The Way the Cookie Crumbles: Are We Really Teaching Math Conceptually?
By Joel Thomas, Rubicon International
You don’t have to attend many PD sessions as a math teacher before you are told that you should be teaching conceptually and not procedurally.
The reasoning is (mostly) sound! [inlinetweet prefix=”” tweeter=”” suffix=””]We want to provide context for student learning so that they retain it longer, and they see how their Math work looks in the real world.[/inlinetweet] Possibly, we want students to get more practice with word problems. Because of these desires, we see the inclusion of more project and inquiry based approaches to learning and more word problems being taught in the classroom.
But, what is really changing conceptually in our instructional approach?
A Tale of 4 Cookies: Teaching Math Conceptually
In an NCTM session called ‘Three Critical Components for Building Bridges Between Concepts and Procedures’, Juli Dixon provided examples of all the ways we pay lip-service to teaching conceptually but actually stay locked in procedures. During her session, Juli offered the following example, which could be used to teach skills around manipulating fractions:
Then, she explained that this question alone is merely procedural, and added the underlined portion.
Immediately, this question becomes one of concept and meaning. Next, she demonstrated how students could easily be shown that these 4 cookies have to be broken down in to halves so no one gets stuck with cookie shards!
But, what do we do with the remaining cookies?
Anticipate the error! 1/5 may look correct, but those pieces are the size of 1/20 of the whole cookie!
She records her numbers as fractions. Clearly a lot of instruction goes into deriving these fractions, but let’s keep with the concept. HERE’S where is starts getting fun!
You see, it’s usually at this point that our “conceptual” teaching strategies stop: you have a cookie, you found the fractions, now we can do “real” math. But, this is actually where the conceptual math becomes so important, and why so many students don’t connect concepts to mathematics.
Usually, once students have written out the numbers, we start using terms like numerator and denominator, least common denominator, or cross multiply and divide. But, once we begin using this language, we’ve stopped talking about cookies! And really, aren’t cookies what we’re all about?
It’s at this point, that we MUST continue discussing cookies in order to solidify the conceptual nature of the lesson. To achieve this, here’s how the teacher’s side of the rest of this lesson might go.
- Okay, so I have a half of a cookie, 1/4 of a cookie, and 1/20 of a cookie. However, if someone asks me how much cookie I have, I can’t just list off those numbers. They just want to know the answer to one question: how much cookie did I get in the end? To figure that out I’m going to measure my bigger cookie pieces using my smaller cookie piece! Cookies, after all, make the best rulers!
- How many of these 1/20 sized cookies can fit in my 1/4 sized cookie piece? This is a great opportunity to model and manipulate a real cookie.
- Good, we can fit 5 of the 1/20 sized cookies into my 1/4 sized cookie; therefore, we know that ¼ of a cookie is equal to 5/20 of a cookie. So now, I have 1/20 of a cookie and 5/20 of a cookie.
- Now how many 20th sized pieces can I fit into my half-sized cookie piece?
- Counting it out, I see that 10 of my 1/20 sized cookie pieces fit into my half-sized cookie piece. Now I know that ½ of a cookie is equal to 10/20 of a cookie.
- Now, I still have three groups of cookies, bu they are all the same size. So, I can give one answer to my earlier question: by combining 1/20, 5/20, and 10/20, I can see that in total, I get 16/20 of a cookie.
Due to time constraints in and outside of class, it’s not reasonable to do a task like this every day. However, it is crucial to always begin new material teaching conceptually before moving into the vocabulary and procedure. Now that I have completed the cookie example, I can reference it when talking about numerators, denominators, manipulating fractions, and (if you were paying attention) even advanced concepts like the commutative and associative properties. And, any English or Science teacher will tell you that vocabulary is most likely to stick when students have a need to use the term. By introducing the concept first, you’ve created that need!
The only other factor keeping you from using cookies to teach all your lessons is your health! Send us your favorite word problems for conceptual teaching!