Questions to Ask Students in Math
For math teachers, tutors and parents who want a student to explain the thinking and not only give the answer, in a lesson or over homework. The questions follow a problem from start to finish: getting started, probing a strategy, responding to a wrong answer without supplying the fix, connecting ideas and representations, asking for proof, and wrapping up the lesson. They work from elementary school to high school, and each has a note on what a good or a worrying answer sounds like and what to do with it.
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The questions
Each question, and why to ask it
Getting started
What is this problem asking you to find?
Why ask it
Have the student answer in their own words with the page turned over. Reading the question back word for word means they do not have hold of it yet, so ask what the answer will be a number of: miles, cookies, a value of x. A wrong start on a word problem is cheapest to catch here, before any arithmetic.
What do you notice about this problem?
Why ask it
Ask it before anyone picks up a pencil and take every observation, including 'the numbers are big'. A student who notices that both numbers are even, or that the graph crosses the axis twice, has often found a way in without being shown one. Follow with 'what do you wonder?' to let the class pose the question itself.
What do you already know that could help here?
Why ask it
Good answers name a fact or a tool: 'area is length times width', 'the angles in a triangle add up to 180'. 'Nothing' usually means the problem looks unfamiliar, not that the student knows nothing, so ask what the diagram or the first sentence reminds them of.
About how big should the answer be?
Why ask it
Get an estimate said aloud or written in the margin before the work starts, and accept a range. It gives the student something to hold the final answer up against. A guess of 5,000 for 48 times 12 tells you the gap is in number sense, which more practice on the written method will not fix.
Have you solved a problem like this one before?
Why ask it
Listen for what the student thinks makes two problems alike. 'It has trains in it' is a match on the surface, while 'it is another one where two things grow at a steady rate' is a match on the structure. If nothing comes to mind, put last week's problem beside this one and ask again.
Which numbers here do you need, and which could you ignore?
Why ask it
The worrying sign is a plan to use every number on the page because it is printed there. Now and then give a problem with one number too many, or one too few, and see who says so. Older students can underline what is given and circle what is asked.
Can you tell me the story of this problem without the numbers?
Why ask it
'Someone has some money, spends part of it and wants to know what is left' shows the student sees the action, and the operation follows from that. When the retelling will not come, the operation is being picked by guesswork or by keyword. Young children can act it out with counters first.
What would you try first, even if you are not sure it will work?
Why ask it
The second half is what unfreezes a student who is waiting to be certain. Accept any honest start, 'I would try some numbers' included, and let it run a few minutes before you comment. Skip it when the pencil is already moving.
What does this word or symbol mean here?
Why ask it
Point at one term: 'difference', 'at least', the bar over a repeating decimal, f(x). A confident wrong meaning, such as reading f(x) as f times x, can explain a whole page of errors, and asking finds it faster than grading does.
Strategy
How did you get that?
Why ask it
A good reply goes through the steps in order. If the telling jumps from one line to another, ask what happened in between, because the skipped step is often the one done in the head and the one that went wrong. 'I just knew' may be true for a young child with a small sum, so ask how they would show a friend who did not know.
Why did you choose that method?
Why ask it
'Because it is what we did yesterday' is honest and worth hearing: the student is matching the lesson and not the problem. A stronger reply points at the numbers. Try it on 1,000 minus 998, where counting up two beats a line of regrouping, and in algebra on a quadratic that factors at a glance.
What have you tried so far?
Why ask it
This is the reply to 'I don't get it'. It moves the conversation from the whole problem to a particular place in it, and anything attempted, even something crossed out, is a start to build on. If the page is empty, go back to what the problem is asking.
Can you show me where this number came from?
Why ask it
Put a finger on one number in the student's work, not on the whole line. Either they trace it back to the problem, or they find it came from nowhere, in which case they have located the slip themselves. Pick the first number that looks off, since every line below it carries the same slip.
Is there another way to solve it?
Why ask it
Save it until there is an answer on the page. A second method that agrees is a check, and one that disagrees is a better problem than any you could write. Leave it out for a student who only just got through the first way.
What tool would help you here?
Why ask it
Offer a choice and let the student pick: counters, a ruler, graph paper, a number line, a calculator. Reaching for the calculator to do 6 times 7 points at a fact that is not yet secure, while reaching for it once the setup is written down is the tool doing its job. Before a test, find out which tools will be allowed, since that varies by class and by exam.
Would a simpler version of this problem help?
Why ask it
Swap the awkward numbers for 2 and 10, or try three cases before tackling n. Once the easy version is solved, ask which steps would carry over unchanged. Seeing that they all do is a strategy that outlasts the unit. Reach for it with fractions, where the right operation is hard to see.
What would happen if this number changed?
