• 121 topics

Math Debate Topics

Questions a math class can argue, from whether zero is even to whether tau should replace pi.

Math Debate Topics

Do we count in tens only because we have ten fingers?

Read it out, then tap a side once for every raised hand.

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Has an answer: prove it 13

  • A price drops 50 percent, then rises 50 percent. Is it back where it began? A) Yes. Show that the two changes cancel out. B) No. Start at 100 and follow the price through both steps.
  • Are all endless sets the same size? A) Yes. Show how to pair any two of them off, one to one. B) No. Show a set too big to be listed first, second, third.
  • Can any number be divided by zero? A) Yes. Say what 6 ÷ 0 equals, then multiply back to check it. B) No. Show that nothing times zero can give 6.
  • Can multiplying make a number smaller? A) Yes. Find a multiplier that shrinks 10. B) No. Show that a product is never less than what you started with.
  • Can two negative numbers multiply to make a negative? A) Yes. Find two negative numbers whose product is below zero. B) No. Carry on the pattern 2 × −3, 1 × −3, 0 × −3 one more step.
  • Do the angles of every triangle add up to 180 degrees? A) Yes. Tear the corners off a paper triangle and line them up. B) No. Draw one on a ball, from the pole down to the equator.
  • Do the prime numbers go on forever? A) Yes. Show how any finished list of primes leads to a new one. B) No. Name the last prime, then explain why none can follow it.
  • Does 0.999..., with nines forever, equal exactly 1? A) Yes. Show that no number fits between 0.999... and 1. B) No. Name a number that sits between the two.
  • Does 1 + 2 + 3 + 4 and so on forever add up to −1/12? A) Yes. Explain the sense in which that value is given to the sum. B) No. Show that the running totals grow past every number.
  • Does a paper strip joined with a half twist have only one side? A) Yes. Draw a line along the middle and see where it ends. B) No. Color one side and show the side that is left blank.
  • Does doubling the sides of a square make four times the area? A) Yes. Draw the big square and count the small ones inside it. B) No. Show that twice the side gives twice the area.
  • Is a square a rectangle? A) Yes. Check a square against the rule: four right angles. B) No. Find the part of the rectangle rule that a square fails.
  • Is zero an even number? A) Yes. Show that zero is 2 times a whole number. B) No. Name the rule for even numbers that zero breaks.

Rules mathematicians chose 17

  • Is 8 ÷ 2(2 + 2) a fair test question? A) Yes. Work left to right and the school rules give one answer. B) No. It can be read two ways, as 1 or as 16, so add parentheses.
  • Should 0 to the power 0 be defined as 1? A) Yes. Algebra and counting formulas stay simple when it is. B) No. As a limit it can come out differently, so leave it open.
  • Should 1 count as a prime number? A) Yes. Only 1 and itself divide it, and old prime tables listed it. B) No. Then 6 is 2 × 3 and also 1 × 2 × 3, and factoring is not unique.
  • Should a billion always mean a thousand million? A) Yes. One word should carry one meaning in every language. B) No. On the long scale the word means a million million.
  • Should a full turn be 400 units instead of 360 degrees? A) Yes. A right angle would be a round 100. B) No. 360 divides evenly by 3, 6, 9 and 12, and 400 by none of them.
  • Should a parallelogram count as a trapezoid? A) Yes. Ask for at least one pair of parallel sides and it fits. B) No. Ask for exactly one pair and the two shapes stay apart.
  • Should an equilateral triangle count as isosceles? A) Yes. Ask for at least two equal sides and every isosceles fact holds. B) No. An old definition asks for exactly two equal sides.
  • Should an hour have 100 minutes? A) Yes. Time sums would then work like any other decimal. B) No. Sixty splits into halves, thirds, quarters, fifths and sixths.
  • Should angles be measured in radians from the first lesson? A) Yes. Arc length is then simply radius times angle. B) No. A right angle of 90 is easier to picture than pi over 2.
  • Should every country write decimals with a point, not a comma? A) Yes. One mark for the whole world would end misread numbers. B) No. Both marks work, and readers know their own.
  • Should mixed numbers like 2 1/2 be retired for 5/2? A) Yes. Next to algebra, 2 1/2 reads like 2 times 1/2. B) No. Two and a half cups means more to a cook than five halves.
  • Should Pi Day move from March 14 to July 22? A) Yes. 22/7 is closer to pi than 3.14 is. B) No. 3.14 is how pi starts, and 22/7 is only an estimate.
  • Should tau, which is 2 pi, replace pi? A) Yes. A full turn would be one tau, a quarter turn a quarter tau. B) No. Pi is in every textbook, and tau would mean relearning them all.
  • Should the × sign go, since it looks like the letter x? A) Yes. A dot or parentheses cannot be mistaken for a letter. B) No. The cross is the sign everyone learns first, so keep it.
  • Should the ÷ sign be dropped for the fraction bar? A) Yes. A fraction bar shows exactly what is divided by what. B) No. The sign fits on one line and every child knows it.
  • Should zero count as a natural number? A) Yes. An empty basket needs a count, and many books start at 0. B) No. Counting starts with one thing, and many books start at 1.
  • Would counting in twelves beat counting in tens? A) Yes. Twelve splits by 2, 3, 4 and 6, and ten only by 2 and 5. B) No. Ten fingers and every book already printed settle it.

