I recently attended a meeting in my state with the supervisors of mathematics from all the jurisdictions in the state. Under the leadership of the state mathematics supervisors, we are all grappling with the transition from our current mathematics standards to the Common Core State Standards. This transition is made more difficult because our state is in the PARCC consortia, which has released very little information regarding the assessments that will steer the new curriculum. The other stumbling block is the existence of the current testing program which is expected to last until 2013/2014. The current testing program assesses content on a lower, skill-based level than what is being hinted at on the new assessments. Principals are very reluctant to transition away from current standards if schools will still receive sanctions due to poor performance on the current tests. I guess every state is wrestling with the same dilemmas.
But, this wasn't the point of this posting. The point I'd like to address is the issue that concerns all of the supervisors; teacher content knowledge. Every time this group gets together the conversation always turns to the topic of professional development and how much content development needs to be done. If you're reading this and are not a math teacher, this is the point where you might get a bit scared, especially if you have children in school. Many teachers have a comfort level with the content they are teaching. They aren't necessarily fluent in the content that comes before or after their course. Is this a problem? It's a huge problem. When a teacher doesn't know how their content fits in the big picture of mathematics, it's difficult to make connections for the students. Many teachers that I've worked with have never taken the time to look back at the curriculum guide that preceded theirs or the one that comes after. Unfortunately, that means that many teachers are teaching every concept as if it's brand new without giving the students any credit for bringing prior knowledge to the table. The scenario that sends me up the wall goes like this: "Ms. Smith, last year my teacher showed me how to do it this way." "I don't care. That was last year and this year you have to do it this way." Heaven forbid that Ms. Smith should look at the methodology, determine it's mathematical significance, and if nothing else, acknowledge how that previous method connects to 'her way'.
With this in mind, I'll describe a recent school visit and a conversation I had with a 17-year veteran teacher. I was observing the teacher's class as they were practicing division of fractions, a subject that very few humans understand. (That's not a smart--- comment! It's true.) The teacher presented a problem and asked a student to show the solution. The problem was 13/12 divided by 10/12. The student went to the board and beautifully showed the algorithm of multiplying by the reciprocal and coming up with the correct answer of 1 and 3/10. (I'm realizing that it's difficult to blog when you need mathematical symbols.)
All was fine until a student raised his hand and said, "Weren't we working with twelfths? How can the answer have tenths?" The teacher responded, "That's just the way it works and that's how it simplifies." WHAT??? It was obvious that the teacher was not any more knowledgeable than knowing the algorithm. I guess that in 17 years she has never been asked that question before or else she's been answering it the same way for 17 years and never bothered to delve deeper into the "why". I won't veer off onto my typical rant about professionalism at this moment.
I had a chance to talk to the teacher after the lesson and asked her about the student's question. She admitted that she had no idea why the algorithm works. I first talked to her about the idea that both fractions in her example had common denominators. The problem was basically asking how many groups of 10/12 could be formed from 13/12. She could see that one group could be formed and she realized that there were 3 "of something" left over. The sticking point is what are those things that are left over. It seems that they are twelfths. But if you had 3/12 as a remainder, the answer would have been 1 and 1/4. Even when I drew a picture for her, it still seems that 3/12 remain. That's the misconception. The remainder to a division problem is always how many items are left based on how a group is defined. When working with whole numbers, most people can define the remainder. Many even know how to change the remainder to a fraction by placing it over the divisor. In this fractional problem, how is a group defined? A group is considered to be ten items. It just so happens that these items are called 'twelfths'. So, there are three items out of ten, making a remainder of 3/10. Students can see this when the fractions have common denominators. They also quickly determine their own algorithm for dividing fractions. They'll find common denominators, then the problem is nothing more than dividing the numerators and dividing the denominators. Because you have common denominators, when you divide them, the quotient is always one. That allows the students to realize that all they have to do is divide the numerators WHEN THE DENOMINATORS ARE THE SAME.
I've used a scenario when developing this concept with students that talks about a manufacturer and a packaging company. The first fraction defines what the manufacturer produces (in the original problem...twelfths). It also tell how many the manufacturer sends to the packaging company. The second fraction defines how many the manufacturer wants in one package (in the original problem...ten). The packager tells the manufacturer that he can make one package but only has three of the next ten to make another package. Students usually have no problem understanding how a problem that started with twelfths can end with tenths.
If students are more sophisticated, the standard algorithm can be developed using complex fractions; meaning a fraction in the numerator and the second fraction in the denominator. In order to simplify the complex denominator, change it to one by multiplying by the reciprocal. That forces the numerator to be multiplied by that same value to maintain equality. At least that explains why the standard algorithm works...not just making students memorize rules with no comprehension.
I'd like to be able to end this posting by saying the 17-year veteran had a clear understanding of division of fractions after our conversation but "the deer was still staring into the headlights" when our time was up. I did leave her by saying that I was always available to talk whenever she comes upon something in her curriculum that she only has a surface understanding of. I also encouraged her to look at the previous course curriculum guide. It just so happens that the previous guide develops the concept of dividing fractions using the common denominator algorithm (because I wrote it :-)). Fancy that...
