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This is related to my previous question What value is there in requiring students to declare the dimensions of an answer when it is already clear from context? , but with a different focus.

A sizeable minority of my primary and secondary school teachers (east-coast US in the late 1980's/early 1990's) insisted that word problems be answered in complete sentences.

Consider the following hypothetical question:

Anne's car travels 225 miles in five hours. What is the speed of Anne's car?

The correct answer, according to these teachers, would be

The speed of Anne's car is 45 mph.

Some leeway in wording was allowed, so an answer of "Anne's car has a speed of 45 mph", "45mph is the speed of Anne's car", or "Anne's car travels at a 45mph speed" could be accepted for full credit, but answers such as "45" or even "45 mph" were considered insufficient and would result at best in lost points and at worst in being awarded zero points for the question.

The explanation that I recall being given was that we were being educated to use math to communicate information to others in real-life contexts, and that context was always necessary. For example, if we walked up to an adult and told them "45 mph!", they would look at us like we were crazy and wonder what it is we were trying to communicate, but if we instead presented them with "The speed of Anne's car is 45 mph", they would instead react, "Wow! I've always wondered about that. Thanks!" and would be sure to lavish us with awards and college recommendation letters.

The school system I spent most of my childhood years in was (at that time) very big into interdisciplinary and cross-curricular studies. We were supposed to write papers about math, do math problems on real history, study the history of science, etc., so my understanding was that the requirement to write answers in complete sentences was an exercise in literacy and not mathematics per se.

So, is there mathematical pedagogical value in requiring students to answer word problems in complete sentences, or is this actually a literacy or communication exercise that has been applied interdisciplinarily to math exercises?

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    $\begingroup$ Just like using units, the hope is that students will think about the meaning of what they're doing. $\endgroup$
    – Sue VanHattum
    Sep 11 at 16:48
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    $\begingroup$ For those interested but not around during this time, the relevant phrase is "writing across the curriculum", and it was not limited to math classes. I don't know when it originated (in a fairly widespread and nontrivial way), but it was certainly well entrenched in mathematical education practices by the early 1990s. Looking up the math education literature from that time will likely give you a lot of pros and cons for its pedagogical value. And yes, there was a lot of push-back at the time. You may have seen joke test questions given for different eras reflecting this. $\endgroup$ Sep 11 at 18:51
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    $\begingroup$ @DaveLRenfro yes, that was the exact term they used! Their implementation, in retrospect, wasn't that good, but I can see how the concept could have been trivially applied in all sorts of scenarios to technically get students writing. $\endgroup$ Sep 11 at 18:55
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    $\begingroup$ @DaveLRenfro and such joke test questions continue on to the present day. Nowadays, a lot of them are social justice-related. For example, "A farmer plants 10 acres of corn on land stolen from Native Americans. His costs are 1/4 of his revenue. Divide into breakout rooms and discuss the land back movement and how the farmer can best check his white cisheteronormative privileges." A few decades earlier, the joke was about excessive New Math abstraction - "Let F be a farm and let A be the cardinality of acres of corn on Farm F...." $\endgroup$ Sep 11 at 19:00
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    $\begingroup$ The explanation you recall seems specious. If someone asked you "How fast were you going?", it would be perfectly normal and reasonable to answer "45 miles per hour" or even just "45". There's a difference between answering a question and making a statement out of the blue. $\endgroup$
    – Barmar
    Sep 12 at 15:05

7 Answers 7

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Yes, there is mathematical pedagogical value in the usage of complete sentences - but this does not only refer to "answers" and not only to "word problems", but to all parts of the solution to any mathematical problem or task.

The reason is that mathematics is not only about numbers, or computations, or equations, or inequalities. It is also (some people might say: mainly) about how to precisely formulate mathematical arguments and propositions. Propositions and arguments are typically composed of complete sentences.

One of the various challenges that I perceive whenever dealing with first year math students at university level is that they find it very difficult to express their mathematical thoughts in a precise and clearly understandable way - and part of this problem is that most of them didn't learn in math classes in school to express themselves in complete sentences.

