What did we do first?
We recently used a common task approach (see pic below) at one of my schools to identify a school wide gap in understanding in the area of multiplicative thinking. We then used concept map building to improve teachers content knowledge and to help close potential gaps while helping to move students from additive thinking to multiplicative thinking. The first common task was given to every student from grade 4 to 8.
Here is a link to the first common task:
What was our second step?
After the the assessment was given all of the math teachers at the school met to moderate mark the assessments. We used the Lawson Continuum to assess what strategies the students used to solve the tasks. We colour coded the strategies based on whether they were counting, additive or multiplicative strategies. See the pic below:
A colleague and I worked together to decide which ones belong under counting, additive or multiplicative. We looked at different research to decide this but some may disagree with us which I would love to hear. We did our best and it probably isn’t perfect so feedback would be welcome.
After the moderated marking was done we discovered that we had a huge gap of students in this school who had not made the transition from additive thinking to multiplicative thinking. We actually had many students still using counting strategies all the way up to grade 8. It confirmed our initial diagnostic work from the teachers at the school who thought this was a significant gap in learning at our school. All of the research we read makes it pretty clear that if students don’t make the transition from additive to multiplicative thinking between grades 3 to 5 then it will impede much of their math learning in junior, intermediate and high school.
Here are some work examples from students in the school:
These students used counting strategies: Composite groups and counting, skip counting
These students used additive strategies: repeated addition
These students used multiplicative strategies, the pic on the right is from the second common task that was done after the PD.
The work in the first two groups of pics was most common throughout the school. We figured out quickly when marking the assessments that an interview was the more effective way to hear/see their thinking. Some students wrote a nice shiny perfect equation but after the teachers talked them they discovered the student had skip counted or used repeated addition. Some though did have auto-recall which is the desired goal for any facts based questions.
What did we do next?
Now that we had all this info about our students, school wide, and knew that their was a significant portion of our students who are still additive thinkers we tackled the third part of the common task cycle. We offered the teachers a PLC day to build content knowledge in the area of multiplicative thinking. We did this by having the teachers build concept maps. We sent this article before the PLC day to have the teachers read it ahead of time.
What was our first step on the PLC PD day?
On the PLC day their first step was to build an initial version of a concept map for multiplicative thinking based on what current content knowledge they had. After they were finished we had a share out and discussion.
Here are some 1st Version Maps before the PD learning (Notice how it varied in how much content was known but overall the webs are fairly small)
Second part of the day was to do some learning about multiplicative thinking. We gave the teachers a choice to choose from these articles below or this site. (The first one is available online and the other two are from NCTM. I think you need to be a member to access them)
After everyone had a chance to read then we had discussion about what people learned. The teachers were able to ask questions, clarify meaning, build on each others ideas and overall start to improve their content knowledge. There was excellent math talk around the learning.
Here some highlights:
- the multiplicative situation has three parts (number of groups, amount in the equal groups and the total)
- the names for the three parts are: factor x factor= multiple
- each of those parts has specific names depending on whether it is multiplying or dividing but maybe using the above three simplifies the situation for kids
- depending on the context you are always looking for one of those parts and the context will determine what operation to use
- there are specific properties for multiplication and division, some carry over from adding and subtracting (associative, commutative and the distributive property, the zero and 1 principal etc.)
- along with the properties the other key ideas of multiplicative thinking are: cardinality, unitizing, part-whole, proportional reasoning, place value
- knowing the situation, knowing the language, knowing the properties, using the array model are key to teaching multiplication and division
- we can no longer teach them as separate entities but as one situation
- there is a sequence for the models used for developing multiplicative thinking, especially the array, the number line is also effective and may be best for kids learning to unitize
What was our last step?
The third part of the day after all the learning was to add to their content maps and explode them out!
Here are some pics of the content maps after the PD learning (You can see by the end of the day the concept maps had exploded out!
Overall I think the teachers found it to be an effective way to build content knowledge. It helped them to better understand the math and to make plans to move their students forward. The maps can also be used to help identify student needs while on-going assessment is happening. These concept maps reminded me of Cathy Fosnot’s landscapes of learning, accept these are co-created by the teacher.
After the PLC learning day we mapped out a plan for the teachers to use to help move their students from additive to multiplicative thinking. At the end of the cycle we did a second common task to see if the students had made progress and help plan where to go next. I like the common task cycle because it identifies school wide trends, helps create common language among the teachers and is based in the student work. Combining the common task cycle with using a concept map to build content knowledge seemed to go together very well. I need to thank Kelli Gates our math consultant in TLDSB for bringing the idea of the concept map to my attention and also participating in the common task PLC cycle.
Here is the link for the second common task we used:
I got the common task cycle idea from Doug Duff. I also love this quote of his..
If you have chance try this idea at your school, I think you will love it!