talking about order of operations, which involves operations and properties of numbers, let\u2019s see what coherence looks like across levels.<\/p>\n 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:<\/p>\n 31 \u2013 15 = (30 + 1) \u2013 15<\/p>\n Using the associative property of addition<\/em><\/strong>, and we can show 30 + 1 = (20 + 10) + 1 = 20 + 11 to explain how borrowing works.\u00a0 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<\/em> between the two amounts), we have to also add 5 to 31 and get 36, and see that 36 \u2013 20 = 16. But how does this work? We are using the additive inverse property<\/em><\/strong> (a \u2013 a = 0) this way:\u00a0 31 + (5 \u2013 5) \u2013 15 = (31 + 5) \u2013 (5 + 15) or 36 \u2013 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.<\/p>\n The distributive property<\/em><\/strong> 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\u2019t have to learn about Dear Aunt Sally since they would realize that they could either do what\u2019s in the parentheses first, or not<\/em>:\u00a0\u00a0\u00a0\u00a0 7(3 + 4) = 7(7)\u00a0\u00a0 or\u00a0<\/em>\u00a0 7(3) + 7(4)<\/p>\n 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<\/em><\/strong> do what\u2019s in the parentheses first?<\/p>\n 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.<\/p>\n 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:<\/p>\n Example:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 3\u00a0 1\/3 \u2013 1\u00a0 2\/3<\/p>\n Let’s look at alternative equivalent representations of the 3\u00a0 1\/3 in the problem:<\/p>\n 3 1\/3 = (3\/3 + 3\/3 + 3\/3) + 1\/3\u00a0\u00a0 or \u00a0 (3\/3 + 3\/3) + (3\/3 + 1\/3)\u00a0\u00a0 or \u00a0 (3\/3 + 3\/3) + 4\/3<\/p>\n If we choose to represent the 3\u00a0 1\/3 as\u00a0 2\u00a0 4\/3, the new problem (2\u00a0 4\/3 \u2013 1\u00a0 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.\u00a0 Since 3\u00a0 1\/3 = (3\/3 + 3\/3 + 3\/3) + 1\/3, we know an equivalent representation is 10\/3. Similarly, 1\u00a0 2\/3 = (3\/3 + 2\/3) = 5\/3. So, 10\/3 \u2013 5\/3 = 5\/3 or 1\u00a0 2\/3. In this method, there is no borrowing at all.<\/p>\n 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 \u2018progressions\u2019 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.<\/p>\n Editor’s Note:<\/em> <\/strong>You can read more on one teacher’s view of all the rules we teach \u2013 and what happens when they get to develop here<\/a>.<\/p>\n =======================================================================<\/p>\n <\/p>\n 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 AV°ÍÊ¿<\/a>. She has trained teachers nationally, and has taught and administered ABE classes for over 30 years.<\/em><\/p>\n Save<\/span><\/p>\n Save<\/span><\/p>\n Save<\/span><\/p>\nCoherence Across the Levels<\/strong><\/h4>\n
LEVEL A<\/span><\/h5>\n
Apply properties of operations <\/em><\/strong>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)<\/strong><\/h5>\n
LEVEL B<\/span><\/h5>\n
Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, <\/em>and\/or the relationship between addition and subtraction. (3.NBT.2)<\/strong><\/h5>\n
Apply properties of operations <\/em>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)<\/strong><\/h5>\n
LEVEL C<\/span><\/h5>\n
Apply the properties of operations <\/em>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)<\/strong><\/h5>\n
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 <\/em>and the relationship between addition and subtraction. (4.NF.3b; p. 62)<\/strong><\/h5>\n
LEVEL D<\/span><\/h5>\n
Apply properties of operations <\/em>as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. (7.NS.2a; p. 71)<\/strong><\/h5>\n
A version of this article first appeared in The Math Practitioner (V18.3. Fall 2013).<\/em><\/h6>\n
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