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I am teaching English in a Japanese primary school and they have a Nepalese child who doesn't understand written Japanese especially so they got me to translate some maths problems for them. I had a lot of those sentence type maths growing in Australia and I wondered what the point of writing sentences that need to parsed by students to extract the mathematics is.

I read somewhere that they think it's real although there would be situations where someone would write a sentence on a situation just to be mathematically analyzed. I can't think of an engineering (my first degree) mathematics situation like that. For instance, in calculus there is no way to have a true sentence problem---it would take many paragraphs to express the many complicated expressions and that would be very difficult for a university student to understand; they don't have that enough time in exams. It's not the same as having an engineering problem to solve---not that we did many.

There are many ways to include context into a maths question (without using sentences with lots of extraneous detail)---diagrams, tables, short phrases (e.g., 240V +-15, 60Hz according a standard)---so that it isn't that hard to linguistically interpret. An applied math problem is not the same as a pure math problem with the same nominal mathematics. For example, $2+2=4$ but the applied maths example is you need $2$ lots of $2$ m of steel and the only available lengths of steel don't include $4$ metre lengths rather only $5$ metre lengths then it's effectively $2+2=5$ (+extra $1$ metre) in metres.

You also have to differentiate between reading and writing. It's a bit easier to write than read---you understand your own writing more and writing summary statements is not what the question is about.

Sure there are some documents in some fields where there are numbers but it's unclear whether pushing primary school children with the simplistic sentence type problems that are normal helps them do such problems. Please explain the value of sentence type problems as against other contextual forms for preparing young children for possible real problems.

Can anyone spot what's wrong with the Paint problem below?

Paint problem

$\frac56$ dL ($1$ dL $=0.1$ L) of paint covers $\frac34$ m$^2$. How far does $1$ dL of paint cover?

It should have shown a paint can and the painting done from it.

What's wrong with the Steel problem below?

Steel problem

$\frac27$ m of steel bar weighs $\frac45$ kg. How much does $1$ m of steel bar weigh?

It should have had a metre length of steel and its weight with an arrow.

What is wrong with the answer key (teacher's copy is shown) is that it doesn't include any details of the context except units.

What is wrong with the context is that primary school students wouldn't have any idea about paint economics or steel usage. These are totally foreign to them and so students know to skim out the numbers. But this is what happens when we insist on having sentence problems---they have to be artificial and superficial otherwise the effort of teaching fractions is very diluted with teaching context and sentence structure.

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    $\begingroup$ Are you asking about the value of so-called word problems? 'Jane has 2 apples and jack has 4 times as many. How many does jack have?' That type of question? $\endgroup$
    – JTP - Apologise to Monica
    Commented May 28, 2016 at 18:59
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    $\begingroup$ All worthwhile mathematical problems start out as language first. In the age of powerful computers, the skill of extracting mathematics from language is arguably more important than the skill of solving the resulting equations. $\endgroup$
    – Austin Mohr
    Commented May 28, 2016 at 19:37
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    $\begingroup$ Well, that's the wrongest claim I've seen today. Have you not heard of specifications, requirements documents, case studies, schedules, budgets, whitepapers, field reports, memoranda, research journals, grant writing, etc., etc.? $\endgroup$
    – Daniel R. Collins
    Commented May 28, 2016 at 22:49
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    $\begingroup$ Without clarification or improvement in the question, this looks like a troll. $\endgroup$
    – paul garrett
    Commented Jun 1, 2016 at 17:58
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    $\begingroup$ 5 days in, OP seems unable to clarify the exact issue. In fact, from his comments on every answer, he seems to be interested more in an argument why every member here is wrong, no matter the advice or nature of their answer. Time to put my vote in. $\endgroup$
    – JTP - Apologise to Monica
    Commented Jun 4, 2016 at 17:49

8 Answers 8

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Consider these 5 problems:

  1. What is $ 35 \div 10 $ ?

  2. We are in the store and want to buy packages of party plates for a birthday party. The plates come in packages of 10. There will be 35 children at the party. How many packages should we buy so that everyone will have a plate?

  3. 10 friends go out to get ice cream. The bill comes to $35. If the friends want to divide the bill evenly, how much should each friend pay?

