Keeping it real . . .

As math teachers, we work really hard to figure out what students know. We confirm where we are trying to take them by regularly consulting our curriculum guides. Then we use backwards design to make our plans to move everyone from their current understanding toward our learning goals.

Carefully selecting and creating problems is a natural part of what we do. If we want students to discover a pattern or relationship, we have to figure out the numbers to use that highlight what we want them to see. If we are developing skills with a certain fluency strategy, crafting the right problem is essential. If we are using a number string so that one problem is now more doable since it has a distinct relationship to a previous one, again the numbers we choose and the sequence of questions are incredibly intentional. Building from known problems to those just out of reach is the careful work of the skilled teacher. 

But what about keeping it real?

As someone that almost always plans and plans and plans in order to intentionally choose the right problem to push thinking forward at an appropriate level of challenge, keeping it real was an idea that was often on the back burner. But having the privilege to see others use real numbers well, has me reflecting on when and where and why keeping it real has a useful place and purpose in our math classes.

Last year I was a guest in a Grade 6 classroom at John Martin Junior High. Students were working towards outcomes in the graphing unit. Discussions about discrete and continuous data, choosing the right graph to match the context, and accurate graph construction were all learning targets. The amazing leader in the room, Ms. Carly Brison gave me a lesson in keeping it real. When students came in after lunch, data was collected with this prompt. How many students are on time? This data was attended to and graphed daily during the duration of the unit. It was posted in a visible space, used to prompt questions about parts of a graph and deciding the scale, and then examined days later for analysis. Brilliant. Just the simple shift of finding personally meaningful data, versus given data, and using it in authentic ways really landed with her students.

Later in the year, Mr. Spencer Roach, was working with ratios, rates and percentages with his Grade 8 classes. Outside we went. Students worked in teams to collect their own data at multiple stations. Skip for three minutes while your partner records the number of jumps, time how long it takes to run around the gagaball court 5 times, record your number of baskets out of 10 attempts. We used our personal data to calculate and compare unit rates. We discussed what data can be represented as a percent, and we made and qualified predictions. The engagement was through the roof! This tiny shift of collecting our own numbers to use, made the activity more relevant, the calculations more meaningful, and the conclusions understandable. 

Next came measures of central tendency in Grade 7. Again, outside we went. The leader, Mr. Liam O’Brien, invited teams to try the long jump, use a hula hoop (hilarious) and time their multiple attempts at 30 m dash to collect their own data. We then made sense of our collected information to calculate mode, median, and mean. What did outliers really refer to? What numbers could or should we ignore? Why? What measure of central tendency was appropriate for each station? What did it really tell us? A master class in keeping it real.

These teachers gave me a lot to consider. Collecting personal data gives us real numbers to use. The results weren’t always pretty or friendly. But this was a useful trade-off for relevance. It also tapped into other important skills like estimating and predicting. This idea of using real numbers in our calculations and discussions – is it just for the data outcomes?

This month I am working at Sir Robert Borden Junior High In Dartmouth, Nova Scotia. As a guest in a Grade 7 class working on percentages, I threw out an idea about keeping it real based on the lessons I learned from educators at John Martin. It was a delayed start day. The absolute worst. Teachers could likely expect just a portion of their class which meant that carefully developed prompts and lessons and any new learning stalled while we waited for an appropriate number of students in attendance. My advice to kick start this uncertain day was this prompt. 

What percentage of students are present today? 

Within a matter of minutes the fantastic teacher, Ms. Carla Jackson wrote the fraction of students in attendance on the board, 18/29. I’m not gonna lie – my heart sank a little. One of my first suggestions as a guest in the room was this ugly number. Not the ideal fraction I would have crafted to make lovely connections between fractions, decimals, and percentages. Not the one I would have picked to discuss using the simplest form or equivalents for help. My multiple methods and talking points evaporated in front of my eyes. What now? Carla and I locked eyes. I’m not sure what she was thinking, but I know I was wondering how to pivot away from this unfortunate fraction. But, that extra minute of silence seemed to give the class the appropriate wait time to consider our prompt. What happened next was a great lesson in keeping it real. Here’s how I remember the discussion that ensued.

One student said, “Well it’s more than 50%”

“How do you know?” Ms Jackson wondered.

“Well 50% would be 14.5 and 18 is more than that . . .”

Another student jumped in, “I was thinking that if 18 kids were here today, and there are 29 total, that means 11 weren’t here. If it was 50% those numbers 18 and 11 would be the same and they’re not.”

Another student, “exactly 50% present is not even possible!”

“Why not?” I asked.

“Because 14.5 students doesn’t make sense!”

Already, the discussion was awesome. I stopped panicking over the ugly number and leaned in.

“So we have more than 50% of the class here today. Can we get any closer?”

Then, another student shared all of this:

“Well 10% of 29 is 2.9
and 2.9 is almost 3
and there’s six 3’s in 18 . . .
But I have to take off 0.6
So that would be 17.4 . . .”

