I hear a lot about “real world” math. While I agree that we need to make sure our students are engaged with their math, I wonder if making things “real world” is the answer to the engagement issue? Take a look at a few examples of problems involving circles:
Problem #1: Find the diameter/radius, circumference and area of some dessert treats.
Problem #2: Which has a greater area?
Which of the two preceding problems would be more engaging to you? For me, the first problem’s context seems quite tacked on. The author has assumed that the math itself isn’t interesting, so they will make it look more interesting by adding dessert treats. Yet, in no way does the context help the student understand anything more about circles or challenge them to think in any way. Most students who would receive a worksheet like this, would ignore the context completely and mindlessly plug in values into their calculator to get the right answers. Dan Meyer has posted many blog posts about Psuedocontexts like these.
The second problem is quite different though. It encourages us to think first… to notice pieces of the problem… to reason informally before any calculations are done. Look again at the picture of the two circular objects. Which has the greater area? What does your intuition tell you? Which one looks like it has a larger area? Problems like this ask us to do much more than calculate, they ask us to start to think mathematically and be engaged in the mathematical processes.
The two problems also differ in what our goals are. The first set of problems aims to make sure our students have a skill (calculating), while the second problem’s goal is more about mathematical reasoning. This difference seems to me to be a really important distinction to make!
Let’s take one last problem involving circles and see how engagement and “real world” math fits in:
Activate (Minds On):
Teacher: Can anyone tell me what you notice here.
Various Students: its a circle, looks like tiles on the ground, I noticed the symbol for Pi, the stones are of various sizes and shapes, some parts are normal colors and others are black or white…
Teacher: What do you wonder? Do any questions come to mind?
Various Students: Why is it there? Where is this from? What is the ratio between white and black tiles? Is the ratio 3.14 times more white? How many tiles are there to make the circle?
Teacher: Let’s explore how many tiles there are. I want you to estimate how many tiles you think there are in your head. First, I want you to think about the range of possible answers you think might work. Write down the lowest number of your estimate (there must be at least ____ tiles) and the highest number (there is no way there are more than ____ tiles). Finally, write down your actual estimate.
Teacher: Who wants to share the lowest possible number there is (50), anyone think the lowest possible number could be a bit higher (100, 200…). Who wants to share highest…. (1000, 900, 600). Many of you think the answer is somewhere between 200 and 600. Write down your exact estimate and we will check in at the end to see how close you are, and whether the answer fits in our range.
Action (Acquire -Apply)
Teacher: With your partner, I want you to calculate how many tiles you think there are (without counting every tile). What information would you need?
Various Students: Diameter, radius, circumference, size of each block…
Teacher: Which of these will help you?
Various Students: Radius…Diameter…
Teacher: With your partner, calculate how many you think there are. Be able to defend your answer.
Students – working in pairs – all find an answer
Students share thinking (Pi x radius squared) – most have same or similar answer (16.5 squared x Pi is about 855)
Teacher – Is this answer reasonable? Was it within our estimated range? (most estimated way less)
Teacher – We calculated how many there should be, but do you think the real number is less or more than this? (here is where the real-life piece comes into effect – the math doesn’t give us the actual answer, but helps us to estimate and make sense of a real situation).
Various Students: “I think less because of the grout between the tiles.” “No, there is grout in the diameter too.” “I think less because the pieces are smaller in the middle.” “Wouldn’t that make it more then if they are smaller?” “Some of the black tiles look a bit larger”… This is where I smile and sit back because the students are driving the conversation! The whole class is engaged in rich mathematical discussion!
Was this problem a “real world” problem? Would it engage your students? More importantly, would it engage your students in the process of thinking mathematically, reasoning mathematically…?
Engagement to me is more than tacking on contexts that we expect our students to ignore to get their answers. Instead, it is about getting our students to be curious… to wonder… to want to find the answer because they have invested thinking before any calculations were attempted.
Great prompts can lead to great discussion and a much deeper understanding of math. I find the real-life component is best utilized when we provide real things (strive for curiosity and perplexity and we will no longer hear “when will we ever need this”). And while making connections to the real world isn’t a bad thing, what we really want is for our students to be making more decisions, thinking more, be in a state of not knowing for a period of time so they actively want to resolve that not knowing. Engagement is more about this active process of thinking and reasoning than making math about dessert treats!
PS – The students calculated 855 tiles, but there are actually only 837 tiles – the real world isn’t as straight forward as school would like you to believe. Math is a tool to help us figure it all out. Start noticing the world through a mathematician’s perspective!