I was recently helping a grade 6 student practice some EQAO prompts at one of the schools I work at. We were looking at some of the multiple choice prompts and going over the ones that the student had misconceptions with. We eventually landed on this prompt:
I think if you have been teaching math for awhile you have probably run into students who consistently confuse mean, median and mode. You may have also heard this question, “What is mean?” or this statement, “I forgot what mean means?”
I have few thoughts on this, could it be because it is often taught procedurally? Or could it be because it is often taught out of context? I know that just because we put problems in a pseudo-context doesn’t mean they will always be effective but I think in the case of dealing with central tendencies like mean, median and mode it is pretty effective. As for the first thought about it being taught procedurally, I think it needs to be addressed too. Both these two thoughts have to come together in my opinion. We need to have students model mean conceptually and put in context so they know why we use mean (or any central tendency).
I am going to share some pictures and ideas how I helped this student get a clearer idea of what mean is and how to model it. When we chatted about it he remembered what mode and median were but really didn’t understand why we use any of the central tendencies. I started by creating a new context for the student by asking him if there were any sports he enjoyed playing. He stated that he played in a soccer league in the summer.
(This picture was taken after we had finished everything in our guided mini lesson, the answer for mean was not filled in until the end)
I started by creating some mock stats for possible shots on goal over a 6 game period. I then asked him what he thought it meant to calculate the mean?
He wasn’t sure, but he said, “The mode of the new data would be 5 because it occurs the most often.”
I asked what that meant and he said, “It means that I got 5 shots on goal, most times out of the 6 games.”
I said, “That is exactly what the mode tells us.” I then said, “We know the mean and mode are related and you told me that mode is the number of shots on goal you got most often in 6 games. Now, how could you use what you know about mode to help you figure out what mean is?”
He said, ” I think it may have to do with my average shots, I think I remember that from last year.”
I think he was starting to access some prior knowledge at this point, he has some idea of what it mean means but hasn’t really understood it or conceptualized its meaning. I let him know that he was correct that it has to do with finding an average but we need to really dig in and model what it means to find the mean of set of data.
We started by modelling the data with snap cubes:
Then I asked him, “If we want to find out what your average shots on goal per game are what do we need to do with the cubes?”
He thought about that question and finally said, “I think it will be one number.”
He couldn’t offer a reason why but I think he was on the right track but couldn’t explain it. I then said, “You are on the right track, it will be one number that describes your mean (average) shots on goal for the six games just like the mode was one number (5). I then asked him to make an estimate based on looking at the data displayed with the snap cubes and using other info he knows.
He said, ” I think it will be five which is also the mode.”
I asked him, “Why do you think that?”
He then said, “If I rearranged the cubes I think they will make equal groups of 5. That makes sense because I want to see how many shots a take most games”
He had started to move towards modeling the mean on his own without me specifically telling him to rearrange the cubes into equal groups or me just teaching him to add all the numbers together and then divide by 6 which is often the procedural way mean is taught. The research says, “When students are first introduced to the concept of mean, they should have opportunities to act it out and explore it concretely.” (Small 2013)
Hereis pic of what happened next after he rearranged the into equal groups:
He moved the cubes into equal groups and then said, “I can make six equal groups of four but have three left over. I know my estimate of 5 is close but it is not going to be 5.” He pondered this for a moment and wasn’t sure what to do next.
I asked him this question, “What can you do with those leftover cubes? You moved all the other cubes into equal groups but don’t have enough to make each group 5.”
He thought about this for a moment and then a light popped in his eyes and he moved the cubes like this:
I asked him, “What does your model represent now?”
He replied, “I think if I was able to cut each cube in half I would have four and half cubes in each group.” ” Does this mean that I would average 4.5 shots per game over these six games?”
I said, “Yes it does mean that. How did you decide it would be 4.5 shots per game?”
He said, “With the cubes it is easy to see, each group is a game, so for each game I have 4.5 shots.”
This was just a quick one on one lesson with this student but I felt an effective way to teach mean. I felt like this student had a much stronger understanding after leaving our little guided lesson. Do I think every student would be able to make some of the jumps he did, no I don’t. Some students may need more explicit guidance.
This idea is not mine it is from Marian Small’s book Making Math Meaningful on page 576. This student still needs more purposeful practice with this concept and we haven’t touched on when to use a certain type of central tendency or when they are meaningful or biased but it is a good start. Like all math concepts we need to start with concrete examples, using models to help students understand the math. I was taught to just add up the numbers and divide by how many numbers are in the set of data with no explanation of what it means or why it works. I hope this post also shines a light on why adding context can make a lot of difference in our students understanding. That original EQAO prompt was just random numbers with a question to calculate the mean. Many students always ask why are we doing this so let’s help them see why we are doing these sometimes random prompts and then maybe we won’t get the question. “What is mean again?”