Why ask it
Alter one thing: double the price, make the slope negative, add a fourth side. Predicting the effect without redoing the whole calculation shows a grasp of the relationship. Having to start over from the top shows the procedure and, so far, no more.
How could you organize what you have found so far?
Why ask it
A page of scattered trials hides a pattern that a table with the trials in order often shows at once. Ask before you suggest the table, and hand over a blank one only if nothing comes. Good for counting problems and anything solved by guess and check.
Can you explain how your partner solved it?
Why ask it
Restating someone else's method takes real listening, and it often shows a student a route they would not have found. 'They did it wrong', said of a method that is merely different, is the worrying reply. Have the pair confirm that both routes land on the same answer.
Mistakes
Does that answer make sense?
Why ask it
If you ask only when the answer is wrong, it becomes a signal and students erase correct work the moment they hear it. A good reply goes back to the situation: 'you cannot hire 4.5 buses, so it has to be 5'. 'I think so' with no reason means the check has not happened yet.
Can you read your work to me out loud, from the top?
Why ask it
Plenty of students hear their own slip halfway through, which sticks better than having it pointed out. Keep a neutral face and do not stop them at the error. If they sail past it, ask them to read that one line again more slowly.
How could you check that without asking me?
Why ask it
Listen for a real method: substitute the answer into the equation, undo it with the inverse operation, compare it with the estimate. Doing it again the same way tends to repeat the same mistake, so ask for a check that takes a different route.
Should the answer be bigger or smaller than the number you started with?
Why ask it
It catches a wrong operation before any arithmetic is checked. A common belief is that multiplying always makes a number bigger and dividing always makes it smaller, which fails as soon as fractions and decimals arrive. Half of 12 written as a multiplication brings that belief into the open.
Which step are you least sure about?
Why ask it
Students frequently know where it went wrong before they know why. Go to the line they name and ask which rule they applied there. If that step turns out to be sound, say so plainly and ask for the next least certain one.
What question did you actually answer?
Why ask it
A wrong answer is often the right answer to a different question: the perimeter when the area was wanted, x when the problem asked for 2x + 1, the new price when it asked for the discount. Have the student reread the last sentence of the problem and set it beside what they wrote.
Can you test that rule with actual numbers?
Why ask it
Made for algebra slips such as squaring a sum by squaring each term. With 1 and 2, the left side comes to 9 and the right to 5, and the numbers do the contradicting so you do not have to. Then ask what the missing 4 could be made of.
You two have different answers. Can both be right?
Why ask it
Have each student explain to the other while you stay out of it. Sometimes both are right, as with 1/2 and 0.5 or an equation with two solutions, and that is worth finding out. Step in only when they have agreed on something false or run out of things to say.
What might someone have been thinking to get this answer?
Why ask it
Put up a wrong answer with no name on it, or one you wrote yourself, such as 1/2 plus 1/3 equals 2/5. Students rebuild the logic ('they added the tops and the bottoms') without anyone being embarrassed. Never use a student's own work this way unless they have agreed.
What will you watch for the next time you meet a problem like this?
Why ask it
Ask after the error is fixed, not before. A useful answer can be seen on paper: 'I will copy the negative sign onto every line'. 'Be more careful' is not a plan, so ask what careful would look like.
Connections
Can you draw a picture or a diagram of this?
Why ask it
A bar model, a labeled triangle or a rough sketch of the graph is enough, and boxes and stick figures count. A drawing with the quantities in the right relationship, the longer bar for the larger amount, shows understanding even when the arithmetic is off. Six carefully drawn apples with nothing connecting them is decoration.
Where would this number sit on a number line?
Why ask it
Use it for fractions, decimals, negatives and square roots. A student who places 3/8 beyond 1/2 'because 8 is more than 2' is comparing whole numbers, not fractions. Ask which two familiar numbers it falls between before asking for the exact spot.
Can you show this as a table, a graph and an equation?
Why ask it
For middle and high school. Ask where the starting value and the rate of change appear in each form. Finding the slope in the equation and not in the table is a sign of a procedure without the idea behind it yet.
What is the same about these two problems, and what is different?
Why ask it
Set two side by side: 3 times 4 and 3 times 40, or y = 2x + 1 and y = 2x - 3. Weak answers stay on appearances ('one has a minus'). Strong ones get to what the difference does: 'the lines are parallel, and one sits four lower'.
Can you make up a real situation this calculation would fit?
Why ask it
Try it with 6 divided by 1/2. A story about how many half-cup scoops fill six cups fits. A story about sharing six cookies between two people is a different calculation, and telling it shows the student is dividing by 2. The story shows what the operation means to the student more plainly than a worked answer does.
What does this number mean in the situation?