Silence. Then
“Wait…what?” 
Some kids said it, many looked confused, but almost all looked curious.

Students were looking at each other trying to make sense of the math they just heard. I was as well. Ms. Jackson, marker in hand, started writing down the ideas in an attempt to make this reasoning visible and understandable by all. 

As she wrote, I asked, “Why did you think about 10%?”

“Because 10% is easy to do. I was trying to see how many 10% ‘s were in 18.”

“And how many did you find?”

“I’m not sure.”

“I remember you finding 6”, I said as I pointed to where Ms Jackson had helpfully recorded this student’s thinking. “What does the 6 mean here?”

“I’m not sure.”

So true. Oftentimes students go down a path based on some great math understanding and get lost in their calculations. Then rather than taking some time to back track and make sense of it, they give up. Is this because they are alone on their thinking path? Maybe they are not used to exploring ideas on their own so they run back to the groups’ collective thoughts and to safety?

Me, “If we can make 6 groups of 10%, what percent would that be?”

Then Ms. Jackson noticed, “What you did here – this is a fluency strategy! What is it called when we alter the problem and then try to account for the change at the end?”

So much good math thinking!
I decided to change course slightly to involve more students in the discussion.

“18/29 is a difficult fraction to work with. Is there a way to change the numbers slightly to make it friendlier?”

One student said, “You could turn the twenty nine into thirty!”

Another student, “You could make the eighteen twenty instead!”

“Should we do both? Or just change the denominator?”

I wrote 18/30 and 20/30.

“Do these changes make the fraction easier to work with?”

Lots of nodding

“Why? How?”

A few students could tell me that 18/30 is the same as ⅗, and that was 60%.
Other students could tell me that 20/30 was the same as ⅔ and that was 67ish%.
Would the actual answer be more or less than these percentages I wondered? 
Ms Jackson found the actual answer using 18/29 and shared it with the class.

I wondered, “What did Ms. Jackson just do with her calculator to find the actual answer?”
“Can we use a calculator?” students asked.
“Yes! But how do you use it? What do you do?”

I noticed many students, with calculators in hand, were doing more estimating and checking some ideas but very few made the connection between fractions and dividing. Fewer still could give me a final answer, even with a calculator. This made me realize that our carefully constructed problems that helped students make equivalent fractions to find the percent were a good starting point, but other necessary connections needed to be made. Ms Jackson and I had some small group conversations with students as we continued to engage with the problem. I asked the few students at my end of the class how you can use a calculator to find the percent. Some said, “divide?” but with a little uncertainty. Others gave me blank stares. 

I asked, then, about ¾ a fraction they all knew.
If you know that ¾ is 75%, how would you prove it with your calculator? What do you do?

Messing around with a known fact to confirm methods is a strategy I use often when students get lost in procedures. Go back to something you know for sure and use that idea for your new problem.

Eventually, we came to our result. We looked at our estimates with slightly altered numbers and decided they were pretty close. We noticed that both the numerator and the denominator could be altered so that they are both “nice” like 20/30 but you alter just the denominator like we did creating 18/30. 18 isn’t super nice, we agreed, but 18 has a relationship to 30 that can be used. When we alter just one part of our fraction it can be easier to decide if our estimate is above or below the actual answer. A great conversation.

This reminds me of the student thinking assignments I completed as part of my course work for my Masters in Teaching Math at Mount Holyoke College. Record student ideas and how you responded. Review later. What do students know? How do you know? How did you respond? How were you able to share student thinking with the group? What learning needs to come next?

After our conversation today, I could tell that most students knew how to find the percent of a number. They could simplify fractions and round numbers successfully. They could confidently relate fractions to 50% in multiple ways. I could also see that we needed to make clear that fractions can mean division. When fractions have a denominator that is not compatible with 100, we can use division as a method to find the percent.

The other thing I realized was this:
Ms. Jackson is very skilled at creating a positive classroom environment where students feel welcome to share their thoughts – even if their thoughts are incomplete. Without this environment, this discussion would not have been possible. She is also great at recording student thinking in a way that makes it visible and accessible for the rest of the class. Responding to student ideas in the moment is tricky business. It requires patience, confidence, and practice. But I think my biggest lesson was in keeping it real. The number was ugly. Ms. Jackson was not deterred. The conversation that resulted was fascinating and fantastic. It helped us understand what students knew and what we still needed to solidify. 

Now don’t get me wrong, we still need the carefully crafted problems. We need the ones made specifically so we can engage with concrete materials. We need the ones that connect prior learning to what is just out of reach. And we need numbers that build and progress in a way that highlights patterns and connections. But we also need to keep it real. Estimation, rounding, wondering, testing and calculating – all of these are possible and probable when the numbers get ugly. 

Thanks Ms. Carla Jackson! In your class, keeping it real is a beautiful thing!