  4. Leah has 35 oz. of juice. She has glasses that hold 10 oz. How many glasses can she fill?

  5. David has 35 large cookies. He can put 10 on a platter. After he has filled all the platters he can, he leaves the remaining cookies on the counter for his family to enjoy. How many cookies are left on the counter?

Note that all of these problems can be solved by solving the first expression $$ 35 \div 10 $$

Yet the solution to problems 2, 3, 4, and 5 are all different. The answers are:

  1. 4 packages
  2. $3.50
  3. 3 full glasses
  4. 5 cookies

The word problems give us a context for how to handle the remainder in problem number 1. Without that context, teaching these different division methods would harder to grasp. It also gives the student a chance to decide in what context they should use each method.

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    $\begingroup$ +1. This is the perfect answer to what seems to be the question in the title. $\endgroup$
    – JRN
    Commented May 30, 2016 at 14:06
  • $\begingroup$ Except the question is sentence type versus other choices of information provision for young children. It is not sentence type versus raw data. You could easily re-express this problems without all the words and still all the context. These are not the same problem. $\endgroup$
    – user2617804
    Commented May 31, 2016 at 9:31
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    $\begingroup$ Perhaps you could clarify the difference each of the following (or at least give an example of): a) sentence type, b)other choices of information c)raw data d) these problems without all the context. I see from the comments that other people felt your question wasn't clear and perhaps you could clarify. Thanks $\endgroup$
    – Amy B
    Commented May 31, 2016 at 12:17
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    $\begingroup$ "You could easily re-express this problems without all the words and still all the context." You may find it easy, but I don't. Could you show us an example of what you mean? For example, could you re-express Amy B's questions without all the words? $\endgroup$
    – JRN
    Commented May 31, 2016 at 14:55
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Professional work in general, and scientific and mathematical work in particular, is done principally in writing. Requirements, specifications, orders, field reports, case studies, journals, grants, etc., are all disseminated and documented in writing. Symbolic mathematical notation is inherently a specialized system of concise writing (arguably, it is nigh-impossible to speak it with proper precision).

Treating mathematical writing as a specialized language, there can be no more important skill than being able to translate from natural language to mathematical language and back again. Without that skill, mathematical notation is entirely useless, an isolated island without meaning. Word problems are precisely the exercise one needs to develop this critical skill.

Consider these words by William Kingdon Clifford (The Common Sense of the Exact Sciences, 1823), which appear at the top of my course syllabi in the next semester:

We may always depend on it that algebra, which cannot be translated into good English and sound common sense, is bad algebra.

Consider, too, George Polya's famous book How to Solve It (1945), which essentially organizes problem-solving into four stages (also reminiscent of the software development life cycle; quote here):

  1. First, you have to understand the problem.
  2. After understanding, then make a plan.
  3. Carry out the plan.
  4. Look back on your work. How could it be better?

Note that every step other than #3 is a reading/writing task. If you dig into Polya's details more, Step #1 includes "Do you understand all the words used in stating the problem?"; Step #2 is essentially translation to an equation; and Step #4 is restating the answer in natural language so as to check for reasonability. The translation between natural language and back is inherent throughout the process.

When the OP says, "I can't think of an engineering situation like that - my first degree.", that's a statement that frankly beggars the imagination, and leaves this writer entirely incredulous and aghast. Surely any real engineering work is initially a problem posed in natural language. Even if we just open a random physics, calculus, or statistics book, we will find that most or all of the problems after a certain point are phrased as natural-language word problems. Let's take a survey from a few, using OpenStax open-source texts as a resource (available for free download here):

OpenStax College Physics, Chapter 2 (Kinematics), Example 2.1:

A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration?

OpenStax Calculus Volume 1, Chapter 4 (Applications of Derivatives), Example 4.1:

A spherical balloon is being filled with air at the constant rate of 2 cm^3/sec (Figure 4.2). How fast is the radius increasing when the radius is 3 cm?