Numbers and infinity 13

  • Are even numbers nicer to work with than odd ones? A) Yes. They halve cleanly and pair up with nothing left over. B) No. Odd numbers have a middle, and every prime but 2 is odd.
  • Are imaginary numbers badly named? A) Yes. The name began as a put-down and still scares students off. B) No. The name is history now, and a new one would confuse more.
  • Are negative numbers as real as positive ones? A) Yes. Debts, depths and temperatures below zero are everyday things. B) No. You can hold three apples but never minus three.
  • Could Roman numerals, which have no zero, run a modern world? A) Yes. An abacus can do the sums while the numerals keep the record. B) No. Place value needs a zero, and long sums need place value.
  • Do we count in tens only because we have ten fingers? A) Yes. Any base would work, and fingers made the choice. B) No. Ten is quick to learn and big enough to be useful.
  • Is "give 110 percent" bad math? A) Yes. Effort has a whole, and nobody has more than all of it. B) No. A price can rise 150 percent, so percents above 100 are real.
  • Is "infinity plus one" a fair answer in a biggest-number contest? A) Yes. Whatever number you name, adding one beats it. B) No. Infinity is not a number you can count up to, so it is out.
  • Is 1/3 a better way to write a number than 0.333...? A) Yes. The fraction is exact and finished. B) No. A decimal lines up on a ruler and compares at a glance.
  • Is an estimate ever better than the exact answer? A) Yes. A quick near answer catches the slip an exact sum hides. B) No. Near is not right, and a bridge needs the real figure.
  • Is infinity a number? A) Yes. Mathematicians compare infinite sizes by clear rules. B) No. It is no point on the number line; it says the line never ends.
  • Is pi a more important number than e? A) Yes. Every circle, sphere and wave carries pi. B) No. Growth, decay and compound interest all run on e.
  • Is rounding a kind of lying? A) Yes. 4.6 is not 5, and small errors pile up. B) No. Every measurement is rounded already, and honest rounding says so.
  • Is zero the most important number? A) Yes. Place value, and so all written arithmetic, leans on it. B) No. One builds every counting number, and zero only holds a place.

Shapes and space 13

  • Can a line with no width exist? A) Yes. It exists as an idea, which is all geometry asks. B) No. Anything you can draw has width, so it is only a story.
  • Does a circle have a side? A) Yes. One curved edge runs all the way around. B) No. A side is a straight line segment, and a circle has none.
  • Does a coastline have a true length? A) Yes. It is a real edge, so a fine enough ruler will find it. B) No. The shorter the ruler, the longer the coast measures.
  • Is a coffee mug the same shape as a donut? A) Yes. Each has one hole, and soft clay turns one into the other. B) No. Shape means lengths and angles, and theirs are nothing alike.
  • Is a four-dimensional cube a real mathematical object? A) Yes. Its corners are just points with four coordinates. B) No. A shape nobody can build or see is only a formula.
  • Is A4 paper better designed than letter paper? A) Yes. Fold A4 in half and the new sheet keeps the same shape. B) No. Printers, binders and habits already fit letter size.
  • Is area more useful to know than perimeter? A) Yes. Paint, carpet and grass seed are all worked out by area. B) No. Fences, frames and running tracks are all perimeter.
  • Is symmetry what makes a shape beautiful? A) Yes. Flowers and snowflakes please us with matching parts. B) No. A perfect match is dull, and the eye likes a small surprise.
  • Is the circle the most perfect shape? A) Yes. Every point on it sits the same distance from the center. B) No. Circles leave gaps when packed, and squares fill a page.
  • Is the golden ratio really the most pleasing proportion? A) Yes. Some artists and architects have built it into their work on purpose. B) No. Many famous claims of finding it are disputed.
  • Is the hexagon the best shape for a floor tile? A) Yes. Equal-area hexagons divide a flat surface with the least total edge. B) No. Squares are easier to cut, lay and fit against a wall.
  • Is the Pythagorean theorem the most useful theorem taught in school? A) Yes. Any straight-line distance on a grid or a screen comes from it. B) No. Percent and ratio get used far more often than any triangle.
  • Should classrooms use globes instead of flat world maps? A) Yes. Every flat map of a sphere has to distort something. B) No. A flat map shows the whole world at once and fits in a book.

