Currently at two of my schools we are working on a PLC that is looking at the planning that has to happen before a lesson or sequence of lessons is delivered. One of the main focuses has been looking at unpacking the curriculum expectation (concept) that the sequence of lessons or lesson is based on. Two statements that are being used a lot lately are “know the learner” and “know the learning”. This post is looking at the later “know the learning”. It is also fairly long so I apologize ahead of time!
We know that we should always start with the curriculum when planning our math lessons but being able to identify the curriculum expectation or expectations that you want a sequence of lessons to be based on is not enough. After you have decided what expectation or expectations you want the lessons to cover you then have to unpack those expectations and really know the learning that is behind them. There is often many mathematical ideas that you need to know and understand to effectively set up learning situations or tasks to help students fully understand a math concept. Many different math people call these by different names: key understandings, key ideas, and sometimes big ideas. They are all basically talking about the same thing although I think big ideas have a different meaning. I use a lot of different resources and you can find these key ideas in all of them. Marian Small’s “Making Math Meaningful” is one such resource that has them laid out in an easy to understand way. When I reference key ideas later blog these same key ideas are called fraction principles in Marian’s book. Cathy Bruce’s fraction pathway research calls them key understandings so you see what I mean by different people calling them by different names.
There is another resource that I think explains what is meant by key ideas even more clearly.“Teaching with Tasks for Effective Mathematics Learning” by Sullivan, Clarke and Clarke has small section in their book where it discusses the role of teacher knowledge in effective task use. It’s on page 15 and 16 in their book. Here is picture of the book and the page which I am referring to.
In this section of their book they refer to two major categories of teacher knowledge that is needed for effective math lessons. The first is subject matter knowledge and the second is pedagogical content knowledge. Subject matter content knowledge is then broken down into two sub-categories which are common content knowledge and specialized content knowledge. Common content knowledge is often what many people have, they can solve the problems based on a concept because of the way they were taught, they know how to do the math! Often this is where many teachers reside, possibly because of many reasons, maybe they are a little math phobic themselves.The second part, specialized content knowledge is this: the knowledge that allows teachers to engage in particularly teaching tasks, including how to accurately represent mathematical ideas, provide mathematical explanations for common rules and procedures, and examine and understand unusual solution methods to problems (Hill et all., 2008, p.378). This is where the understanding of the key ideas for a concept lay, along with the other pieces laid out in the definition of specialized content knowledge.
Here is an example of where this understanding of key ideas comes into play in the classroom. Recently in a grade 4 classroom we were doing this guided math task. The students had red and blue coloured tiles and the teacher was asking them to build shapes that represented the fractions being called out. These work examples are from when 1/4 fourth was called out. Here are three of the student solutions that came after 1/4 was called.
At first glance two clearly stick out as representing 1/4 but one is not so clear. I want to focus on two key ideas happening here ( there are others). If the teacher doesn’t understand clearly or has not planned for, then one student possibly could be told theirs is incorrect or a quality teaching moment could get looked over. The teacher needs to see what is evident in these solutions and name the math for all students to see. All of these solutions represent 1/4 because of the key ideas that fraction pieces have to be equal in area and not necessarily the same shape and that the pieces don’t have to be adjacent to one another, which allows for the picture on the right to also represent 1/4. Many teachers do not understand these key ideas or specialized content knowledge and it is so important that we do!
The teacher I was working with here was the first to admit they didn’t understand either of those key ideas before we started planning for this sequence of tasks. We started with this curriculum expectation: represent fractions using concrete materials, words, and standard notation, and explain the meaning of the denominator as the number of fractional parts of a whole or a set, and the numerator as the number of fractional parts being considered. As you can see there is a lot of math understanding in that expectation. If you just pick a task or sequence of tasks you think will help students learn this expectation without planning and understanding the math behind it (unpacking) then you are going to miss opportunities like what came up in this teachers class or possibly teach misconceptions.
When we unpacked this expectation in planning we identified the key ideas, two I mentioned above. The teacher was ready then when the task was done in class. She was able to see that all of these representations of 1/4 were accurate and then was able to connect the representations (on the fly) together and point it out to the class that they all represent 1/4 by allowing the student to explain their thinking about why the picture on the right also is accurate. There was quite a debate in class between students about the third representation above on the right. Only one student made one like that and they justified their work helping to teach a key idea to the whole class. This opportunity would have been totally missed without the planning ahead of time. There are other key ideas involved in this expectation but I focused on these two for this blog. I think as instructional coaches it is so important for us to help work with teachers to help develop this specialized content knowledge.