OpenStax Introductory Statistics, Chapter 9 (Hypothesis Testing with One Sample), Example 9.21:

In a study of 420,019 cell phone users, 172 of the subjects developed brain cancer. Test the claim that cell phone users developed brain cancer at a greater rate than that for non-cell phone users (the rate of brain cancer for non-cell phone users is 0.0340%). Since this is a critical issue, use a 0.005 significance level. Explain why the significance level should be so low in terms of a Type I error.

Anywhere we look at core prerequisites for any standard engineering curriculum, we find that the work comes to us in written, natural-language form first, and developing the capacity to parse and translate that language to mathematical language is essential.

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  • $\begingroup$ That they have those problem statements in textbooks doesn't prove your case- only show the love of sentences for textbook writers- many are quite unrealistic about field use of technical skills. If you could link the simplistic problems in younger years to aide solving such questions leading to support in real professional use of maths then you'd have an excellent answer- as it is you answer is just OK. I suspect a lot of real use would be of analysis of appendices, tables and graphs. $\endgroup$
    – user2617804
    Commented May 31, 2016 at 13:16
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    $\begingroup$ @user2617804: If these cases are so universal in any core textbooks, then how could you have not encountered them in your engineering degree? Your story doesn't make any sense; you refuse to clarify your negative remarks with good examples; and so I have a strong suspicion that you're trolling us. $\endgroup$
    – Daniel R. Collins
    Commented May 31, 2016 at 16:45
  • $\begingroup$ They aren't my core textbooks- all discarded long ago. How to Solve It 1,2,4- isn't necessarily a reading and writing sentence problem- could easily by contextually expressed without sentences. Sentences slow you down. $\endgroup$
    – user2617804
    Commented May 31, 2016 at 20:23
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    $\begingroup$ "Calculus [...] never has sentence maths problems." You mean like "Find the maximum area of a rectangle with a given perimeter?" $\endgroup$
    – JRN
    Commented Jun 1, 2016 at 11:10
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    $\begingroup$ Or, like, the example in the answer above. $\endgroup$
    – Daniel R. Collins
    Commented Jun 3, 2016 at 3:14
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I think you have a very different understanding of what "mathematics" is than I do.

Consider the following questions:

  • What kinds of geometric properties are held by the figure you get when you join the midpoints of adjacent sides of a polygon?
  • Under what conditions on the parameters $a,b$ does an equation of the form $a^x = x^b$ have a rational solution?
  • How many different ways are there to make \$3.45 using only three different kinds of coins?
  • If you know a 5th-degree polynomial has real solutions only at $x=3, x=5$ and $x=-2$, what else can you say about the polynomial?

I am not sure how any of these questions could be expressed without using "sentences". Do you consider these valuable mathematical questions?

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  • $\begingroup$ Easy for point 1 draw a diagrammatric representation of a polygon and split the sides in half and then isolate said figure and geometric figures? $\endgroup$
    – user2617804
    Commented Jun 2, 2016 at 9:32
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    $\begingroup$ That would illustrate the object, but how could it convey the question? $\endgroup$
    – mweiss
    Commented Jun 2, 2016 at 23:03
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There are three direct benefits, as far as I can tell.

  1. Word problems answer questions like "why do I need to know this". If you have a student who thinks that learning math is pointless then you can use word problems to help them understand why math is important

  2. For some students math is very difficult when's it's just abstract numbers. For more linguistically oriented students the word problems may help them visualize the problems and gain an understanding of the underlying mathematical concepts. For more mathematically oriented students word problems may be distracting and more difficult than simple equation - based questions, meaning that they will be forced to develop their linguistic reasoning skills.

  3. As others have pointed out, the ability to study through excess information to find the actual salient details is a valuable skill.

Just a note: for points 1 & 3 it may be best to tell the student directly that these are part of the reasons for word problems. If you just give students the questions without clarification they may just think that word problems are pointless and irritating.

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  • $\begingroup$ Its a bit different to add few descriptive words or images than say 10 introduction text on a factory that produce 20000 apples in first year and 10 lines of text on factory improvements so it produces 30000 in second year and then more text of line executive talk leading to what was improved production rate. $\endgroup$
    – user2617804
    Commented May 29, 2016 at 7:48
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I think there's a very basic answer to this. It's the very origin of math and numbers. Try to teach a child that 1+1=2 without the visual (or tactile) example of using fingers to show the concept of 1 and 2, and how the oneness of a single finger has something in common with the oneness of a single block. It's only when this sinks in, do we get comfortable with the multiplication table, although we still like to use physical aids to show how say, 3 rows of 4 eggs and 4 rows of 3 eggs are the same dozen eggs. Can I explain the commutative property of multiplication without eggs? I suppose, but why would I want to?

