Start With What Makes Sense

I have a simple premise. How do you teach a student anything? Start with what makes sense. Think about what is already intuitive to students. What situation they are already familiar with is this similar to? If we start with the stuff that feels obvious and grounded — the “of course that’s how it works” kind of logic — then even the more abstract concepts have something solid to stand on.

Hear me clearly: I am not saying “start with what students already know” (i.e. the skill you taught them yesterday or last year). I’m not saying “activate prior knowledge,” though this is certainly a useful teaching strategy.

What students know is in flux. What students can remember from the previous day, week, or year is volatile and flimsy. What students understand is sturdy. If you understand, for example, that area is just counting little squares that fit into a space, you naturally see why the area of a rectangle is base × height. You don’t have to memorize a formula — it makes sense because multiplying the number of rows by the number of columns gives the total number of squares. This kind of understanding sticks, and it’s exactly the kind of intuition we want students to build around every topic.

The Difference Between Knowing and Understanding

When I say ‘start with what makes sense’, I mean to start with what students own, what students find intuitive and logical. What is obvious to them. It doesn’t need to be recalled or re-taught. Students won’t have to look back at their notes to find it. After a while, the skills we teach in school will become part of this body of knowledge that makes sense, but this takes time.

For example, consider the distance formula:

Students may know this formula because you taught it to them yesterday. They may have memorized it and can reproduce it accurately. But does this formula make sense to students? It depends on the approach we use to teach it and also on the opportunities we give them to grasp its logic. The reason I don’t have to look up the distance formula anymore is because I understand that the true distance between any two points is dependent on the horizontal and vertical distance between them as defined by the Pythagorean Theorem, and that those horizontal and vertical distances are determined by finding the change in, or difference between, the x and y coordinates.

Building on Intuition, Not Just Skills

Students may come to us with knowledge gaps, or the infamous “learning loss.” But not a single student comes to us devoid of intuition and logic. Not a single one. One of my core beliefs is this: Every topic can be taught at a developmentally appropriate level in ways that are accessible and sensible to students and authentic to the discipline. How does this change the way we teach? This belief keeps me from the “my kids won’t understand this, so I’ll just have to teach them the procedure or show them the steps” mindset. Instead I ask myself: what about this concept already makes sense to students? What part of this is intuitive? How can I present this topic as a logically satisfying extension of something students find familiar and obvious?

We often expect that the only knowledge students bring to the table is the thing we taught them yesterday, last week, or last year (which we hope they remember). But students bring so much more – their experiences being a human in the world for starters! So instead of relying only on the facts and skills they were taught in their previous math class, we should start with what already makes sense to them. We should build on intuition, not just on previously learned skills.

Building on previously learned skills sounds something like: “Yesterday you learned that to calculate slope, you do y2-y1 over x2-x1. Today you’re going to find the slope and then use that along with the y-intercept to write an equation.”

Building on intuition sounds like: “Yesterday we talked about how quickly one thing changes compared to another, which we called the rate—like miles per hour or dollars per topping. Today we’re going to see if we can find a relationship between those two quantities that always works, and that will allow us to determine one quantity simply by knowing the other. Like being able to choose any number of toppings and then figuring out the cost of the ice cream.”

When we focus only on previously learned skills, we’re assuming every student has perfect recall of what was taught yesterday, last week, or last year. Spoiler: they don’t. Some students will have missed class, forgotten steps, or they may struggle to interpret the abstract symbols they see in their notes. For them, a lesson built solely on assumed prior skills can feel like climbing a cliff with no footholds. That’s why so many students fall behind in ways that seem unfair — not because they can’t learn, but because the lesson never starts where they actually are: with what makes sense.

Case Study: Starting With What Makes Sense About Elimination

So what does this look like in practice? Let’s look at an Algebra 1 lesson on elimination called Gas Station Snacks. The activity is designed to get students thinking about the rationale behind elimination—what it is and why it’s a desirable strategy.

The Activity starts with this question.

We’ve taught this lesson dozens of times, and students always have a strategy for solving this first question. Ask any member of a group to explain their thinking and they’ll quickly point out that the only difference between the two purchases is that the second purchase has two more Combos, and those two extra Combos must be responsible for the $4.58 price difference. Each Combo, then, has to cost half of that – $2.29. This feels natural and obvious to students.

“What made that problem easy?” we always ask. Only one of the items was different. Ah, yes. So the only variation was due to the Combos, and it was easy to pin the extra cost on the Combos. How neat and tidy! I love it when it works out like that.

