From number strings to math talks, numeracy routines have made big in-roads in elementary math classrooms. And with good reason. They’ve proven to be effective instructional strategies for helping students develop mathematical flexibility and computational competence.
However, numeracy routines haven’t yet become a mainstay within middle school math classrooms. Why?
Often, there is a misconception that numeracy routines (such as number strings and math talks) are only for elementary classrooms. The belief is that these routines can be useful for developing strategies around whole number computations but lose their relevance in the middle school world of rational numbers, algebraic reasoning, statistics, and geometry.
In this blog post, I want to deconstruct this myth by considering why we should use numeracy routines in middle school math.
What is Mathematical Flexibility and Why Do We Need It?
One of the hallmarks of number routines is their ability to help students develop multiple approaches to solving a particular problem.
Take for example the problem:
297 + 128
Yes, students could use a traditional addition algorithm to determine this sum, but they don’t have to—and this approach wouldn’t necessarily be the most efficient pathway.
Instead, students might think about:
- Starting at 297, then adding 100, then 20, then 8.
- Or… Starting at 128 and adding 300, then subtracting 3 to compensate for the 3 too many they added.
- Or… Taking 3 from the 128 and giving them to 297 to create an equivalent (and easier to think about) sum of 300 + 125.
These three approaches represent three different ways to think about determining this sum.
Having access to a wealth of strategies helps students develop mathematical flexibility, whereby students can determine an efficient strategy to use for a particular problem with particular numbers.
Middle School Students Need Mathematical Flexibility, Too
This same flexibility can be incredibly helpful in middle school, as well.
Not just in questions that are rational number equivalents of the problem above, such as:
2.97 + 1.28
(You may want to pause here and think about how you could reason through this decimal sum in multiple ways, like in the example above.)
But also in questions that reach into other middle school math topics. Take, for example, percents.
Say you were asked to determine the following percent:
44% of 25
How might you do this mentally (or with reasoning)?
You might start by saying:
- Well, I know 10% of 25 is 2.5.
- So, 20% would be double that, so 5.
- 40% would be double that, so 10.
- That gets me pretty close to 44%! I just need 4% more…
- Well, if I know that 40% is 10, 4% must be one-tenth of that, so 1!
- So, 44% of 25 is 10 + 1 = 11.
(As you’re reading through this thinking, how are you visualizing it? How might you record this thinking visually to make it accessible to other learners? We’ll save that conversation for a future blog post.)
New question to think about: How would you reason through this percent problem?
25% of 44
You might use some partitioning and iterating like described above…or you might recognize that 25% is equivalent to ¼, so you’re being asked for ¼ of 44, or 44 divided by 4. Oh, that’s 11!
So, 44% of 25 and 25% of 44 are equivalent? Yes. (Do you see why? There’s some really interesting mathematics lying underneath these two problems.)
But, how does knowing this equivalence help us? Well, when we are thinking about a percent problem, it might be helpful to think flexibly about the numbers in the problem.
For example, would I rather think about:
27% of 30
30% of 27
This is mathematical flexibility. The ability to contemplate before you calculate and then select a tool, strategy, or next move that helps you reason more efficiently or thoughtfully about the problem.
Where Else Does Mathematical Flexibility Show Up in Middle School Math?
One more quick example of where mathematical flexibility comes up in middle school. Say you’re determining the volume of a pyramid and you’ve substituted the area of the base and the height of the pyramid into the volume formula to yield this expression:
- How would you compute this product?
- How else could you compute it?
- Are all of your computation pathways equally efficient? Equally difficult?
If we’ve developed our mathematical flexibility, we might see that it is more efficient to think about this product like this:
Here we’ve used the commutative property strategically to rewrite the product in an easier way to think about and compute.
Now we might see that ⅓ of 15 is 5, so this product is 5 x 8, or 40.
Mathematical flexibility doesn’t end there, though. There are countless other topics in middle school math—from comparing numbers to comparing ratios, from solving problems to solving equations—where mathematical flexibility empowers students to think more strategically, intentionally, and fluidly.
How Do We Foster Mathematical Flexibility?
The important idea here is that middle school math problems often benefit from strategy and this strategy is evidence of mathematical flexibility.
This mathematical flexibility is not innate, nor is it the exclusive domain of students identified as gifted or talented. Rather, mathematical flexibility can be developed—both for ourselves as math teachers and for our students as young mathematicians.
One proven way to develop mathematical flexibility—even especially at the middle school level—is through numeracy routines. Check out the video recording of my recent webinar to explore two such routines that will help you (and your students!) develop mathematical flexibility.
Free Webinar: Join Michelle for a demonstration on how to use these targeted sets of related problems to build a strong foundation of number sense and mathematical reasoning in your 6-8th grade students.