It’s Inevitable

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Let’s face it…students coming to us using k-c-f (invert and multiply) to solve division of fraction problems is inevitable. There are too many teachers who rely upon this method as the sole method for dividing fractions. From my experience with teachers, many of them believe you cannot divide fractions, you can only multiply them, which makes invert and multiply a logical method to use.

Does this mean you should continue on with this trend of misunderstandings when students use this within your classroom?  If you’re teaching for correct answers only, I’m sure there was a resounding “yes!” to my question. However, if you teach for understanding which allows students to build upon their knowledge, apply concepts to new information and see patterns and structure you probably paused to ponder what other methods are there.

The black hole of resources within nzmaths (Nzmaths: Lesson on Dividing Fractions) houses a wonderful set of lessons that help guide students through developing an understanding of dividing fractions. Towards the end of the sessions students are encouraged to look for patterns when dividing whole numbers by fractions.

Two things I added when implementing these sessions with students were beginning with 1 whole and asking students to create a visual representation.

Because many teachers do not conceptually understand the invert and multiply method, I will not rob you of an opportunity to make sense of it for yourself. Try it out and leave a comment.

Never Wave My Flag

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Don’t get me wrong, procedural fluency is very necessary. However, if procedural fluency is your only means of understanding you have a huge problem. This problem extends pass performing well on a state assessment. When procedures fail you, what do you have to rely on but conceptual understanding. An Englishman put it this way, “The only place procedural understanding brings success with problem solving is in American classrooms. In all other cases, we must rely on our conceptual understanding.” (I hope I quoted you accurately Phil.).

So when I discovered that majority of my students were procedurally fluent with converting fractions to decimals and didn’t understand at all why the rules they applied so successfully even worked, I couldn’t turn a blind eye. In the midst of feeling like I was ready to give into the ways of rules and tricks without understanding, I took a stand. I would not give up on them and I would not give up on my beliefs.

This lesson was implemented.

I thought it would be the bridge to understanding for my kids. I figured if I started from the bottom, the basics, with modeling things would click. What I discovered was they had minimal experience with modeling. Most didn’t know where to begin with setting up a model to demonstrate division. They didn’t get the idea of the physical manipulation of the blocks showed the division. Instead they thought they needed to represent the dividend, divisor as well as the division symbol. I also discovered they were limited in their reasoning abilities due to desire, experience or cognition. Needless to say, of the papers I’ve reviewed, no one was able to make the connection between why the rule in which they are proficient works.

I’m not giving up though. My next steps will be more explicit. Not explicit in telling them why it works. But explicit in my questioning as we model the division of 1 divided by 4. “What can we do to physical put 1 whole into 4 groups?” Students should conclude the whole can be exchanged for ten tenths. Draw the model. “What number is represented by the model?” My goal is for students to see adding the decimal point and zero is the ten tenths needed to physical put 1 whole into 4 groups. So the problem doesn’t become ten divided by 4 but actually 10 tenths divided by 4. Which is still 1 divided by 4.

I know moving forward it will be important to spend time working with modeling with manipulatives . I’ll probably end up using some fifth grade tasks from the GA math frameworks to help activate prior knowledge through the use of modeling with manipulatives.

Hand On the White Flag

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Fighting the “good fight” can be a rewarding thing. To know you are a part of molding thinkers and mathematicians brings great feelings of a job well done. You’re fighting for all the little guys and gals that others may have been unequipped to do so.

When you are a middle school math teacher you have two options as I see it. You can carry on the tradition of rules and tricks minus the understanding. Or you can provide bricks for a shaky foundation put in place by the most well meaning teachers. I chose the latter. And so far this year, my students have not been resistant to the remodeling of their math thinking. Until today…

After conducting a number talk yesterday to discuss changing a fraction to a decimal, we as a class identified two strategies we could apply for this. One being, find an equivalent fraction with a denominator of 10 or 100 and the other divide the numerator by the denominator. We attempted to apply those strategies today bringing to light the old issue of being fluent with a procedure but not understanding why the procedure works.

Student: to change 5/6 to a decimal, you…well you can’t divide 5 by 6 so you put a decimal and zero. And 50 divided by 6 is…
Me: actually, you can divide 5 by 6 and we are getting ready to do it.
Student: well just add a decimal and zero
Me: Let’s think about what number we can multiply 6 by to 5.
Random students: 1. 0. -1

I inquired why do we need to add a decimal and zero and the only response given was, “Because you can’t divide 5 by 6.” Twenty minutes we spend on looking at numbers that would give us less than or just about 5. Many students lack the concept of fractions and decimals being part of a whole. Some wouldn’t budge from their add a decimal and zero rule and in turn decided not to make sense of the quantities. I could tell there were many that had just checked out. Those bricks I was trying to help lay sat squarely on my chest. I put my hand on the white flag, grasped it tightly and had a private conversation with myself.

Was it worth the struggle the students were facing as their rules of old were challenged? Was it worth the terrible feeling of knowing your students, a large majority of them, just weren’t getting it? I wanted to say, “yes, we just add the decimal and zero” and follow the steps of long division as I waved my white flag.

I left work feeling defeated, conflicted, and lost on what to do. In deciding to go for a walk after having such a tough day, a song came to mind. “Started From the Bottom” by Drake which discussed his struggles coming up before making it big as a rapper. Then it hit me, start from the bottom, the basics and strategically work my way up to what number I could multiply 6 by to get 5. I’m working a bottom up lesson to implement tomorrow. I’m hoping this bottom up remediation approach will help students make the connection between the procedure and why it works.

Started from the bottom now we’re here
Started from the bottom now my whole team ________here
Started from the bottom now we’re here
Started from the bottom now the whole team here, _______
Started from the bottom now we’re here
Started from the bottom now my whole team here, _______
Started from the bottom now we’re here
Started from the bottom now the whole team ______ here

– Drake, “Started From the Bottom” lyrics