Who gets to be challenged in class? How does the work we give students impact their status (or how they perceive their status) in the classroom?
You’ve probably met a student like Calvin before. Calvin wants everyone in his third grade class to know he’s good at math. (They know.)
He is good, too. He looks for patterns, and makes connections. Someone taught him how to add fractions with unlike denominators, too, which looks like a pretty cool party trick to his eight year old classmates.
Calvin wants a challenge in math class — but if something is too challenging, he bawks at it: “I don’t like this! Give me a better challenge.” (Sigh.)
Calvin’s third grade class is currently studying fractions. On my first day in the classroom, Calvin finished a task much earlier than his peers. “Ugh,” he lamented in a stage whisper — loud enough for everyone around him to hear. “This is so easy. It’s boring.”
In a pinch, I pulled out my phone and showed him a Yohaku puzzle that involved fraction addition/subtraction. They’re a lot of fun, and I’ve used them in classrooms from first grade up to 7th!
To solve a Yohaku puzzle, fill in the empty cells such that they give the sum (or product) shown in each row and column. This is an additive puzzle.
Calvin was thrilled!
Opening Up The Challenge to Others
The next day, I brought a packet of these problems for Calvin, and offered them to anyone who expressed interest, too. Of course, I forgot: the class was working to understand equivalence. The only students ready to work on fraction addition (with unlike denominators) were ones with experiences outside of school. Students that were not familiar with this content puzzled for a little while, wrote some incorrect numbers, and grew bored.
Although the puzzle was great for Calvin, and pushed his thinking in appropriate ways, it wasn’t a fair challenge for all. Accidentally, I had elevated Calvin’s status — and Calvin didn’t need any help elevating his status. He was happy to announce his superiority to others.
I had to rethink things.
- When do we give “challenging” tasks or puzzles to students? Is it only after they’ve completed ‘other’ work?
- What’s the goal of this “challenge work”?
- How could I provide a challenge that would push Calvin’s thinking, but also the thinking of students who only had experience with third grade content?
The Striped Snake Barbecue
Enter: Math Pickle’s Half Fraction Snake Challenge, aka the Striped Snake Barbecue (@gamesbygord)
This puzzle is about equivalent fractions and halves. There are 3 customers at the Snake BBQ: one that wants a portion of snake that is 1/2 blue, another that wants a portion of snake that is 1/2 red, and one that wants a portion of snake that is 1/2 yellow. Your job is to make two cuts in each snake to partition the snake into 3 portions, one for each customer.
I launched the puzzle with a “notice/wonder” (“what do you notice? What do you wonder?”) about the first snake on the page.
- There are three colors used.
- There are 4 of each color.
- There are 12 squares in all.
- There doesn’t seem to be a pattern. It goes yellow, red, blue, red yellow, red blue?
- That might be a pattern. It’s yellow and blue with red between them. But the numbers don’t make any sense.
- There are always 2 blues in a row.
Then I launched into a story about our Snake BBQ restaurant, and the conceit and constraints of the puzzle. Students were intrigued.
Students worked using “SmartPals” (transparent dry erase pockets). I could see this working with the puzzles tucked into sheet protectors, or even with paper and pencil. The colors make it easier to access, but I imagine it could be done in grayscale, too. There is a black & white version available on MathPickle.com.
Students preferred to work parallel to peers. They mused aloud.
“I cut here, so that the first part is 1/2 blue.”
“But then the rest of the snake is impossible.”
“Maybe this is the impossible snake?” (One of the six snakes on the page is “putrid,” and doesn’t work.)
“This snake is 1/2 red AND 1/2 blue. I’ll give it to the red customer.”
“Oh, this part IS 1/2 yellow! There are 3 yellow and 3 not yellow. Half.”
“Is it okay if my 3 yellow aren’t in a row? Is it still a half if they’re broken up?”
“Yeah. I think so.”
Students who completed the first challenge were given 3 “mythical circle snakes,” with no heads or tails. Students could then make 3 cuts, partitioning the circle snake into 3 portions, to appease the customers.
“The circle snake is harder! I don’t know where to start!”
“There are more squares in a circle snake!”
To close the problem, we discussed strategies for proving that the fractions are equivalent. This is precisely where we are in our fraction unit! How perfect!
When and how do we “challenge” students?
I love Yohaku puzzles. (I might write about them another time.) But the fraction addition Yohaku weren’t the right choice for this particular third grade class.
Meanwhile, the Striped Snake Barbecue was accessible to just about everyone in the class. Some students already had a good sense for equivalent fractions, and were able to reason that 4/8, 5/10, etc. would be worth 1/2. Other students needed some extra supports, like red, yellow, and blue cubes that he could organize into equal piles. If the number of cubes for the color you want (e.g. yellow) is equal to the number of the other cubes (e.g. red and blue), then your snake is 1/2 yellow. This allowed them to draw on 1st and 2nd grade understandings of halving.
There are still days when I offer something to students that I know the other students in class probably cannot access. It’s a decision that I know comes has the potential to yield negative consequences for our classroom community. So, increasingly so, I try to find challenges that are more open — like the Inaba Place Value Puzzles, which I have used in 2nd graders, 3rd grade intervention, and with 6th grade accelerated students! It’s all in the framing (the launch AND the close), the math you draw out for students, the way students interact with the puzzles (including available tools).
What’s in a word?
Maybe thinking of these as “challenges” is not even the right word or framing for us, as educators. It implies that there’s “regular work” and then there’s “extras.” Students only get to work on a ‘challenge’ after they complete some other required work. Some days this happens, but some days, like with the Striped Snake Barbecue, it just becomes the task for everyone. It’s not extra or for enrichment. It’s just how we learn together.
- Yohaku Puzzles
- Half-Fraction Snake (Striped Snake Barbecue) from MathPickle.com
- Posts about Inaba Place Value Puzzles (Feb 2017) (Nov 2018)