Math talk is all the rage in our dialogue about improving student understanding in math classrooms. Getting students to verbalize their reasoning and moving beyond math learning being just about “getting the right answer”. Encouraging students to use math language, to agree and disagree with each other, to have their own ideas and strategies and be able to back them up with reasoning. One tool that kept popping up during NCTM was the idea of “Which one doesn’t belong?”
To get the idea of the exercise, look at this picture and identify which shape is different and justify why.
Now to follow through with developing my own math talk practices, I am going to share both why I agree and disagree with this type of activity.
I do essentially appreciate this type of reasoning activity, but I believe there is one key element that is quite problematic. The potential of equating difference to not belonging or being wrong.
Now it might be my feminist upbringing kicking in, and the lyrics to an Ani DiFranco song dancing in my brain, but I think those of us in working in education don’t always realize the power of our words. We can’t just wear a pink shirt one day out of the year or go to the yearly anti-bullying assembly. We must be conscious of the struggles our students may be facing and think about the language we use daily. Whether it is because of mental illness, body image, gender identity, sexual orientation, cultural identity, religion, physical ability, or any number of other things, our students often can feel different and alone in their struggles and we as adults in their lives need to step up and think about what we might be doing that is alienating.
I am not saying don’t do these types of activities. I am just saying be conscious of the words you choose. Generally they are thought of as “What One Doesn’t Belong”. I frame them as “Which one is different?” Take the negative connotation away, take the “othering” away, and focus on the fact that everything has characteristics that are both similar and different to other things.
Once you take the problematic language away, the possibilities of these activities are endless.
Christopher Danielson has been complaining for years (his blog confesses to it) that math books for kids are awful. The opportunities for engaging in imaginative and creative thought in literacy when reading storybooks are unlimited. But what do we get for developing math concepts with young children? We get counting (1...2…3...4...5… with the respective number of objects on a page in consecutive order). We get shapes (one page of rectangles, a completely different one for squares, triangles almost always equilateral and usually with a vertex oriented “up” on the page). We rarely get anything to have a conversation about or to spark our imaginations.
So Danielson has written a “better shape book” called “Which One Doesn’t Belong” (Grrr… but moving on). The idea of the book is that young children can see, think, and reason about similarities and differences without us telling them what is “right” or “wrong”. They may see things we didn’t, they may make connections we weren’t expecting, they may use language we don’t anticipate, but there is a chance for a rich conversation.
Back to the example from before (which is from the book).
Which one is different?
Well hopefully you can see why one, two, wait maybe all four of them could be seen as different.
The top left is different because it only has three sides. The bottom left isn’t filled in red. The top right seems to have 4 angles of equal size. The bottom right is resting on it’s bottom, while the rest are “standing” on a point.
The idea is that all of the shapes have similar characteristics to each other, but also are unique in some way.
Let’s look at another one… This one comes from a website developed by a teacher who liked Danielson’s idea and ran with it. Now tons of people submit ideas and there are a whole bunch for you to use.
This time we are going to step it up a notch and try to think of two things that are different about one of the numbers.
43 - not a perfect square AND a prime number 16 - an even number AND the only number that if you add 4 you get a zero in the ones place 9 - a single digit number AND the only one where the sums of the digits do not equal 7 (1+6=7, 2+5=7, 4+3=7) 25 - the only number that the product of the digits is not a multiple of 3 (9, 1x6=6, 4x3=12, 2x5=10) AND the only number that has 5 as a factor
Whew that was tricky.
Final example is an incomplete set. So you have to come up with a suggestion for the final graph. (Wait don’t give up just because you don’t remember anything from grade 10 math class. Many of your students are going to be feeling the same way you are right now when you teach them a new concept that you might think is so easy. Work through the struggle and use the knowledge and language you do have. There is NO wrong answer).
Just to note we are talking about the red curvy lines, the horizontal and vertical lines are reference lines.
1. Each red line opens up (or would catch water if we poured some in from the top), so a line that opens down would be different.
2. Each red line crosses the vertical line line above the horizontal line (or has a positive y intercept), so a line that crosses below would be different.
3. The top left red line is above and below the horizontal line (has a range of positive and negative values), so we could create another line that is only above the horizontal line, therefore the top left is the different one.
As you can see this type of activity opens up so many possibilities for discussion. So just a quick recap as I realize this post may have gone on for too long.
1. Be careful about the language you choose on a daily basis. Frame these problems as “Which one is different?”
2. Look for the unique characteristics of each object.
3. Look for two characteristics that make each object unique.
4. Have students create an object that fits the incomplete set.
Websites to check out for more…
Whew you made it through! Thanks for sticking with my long winded thoughts. You get rewarded with more San Francisco beauty (on top of all the awesome math you just experienced of course).