Chance and data 15

  • Can a survey of 1,000 people speak for a whole country? A) Yes. A well-drawn random sample can mirror the whole. B) No. Who agrees to answer is never quite random.
  • Does a shuffle that plays three songs by one band prove it is not random? A) Yes. A true shuffle should spread the songs out. B) No. Random runs clump, and people expect them to spread out.
  • Is "1 in 4" clearer than "25 percent"? A) Yes. You can picture one person in four. B) No. Percents share one scale, so any two compare at once.
  • Is a coin toss the fairest way to settle a tie? A) Yes. Neither side can argue with one chance in two. B) No. A tie should be broken by something the players did.
  • Is a link between two things ever enough reason to act? A) Yes. Waiting for proof of cause can mean waiting too long. B) No. Two things can rise together with a third behind both.
  • Is a one in a million chance the same as never? A) Yes. For any plan you make today, treat it as zero. B) No. Give it millions of tries and it happens every day.
  • Is a pie chart ever the best chart? A) Yes. Parts of one whole are what a pie shows at a glance. B) No. Slices are harder to compare by eye than bars.
  • Is a random draw fairer than letting captains pick teams? A) Yes. Nobody stands waiting to be picked last. B) No. Chance can stack one side, and captains can balance them.
  • Is any real die perfectly fair? A) Yes. Fair enough that no player could ever tell the difference. B) No. No real cube is perfectly even, so one face is favored.
  • Is the mean a fairer average than the median? A) Yes. It uses every value, so nothing is ignored. B) No. One huge value drags the mean, and the median stays put.
  • Is the mode ever the best average? A) Yes. For shoe sizes or favorite colors it is the one that fits. B) No. It ignores most of the data, and there can be two of it.
  • Is there such a thing as a lucky number? A) Yes. If a number makes you play bolder, it has changed the game. B) No. Dice have no memory and no favorites.
  • Should a weather forecast give a percent instead of rain or no rain? A) Yes. Seventy percent tells you how sure the forecast is. B) No. People want to know one thing: umbrella or not.
  • Should the lowest test score be dropped before averaging? A) Yes. One bad day should not define a term. B) No. Every score is evidence, and dropping one flatters the average.
  • Should the scale on a graph always start at zero? A) Yes. A cut-off axis makes a small gap look like a cliff. B) No. Starting at zero can flatten a change that matters.