Why ask it
Point at one number: the answer, or the 2.5 in y = 2.5x + 10. 'The cost of each mile' connects it to the context. 'The slope' names it correctly and stops short, so ask 'the slope of what?'
Can you say that again using the math words?
Why ask it
Use it after 'the top number', 'the slanty line' or 'timesing'. Accept the idea first and ask for the vocabulary second, so the student is heard before being corrected. With younger classes, keep the words on the wall to point at.
What pattern do you see, and what would the tenth one be?
Why ask it
Going straight to the tenth term separates two ways of seeing. A student who counts on step by step sees how each term grows from the one before, and one who jumps there has found the rule. With young children use shapes or skip counting, and with older ones ask for the nth.
Which one of these does not belong, and why?
Why ask it
Show four numbers, shapes or graphs chosen so that a case can be made for each, such as 9, 16, 25 and 43. There is no single right answer, which draws in students who rarely volunteer. Listen for reasons built on properties: 'the only one that is not a square', 'the only even one'.
The answer is 12. What could the question have been?
Why ask it
Everyone can offer something, from 6 plus 6 to the number of edges on a cube, so it suits a class with a wide spread. Run a second round with a condition: 'it has to use a fraction' or 'it has to be a word problem'. Ten replies that are all additions tell you which operations feel safe to this group.
Prove it
How do you know that is right?
Why ask it
'Because I put it back in and both sides came to 14' is reasoning. 'Because the calculator said so' or 'because it is in the back of the book' rests on authority, so ask what they would say to someone who had neither.
Will that always work?
Why ask it
Ask it when a student states a shortcut drawn from a few examples, such as 'to multiply by 10 you add a zero'. Send them looking for a case that breaks it, and 2.5 times 10 will. A rule that survives the hunt can go on the wall along with its limits.
Is that always, sometimes or never true?
Why ask it
Give a statement: 'the sum of two odd numbers is even', 'a shape with four equal sides is a square', 'x squared is bigger than x'. 'Sometimes', with the cases sorted, is often the most thoughtful reply. An 'always' backed by three examples needs to be pushed for a reason.
Can you show me one that fits and one that does not?
Why ask it
Ask it when a student recites a definition: a prime number, a function, a parallelogram. The one that does not fit is the harder half, and a near miss, such as 9 for a prime or a sideways parabola for a function, shows the definition is understood and not only memorized. If both come straight off the textbook page, ask for a fresh pair.
How would you convince someone who does not believe you?
Why ask it
It raises the standard from 'I think' to an argument another person can check: a drawing, a general reason, an example chosen to be hard. In class, let a partner play the doubter and say which part did not persuade them.
Do you agree or disagree with that, and why?
Why ask it
Put it to the class as soon as one student makes a claim, before you have said whether it holds. A good reply gives the other student's reason back in new words or adds one. A bare 'I agree' is an invitation to ask which part.
Why does that rule work?
Why ask it
Aim it at procedures learned by heart: invert and multiply, carry the one, cross-multiply, move the decimal point. 'That is just how you do it' marks a rule held in memory alone, which is easy to misapply later. Choose one rule per unit to open up, because nobody can do this for every rule every day.
What are you assuming?
Why ask it
A high school question. Dividing both sides by x assumes x is not zero, a travel problem assumes a steady speed, and a diagram may only look as if it has a right angle. Naming the assumption is what lets a student say when the answer would stop holding.
Could there be more than one answer?
Why ask it
Good for x squared equals 16, rectangles with a perimeter of 20, or ways to make 50 cents. Stopping at the first answer suggests the student expects one answer per question. Follow with 'how do you know you have found them all?', which is the harder half.
Wrap-up
How would you explain today's idea to a classmate who was absent?
Why ask it
Two or three sentences on a slip of paper at the end. An explanation that is all steps ('you flip it and multiply') tells you the procedure landed, and one with a reason in it tells you more. Read the slips that evening and open the next lesson with the clearest one.
Which problem made you think hardest today, and what made it hard?
Why ask it
'Too many steps', 'I did not know where to begin' and 'the fractions' call for three different responses from you. If most of the class names the same problem, it deserves ten minutes tomorrow. A student who says nothing was hard may need something harder.
What was your most useful mistake today?
Why ask it
Go first with a real slip of your own, which sets the tone better than a poster about mistakes. The answers worth having name the error and what the student does now in its place. Keep it on paper in a class that is not yet at ease with being wrong in public.
What would you ask if we had five more minutes?
Why ask it
Collect the answers in writing and sort them at your desk. Three slips with the same question give you tomorrow's opening. A blank slip from a student whose work showed trouble means a private check-in the next day, not that all is well.
Can you write a problem of your own that uses today's math?
