mental math skills
40 x 6
300 + 150
120 - 45
Are these problems your students could successfully solve mentally? If you’re thinking “no way!” you’re not alone. This past year at my school we implemented some new testing that had students doing mental computations - NO paper and pencil. And students struggled. And we all know what happens when students struggle - teachers panic!
There isn’t really a good reason that students shouldn’t be able to do these computations in their head. So why aren’t they? There’s actually a few reasons.
That pesky traditional algorithm
I know, I know! HEAR ME OUT. I often tell people I am anti-algorithm, but that’s not all the way true. I tell people that to prove a point. The standard traditional algorithm (that is only used primarily in the U.S. - look it up) is often taught exclusively and way too early. 2nd and 3rd grade is way too early to be forcing students into a standard traditional algorithm. It may help them get solutions “faster” but it’s actually hindering their ability to understand place value and do mental math.
Let’s think about it.
Take the problem 46 + 32. Traditionally, we prompt students to immediately negate place value thinking and focus on digits. We may even have them draw a line between the 4 and 6, 3 and 2.
The students are NO longer working with the beautiful number 48, which has it’s own wholeness. It is 4 tens and 8 ones (or 3 tens and 18 ones or 2 tens and 28 ones…on and on), it is ten less than 58 and 1 more than 38. 48 has VALUE. But we want students to think of it as a 4 and an 8 (and wonder why they struggle with place value). We may even call it 4 tens and 8 ones…but that’s not enough for students to develop a conceptual understanding of the wholeness of those numbers and what they are.
When students add the 4+3 to get 7 and the 6+2 to get 8, resulting in 78 (even though 7 and 8 make 15…) they are only thinking about DIGITS.
What is the largest number students dealt with here? The largest number they THOUGHT about was 8. We are keeping our students from thinking about number bigger than 8.
Even with “regrouping” like in 57 + 34, students are never thinking bigger than 9. They are prompted to add 7+4 which is 11 BUT that 11 is immediately broken into it’s digits with 1 on the bottom and 1 on the top and now students are adding 1+5+3.
With the traditional algorithm, students are never thinking bigger than 9.
And yes, if you’re someone who teaches the algorithm with base ten blocks then your students are seeing a representation of the process that is happening. But if they haven’t done the deep work on place value and thinking flexibly with numbers, the concept of those ten blocks actually being TEN or 4 of them being FORTY may not be hitting home. Plus, how often are they writing the numbers out to correspond with those blocks? How long do we spend with manipulatives? All of these things are factors.
The point, though, is if we introduce and only use the traditional standard algorithm with young students, they are never really thinking bigger than 9. Which is probably why they can’t mentally add 42+72. They have never thought about 42 as it’s entirety. They have never thought about what a 40 and a 70 would make (not a 4 and a 7).
Probably the worst thing about this is once we teach the standard algorithm, typically in 2nd grade, we give ourselves a happy dance, assume students can carry that knowledge on over and start working with BIG numbers. Have you ever seen a kid do this?
Or just generally have no idea what to do with that 12?
This student doesn’t understand big numbers. And without an understanding of big numbers they aren’t going to be able to mentally add 300 and 200 (I’ve seen it). They’re definitely not going to be able to add 350 and 150.
We cannot just assume students have a strong understanding of large quantities - even if they did with smaller ones.
2. The paper-pencil push
Another reason students aren’t able to mentally compute is pretty simple.
We don’t let them.
Think of the times you have to calculate in your life, like at the grocery store. What do you do most often?
A. Pull out pencil and paper
B. Think mentally
C. Pull out the calculator that everyone always has on them because phones
Let’s be honest. C is the most common answer and A is the least. In a world where students are most likely always going to have a calculator on them, we don’t need to turn them into one.
Calculators use algorithms.
It’s how computers are so efficient and why we all hate Facebook, but never leave. ALGORITHMS.
But the human brain is capable of so much more. That’s why we believe teachers will never be replaced by computers. Nor will doctors, or artists, or mathematicians.
BUT WAIT. Why do we have mathematicians if we have state-of-the-art calculators? Because the human brain is an amazing pattern-weaving, thought wielding, critical thinking machine. And if all we use it for is to do what the dang calculator can…then we are missing the mark.
TAP into your students human potential. THEIR potential to think, solve, and create. So what if their strategy is long or weird or crazy. If they are thinking about numbers flexibly and beautifully, they have beat the calculator. They will have unlocked the true beauty of math. And I haven’t met a student who doesn’t fall in love. Like, head over heels, doing math at recess, wanting to start a math club-in love.
My students spend the first few weeks of school constantly frustrated at me when I don’t let them grab pencil and paper. Kids want to grab pencil and paper for 53 + 11. “What do you need a tool for!? You have a brain! THAT’S your best tool!” is what I always tell them. We have to stop reinforcing the belief that students need paper and pencil or whiteboards and markers to DO math. It’s simply not true. BUT the more you pull it out, the more true it will become.
So what can we do!?
The answer to this is so simple but also the hardest thing in the world.
We have to change.
We have to start by letting go of our own preconceived notions of what math is, what math education is, and how we were taught.
And we have to be okay with saying
“The way I was taught might not have been the best way there is”
Because the world is different now…but math really isn’t. If you’ve been with me for a while, you know I get so frustrated at the term “new math”. It’s insane to think that anyone could’ve made math new. Math is nature - math is patterns. Math is THINKING. If you think thinking is new then…I’m scared for you.
But the world is different in that we don’t really need to be trained in a streamlined way of calculation to do a job. Students will have calculators in their whole adult life. What we truly need is problem-solvers and people able to think flexibly and differently. But in my experience, students that are strong mental thinkers can actually eventually do it faster than you punch it into a calculator (trust me, we have tested it). So if speed is your hang-up…time to find a new one.
Once you let go of that notion that there is only one way to do math, you have opened yourself up to helping your students immensely.
Then it’s just time to DIVE into numbers. Have students sit in a circle and count by 1’s, 2’s, 8’s, 200’s, 30’s, whatever. Write the numbers on the board and have students find patterns.
—>Encourage students to ALWAYS break numbers into their parts (expanded form) in various ways
267 can be 1 hundred+4 tens+27 ones OR 14 tens and 67 ones. It can be made in SO MANY ways.
—>Have them add the parts, take parts away, PLAY with numbers.
—>Never let them call the 5 in 752 a “5”. Always interrupt them and say “that’s weird…I didn’t know 7 +5 + 2 was 752…” until they automatically call it 50 because they’re sick of you (hi students! ;p)
—>Have your students build vertical number lines with smaller and then larger numbers. Ask them to relate the numbers, which are close and which are far. Is 200 closer to 0 or 900? Where does 200 live in relation to these numbers?
Start helping your students have a giant number line constantly running in their brain.
If you’re in a position to, please please please push off the standard traditional algorithm until upper elementary grades when students can construct it well. Try it and see the difference and then advocate for it.
Daily number sense routines are not that difficult to implement. The best news is, you only have to do a little at a time! Start a few here and there and watch the ball start to roll.
Check out my 2nd-3rd grade warm-up resource here!
And most importantly, never stop learning.
Many of us were not given opportunities to build our own number sense. I wasn’t until college. Find a way to grow in your own relationship with mathematics and watch it inspire your students.
As always, if you have any questions please reach out to me at AubreeTeaches@gmail.com
And HAPPY TEACHING!