But of course, we have to up the ante a little. It’s not enough to start and stop with what is already obvious to students. We have to apply that intuition to arrive at something new that will, over time, also feel sensible and intuitive.

Immediately students realize that we are playing a different ball game now. BOTH the M&Ms and Starbursts are different. It’s unclear how much each item is contributing to the total price.

So the question becomes: how do we apply the intuition of the first question to this harder second question? This is where we use some carefully crafted prompts. Parts a, b, and c of question 2 might at first seem out of sequence. Don’t we have to isolate the cost of each item first and then find the cost of each of those purchases? We don’t actually. We can use the two given purchases to figure out the costs of lots of other purchases, using, you guessed it, logic and intuition.

In part a, students realize that 6 M&Ms and 5 Starbursts is simply the combined total of both purchases, so that must cost $8.34 + $6.15 = $14.49. In part b, they realize that 4 M&Ms and 6 Starbursts is exactly double the second purchase, so it must cost double $6.15, or $12.30. In part c, they determine that 2 M&Ms and 1 Starburst must cost half of $8.34, since it’s simply half of the first purchase.

Once students have these extra purchases figured out, it’s time to tackle the original question: how much does an individual item cost? If students get stuck, we can use a few focusing questions. Note that these questions do not prescribe a method or take them through the steps. They just help students notice and make connections.

• How could you use these purchases to figure out the cost of a pack of Starbursts?

• What about the Combo-and-gum problem made it easy to solve? Could we do something similar here?

Using either or both of these questions is generally enough to get things to “ping” for them. They could compare the first purchase with the purchase from 2b, or the second purchase with the purchase from 2c. The important thing is that we didn’t just ask them for the cost of 4 M&Ms and 6 Starbursts and then jump to question 3. That would have felt random. First, we had to establish the idea that it’s possible to generate many more purchases from the two we already had, and then the goal becomes to determine which purchases are helpful in isolating the price of an individual item.

This is problem solving. We start with what strategies we have, then we identify what strategies are useful. It’s the difference between what we could do and what we should do. The former without the latter can lead to spinning our wheels. The latter without the former often results in mimicking rather than sense-making.

The Danger of Teaching What Doesn’t Make Sense

So far I’ve been making the case for starting with what makes sense. But what happens when math doesn’t make sense to students? The truth is this: if math feels arbitrary, students check out. Our brains are wired to make sense of patterns and construct logical stories. When those connections aren’t there, when there is no thread to follow, students give up.

One of my FAVORITE student complaints is, “Miss, this doesn’t make sense!” Underneath that frustration is something really important: an expectation that math should make sense. That expectation is at the very heart of the mathematical discipline. What scares me is when students stop expecting things to make sense–when they’ve had enough experiences of being asked to memorize and grind through procedures without reason that they no longer trust their intuition.

That’s the real danger. There is nothing more harmful to students’ sense-making than requiring them to suspend disbelief. When we ask them to set aside what they know about how the world works just to survive a fake math world, they disengage. And honestly? They’re right to. Math should make sense.

Why Contrived Problems Undermine Sense-Making

You may think that I'm about to make an argument that all problems have to be “real-world.” I'm not. I love a good fantastical problem about opening and closing 100 lockers. In this case, the concrete context is simply a tool to give footing to abstract ideas. What I am arguing against is the “word problems” or “application problems” that defy everything we know about how the world works.

As a very simple example, consider this problem:

Janet bought 5 drinks at Starbucks for her 4 coworkers and herself and spent $27.50. How much did each drink cost?

At first glance, this seems realistic. People buy coffee for coworkers all the time. But the ruse here is in the math: the assumption that each drink would cost the same. That’s incredibly unlikely. If we want students to wrestle with division, we need a context that highlights equal parts or fair sharing. There is no reason to think the 5 drinks would cost the same amount, yet there is no other feasible operation for students to work with here. So they have to suspend disbelief and accept that “total ÷ number of items” is the rule.

Over time, this is also what leads to students overgeneralizing “key words” and solving problems mindlessly based on rules like “when you see the word total, that means add.” It’s because they’ve been forced to turn off their thinking too often, treating math like a set of arbitrary steps instead of a logical story.

An Invitation to Teach Differently

Students don’t come to us empty-handed. They bring intuition, logic, and the expectation that math should make sense. Our job is to honor that. When we build on what already makes sense, we help students see math as a coherent, logical story that belongs to them—not just a set of rules they have to memorize.

So here’s the invitation: Don’t ask students to set aside what they know about the world. Start with what it is obvious and logical, and help them see how far their intuition can take them. That’s how we help students trust that math makes sense—and that they can make sense of it.