I'm specifically addressing "what is the point of writing sentences that need to parsed by students to extract the mathematics?"

Because in grade school and high school, the math is meant to apply to real life, not to be abstract. In my experience, it's just the opposite (of your objection), students will be more engaged when a problem is offered in a way that occurs to solve a real situation. "shirts cost \$X, pants cost \$Y, how many of each did Jane and Jack buy given $Z?" This type of problem is preferable to only having lists of 2 equations in 2 unknowns. More than engaging the student, it helps them apply math to situations they are likely to encounter on their own. In high school, we have the issue of "when will I need the law of cosines in real life?" and I struggle at times to honestly say that more than a select few careers with call for its use. (That's another issue/question). But - in primary school, nearly every last bit of math turns into the math we actually should master to understand our own lives. How to balance my checkbook, go shopping and compare unit cost, handle my finances, order floor tiles for my bathroom project. Strip out the words from the math, and you find you've produced an adult who, as a child, could add 8% of a number to itself, but never made the connection this is how you calculate sales tax on a purchase.

To come full circle, when I proctor exams (in a high school), I'm sensitive to the students for whom English isn't a first language, and I tell them that for word problems, they can come ask for clarification on questions where their math is fine, but the vocabulary is an issue. In your case, this is what you need to do, to help translate the words into the student's native language.

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  • $\begingroup$ You disproved your own point. You show you don't need sentences to add context- you can show eggs to count, you can draw a diagram. $\endgroup$
    – user2617804
    Commented Jun 4, 2016 at 13:00
  • $\begingroup$ Thank you for ignoring the context of that example and the spirit of the entire answer. $\endgroup$
    – JTP - Apologise to Monica
    Commented Jun 4, 2016 at 13:22
  • $\begingroup$ It is possible that the OP does not completely understand the answers because of unfamiliarity with English. $\endgroup$
    – JRN
    Commented Jun 5, 2016 at 4:22
  • $\begingroup$ @JoelReyesNoche - I see the edits made over night and commented. $\endgroup$
    – JTP - Apologise to Monica
    Commented Jun 5, 2016 at 11:03
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    $\begingroup$ It is now clear to me that the OP is just trolling us. It would be best to stop responding to the OP. $\endgroup$
    – JRN
    Commented Jun 11, 2016 at 2:07
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Perhaps an example would clarify what you mean.

Question 1:

What number is obtained after reducing 40 by 5%?

Question 2:

Farmer John has lately been having problems selling his corn. He needs to start selling more of them soon, or they will start to go bad. It seems that the retail price of 40 cents is too high, so he decides to start offering a 5% discount. What would the new price of the corn be?

If I understand you correctly, you feel that these two questions are mathematically equivalent (that is, they test for the same mathematical skills and understanding).

It seems that you believe that the second one is inferior compared to the first because it has a lot of irrelevant (extraneous) text that serve only to distract and confuse.

But another way to look at it is to say that the second question provides context and relevance by presenting the math in a (supposedly) real-world situation.

I would argue that sometimes these "irrelevant" texts can sometimes actually aid the student with the math.

For example, compare:

Question 3:

$812\div 4=$

Question 4:

Alice, Bob, Charlie, and Dan are a close group of friends. Alice recently won \$ 812 in a lottery and would like to share the amount equally among the members of the group. How much should each friend get?

It so happens that many students answer question 3 incorrectly. (They answer "23," a split-dividend error.) But if you were, say, Dan, and your friend Alice wanted to divide \$ 812 equally among the four of you, would you be satisfied with getting \$ 23?

