In the course I am currently teaching at Drexel University, we have been focused on the importance of addition and multiplication properties as students progress through mathematics. Particularly the idea of inverse numbers (additive inverses and multiplicative inverses),the additive and multiplicative identities, and the commutative and associative properties of addition and multiplication. A strong foundation in these concepts, which starts in elementary school and builds as students progress to more abstract and complex math concepts such as proportional reasoning, solving equations, composition of functions, and working with matrices, is really important. In fact, if we spent more time using precise language and justifying our reasoning with properties consistently, as we model and help students learn and discover, there would be a lot less confusion and much more connection of prior knowledge to ‘new’ concepts. Instead, we often provide a short-cut, or a ‘trick’ (with a cute acronym like KSP (keep, switch, flip) or ‘Cross-multiply-divide’ with no basis in the true mathematics. Students focus on memorizing isolated rules versus connecting new concepts and seeing learning as just an extension of prior knowledge.
Let me try to explain what I mean by providing a sense of prior knowledge and how it connects to more abstract concepts:
- Additive identity: 0 and the Additive Identity Property: a + 0 = a (5th/6th grade)
- Additive inverses create the additive identity – so -b + b = 0 or -c + c = 0 (5th/6th grade)
- Multiplicative identity: 1 and the Multiplicative Identity Property: b*1=b or 1*b=b (5th/6th grade)
- Multiplicative inverses create the multiplicative identity (i.e. a 1) = d * 1/d = 1 or -1/f *-f = 1 (5th/6th grade)
- Addition and multiplication are commutative (switch the order and you get the same solution) (1st/2nd grade) and associative (switch the grouping and you get the same solution) (3rd grade)(this explains why we want to change subtraction to addition of the additive inverse number, and why we change division to multiplication of the multiplicative inverse (reciprocal) – so we can USE THE PROPERTIES!!!
Understanding the above, then makes solving equations easier – and we don’t need to avoid equations with fractions or decimals, because the properties apply to these rational numbers as well.
Example: -5 = (1/3) x – 8
- Change the problem to addition of the additive inverse: -5 = (1/3)x + (-8)
- Add 8 to both sides (commutative property – can add in any order) because adding additive inverses (8 and -8) make zero (additive identity property) -5 + 8 = (1/3)x + (-8) + 8
- Group the inverses (associative property) and solve: -5 + 8 = (1/3)x + (-8 + 8) which is equivalent to 3 = (1/3)x + 0 equivalent to 3 = (1/3)x
- Use the multiplicative inverse property (multiplying by the reciprocal will create a 1) and multiply by 3/1 on both sides: (3/1)*3 = (3/1) *(1/3)x (commutative property allows us to multiply in either order on both sides).
- The multiplicative identity property says 1 times any number is itself, so we end up with 9/1 = 1 *x or 9 = x
*Note – we did not use subtraction or division at all – we used the understanding of inverses, identities, and addition/multiplication properties to explain. No tricks, and working with actual numbers (so fractions and integers) with justification for all steps.
Example: Solve the proportion 3/16 = x/20
- This is really an equation where the quantity x is being multiplied by 1/20. Understanding that I can use the multiplicative inverse to multiply by the reciprocal to make a 1, I multiply both sides by 20/1:
- 20/1 * 11/12 = x/20 * 20/1 (commutative property lets me multiply in either order on both sides)
- I can even decompose my multiplication and think about making ones through the same understanding: 4*5*3 /4*4= 1*x
- 15/4 = x/1 or x = 3.75 (multiplicative identity)
- Note – the trick we often tell students to memorize is ‘cross-multiply and divide’, but if instead we focused on just applying their understanding of multiplicative inverse and making those 1 pairs, there would be less confusion, less forgetting the ‘trick’, and less applying that trick to other problems where it is in appropriate.
Obviously I can’t demonstrate a whole course of study in one blog post – what I am really emphasizing here is how important consistent mathematical vocabulary and use of properties is, instead of acronyms, short-cuts, tricks, mnemonics, etc. that we often give students with no basis in understanding. Instead of seeing math as a connected whole, building on to prior knowledge as they move through the grades and topics, we treat it as isolated topics with no connection. It’s no wonder students think every year they are learning something new. If last year when they worked with division of fractions their teacher taught them to “Keep, Change, Flip”, and this year the teacher is talking about Ketchup Covers Fries or KSF….no wonder they are confused. None of these are grounded in the properties and vocabulary of mathematics.
What we should be doing instead is focus on applying properties and using the mathematical language/vocabulary/properties right from the very beginning and ALL THE TIME. So instead of disconnected acronyms of KSF or KCF, they focus on extending their understanding of additive inverse, inverse operations with the inverse number and division of fractions ends up being just an extension of what they did with subtraction of integers – i.e. use your inverse operation with the inverse number. So dividing with rational numbers is just multiplication (inverse operation) by the multiplicative inverse (i.e. reciprocal), similar to subtraction being addition (inverse operations) with the additive inverse (opposite signed number) – same general idea, same vocabulary, and just building on prior knowledge.
Let’s stop dumbing down mathematics and use the words and properties that truly allow students to connect and look for those patterns and develop their own understandings and rules. Let’s get away from tricks and mnemonics as our ‘teaching’ method – instead, let students figure that out themselves through the use of precise math language and application of properties. Let’s start in elementary school. Use precise mathematical language (along with clarifying words of course, but always with (not instead of) proper mathematical language/vocabulary/properties).
Think about it – we wouldn’t change the Spanish word for grandmother (abuela) or the French word for bread (pain) to other words, because then how would we communicate and be understood by others speaking those languages? Why is it okay to change the words or use different words or tricks, instead of the using the math language and properties? No wonder students are often so confused or why teachers think they have to ‘reteach’ things every year – if we are not consistent with students in using mathematical language, we are in fact talking a different language to them. No wonder they so often seem lost and frustrated.