- Episode Highlights
- Transcript
“Model with mathematics” sounds straightforward… until you try to define what it looks like in a real K–3 classroom. Is it about teacher modeling? Using manipulatives? Solving word problems? This episode takes a closer look, and might just shift your perspective.
In this episode, we take a surprising turn into what Math Practice 4 really asks of us and our students. We’ll explore how to help kids see math in the world around them, use it to make sense of real-life situations, and develop the kind of number sense that sticks far beyond the classroom.
It’s time for math to feel more purposeful, more connected to kids’ lives, and more joyful to teach. This episode will leave you with a fresh perspective, and a few practical routines to get started.
Links mentioned in this episode:
Math Practice 4: Model with Mathematics
Hello, Meaning-Makers! Welcome to the fourth episode of the Meaningful Math Podcast. I’m excited to be back today to continue our discussion about the Math Practices. Today, we will dive into Math Practice 4, and its explanation might actually surprise you a little. Let’s get started!
What Does It Really Mean to Model with Mathematics?
When you hear the phrase “model with mathematics”, it might make you think of traditional modeling where the teacher explains a concept, shows how to achieve a desired outcome while thinking aloud, and gives examples of what is expected.
Typically, the teacher and students practice it together, and then the students try it by themselves. This process is often referred to as “I do, we do, you do”.
Or, this phrase might make you think of students building concrete models, such as using base ten blocks to represent a quantity. This is actually referred to as modeling mathematics, which I know sounds almost exactly the same as “model with mathematics”, but I promise the nuance makes a difference.
Neither of these concepts is the focus of Math Practice 4. Instead, this practice is all about encouraging our students to make sense of the world around them using mathematics.
Why Word Problems Alone Aren’t Enough
Now, you might be thinking that this can easily be achieved by having students solve word problems. After all, those have real-world contexts, right?
While that may be true, word problems provide a real-world scenario for our students. Math Practice 4 calls on STUDENTS to be the ones to do that work. We want students to look at the world around them and think:
- I see math all around me!
- How can I explain phenomena (something unique that I observe) using math?
- How can I solve this problem or answer this question using math?
Making Math Purposeful, Curious, and Creative
Let’s take a look at some scenarios that might prompt students to think this way:
- A student notices there is a grid pattern on the window and thinks, “That looks like an array! I see 3 squares across and 3 squares down. There must be 9 squares in the grid because I know 3 x 3 = 9!”
- A class of 20 students has one person absent. When it’s time to break into partner work, a student comments that there will be one group of 3 since 19 is an odd number.
- A student sees a butterfly and thinks, “That butterfly is beautiful. Its right wing is a mirror image of its left wing. That’s symmetry!”
- The class is lining up for lunch. The teacher tells the students to line up behind the student who is the Teacher’s Helper for the day. The students know this means that the Teacher’s Helper will be first in line and ahead of everyone else.
- A student is baking cupcakes for a birthday party. The recipe is for 12 cupcakes, but he has 20 people coming to the party. He thinks about it and decides to double the recipe, and he concludes that there will be 4 extra cupcakes.
- Two students are building towers out of blocks. Their towers are the same size, but one student wants her tower to be the tallest one, so she decides to add more blocks to hers.
In all of these situations, the connections between math and the real world were observed by the students, and they occurred organically. The very best case scenario is that students will not need to be told to use math in their daily lives – it will just happen naturally.
We want our students to see that they can view situations and questions like the ones I previously mentioned (along with countless others) from a mathematical perspective. We want students to experience how math can help them make sense of the world around them.
My favorite thing about Math Practice 4 is that it makes math purposeful and approachable for our students, as they experience the real-world application of their learning in authentic ways.
Our hope is that this endeavor will be exciting and interesting for our students and that it will spark their curiosity. Children are naturally curious, so we want to do everything we can to continue to foster that curiosity. This is especially true in math, which is traditionally viewed as process-oriented and answer-driven with little room for curiosity to emerge.
Using the “Notice and Wonder” Routine to Spark Mathematical Thinking
Now, you might be wondering, “How can I help my students see math in the world around them?” One way you can do this is to incorporate a math routine called “Notice and Wonder”.
With this routine, you display a photo and simply ask your students, “What do you notice? What do you wonder?” Don’t specify that their responses need to be math-related – just see where the conversation goes.
It’s intentionally very open-ended for a couple of reasons. The first being that it’s an easy entry point for our students. Every student can find SOMETHING to share about the photo, whether it’s math-related or not.
The second reason is that it allows room for creativity, and we want our students to experience math as being creative. There is no limit to what students can share, but inevitably, someone will share something math-related, and the conversation will take off in an exciting mathematical direction, maybe one that you couldn’t even have anticipated!
During this routine, I like to record students’ thinking on a t-chart. Think back to our last podcast episode and consider how the strategies for facilitating mathematical discussions could work well with this routine.
Sidenote – a quick Google search of “Notice and Wonder” can lead you to some amazing websites with collections of photos to choose from. You might even intentionally select a photo that has some connections to what you’re working on (e.g., a photo of two different-sized animals if you’re working on comparing lengths, a photo of dice if you’re focusing on subitizing, an array of doughnuts if you’re working on multiplication, etc.).
Encouraging Students to Ask: Does My Answer Make Sense?
As students model with mathematics, we want to encourage them to think about if their answers, ideas, or results make sense in the context of the situation.
An example I use to illustrate this point involves checking out at the grocery store. We’ve all done it – we load the items onto the conveyor belt, fumble through our wallet to find our credit card, and swipe it without thinking as soon as the cashier reads us the total, anxious to hurry up and get out of the crowded store.
But what if an error was made? Would you be able to catch it? This happened to me once when I was leaving Target. As I walked out, I was thinking, “Gosh, that total was higher than what I was thinking it would cost.”
Sure enough, after a quick check of my receipt, I saw that the cashier had scanned the laundry detergent twice, making my total almost $20 higher than it should have been.
Because I knew the approximate cost of the items I bought, I was able to tell that I overpaid. I couldn’t tell you exactly how much the total should have been or exactly how much I overpaid down to the penny, but I had a ballpark idea.
I thought to myself, “Does this total make sense given what I know about the cost of the items I bought?” That general awareness was made from my ability to round numbers, add them mentally, and compare that amount to the total I paid.
All of this work is an example of modeling with mathematics because I made sense of the situation using math reasoning.
Strengthening Number Sense Through Modeling
The ability to assess if answers, ideas, and results make sense applies to all areas of math. Every time students solve a problem, we want them to constantly be thinking, “Does my answer make sense with the information I have and what I know about the world?”
I’m sure you’ve experienced this before – a student gives an answer to a subtraction problem that is larger than the minuend. We want the student to see that answer and think, “There’s no way this can be right. It doesn’t make sense that the difference would be larger than the minuend! I need to go back and see where I went wrong.”
This type of thinking is a reflection of the student’s number sense. When students model with mathematics, they are utilizing and strengthening their number sense. This is a skill that they will carry with them forever!
Reflecting on Math Practice 4
Meaning-makers, how can you encourage your students to model with mathematics? What are you already doing in your classroom that’s working? What are your big takeaways from this episode? What questions do you still have? I can’t wait to dive into Math Practice 5 with you next! Until then, have a great one!
