My last post, Multiplying Fractions: A Desmos Area Model, blended conversations from the same lesson run in two different classrooms. Although the lesson flowed similarly in both rooms, there was one student who viewed the presented problems differently. Gabe has an artistic eye and his method of working through the math problem was unique to both classes. On the other hand, Gabe’s mood was quick to change. One misstep would cause frustration and self doubt to settle in. These characteristics provided me an interesting challenge. How do I honor his method while reducing his frustration level? (All names have been changed)
Me: Draw a square and shade in half of it.
I didn’t want to influence how student’s shaded their square, therefore I waited until they finished before displaying the Desmos model. Gabe was the only student to draw a diagonal line. (Pictures have been recreated).
Me: Circle half of your shaded region.
Me: What fraction does the circled section represent?
Up until I asked this question, Gabe was smiling and seemed to be having fun. My question uncovered a misconception in Gabe’s thinking. His answer was 1/3. One piece was circled out of the 3 sections. When he realized that 1/3 was not the correct answer, he became upset. His lighthearted demeanor turned heavy. I heard his group talking about equivalent fractions. My attempted to connect the group conversation to his picture was not successful.
Since this was my first time teaching this class, I stepped away. Gabe needed a chance to regain composure and I needed to check in with other students. After circulating, we regrouped whole class. Most students had either 1/3 or 1/4 as their answer. The class discussion addressed two points: 1) How someone would conclude the answer of 1/3 and 2) The need for equivalent sections (which generated the answer of 1/4).
I kept my eye on Gabe as we worked through the second situation. His unique viewpoint intrigued me and his frustration level concerned me. I challenged myself to make Gabe smile before class ended.
Me: I found 1/4 of a pan of brownies in the office. Draw a square and shade 1/4 to represent the remaining brownies. Most students, including Gabe, drew the same picture.
Me: I decided to give 1/2 of the remaining brownies to Ellie. Highlight, circle etc… the section given to Ellie.
Whereas the majority of the class drew the version shown in Figure 2, Gabe sectioned his picture differently (Figure 3). Gabe was still visibly upset but continued to be an active participant.
Figure 2 Figure 3
Me: What is the fraction size of the brownie given to Ellie?
This time around, students, who had the answer of 1/3 before, remembered our discussion of equivalent sections. I watched them add more details to their pictures leading them to an answer of 1/8. This included Gabe. Even though students arrived at the same answer of 1/8, their strategies differed. If you’d like to read about the various student strategies click here.
My time as a guest teacher was wrapping up. Although, I didn’t have enough time to discuss a different situation, I did have enough time to present the class with a related challenge.
As mentioned earlier, I had been keeping tabs on Gabe waiting for a chance to positively reinforce his work. The opportunity presented itself when I observed Gabe sectioning his drawing into eighths. He, again, had a unique solution. I asked him if I could show his picture to the class and he replied, “yes”. Still no smile.
Me: I’d like to show you a picture one of your classmates created. I drew Gabe’s pre-sectioned picture on the white board. Question, how would you create equivalent sections for this picture?
The class liked the challenge and got right to work. Lot’s of hands raised, including Gabe’s, in hopes of providing a solution. I called on Gabe. He applied his solution to the white board drawing proudly and walked back to his desk smiling.
Now that I’ve stepped out of the classroom to be an Instructional Technology TOA, I use guest or co-teaching opportunities to maintain and hone specific skills, such as (but not limited to):
- Incorporating different strategies into the class discussion
- Finding connections between different strategies
- Tinkering with question types that require students to do more of the thinking.
- Inviting all students to the math party
With Gabe, I focused on #1 and #4.
1. Incorporate different strategies into the class discussion
Once I introduced the Desmos area model (pictured on the right), most students gravitated towards that arrangement. Gabe didn’t. He continued to section his square differently reinforcing his individuality and making highlighting his approach a priority. Exposure to different strategies not only honors student thinking but also provides students with references for future problem solving tasks.
4. Invite all students to the math party.
I first heard this belief at a Dan Meyer workshop around 3 years ago. The Low Foor- High Ceiling idea was a prevalent theme throughout the series of activities. Math should be inclusive. It’s up to the teacher to design a lesson that allows every student entry to the task regardless of their mathematical level. (link to Dan Meyer’s Ted Talk)
The frustration Gabe exhibited wasn’t productive struggle. He was starting to withdraw or leave the party. To his credit, Gabe never left, however he wasn’t fully enjoying himself.
At a party, the host touches base with all the guests, makes sure they feel welcomed and connects them with other party goers. The teacher is the host of the classroom party. As the guest teacher, it was my responsibility to make sure Gabe felt welcomed and connect him with his classmates. By being attentive to Gabe’s progression through the second problem, I found my opportunity to draw him back into the party.