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How Can We Anticipate to Fuel Sense Making? Stretching Trees

by Jon Orr

Are you looking to avoid “Lesson Flops” and bring on a “Lesson Successes?” I sure do. That’s why I plan with anticipation of student thinking in mind.

I want to share a lesson I co-created with Brian McBain and teachers at Wallaceburg Secondary School and how anticipating student thinking helped avoid those flops!

Let’s run through the lesson first, then I’ll give you a window into how we planned it.

In random groupings students went to their wall space and were presented this first task.

I have to admit when we planned it we anticipated everyone to draw Christmas trees but after showing the image they all drew a variation of the one above.

Drawing the trees was no biggie since our new amazing whiteboards from Wipebook.ca (wipebook.com) has grids on them. Students counted up 20 units and drew their very best tree! Onto the next part of the lesson.

Draw another tree that has a height that is less than 50% of the first tree’s height.

Here is a typical drawing from my students.

In small groupings and also as a whole class I asked and discussed “How do you know the height is less than 50% of the original tree?”

And then we moved onto this…

Draw a tree with a height that is more than 50% of the original tree’s height.

With this prompt we wanted to dive into the answers a bit more. “How do you know it’s more than 50%?” “How can we verify that 16 units high is more than 50%?” Also with this we had students drawing trees higher than 100%. We paused the class and verified and shared out the different tree heights around the room.

I prompted them to draw a tree that was exactly 30% of the original tree’s height.

This is where I was super interested to see how they would solve this. Their solutions were going to fuel the discussion going forward (Check below to see how we anticipated what they would do).

Most groups of students used the grid and found a unit rate. Can figure out this strategy?

This group knew that 10 units would be 50% so they took the 50% and divided it up into 10 units giving 5% per unit. Then they counted up by 5s until they reached 30% and got 6 units high. Other groups took the whole 100% and divided it by 20 to get 5%/unit. As a class we gathered around these solutions and explained the strategy. Any group that was stuck went back to their boards to use this newly presented strategy and the other groups pushed forward with this new prompt.

Groups progressed through this prompt at different times, but when they were ready I gave them this one: “Draw a tree with a height that is exactly 62% of the original tree.”

This is where the struggles happened. Again we were interested in HOW students solved this problem. Most new that 60 was going to be 12 units high….and then just estimated from there how high the tree would be. Some did guess and check to narrow down how high exactly 62% was. This was exactly what we had planned. We had wanted and led the students here to create this struggle so that we could step in and teach them a strategy!!

We used a double number line: One side showing percent from 0 to 100 and the other side showing the heights of the tree. But instead of a horizontal number line we tipped it up and made it vertical!

We had a discussion on proportions: “Is this a proportional relationship?” “How do we know?” Yes….so we can apply a proportional strategy to solve this. After that the students had a new and improved strategy to try the next few prompts:

Draw a new tree that has a height that is 17% of the original tree’s height.

and then,

And then we switched to a new “starting” tree.

And kids drew this.

After I felt that groups were comfortable, their next task was given out (which stretched into day 2). We changed the scenario from trees to colouring.

Want to get the PDF with all the image prompts and handouts? Click Here.

When reflecting back this lesson was not one of the “flops” it was a “success” and most of the credit has to go to planning with anticipation in mind.

Anticipating to Fuel Sense Making

When Brian, the team, and I set out to design this lesson we were looking for a way for students to feel like they weren’t learning something new. That they could take the idea of percent and just use it like they have already solved proportion problems. We also wanted students to follow the Hero’s Journey and feel that there was a definite need to use a proportion strategy.

Here is what the early stages of the planning process looked like. Yep, scribbles in a journal. We spent a lot time thinking about the right progression of prompts so that we could maximize student work and use their strategies to push learning forward.

We also spent a great deal of time planning out the different strategies we thought students would use to solve the original prompt “Draw a tree that is 30% of the original tree’s height”

We outlined the strategy of finding the unit rate of 5% per unit, we thought many students would already know the “rule”: Turn the % to a decimal and multiply (But no one did do that in my class). We thought it was possible for them to create a proportion. We thought some groups would try a guess and check strategy. Like: “I think the height is 7. Let’s see if 7 out of 20 is 30%.” Only a few groups did this.  We ranked each strategy in order of most likely to least likely.

Anticipating their solutions and strategies puts me in a better position to understand their thinking and help shape that thinking. For each possible attempt I need to be ready to provide feedback to help them achieve our goals.

We take for granted how much time is needed to prepare and anticipate adequately. It takes time to make this happen, but that time is worth every minute. Especially if it puts me in a better place to understand what my students are thinking.

This has been my assessment goal: Understand their thinking in order to push them further. That’s it! That’s the main idea.

