In Math Class

What Actually Works When Teaching Math

I have been teaching math for a while now, and the biggest thing I have learned is that most students do not struggle with the concepts themselves. They struggle with the way things are presented and the gap between abstract notation and actual understanding. Let me walk through what I do and what I have seen fail. When I introduce a new topic, I never start with the formula. I start with something tangible. Fractions? I bring actual pizza slices or paper circles and cut them up. Ratios? I mix different colors of beads and count them out. The point is that students need to see the thing before they see the symbol for the thing. I have watched too many kids memorize "a over b" without having any idea what that represents in the real world. There is a specific case where this became obvious. I was teaching proportional reasoning to a group of ninth graders, and one student kept getting the answers wrong on paper but solved the problem perfectly when I asked her to imagine she was dividing candy bars among friends. The workaround was simple: I stopped giving her the abstract version until she could explain the answer in plain language first. Once she could say "three people share two bars so each gets one and a half," the equation 3x = 2 stopped being scary. She could derive it herself.

The Core Approach to In Math Class

Here is the method I use and recommend. It is not fancy. It is not research-backed in any dramatic way. It is just practical. Step one: Introduce the concept through a physical or visual example. Make it concrete. Give students something they can manipulate or picture. Step two: Have them solve a few problems using that concrete model before moving to anything symbolic. This takes time. I usually spend one full class period on this. Students who rush ahead here end up confused later, and I see that pattern repeat every semester.

Step three: Introduce the notation. Show them how the symbols connect to what they already did. This is where the "aha" moment happens, but only if steps one and two were done properly. Step four: Practice. Not a ton of repetitive problems, but a range of them. Different numbers, different contexts, some word problems. The variety prevents students from just memorizing a procedure without understanding it. Step five: Check for understanding frequently. I do quick exit tickets every other day. Three questions. They tell me whether the class is ready to move on or whether I need to revisit something.

A common pitfall most teachers overlook

Most educators focus on getting students to the right answer. I have found that getting students to explain their thinking is more important. When a student says "I multiplied because the problem said more," you know they do not understand multiplication. When they say "I multiplied because I needed to find the total number of items in five groups of seven," you know they do. The difference is everything. I use a simple technique for this. After a student gives an answer, I ask them to explain it to the person next to them. Two minutes. They have to teach their partner. You would be surprised how many students cannot explain their own work when forced to put it into words.

What Does Not Work

Memorization without context. I am not saying it never helps. Arithmetic fluency matters. But memorizing procedures like FOIL for multiplying binomials without understanding what those letters represent leads to students who can follow steps but break down when the problem looks slightly different. Homework overload. More is not better. I assign maybe five or six problems per night, and they are usually mixed from different topics. Spaced repetition beats cramming every time. I have seen students drown in thirty problem sets and retain less than students who did six problems regularly over two weeks. Group work without structure. Pairing students up and saying "work together" does not help unless you give them a specific task with roles. I use roles like "explainer" and "checker" and rotate them. Without that, one student does all the work and the other zone's out.

Dealing with Different Skill Levels in One Room

This is the hardest part. You have students who finish in ten minutes and students who need double the time. I keep a stack of challenge problems that are genuinely interesting, not just harder versions of the same work. Things like "explain why this shortcut works" or "find three different ways to solve this." For students who need more support, I provide scaffolded worksheets with worked examples and smaller steps. One edge case I ran into last year involved a student who could do algebra but failed every geometry proof. Standard algebra instruction was not working. What worked was having her draw the diagram first and label everything she knew, then write out each statement as a separate sentence before attempting the formal proof structure. She needed to slow down and externalize her thinking on paper. Once she did that, proofs became manageable.

A Note on Technology

Tools like Desmos and GeoGebra are useful, but they are tools, not solutions. I have seen teachers replace conceptual teaching with screen time and wonder why students still do not understand. Use technology to visualize, not to substitute for explanation. A graphing calculator showing a parabola is worth more when the teacher has already walked through what makes it a parabola. If you are looking for resources, Desmos has free activity builder at desmos.com/activity. It is not a complete curriculum, but it is a solid supplement. Khan Academy is fine for extra practice, though the explanations can be thin on the why.

Putting it all together

The essence of effective math instruction is simpler than most people think. Concrete first, then symbolic. Practice with variety. Check understanding constantly. Adapt to the students in front of you. It sounds obvious, but the execution is where most places fall short. I have sat through math classes where the teacher spent forty-five minutes writing on the board and ten minutes checking if anyone was following. That is not teaching. That is broadcasting. The students who benefit most are the ones who get to connect the abstract to something they can hold in their hands, even if that thing is just a drawing on a piece of paper. Everything else builds on that foundation. Skip it and you are building on sand.