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    $\begingroup$ I agree. Written math is about communication. And we humans usually communicate in complete sentences (at least in writing). The right words here and there between all the "real math" can do wonders for understanding someone else's mathematical arguments. And learning this communication is a skill that takes practice. $\endgroup$
    – Arthur
    Sep 13 at 11:25
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    $\begingroup$ What I find interesting, and compelling, about this approach is that it applies directly to quality commenting in source code. Don't comment obvious steps, but do explain (e.g. referencing a theorem you used) anything more complicated on the way to the answer. $\endgroup$ Sep 13 at 11:34
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    $\begingroup$ @Davos: "math is not even remotely about 'complete sentences', it's about symbol manipulation"" Pardon me? Every math book I've had a look at since I started studying maths about 15 years ago is full of arguments and results which are all phrased in complete sentences, and the same is true for every math article I've ever seen. Whenever I discuss mathematics with colleages (and I do so all the time, as it is part of my job), we do so in full sentences, no matter whether verbal or in writing. I've never heard any mathematician claim that math were about "symbol manipulation"; [...] $\endgroup$ Sep 15 at 12:28
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    $\begingroup$ if it were, I would have abandoned the subject long ago, because I can hardly think of anything more boring than symbol manipulation. The second part of your argument ("let alone a defensible induction...") is a straw man because I never said that "repeating the question in slightly different form leads to better math learning". I said using complete sentences is important in "all parts of the solution to any mathematical problem or task". Explicitly stating the answer to a question, e.g. in the last sentence of the solution, is one single instance of such a "part of the solution". [...] $\endgroup$ Sep 15 at 12:31
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    $\begingroup$ Of course, focussing on this instance alone will hardly have an effect (it might even have a negative effect, since students will find it ridiculous if you typically don't care for complete sentences, but in this particular case you suddenly do). So I'm arguing for a culture in math education where complete sentences are valued as the common mean to express mathematical reasoning. [...] $\endgroup$ Sep 15 at 12:33
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I do think there is value in expecting students to give answers in correct English. It will certainly help when they start to face longer and less structured questions.

However, the example you give is entirely inappropriate. When you are asked a direct question, it is correct English to simply respond with the answer. You do not need to prefix it with "The answer to your question is", or words of like meaning. Requiring this sort of thing just becomes an artificial constraint designed to trip people up, rather like Jeopardy.

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    $\begingroup$ Indeed, I remember wearing out my hand writing out answer sentences that were just rephrased versions of the question word in the problem. "The number of marbles in the jar is...." "The number of cows on Mary's farm is...." "The length of the USS Dreadnought is... feet." $\endgroup$ Sep 12 at 13:47
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    $\begingroup$ I'm not completely convinced by the second paragraph. I agree with you that answering to the question "What's the speed of Anne's car?" by "45 mph" is entirely fine if you are only supposed to give the right answer to the question. However, if this is an exercise for students, one might in many situations want them to not only give the right answer, but to also explain how they found it. In this case, one would again need a longer sentence to point out what is the answer to the question (e.g. "Since 225 miles / 5 hours = 45 mph, the speed of Anne's car is 45 mph"). $\endgroup$ Sep 13 at 8:33
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    $\begingroup$ @JochenGlueck Yes, I agree that there is merit in requiring the student to show working, and I agree that if you do so, this changes how much you need to add to make it correct English. But my point was that "the speed of Anne's car is 45mph" (as given by OP) is no better than simply "45mph". Similarly, if reasoning was required, I think "45mph, since that is 225 miles divided by 5 hours" would be sufficient. $\endgroup$ Sep 13 at 8:54
  • $\begingroup$ @EspeciallyLime: Thanks for your response. I agree. $\endgroup$ Sep 13 at 8:57
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    $\begingroup$ If one is given a set of equations and asked to solve for x, and the equations can only be satisfied if x is 5, a correct answer would be "x=5". In a word problem, the same principles would apply, though I would think using pronouns in an answer would be reasonable, e.g. "It's moving at 43mph" if it's clear what's being referred to. $\endgroup$
    – supercat
    Sep 13 at 23:09
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An example of a problem where phrasing the answer as a sentence might prevent mistakes and encourage understanding is:

Alice goes to the store with \$2.00. A gumball costs \$0.80. How many gumballs can Alice buy?

A careless student might write

$\$2.00\ /\ \$0.80 = 2.5$

but writing out

"Alice can buy 2.5 gumballs" might make them think again. Relating the numbers to an assertion about reality could plausibly help students check that the answers they obtain are reasonable.

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    – Sue VanHattum
    Sep 14 at 15:34
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Jochen's answer is very good. To add to it, here's the reasoning I was given as a student: answering in full sentences makes students think about the answer in different psychological context. This can help them catch mistakes.

For example, imagine the student answering the question had made the common mistake of dividing the hours by the distance instead of the other way around, and got an answer of 0.0222. Writing the answer as a full sentence ("The speed of Anne's car is 0.0222mph") might well prompt them to realize that they've made an error.

