Same But Different is a powerful routine for use in math classrooms. The activity of same but different is an activity where two things are compared, calling attention to both how they are the same and how they are different.

This apparent paradox is the beauty of the activity. I would like to highlight the precise language of same BUT different. In this analysis, instead of making a choice – am I going to prove that these are the same or am I going to prove that they are different – students are considering how two items can be both. This is a critically important distinction.

I learned about the idea of same but different years ago when working with a speech and language pathologist who specializes in executive functioning disorders for teaching children grayscale thinking. Grayscale thinking is the ability to see the world as having some middle ground versus being a world of black and white. For example, there is something between happy and sad, or in between short and long, etc. People who have trouble with grayscale thinking tend to be rigid and get stressed when life doesn’t fall into one category or another. “The problem with black and white thinking is that it usually does battle in a world that is nuanced and gray,” says Byron Williams in his article on the subject. The routine of same but different helps call attention to the features of things – and how we can connect something new to something we are already familiar with, thus helping to develop flexible gray scale thinking.

For example, think of a young child who only likes to play with HIS truck. We can compare his trucks to someone else’s truck and notice all of the ways the two trucks are the same. We can hone in on what it means to be a truck – 4 wheels, big, an overall conversation about “truckness.” With this in mind, we can say to an anxious child, “So these are both trucks. This is your truck, and this is your friends truck. They are different, but also the same. They are trucks.”

We all know someone who struggles with this type of thinking. For me, this was my son. Such conversations became the norm always pointing out how things were the same but different, such as, “We are at the beach! You know all about beaches. There’s sand, and water. We swim. But THIS beach is different. There’s a dock we could swim out to!” We were calling our son’s attention to the common features of things, making things comfortable while helping him form a connection to something new. The amazing thing was that through using this language explicitly and practicing this repeatedly, he began to do the same. He began to attend to categorical thinking and the features of things and learned to make links and connections. His world became less black and white.

I believe having  this insight from the point of view of speech and language pathology we could profoundly impact mathematical reasoning with the routine of same but different.

One of the reasons students struggle in math is that they fail to make any connections. For some children (those lacking grayscale thinking) every concept they learn is its own entity without any connection to the larger network of mathematical ideas. Just like the young child who only likes his trucks, someone who has poorly developed number sense might see each number as its own thing, and not part of the larger number system. Ask a young child where they can find 8 on a number line to 10. If they don’t race towards the end of that number line knowing it’s near ten, but instead start at one and make their way up the number line, that could be an indication that they are lacking a systematic understanding of the counting system. They can only locate 8, for example, by considering each consecutive number starting at 1.  A mathematical conversation using the language same but different that calls attention to how a new concept in math is the same as that other familiar and comfortable concept but different in a specific way could be a tremendously useful conversation in growing that network of connections. I believe this could also reduce anxiety as children become the sense makers in the conversation.

All roads always lead back to math.

In learning about this routine for helping with executive functioning concerns, light bulbs were going off for me. I thought how same but different is basically an exercise in understanding equivalence. It has always bothered me that students are told 4/8 is the same as 1/2 … well yes and no! They do have the same value, but the pictures in my mind of these two fractions are very different. Sure, they are the same but they are also different. And Place Value – one ten is the same as ten ones, but it is also something different.  There’s the link from fractions to decimals and percents to consider. I realized the possibilities were endless.

Thinking about early number, we can look at different images for any number – say 4 – and notice that while all of these images of 4 are the same in that they are 4 items, they are different in their arrangement. For place value understanding, this idea gets at the crux of its complexity. We introduce children to a new unit called a ten that is simultaneously two things – one ten and ten ones. One ten is the same value as ten ones, but it is different in that it is one ten! Confusing, right? By calling children’s attention to each new concept and how it is the same in some way to something they already know, but different in a key way, we help to grow their network of connections and thus foster a robust understanding of concepts. We can continue to play this game walking through topics .. from counting to subitizing, from thinking in ones to thinking in tens, from understanding addition to understanding subtraction, linking addition to multiplication, division to fractions, fractions to decimals to ratio … it goes on and on and on!

When using the same but different routine students are forced to look at the features, the characteristics, the defining qualities of what we are comparing. They notice the overlap of ideas as well unique distinguishing qualities. But specifically saying, “these two things are the same” has a way of lowering cognitive tension and beginning the building of a bridge to understanding. Using this explicit language with students (How are these the same but different?), I see an opportunity for a deep impact on student learning. By using the images on this website, as a routine, we have an activity ready for use in the classroom, but at a more important deeper level, we are teaching a way of thinking: grayscale thinking, categorical thinking; building a network of ideas but also an approach to how we learn and think about all of mathematics.