How math is taught 23

  • Is "When will I ever use this?" a fair question in math class? A) Yes. A reason to learn is the least a lesson owes. B) No. Practice in clear thinking is the use, whatever the topic.
  • Is being fast at math the same as being good at it? A) Yes. Quick recall is what skill looks like. B) No. A hard problem can take a mathematician years.
  • Is one hard problem better homework than twenty easy ones? A) Yes. Wrestling with one idea teaches more than repeating one step. B) No. Skills need repetition, the way scales do in music.
  • Is there such a thing as a math person? A) Yes. Some people take to numbers the way others take to music. B) No. Practice and good teaching explain most of the gap.
  • Should a formula sheet be allowed in every math exam? A) Yes. Exams should test using a formula, not storing it. B) No. What you carry in your head is what you can think with.
  • Should a right answer with no work shown get full marks? A) Yes. Right is right, however the student got there. B) No. The reasoning is the math, and the answer is only its last line.
  • Should a sound method with a wrong answer earn most of the marks? A) Yes. One slip in arithmetic should not erase good thinking. B) No. Outside school a wrong number is a wrong number.
  • Should a student who is ahead in math jump to the next course? A) Yes. Waiting for the class to catch up teaches only patience. B) No. A skipped topic leaves a gap that shows up later.
  • Should calculators wait until students know their times tables? A) Yes. A tool is only safe once you can spot its wrong answers. B) No. A calculator frees slow reckoners to do real problems.
  • Should coding count as a math credit? A) Yes. A program is logic and variables put to work. B) No. Code is tested by whether it runs, and math asks why it must.
  • Should geometry be taught before algebra? A) Yes. Shapes you can see make a gentler start than letters. B) No. Algebra is the tool the rest of school math is built with.
  • Should geometry class still teach two-column proofs? A) Yes. Giving a reason for every step is the habit worth forming. B) No. Real proofs are paragraphs, and the columns hide the idea.
  • Should graphing by hand give way to graphing software? A) Yes. Software lets you test many ideas in the time of one sketch. B) No. Plotting points yourself is how a curve starts to make sense.
  • Should high schools teach statistics before calculus? A) Yes. Everyone meets data, and few meet a derivative. B) No. Calculus is the gate to science and engineering.
  • Should long division by hand still be taught? A) Yes. It shows what division is doing, digit by digit. B) No. It is hours of drill for a sum a phone does at once.
  • Should math be taught through its history? A) Yes. Seeing how an idea was found makes it easier to hold. B) No. Stories take time from practice, and practice is the point.
  • Should math tests be timed? A) Yes. Fluency is part of knowing, and a clock shows it. B) No. Speed measures nerves, and careful thinkers are still right.
  • Should mental arithmetic be practiced every day? A) Yes. A little daily practice keeps number sense sharp. B) No. That time is better spent on one good problem.
  • Should money math, such as interest and budgets, be a required class? A) Yes. Everyone will meet a loan, and few will meet a cosine. B) No. Algebra already covers it, and a new class crowds out proof.
  • Should passing algebra be required to graduate? A) Yes. Reasoning with unknowns is basic thinking, not a specialty. B) No. One course should not stand between a student and a diploma.
  • Should the order of operations be taught with a rhyme like PEMDAS? A) Yes. A memory aid gets the order right from day one. B) No. It can be misread as adding first and subtracting after.
  • Should times tables be learned by heart? A) Yes. Instant facts free the mind for the harder step. B) No. Knowing why 6 × 7 is 42 lasts longer than a chant.
  • Should word problems be about real situations only? A) Yes. Nobody buys 60 watermelons, and real numbers earn attention. B) No. A silly story is fine, because the thinking is what counts.

What math is and what counts as proof 19

  • Do numbers exist when nobody is counting? A) Yes. Seven was prime before anyone wrote it down. B) No. Numbers are a tool people made: no people, no tool.
  • Is a definition something you can argue about? A) Yes. Whether 1 is prime was a choice, and choices have reasons. B) No. A definition is a label, so argue about the facts instead.
  • Is a picture enough to prove a theorem? A) Yes. A good diagram shows at a glance why it must be so. B) No. A drawing shows one case and can fool the eye.
  • Is a proof that only a computer can check a real proof? A) Yes. In 1976 a computer-aided proof settled the four color map problem. B) No. A proof that nobody can read explains nothing.
  • Is a proved theorem certain forever? A) Yes. A sound proof does not wear out with age. B) No. One proof of the four color problem stood 11 years before it fell.
  • Is a short clever proof better than a long plain one? A) Yes. A proof you can hold in your head is one you understand. B) No. A plain proof shows every step and hides no trick.
  • Is beauty a good guide to truth in math? A) Yes. Simple, elegant ideas have a way of turning out right. B) No. Some true results are ugly, and taste is not evidence.
  • Is chess a kind of math? A) Yes. It is all rules, logic and counting moves ahead. B) No. Math proves things, and chess only wins or loses.
  • Is it still math if you only follow the steps? A) Yes. Recipes are how most math gets used every day. B) No. Without the why, it is only copying.
  • Is math a language? A) Yes. It has symbols, grammar and readers in every country. B) No. A language can tell a joke, and math can only state.
  • Is math invented rather than discovered? A) Yes. People chose the symbols, the rules and what to study. B) No. The primes were there before anyone had a word for them.
  • Is math more an art than a science? A) Yes. Mathematicians prize elegance and build from pure thought. B) No. It tests claims and throws out what fails, as science does.
  • Is math the only subject where an answer can be completely right? A) Yes. A proof leaves no room for a second opinion. B) No. A spelling or a date is right or wrong too.
  • Is math worth doing when it has no use? A) Yes. Ideas studied for their own sake can turn useful later. B) No. With real problems waiting, pure puzzles are a luxury.
  • Is proof by contradiction less satisfying than a direct proof? A) Yes. It shows a thing cannot be false without showing why it is true. B) No. Showing the opposite is impossible is as solid as proof gets.
  • Is sliding four triangles a nicer Pythagorean proof than algebra? A) Yes. Move the pieces and the truth sits in front of you. B) No. Algebra shows every step and trusts no picture.
  • Is solving a puzzle the same as doing math? A) Yes. Spot a pattern, test it, explain it: that is the job. B) No. Math goes on to prove it for every case, not only this one.
  • Will computers one day do all the proving? A) Yes. Software has already checked a long proof step by step. B) No. Choosing what is worth proving takes human taste.
  • Would aliens have the same math we do? A) Yes. Two and two make four on any planet. B) No. They might count and picture things in ways we cannot guess.

