Want to encourage independent math thinking? Help your kids learn how to model with math & visual tools as they seek to solve problems. Find ideas & free resources to help below!
Welcome back to my series on the Standards for Mathematical Practice! In each article I’m trying to unpack these standards in a way that makes sense so that you can help your students develop these math habits. This will help them to move beyond memorization and drill to meaningful problem solving.
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I have enjoyed reading and thinking through different ways we can foster these habits in our classrooms and encourage kids to think deeply about math (and not be afraid to make mistakes!).
Today I want to focus on two more practices that we want students to work towards/be comfortable doing:
1. Model with mathematics and
2. Use appropriate tools strategically.
There are some who disagree about what exactly modeling means and looks like in the math classroom, but I see it as being able to think about real life situations in math language.
In other words, seeing a situation and being able to describe it using math symbols or expressions or write an equation to represent it. This means kids can apply what they know about math to actual real world problems.
Making Sense of Real World Problems
This means more than simply picking out key words in a word problem. It means, as the standards state, students “are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later.”
This goes back to our discussion of perseverance in problem solving, feeling comfortable talking about or explaining their thinking, and then revising their thinking or strategy as necessary.
It also means that students are able to take their solution and consider it in the given context. I always tell my students to ask two things once they’ve solved a word problem:
1. Does your solution answer the question being asked? (sometimes the math is accurate, but doesn’t actually answer the question)
2. Does that answer make sense, given the context?
As an example, I was recently tutoring a couple of 8th graders. We were discussing the following problem:
“John drove the last 240 miles of his trip at 60 miles per hour. If the whole trip was 480 miles and took 10 hours, how fast was he driving for the first leg of his trip?”
One student immediately recognized this as a distance, rate, time problem and wrote the following equations:
240 = 60t, t = 4 hours
480 = 10r, r = 48 mph
She knew which values were related and how to correctly use them in the distance formula, but when I asked, “What do those values represent? Do they answer the question?” She realized that although all her math was correct and represented different aspects of the situation, we hadn’t actually answered the question being asked here. So we had to dig a little deeper to explore how we could use what we know to find the speed of the first 240 miles of the trip.
We noticed that the distance was the same (240 miles) and the time spent driving the first 240 miles was 6 hours (the whole trip – the second half: 10 – 4 = 6). We could then find the rate–40 mph.
So then we could discuss the second question–does this answer make sense in the context? They decided, yes, this is reasonable because the first half of the trip took longer (6 hours), so therefore they must have been driving slower.
Rather than simply plucking out numbers and sticking them into an equation (d=rt), we were able to unpack the situation and ask what all the different numbers represented, and then reason logically about the final solution.
Use Visual Tools & Models
Part of being able to make sense of a real world problem and describe it or model it with mathematics is knowing how to use mathematical tools to show the situation or make sense of the problem. (For the above problem, we drew a picture and labeled all the information we did know and didn’t know yet).
For any given situation, there are probably half a dozen math tools that could be used to model the problem or solve it. Rather than telling students which tool to use, we want them to feel empowered to choose a tool (whether a hands on manipulative or a visual model) that makes sense to them.
Possible math tools students can use might include hands on manipulatives like base ten blocks, fraction tiles, counting bears, a ruler or protractor, pattern blocks, or a rekenrek.
These tools don’t necessarily need to be something you hold in your hand–it might be a number line (or open number line), a ten frame, or drawing a chart or table on a piece of paper to help them organize the information. It might be drawing a picture so they can visualize the situation, or for older students, it might mean knowing how to use their graphing calculator to model, graph or look for a pattern.
Depending on the tool, it might require some explicit instruction from you so that students know how it works and how to use it correctly. But once they’re familiar and know how to access these tools in your classroom, it’s more matter of providing time for them to put them to use.
So again, whether it’s solving and applying math to real world problems, or choosing appropriate tools, the best way to help kids grow in their confidence and proficiency is to provide meaningful practice with these skills and tools.
Resources to Help You Get Started:
Ready to get started? The following resources will help you find real world problems and tasks for your students, as well as tips for finding and using hands on or visual tools to model with mathematics. I hope this is helpful!
- Explore real world math tasks here
- Help kids see math in the world around us by reading, Math Curse | includes printable extension activities
- Read my Ultimate Guide to Hands On Math
- Find a huge list of online math manipulatives here | helpful for virtual learning!
- Grab a free set of Printable Math Manipulatives | Create individual toolkits!
Here’s another helpful tip from reader, Deborah:
“As a former Math Instructional Coach, I noticed that when we ask students to explain their thinking, they immediately think that their answer is incorrect. So I changed the way I asked for their explanation: “I agree with you, but could you explain …..why you knew that you needed to multiply___ and _____, what words in the problem made you set up your equation in that way, why you put parentheses around 4 x 22, etc.”
Ready to Dig Deeper?
If you are enjoying this conversation and want to learn even more about modeling with math and using visual, hands on tools, I go into much more depth than I had time for here in my courses, Making Math Visual and Making Sense of Word Problems.
Although all of my courses are available separately, they are truly designed to work together to provide a foundation for math teaching in K-8. All together, the 6 courses include 8 hours of professional development, which you have lifetime access to. If you work through one course per week, you could easily complete all of them.
FREE Math Practice Standards Poster Set!
Want to explain these math habits to your students? This fun and free set of posters provides a springboard for discussion as well as a visual reminder all year long.
There are two versions included in this pdf file. One includes a cursive font and the other includes only manuscript fonts so you can use these with younger students.
I’ve also adjusted the wording of some of the standards so that is uses more kid friendly language.
Grab this free poster set by entering your email below!
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