After my son had collected candy at a parade he wanted to eat some. I told him he could choose one thing from his bag. He chose a clear zipper bag with 2 packages of Starburst in it. Each package contained 2 Starburst. He chose one, but he actually got to eat 4 candies.
If you are familiar with Christopher Danielson’s work, you have most likely had conversations around units and the variety of units that can exist in one situation, like the example above. If not, I will discuss his book How Many and the hashtag #unitchat in a future post. It wasn’t until the past few years, however, that I realized there is more to it than just seeing the different units. Students need to be able to coordinate these units.
A few years ago, the MTMS Journal published a number-sense article about coordinating units and an app called Candy Depot. The article highlights the role of units coordination in fractions and multiplicative reasoning, and it discusses an assessment tool for determining stages of student development. The assessment has been a powerful tool in my classroom. By deepening my understanding of units coordination and assessing students’ abilities in this skill I have a new perspective on math instruction, student misconceptions, and strategies that support student progress.
According to Anderson Norton, "Units Coordination refers to students’ abilities to create units and maintain their relationships with other units that they contain or constitute." (111) Students begin coordinating units as they work on counting principles. The Math is Visual website has some great examples of this. In hierarchical inclusion, students not only need to understand the unit being counted, in this case candies, but that the 3 candies already counted have become a composite unit.
Unitizing and skip counting create even more units to coordinate: individual candies, a composite unit of 3 candies, counted candies, and uncounted candies (You can see where the multiplicative reasoning starts to come in, can’t you?) Episode 58 of the ZPD Podcast is a great episode about the use of units coordination in skip counting.
At the heart of all of this is the idea that there are many units that students need to keep track of as they work to understand number relationships in math problems. Depending on the problem, the unit attached to one item or group can change and students need to be flexible in this understanding. This idea is illustrated well in this tweet:
The assessment laid out by Anderson Norton, Steven Boyce, and Jennifer Hatch in the original MTMS Journal article is a way to assess how many levels of units students are able to coordinate at a time. (113)
So, how do you assess how many levels of units students are coordinating? With a bars task.
The assessment gives students series of 3 bars followed by questions about their relationships:
- How many times does the medium yellow bar fit into the long red bar?
- How many times does the small blue bar fit into the medium yellow bar?
- How many times does the small blue bar fit into the long red bar?
Students at stage 1 use the size of the bar to determine answers and struggle when the bars are not drawn to scale. Students at stage 3 use multiplicative reasoning, flexibly move between any 2 bars using relationships already known, and determine answers even if a bar does not evenly fit in another.
It has been very eye-opening for me to see all the different levels of thinking in my classroom.
As you begin your journey with understanding units coordination, I suggest these great resources:
I challenge you to start noticing the units in the tasks your students are working on. I will be posting mine on Twitter with the hashtag #unitscoordination. Join me!
Norton, A., Boyce, S., Phillips, N., Anwyll, T., Ulrich, C. & Wilkins, J.L.M. (2015). A Written Instrument for
Assessing Students’ Units Coordination Structures. IEJME-Mathematics Education, 10(2), 111-136.