How do we engage adult learners in the seemingly simple yet complex mathematical ideas of the primary grades? While teachers can examine the mathematical focus of K-2 tasks, and think about what they might look for as they observe, what questions they might ask to assess understanding, or consider how they might support or extend the targeted math ideas, playing *Counters in a Cup* or solving a *How Many of Each?* problem is not exactly an engaging math task for adults.

A few weeks ago, I attended an *Investigations* Workshop. I observed a group of K-5 teachers and coaches immersed in a counting task, sharing their strategies, and then connecting how their experience brought out the complexities of counting. As I watched, I was once again reminded of why *How Many Butterflies?* is one of my touchstone tasks for engaging adult learners in the mathematics that is central to the primary grades. Participants are instantly drawn to the large, laminated “Butterflies” poster as they find a butterfly they recognize or point out ones that are especially unusual. The display is interesting in that there is variability of color, size, pattern and orientation. Very quickly, a comment of “Wow I never knew there were so many types of butterflies!” is followed by the question of “I wonder how many there are?” and the investigation begins.

There are several factors that make this counting task challenging: the butterflies are randomly arranged on the poster so counting them in an organized manner requires a plan. The butterflies are stationary. You can’t pick them up and move them as you count or organize them into convenient sized-groups, so determining how to keep track of what has been counted and what’s left to count becomes a decision. Since people are working in small groups, negotiating a plan of action becomes part of the task and once the plan is enacted, it is not unusual to hear comments such as “Could you count more quietly, I’m getting distracted” or “No, I think that butterfly is on my side of the poster and I counted it already.” Inevitably someone catches wind of an interesting counting strategy being employed by another group and convinces their group to abandon their original idea and shift to a different strategy. Rarely do all groups arrive at the same answer. In fact, it is the result of different answers that spurs people into recounting and double-checking their work. These comments and experiences are remarkably similar to what you might see or hear when watching K-2 students engaged in similar work.

The debrief of the task brings out all of the complexities about counting that are in fact central mathematical ideas in the primary grades. These ideas include knowing the rote counting sequence and assigning one number to each object, keeping track of what has and has not yet been counted, and knowing that if you count smaller sub-groups and add those groups together, the total represents the number of butterflies on the poster. Primary teachers often reflect on how adults’ knowledge about counting and quantity influences their experience, citing things like “If we all have the same poster then we should all get the same number” and “If our group counts and then double-checks, the number should be the same” as being obvious to adults or even to older students. These ideas however, are not obvious to our youngest learners and are important parts of the mathematics they are working on. Inevitably upper grade teachers talk about their own “ahas” about the complexities of counting and reflect on how they can see aspects of these same ideas about counting by 1s and groups coming into play in the later grades in multiplication and work with fractions and decimals.

I am always searching for and playing around with K-2 games and activities that can be adjusted (e.g. adapting a material) or extended (e.g. changing the question) in such a way that the underlying math investigation shifts from being just right for a K-2 student to engaging for an adult. Part of the engagement is that the task is interesting and challenging, but equally important is that it offers teachers a new look at the math they teach and reaffirms the importance and complexities of even the “simplest” mathematics.

### Karen Economopoulos

*Investigations*Center for Curriculum and Professional Development. Her work on

*Investigations*spans all three editions; she oversaw the development of K-2 for the 1st edition, and co-directed the development of the 2nd and 3rd editions. Karen works extensively with classroom teachers, administrators, and school districts in the area of curriculum reform. She is a former K-2 teacher with a special interest in supporting the range of learners in elementary classrooms.

#### Latest posts by Karen Economopoulos (see all)

- Counting is More than 1, 2, 3: Engaging Adults in K-2 Mathematics - August 8, 2018
- “This is Only Getting Better!…And Harder!” - June 12, 2018
- Counting is Serious Business - February 26, 2018

Love this! PD with adults using just the I3 materials can be challenging and as you state sometime less than engaging. Where can this poster be purchased?

Hi Kristen, As far as we know, the butterfly poster we use is no longer commercially available. To do a similar PD activity, look for a poster (or other material like an interesting sheet of wrapping paper) with more than 100 objects, that are not arranged in an “organized” way, and where there is some variety that makes the task interesting and distracting at the same time. And as Chris Mainhart points out in his comment, think through the important aspects and complexities of counting that you want to raise in the discussion. I would love to hear how it goes if you decide to try it with teachers!

I love it Karen! I’m so pleased to know that the butterflies continue to live on. This task has been and will always be a powerful way for teachers to understand the complexity of counting (as seen through the eyes of young students).

I love this idea for teacher pd! Also, I just found the poster available on bananaroad.com for $9.99. The link is

https://bananaroad.com/products/butterflies-types-lepidopterology-27×39-poster