Unsolved: back your hunch 8

  • Can a box have whole-number edges and whole-number diagonals throughout? A) Yes. Boxes that miss by a single diagonal are known. B) No. Long computer searches have not found one.
  • Can every even number from 4 up be split into two primes? A) Yes. Every one tested, up to four quintillion, has worked. B) No. A pattern that holds a long way is still not a proof.
  • Do pairs of primes two apart, like 11 and 13, go on forever? A) Yes. Pairs with hundreds of thousands of digits have been found. B) No. Primes thin out, so such close pairs may one day stop.
  • Does every string of digits turn up somewhere in pi? A) Yes. As far as anyone has looked, the digits behave like random ones. B) No. Nobody has proved it, and looking random is not proof.
  • Does the 3n + 1 rule bring every counting number down to 1? A) Yes. Halve evens, triple odds and add 1: every start tested falls. B) No. Tested is not proved, and some huge start may never land.
  • Is any odd number equal to the sum of its smaller divisors, as 6 is? A) Yes. Nothing proves that one cannot exist. B) No. None has turned up, and any would be over 1,500 digits long.
  • Is pi plus e an irrational number? A) Yes. Both are irrational, and a neat cancellation would be a fluke. B) No. Two irrationals can add to a whole number, and no proof exists.
  • Is there always a prime between one square number and the next? A) Yes. It holds for every pair of squares checked so far. B) No. Gaps between primes grow, and one may outrun the squares.

How it works

Of these 121 math debate topics, 76 suit middle school and 45 high school. Many turn on a definition, a way of teaching or an unsolved problem. Thirteen have an answer; on those, each side's line says what to show.

Read the full guide

The thirteen cards that have an answer

They sit in Has an answer: prove it. The verdicts: zero is even; 0.999... equals 1; a square is a rectangle; the primes never end; dividing by zero is undefined; two negatives multiply to a positive; multiplying by a half shrinks a number; a price cut 50 percent and then raised 50 percent ends a quarter down; doubling a square's sides gives four times the area; the half-twist strip has one side; a triangle drawn on a ball adds up to more than 180 degrees; some endless sets are bigger than others; and 1 + 2 + 3 + ... has no ordinary sum.

  • On a card from Rules mathematicians chose, ask each side what would break if the other definition won.
  • The eight Unsolved cards have no known answer: nobody has proved that the 3n + 1 rule always reaches 1, so vote on a hunch.

Questions and answers

What are good math debate topics for middle school?

Under Topic, Middle school holds 76 cards that need no math past arithmetic, shapes, graphs and chance: Should an equilateral triangle count as isosceles? Does a circle have a side? Is rounding a kind of lying? Should times tables be learned by heart?

What are good math debate topics for high school?

High school holds 45 cards that lean on algebra, proof, limits or high school courses: Should tau, which is 2 pi, replace pi? Should 0 to the power 0 be defined as 1? Is a proof that only a computer can check a real proof?

What are debatable math topics?

Ones that turn on a choice, not a calculation. Zero is a natural number in some textbooks and not in others, lists of primes included 1 as late as 1956, and 8 ÷ 2(2 + 2) can be read two ways.

What are math education debate topics?

How math is taught holds 23: Should calculators wait until students know their times tables? Should math tests be timed? Should a right answer with no work shown get full marks? Should high schools teach statistics before calculus?

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