Anticipating their thinking will always put me in a better position to fuel their sense making.

Read More: Fuel Sense Making & Black Box Defrost

How Small Nudges Could Have A Big Impact On Math Education

by Jon Orr

Do you ever notice that many decisions are secretly being made for us? You probably missed them like I did.

Consider this: Last Friday, and much like every Friday our math department headed out for a quick snack and recap of the week at one of our favourite restaurants. We enjoyed stories from a week’s worth of lessons and working with students while looking forward to new stories for the upcoming weekend. Like every Friday when it’s time to leave I get the bill. What do you notice?

The tip was suggested for me!

I didn’t have to think too hard about what to leave. The restaurant has made it real easy for me to include that good tip. How many more tips do you think this restaurant earns compared to other restaurants that don’t have this feature? A bunch more is the answer.  

Here’s another example of a subtle suggestion that has a big influence on our decisions. Have a look at this image taken at the University of Pennsylvania.  

We have to do a double take and think about what side to place our trash in. We are pushed to consider our trash placement instead of just tossing it in a bin. A subtle suggestion that influences our decision. Recycle or Landfill?

Both of these scenarios didn’t just happen by accident or without careful thought, both are using a branch of behavioural economics to influence decision making called Nudge Theory.

From wikipedia: “A Nudge is a concept in behavioural science, political theory and economics which proposes positive reinforcement and indirect suggestions as ways to influence the behavior and decision making of groups or individuals.”

If we go back to the tipping example: the suggested tip amount on the bill is a small indirect suggestion for me to give a good tip. The restaurant has made it real easy to leave that tip (they did the math for me). Also notice that they didn’t provide tip amounts less than 15%. They nudged me to make a tip of 15% or greater! They nudged me to choose the behaviour that was favourable to them.

In the garbage bin example the creators drew your attention to the negative impacts of putting items in the garbage side. If you toss on the garbage side that trash goes directly to the landfill. They nudged you, very subtly, to think before you toss trash.

The basic idea behind a successful nudge is to make it very easy to do the favourable behaviour and hard to do the unfavourable one. It was easy for me to make a tip of 20% and harder if I wanted to tip less. It was easy for us to make a decision to recycle because choosing the alternative was something we generally want to avoid: filling landfills.

Nobel Prize winning economist Richard Thaler along with Cass Sunstein brought Nudges into the forefront in 2008 with their book: Nudge: Improving Decisions about Health, Wealth, and Happiness and I learned about it from the Choiceology Podcast episode: The Big Impact of Small Changes by Dan Heath. In that episode Heath outlines a great example of a nudge: the power of the opt-in versus the opt-out.

In Austria, more than 90% of their citizens are organ donors. In the neighbouring country Germany? Less than 15%. Are Austrians that much more conscience about organ donation? Nope. In Austria citizens are auto-enrolled in the program and have to opt-out if they would rather not be a donor. In Germany you are not auto-enrolled and have to opt-in to be a donor. That’s a nudge! A small subtle change can have big outcomes in decisions. Auto-enrolling capitalizes on our laziness factor. It’s easier if we do nothing compared to doing something.

“It’s a question of putting the best outcome along the path of least resistance and letting your automatic system do its thing.” Mark W Riepe writes regarding nudges.

What implications does Nudge Theory have in the math classroom?

 Nudging in Math Education

In my opinion one of the most poorly designed calculator features on my smartphone is the percent button.

Students often misuse the % button the calculator and are not sure what is going one behind that it. It’s like a black box of percent calculation. Many misconceptions arise in my grade 9 applied class when students use this button without knowing the math behind it.

Compare that to what Desmos is doing.

Desmos has auto-added the word “of” after the percent symbol. There is no way for a student to delete the word of. It’s stuck there. It forces us to think about how we use that button. Both teachers and students are now auto-enrolled in finding the percent of a number before doing anything else. It’s an amazingly small change that has a huge impact on learning. They have made it easy to do the right thing. And impossible to do the wrong thing. That is a nudge!

Have you seen these charging stations for phones in classrooms?

Teachers are making it super easy for students to choose to put their phone away if it’s a distraction to their learning. A student will gladly get the juice and put that phone away for an hour. Nudge!

I want students to regularly self assess their learning in my classroom. I want them to be more accountable to get what they need. I make it very easy for them to see their own progress on the learning goals in our course. By auto-enrolling them in their Freshgrade portfolios student can access all the learning goals anytime and work towards showing improvement on them! It’s my way of giving them a nudge to make good decisions regarding their assessment.

What kind of nudges are you trying?

What are you doing to make the right behaviours easy and the wrong ones hard? What are you doing to affect choices your students make in your classroom?