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  • $\begingroup$ In my experience as a TA -- at the college level! is that most students never stop to evaluate the reasonableness of their answers. But we can always hope. $\endgroup$ Sep 13 at 11:31
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    $\begingroup$ @CarlWitthoft My experience as a student -- at the college level! is that many professors and textbook authors never stop to evaluate the reasonableness of their questions. Richard Feynman's book Surely You're Joking Mr. Feynman includes an example from an elementary math textbook that asked the question "What is the total temperature of [all the stars in the sky]?" which is a terrible question because the concept of "total temperature" is nonsensical. Or endless questions from physics professors about pitchers who can miraculously pitch baseballs at 80% the speed of light. $\endgroup$
    – Andrew Ray
    Sep 13 at 20:11
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    $\begingroup$ @AndrewRay Seeing as the last item in your comment was well-covered in the very first What-If , I think you are NO FUN AT ALL :-) $\endgroup$ Sep 14 at 12:24
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I do think this has (some) pedagogical value, specifically for word problems.

Word problems are designed to give students practice applying math to real-world situations. Doing this effectively means you want the students to understand the situation being described, and think about how math can apply to it. (This also requires that your word problems be realistic; if the real-world answer is "that's nonsense", it's a bad word problem for a math class.)

What you don't want is for the student to scan the problem looking for two numbers, and a mathematical operation to apply to them. Requiring the student to restate the situation at hand, and how the number they're giving fits into that situation, is one way to tell if they've read the problem thoroughly, which is a component to understanding it.

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The answer to What is the speed of Anne's car? is simply 45 mph

However, math problems such as this follow a very simple pattern of pick up every number and do something with them, so with a simple scan of the text, the answer must be one of

  • 230
  • 220
  • 1125
  • 45

(-220 or 0,022… would be 'unlikely' answers)

which is exactly what some students will do: picking one of them seemingly at random, without paying any attention to the wording.

This can lead to interactions with certain students such as:

  • Teacher: What's the speed of Anne's car?
  • Student: 230!
  • Teacher gives look of "this is not the right answer"
  • Student: I meant 220!
  • Teacher: Are you sure?
  • Student: 1125!, 1125!

A full answer of

The speed of Anne's car is 45 mph.

requires from the student not only to read the numbers and guess the right operation (perhaps all the tasks today are divisions?), but also to parse the subject of the sentence (Anne's car), the property being sought (its speed) and build a full sentence out of it.

(Omegastick points out that an implausible value associated to a sentence might prompt them to realize that they've made an error. But I wouldn't hold my breath. I think their parents are more likely to notice from the context that the car is probably not going at 0.022 mph. If the students suspect it may be wrong, that probably stems from the answer not being an integral number.)

As a nitpick, I should note that the above answer is still not complete. If we get fancy, we should say that

The speed of Anne's car is 45 mph on average.

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    $\begingroup$ But they don't build a sentence out of it. They just regurgitate the original sentence (with the interrogative phrase filled in). $\endgroup$ Sep 14 at 8:46
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I don't think there's a mathematical reason for this, but teachers may view this as part of an interdisciplinary approach. Requiring students to answer in complete sentences reinforces what they're learning in language class, by having them write complete, grammatical sentences.

I recall similar requirements in other classes. For instance, in History class, if asked when the US declared its independence, you were expected to say something like "The Declaration of Independence was signed in 1776" rather than just "1776".

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    $\begingroup$ It's practice writing complete sentences. $\endgroup$
    – Barmar
    Sep 13 at 1:13
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    $\begingroup$ @Araucaria-Nothereanymore. I admit that it's been 50 years since I was in elementary school, but I recall learning things like sentence structure and grammar in English classes. $\endgroup$
    – Barmar
    Sep 13 at 1:15
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    $\begingroup$ You studied grammar in English classes? You're the only person that ever did (although some of us got a couple of fetishistic rues about not ending sentences with prepositions and the like).! I much prefer your cogent and insightful comment under the original question! ;-) $\endgroup$ Sep 13 at 1:18
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    $\begingroup$ @Araucaria-Nothereanymore. Like I said, it's been a long time, so my memory is vague. I remember diagramming sentences in high school, I can't remember specifics of what we studied in grade school other than spelling tests. But I'm pretty sure writing simple sentences was involved. $\endgroup$
    – Barmar
    Sep 13 at 1:22
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    $\begingroup$ @Araucaria-Nothereanymore. it's more like the English teacher insisting that any essay with math in it have the correct math (as well as proper grammar and spelling). Which BTW I fully support $\endgroup$ Sep 13 at 11:32

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