If you’ve taught math at any level for long enough, you’ve come across the question or idea behind the question of algorithms. More specifically, algorithms or strategies for solving computational problems.

So it begs the question - what IS an algorithm?

In terms of mathematics teaching, an algorithm is a way to compute that always works and is efficient. When most people are talking about algorithms in regard to elementary mathematics, they are talking about the traditional algorithm. The one where people “borrow”, “carry”, “regroup”, “trade”, “add a 0”, etc. This is an important distinction because it isn’t the standard/traditional algorithm everywhere…in other countries there are other dominant algorithms.

We use a lot of algorithms in elementary math classrooms. What it really means is that we are showing kids the most efficient way to carry out computations. Which sounds nice - if our goal is to have our students be as effective at mathematics as a computer.

But there are two flaws here. One, that our students won’t ever be able to carry out an algorithm as well as a computer. Ever. And two, that perhaps that shouldn’t be our goal at all..

If we establish that we should help students become critical thinking problem solvers, which the mathematical process standards call them to be, then we need to think about moving past a standard algorithm.

So the natural other side of the coin is strategies. And we hear a lot about computational strategies, positive and negative, in this era of “new math” (which is an oxymoron…)

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Here’s the things that I think makes people so frustrated with this “new approach” to math. We knew that we wanted students to be more flexible and not compete with computers. But curriculum companies, writers, and even teachers weren’t sure how that would look or come about. So we collected all of these strategies for computing (that work every time) and started forcing students to practice them all.

We changed the algorithm but we didn’t really change the big picture, the way we were approaching mathematics teaching. In fact, the way we might approach teaching in general. Moving away from passing on knowledge and towards helping students discover patterns and build on their own schema. Instead of teaching the one standard algorithm, we were now teaching 5-6 different ones. Students were still being taught to follow steps, repeat the teacher, and do pages of practice but in so many different ways. They were rightfully overwhelmed, which led to parents being frustrated, and teachers being confused.

A strategy, like open number lines or partial sums, is really only a strategy if the student has come to make sense of it. I purposefully don’t use the words “discover it” here even though it’s what I mean, because people inevitably say “students aren’t just going to discover all these different ways!” and that’s true.

Students don’t need to pluck a strategy out of thin air to have discovered it. They need to adopt it and make it make sense to them. An open number line is just a tool to a kid who has intimate number sense. Partial sums comes naturally to students that have internalized value of numbers. Using friendly numbers might come up from one student and then other students understand it, take it, and use it for themselves.

The point is that the student didn’t learn the “strategy” from a song, poem, poster, copying the teacher, doing a worksheet, etc. If they did, then that strategy is now actually just an algorithm. It’s a set of steps someone else came up with and handed down for others to use to get answers. And it’s very good for that…but only that.

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So how do we help students create strategies? It all starts with our classroom mathematics community.

For students to begin discovering and adopting mathematical strategies, they need to know numbers really really really well. Spending time on knowing number instead of producing computers (in this sense “kids who computer fast”) can actually pay off a lot in the end. Playing number games, encouraging mental math, and number talks are all things that will help students know number. When they know number and the value of numbers then they can start with lay with those values by putting them together, taking them apart, comparing them, multiplying them, and more.

Kids also need to have a mathematical community where their ideas and attempts are valued, sharing is a necessity, and mistakes are okay. A common misconception here is that we are going to let students get wrong answers and it’s alright as long as they tried and used a strategy. This just simply isn’t true… I don’t know a single math teacher that would tell you they are just okay with a student leaving them thinking 7+8 is 22 or anything else that is a wrong solution. But students need to know, understand, and believe that their math brain grows from the process more than the answer. That isn’t really the way math class has ran for a long time… If students feel comfortable and confident to speak out, even if they’re unsure, the whole class is more likely to come into contact with strategies that they understand and can use for their own problem solving. They’re also more likely to play around with numbers, ideas, ad strategies, which can often lead to some amazon discoveries.

Students need to know that you are not the all-knowing math god in the room. For students to build strategies, they need to start looking at their own abilities and the abilities of their classmates. Otherwise, they just turn to you. Just tell me. And this is tricky if you have older students that are used to this type of schooling. It will take a while, which will make you want to give up and just tell them. This happened to me in moving to third grade. But I didn’t give in, and something amazing happened. Students stopped turning to me. They knew I wasn’t going to give them answers and I continued to give them really engaging, puzzling, real-world tasks that they desperately wanted an answer to. They were forced to start figuring something out.

Strategies were born. Strategies were shared and adopted. It was that easy.. but it all started with setting up a math community that led us there.

So I hope it’s clear how the answer to helping our students become thinkers might lie more in our approach to teaching and our class community than the things we are teaching them. And that by allowing more time and freedom for these things to happen, we can empower our kids to beat those computers with their amazing human brain capabilities.

Because after all….math is natural.

If you teach 3rd grade, you’re in extra luck. I have worked to create a resource that uses the 4 components of quality math lessons (video here) and encourages students to build strategies for multiplication. It’s 3rd grade multiplicative thinking unit - yes an entire unit! - called Math is Natural. Check out a freebie lesson HERE!