Lately I have been working with some teachers in my schools to help them craft more effective learning goals. We have been looking at learning goals that are more precise which express what you want the students to understand. The idea is to move away from learning goals that are just the curriculum expectations reworded in a “I can or We are learning to,” statement. Before I go into more detail and provide some examples of the work we are doing I just want to give some background on where this journey started.
Early this year I was given the book “The 5 Practises by Smith and Stein” by one of my principals. I had also seen it used in an early adobe connect PD session back in January. If you have not seen or used this resource I suggest getting it immediately, it is one of the best math resources I have had the pleasure of reading. Not only did it lead to a deeper look at learning goals but it will change the way you plan math, especially your consolidations. The 5 Practices has a specific chapter (Chapter 2) on creating learning goals and selecting tasks. This sparked my interest and I wanted to help the teachers I work with and myself to craft more effective learning goals. I also had the opportunity a few years ago to work with Marian Small at a board PD session where she also spoke about the need for more precise learning goals that focus on an idea/understanding the students need know. She also explained that many of our expectations have multiple understandings that the students need to know in order to fully meet the expectation. The idea was not new to me but the 5 Practices book made it much clearer what we needed to do to accomplish this and added in the extra part about being very deliberate with the task you choose that will allow students to meet the learning goal/goals you have chosen.
I am going to start with an example of what may have been a learning goal that I may have used in the past or a type still commonly used. Please remember this is my take on this process and I am not saying it is the only way to make learning goals or that it is the correct way. It’s a way that I think makes a lot of sense has helped me and seems to be helping teachers I work with focus their lessons and tasks. I also have got positive feedback from the teachers after they have seen this modelled. This example and task that the pictures are from is a grade 3 classroom I recently worked in. The curriculum expectation the teacher was using was determined through her initial diagnostic assessment as being a need for the whole class.
Expectation: divide whole objects and sets of objects into equal parts, and identify the parts using fractional names (e.g. one half; three thirds; two fourths) without using numbers in standard fractional notation.
In Ontario students are not introduced to standard notation until grade 4, which could be a whole other blog on my opinion on that! To be honest we did use some standard notation in the lesson. While speaking at OMAE Cathy Bruce actually talked about using standard notation earlier than what the curriculum says it should be introduced. She thought it wasn’t a problem especially if you feel your students are ready for it. You are not evaluating the use of standard notation but using it to further the conversation on fractions and notate student thinking. As you can see with that expectation there is a lot of key ideas the students need to understand in order to meet that expectation.
We may have seen this learning goal used in the past: We are learning to divide whole objects and sets of objects into equal parts. This is just the expectation rehashed and in my opinion too big! The teacher I was working with and I talked about what are some ideas they need to do to truly meet this expectation. Here are a few that we came up with:
- understand the different representations of fractions (part-whole continuous, part-whole discrete, fraction as measure etc.)
- understand what the fractional names mean (unit fractions etc.)
- understand equipartitioning and that it is equal area not the same shape
- understand that the fractional parts don’t have to be attached or adjacent.
As you can see there is a lot to understand in this one expectation (I am sure we missed some too!) We decided they needed to truly understand equipartitioning first as it relates to part-whole continuous (area models) and fraction as a measure from 0 (number lines). Although we didn’t address fraction as measure with this task.
This is the new learning goals we came up with:
We felt this learning goal is very specific with the idea of the parts being equal area (equipartitioning) but not necessarily the same size or shape. I usually like to link to big ideas which in most cases there is more than one big idea that could be used. We decided to link to “Fraction parts do not need to be attached or beside each other”. Now the students have clear idea that they will understand after they complete the task or series of tasks often it takes more than one visit to deeply understand a key understanding in math.
After crafting this learning goal we set out to pick a task that we thought would allow them to understand the learning goal when completed. Here is the picture of the task:
This is the Minds On we used, which we did using a number talk format. Having the students defend their thinking and build on each others ideas.
Here is a picture of the task we used and some pictures of finished student work.
These two students actually went further and labelled their parts even though the task never asked them too!
During consolidation we used three examples of student work and put them up on the whiteboard. I asked the students this question: What do you notice about how these three students have divided up their brownies?
After I asked that question I gave the students time to talk to an elbow partner and then they shared. The students had excellent comments.
One student said “The friends that are getting brownie’s from the middle pan are all getting the same size piece which is fair.”
He then went on to say that “The middle one is fair but I want a piece from the left pan, actually I want piece 5 from the left pan”
LOL. This set off a debate about fairness.
Another student said “I think the middle one is divided up equally but I think the one on the right is pretty close to being fair.”
I thought this was a great point. I then asked the class what they thought about her comment, they all agreed with her that it is almost equal but is not as equal as the middle solution.
I then asked “Why they thought that could have happened?”.
One student answered with “I think the middle person folded very carefully and that the person on the right didn’t use a ruler to draw their lines and that may have helped.”
All of this was great conversation so we wrapped up discussing the two solutions with misconceptions and why they are misconceptions (not equal area parts). The students seemed to be able to clearly see the difference between the middle one and the other two solutions. They even saw that it may just be a fine motor issue with the one on the right which is common with younger students. If this happens I feel you then have to use your judgement about whether they don’t understand equipartioning or if it is a case of fine motor issues.
The students didn’t use any partitions that were equal area but different shapes although we didn’t really expect them to yet. That may be were this teacher goes next with this idea and use a task that builds on this task and reinforces the idea that the parts have to be equal area not shape. We used this simple exit card for formative assessment as the ticket out the door. There were also tons of formative assessment opportunities during the task!
We were happy with the results, all but three students out of 19 wrote or drew on their sticky note that the most important idea was that when dividing up whole objects the parts have to be equal in area.
I know this was a very long post but hopefully helped show how crafting a quality learning goal based on understanding and then choosing a task that will help meet that learning goal gives the students a clearer picture of what they are to understand and helps the teacher guide the consolidation to create a storyline out of the solutions to help all students understand the learning goal.