Good Questions For Math Teaching

What Actually Makes a Math Question Worth Asking

Most math questions teachers use are too narrow. They ask for a single procedure, a plug-and-chug answer, something that can be guessed from the setup alone. Students learn to recognize the shape of a problem and fire off the matching algorithm. That works fine on a quiz but falls apart when they need to think. I spent years writing exams before I stopped trusting my own instincts about what makes a question good. The thing I learned the hard way is that a strong math question usually has two properties: it forces a decision at some point, and that decision isn't obvious from scanning the problem statement. If the student can read the question and immediately start working without any actual choice being made, the question is doing about as much as a worksheet drill.

Good Questions For Math Teaching

Here is how you actually build them instead of finding them online and handing them out. Start with a concept. Let us say you are teaching quadratic equations. Most teachers end up with something like "solve x squared minus 5x plus 6 equals zero." That question tests one skill. It does not test understanding. Instead, flip it. Give the roots and ask what the equation could be. Or give two different forms of the same equation and ask why they look different. Or show a solution that contains an extraneous root and ask where it came from. The math is identical. The thinking required is completely different. I ran into a real problem with this when I was building a unit on rational expressions for second-year algebra. I wrote a question where students had to simplify a complex fraction and explain why a certain value could not be part of the domain. Three students simplified it correctly, then wrote "x cannot equal 1 or negative 2" without checking whether those values actually made any denominator zero. They had memorized the procedure for finding excluded values but had no idea what they were doing. I had not caught that gap in my original question design. The fix was adding a part that asked them to substitute both values back into the original expression and show exactly what broke. Once they saw the division by zero happen in front of them, the rule stopped being arbitrary. That question stayed in my bank after that.

Another thing most people miss is that difficulty and depth are not the same thing. A question can be incredibly hard because the arithmetic is tedious, or because it requires five steps that a student might drop. That is not a good question in the sense that matters for teaching. Good questions are accessible at the entry point. A student who knows nothing about the topic should still be able to engage with it, make a guess, try something, and get somewhere. The depth comes from where the question can take them if they keep going. Look at a problem like asking students to find the area of a triangle given only the coordinates of its vertices. Some will use the base-height formula after finding side lengths with distance. Others will use the shoelace formula. Some will draw a rectangle around it and subtract. The question works for anyone who knows what area means. The range of valid approaches is the point. If you only accept one method, you are not testing the concept. You are testing whether you showed that exact method recently. There is also a trap with context-heavy word problems. Teachers love wrapping math in real-world stories because it feels more engaging. But if the story requires outside knowledge to solve, you are no longer testing math. I once gave a problem about comparing cell phone plans that assumed students understood monthly fees, per-minute charges, and overage structures. Half the class got stuck on the interpretation, not the algebra. The question was testing their life experience, not their ability to set up and solve a linear system. I rewrote it to remove the industry assumptions entirely. The math stayed the same. The barrier to entry dropped dramatically.

When you are evaluating whether a question is good, ask yourself these things: can a student enter it without having seen the exact same problem before, does it have more than one reasonable path to a solution, does it reveal something about the student's thinking when they get it wrong, and does getting it right require understanding rather than pattern matching. If the answer to any of those is no, the question is still usable. It just belongs on a practice sheet, not in a unit assessment. Using diagnostic questions for testing is a common mistake. Diagnostic questions should have wrong-answer options that tell you what the student is confused about. A multiple choice question on factoring where the distractors are common sign errors gives you more information than a blank response line, because every wrong answer maps to a specific misconception. I started using that format for my end-of-unit checks and cut my grading time roughly in half while actually learning something from the results. One more thing that is worth noting. Good questions take time to write well. If you are trying to build a full bank of them, expect to spend about twenty to thirty minutes per question when you are starting out. The first batch will be rough. The tenth will be better. The key is that you keep every question that made you pause while writing it. If a question surprised you or made you think about the topic differently while you were creating it, it will do the same for your students. That is about as reliable a signal as you are going to get.