Why ask it
Writing a problem that works asks for something solving does not: the numbers have to come out, and the question has to match the operation. The problems that cannot be solved, or that turn out to need a different operation, tell you the most. Have partners swap and solve them at the start of the next lesson.
Which of today's problems could you teach to someone, and which do you still want help with?
Why ask it
Ask for problem numbers in two columns. A list points at specific work, which a rating from one to five cannot do. Compare the 'could teach' column with what is on the page, since confidence and accuracy do not always match.
How to ask math questions that get students explaining
Practical guidance for the conversation itself
Before the lesson or the homework
Work the problem yourself first
Solve the task before the students do and note where you expect them to stall and which wrong answer is the tempting one. Then write two or three questions from this page next to those spots in your plan. A question chosen for a particular problem lands better than one reached for in the moment, and it stops you from falling back on 'does everyone get it?'
Pick a task worth talking about
Twenty exercises of the same kind leave little to ask about beyond the answers. One problem that can be solved in more than one way, or that has a believable wrong answer, gives every group on this page something to work on. If the textbook only offers drill, take one exercise and ask for two methods, or change one number and ask what happens.
Fit the wording to the age
The groups run from elementary school to high school, but the words should change. A six-year-old with counters hears 'how did you count them?', and a sixteen-year-old hears 'what are you assuming?' Both are being asked for reasons. With young children, let showing with blocks or fingers count as an explanation.
For parents and tutors
You do not need to know the method your child was taught, and three questions will carry most homework sessions: what is it asking, how did you get that, and how could you check? Ask to be taught the method as if you were the student. If it differs from the one you learned and the explanation breaks down, send a note to the teacher asking how it is taught in that class before you teach your own way.
While students are working
Wait longer than feels natural
After a question from Strategy or Prove it, count slowly to five in your head before you say anything else. The silence feels much longer to you than to the student, who is using it to think. Tell the class at the start of the year that you will wait, so that a pause reads as thinking time and not as a sign that someone is in trouble.
Ask about right answers as often as wrong ones
If 'how did you get that?' only ever follows an error, students learn to hear it as a correction and start erasing. Ask it of the first correct answer of the day. It also shows you the students who reached the right number by a method that will fail on the next problem.
Turn their question back, within reason
'Is this right?' can be met with 'how could you find out?', and 'what do I do next?' with 'what have you tried?' The limit is a missing fact. If a student does not know what 'perimeter' means or which coordinate is written first, no question will draw it out of them, so tell them and move on to the thinking.
Let partners talk before the class does
In a classroom, give thirty seconds to tell a neighbor before you take answers from the room. More students get to put an idea into words, and the ones you then call on have had a rehearsal. It also lets you walk around, listen, and choose whose explanation to hear first.
Keep the pencil in their hand
The moment you write on a student's page, the next line of thinking is yours. If something needs showing, use a separate sheet or a different example with other numbers, then hand the original problem back untouched.
Reading the answers
Steps are not yet reasons
'First I did this, then I did that' tells you the student can carry out the method. 'I did this because...' tells you why it works for them. Both are worth hearing, and the questions under Prove it are how you move from the first kind of answer to the second.
Treat "I don't know" as a place to start
It usually means 'I do not know all of it'. Ask which part they do know, or go back to the first question in Getting started. A student who can say what the problem is asking has already stopped not knowing.
Hear a wrong answer through to the end
Wrong answers nearly always have a logic, such as calling 0.25 bigger than 0.3 because 25 is bigger than 3. If you cut in at the first error you fix one problem. If you hear the reasoning out, you find the idea that will cause the next ten, and that idea is the thing to teach.
Keep a note of what you hear
Carry a class list and jot a word or two beside a name when an explanation shows something: a neat method, a shaky idea, a word used wrongly. By the end of a lesson a few of the same notes will repeat, and those decide where tomorrow starts.
Mistakes to avoid
Narrowing the questions until the answer falls out
It starts with a good open question and ends with 'so what is 6 times 4?' Each smaller question felt helpful, but you did the reasoning and the student supplied the last number. When you notice the questions shrinking, stop, go back to 'what have you tried?', and give it time.
Hiding an instruction inside a question
'Shouldn't you find a common denominator first?' is telling with a question mark on the end. Students follow it and learn nothing about when to do so. If you mean to tell, tell. If you mean to ask, ask something you do not already know their answer to.
Questioning a student who is thinking well
Someone deep in a problem does not need to be asked how it is going. Watch for a moment first. Ask one question when there is a stall or an answer on the page, then walk away and come back in a few minutes to hear what became of it.
Taking the first correct answer and moving on
When the quickest student answers and the lesson rolls forward, everyone else learns that their thinking was not needed. After a correct answer, ask who got there another way or who can say why it works, and hear two or three voices before confirming anything.