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  • $\begingroup$ False comparison there- You show a picture of 812 dollars lottery winnings and arrows to show its division to four friends so you aren't discriminating against children with slight reading, information processing or English problems. $\endgroup$
    – user2617804
    Commented May 30, 2016 at 8:25
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    $\begingroup$ Okay, now it seems that I really don't understand what you're trying to ask here. Are you saying that mathematics should be taught without any kinds of words at all (whether written or spoken)? $\endgroup$
    – JRN
    Commented May 30, 2016 at 9:33
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    $\begingroup$ If so, then how do you propose asking a question such as "What number is obtained after reducing 40 by 5%?" $\endgroup$
    – JRN
    Commented May 30, 2016 at 9:34
  • $\begingroup$ And the value of such no context adding question is? Might as well have it in straight maths for such a worthless question. $\endgroup$
    – user2617804
    Commented May 30, 2016 at 12:37
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    $\begingroup$ @user2617804, I'm sorry, but I still don't see your point. Are you saying that "What number is obtained after reducing 40 by 5%?" is "a worthless question" because it is expressed in words? Could you give us an example of a "worthy" question so that we could all understand what you are trying to say? $\endgroup$
    – JRN
    Commented May 30, 2016 at 13:56
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The point to word problems is scalable testability. You need to state some problem once and can test thousands of students, compare their answers and produce impressive statistics. Whether this has something to do with real-life situations, where "you have to science the shit out of this" (quoting The Martian) is debatable.

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  • $\begingroup$ Why sentence problems when numbers with units would allow the same testability- its what students have to extract from the sentence anyway. $\endgroup$
    – user2617804
    Commented May 29, 2016 at 20:34
  • $\begingroup$ I did not state what was being tested. It is something more than pure numerical or algebraical ability, but (at least in my opinion) distinctly less than what is necessary to successfully transform real-world situations into mathematical notation. $\endgroup$
    – Stefan Schmiedl
    Commented May 30, 2016 at 11:04
  • $\begingroup$ Except that sentences have a limited relationship with real situations- sometimes yes, sometimes no. I read testability as the ability to have a test that can conducts on lots of student that can be repeated and be not prone to error in marking, the interpretation by students. Basically you can run a test on the material/ style of problem. Can that do with diagrams. $\endgroup$
    – user2617804
    Commented Jun 4, 2016 at 13:07
  • $\begingroup$ You can use diagrams. But then you're not testing the same cognitive processes as with word problems. Personally, I'm not sure if all those tests out there are testing what their authors meant to test. $\endgroup$
    – Stefan Schmiedl
    Commented Jun 5, 2016 at 13:16
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I work in a highly technical field, and (realising this is an unpopular view) regard this type of question as an attempt to create bias in favour of (a) native English speakers, and (b) less mathematically able students. The 2nd point may be slightly less intuative, but frequently this question style is ambiguous when parsed in a litteral/formal way, and also the padding can be very distracting to anyone on the autistic/aspergers spectrum.

If someone asks me to solve this sort of problem in real life, I will insist they ask a well defined problem rather than give me their life story. If they can't focus, I will not try and work out what they want.

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  • $\begingroup$ how does this attempt to answer OP's question? $\endgroup$
    – JTP - Apologise to Monica
    Commented Jun 4, 2016 at 17:50
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    $\begingroup$ Re: Last paragraph, would you agree that it's important for your interlocutors to be able to translate their natural-language problems into well-defined mathematics? Clearly someone has to do that. $\endgroup$
    – Daniel R. Collins
    Commented Jun 4, 2016 at 18:08
  • $\begingroup$ I'm not sure that always applies. Maths or logic problems and natural language problems can overlap, but it's a mistake to mix the two together. Or rather, the first part of the process is so trivial that it shouldn't be mixed in all the time. $\endgroup$
    – Sean Houlihane
    Commented Jun 4, 2016 at 18:34
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    $\begingroup$ Why do you think that word problems are "an attempt to create bias in favour of [...] native English speakers"? Do you think all word problems are in English? $\endgroup$
    – JRN
    Commented Jun 5, 2016 at 1:29
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    $\begingroup$ "I will insist they ask a well defined problem [...]" And how do you propose to teach them this skill (of converting natural language to mathematical language) without using any examples involving natural language? $\endgroup$
    – JRN
    Commented Jun 5, 2016 at 1:33

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