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While teaching students mathematics, I have noticed that some seem sloppy in the way that they write down their work.

For example, a student is given a question: What is the area of the rectangle?

   4     3
┌──────┬─────┐
│      │     │ 2
└──────┴─────┘

(sorry for the terrible word picture, the original was a picture.)

The student writes: $3+4=7 \times 2 = 14$ This is clearly wrong, though they got the correct answer.

I can also see their thought process. They first added 3 and 4. Then multiplied by 2. However what they wrote is wrong, since $3+4 ≠ 7 \times 2$.

Can anyone cite any research to show what effect this has on the progress of the student? Or share personal experience?

Note: as this is not a question about units, I have not included any.

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    $\begingroup$ Related to the specific example, but not duplicate: matheducators.stackexchange.com/questions/7964/… $\endgroup$ – Tommi May 14 '18 at 12:10
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    $\begingroup$ Intended is $\,(3+4=7)\times 2=14.\,$ Teachers should explain why the parentheses are required. Whether or not one condones such notational (ab)use (value-annotated expressions) is a matter of taste. In some contexts it greatly aids comprehension if used competently. Of course there are better ways to annotate partial-evaluations when evaluating nested expressions, A prototypical application is in Horner's nested form for polynomials, e.g I exploit Horner partial evaluation to give quick universal divisibility tests using modular arithmetic $\endgroup$ – Bill Dubuque May 14 '18 at 18:38
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    $\begingroup$ I think this sort of imprecision can lead to errors in the future. Especially when things get more complicated (e.g. moving things around and dividing and multiplying to solve one equation for one unknown.) My advice is to push for clear exposition. It ends up helping the person when things get more complicated. $\endgroup$ – guest May 14 '18 at 21:00
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    $\begingroup$ Suppose you were an elementary school teacher, and your 7-year-old student wrote a 50-word essay that included the sentence "I like kats n dogs." You might praise the student for some aspects of the essay, but part of your job would be to tell the kid that "cat" is spelled with a "c," and "n" is not a way to write "and." If you don't do this, you are doing this kid a disservice. $\endgroup$ – Ben Crowell May 16 '18 at 5:35
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    $\begingroup$ Note that this is exactly what you would enter into a four-function calculator to do this calculation (press $3$, then $+$, then $4$, then $=$ to read out a $7$, then $\times$, then $2$, then $=$ to get $14$). So it appears that the student is either showing their work by writing down their calculator inputs, or believes that the right way to write an equation is total left-to-right as a four-function calculator would express. $\endgroup$ – TheHansinator Mar 11 at 23:17
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This example strikes me as resembling a phenomenon I've seen where a problem is of the form

Given A, find C

where the main approach to actually solving the problem is:

  • Using A, obtain B
  • Using B, obtain C

however, the student seems entirely incapable of even conceiving of doing these as separate steps. Any attempt to help them obtain B just results in them trying to arrange the advice into yet another guess as to a formula formula for computing C. Even if you explicitly ask them what B is!

I find it quite plausible that this difficulty could be the product of the very sort of sloppiness you describe.

The sort of formula you write is a sort of run-on sentence starting with "find the width" and running into "find the area". Imagine, now, that their teachers correct this problem by showing how to combine the two run-on sentences into one compound sentence: $a = (3+4) \times 2 = 14$, but never teach the student to separate things into two separate sentences; e.g. $w = 3+4 = 7$ followed by $a = w \times 2 = 14$.

So now, you have a student whose only means of expressing a complex mathematical idea is by doing so all at once — they simply don't have any experience in expressing the individual ideas separately.

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    $\begingroup$ I think this is an astute observation. I've had college students expressly say that they're not comfortable doing anything with a problem until they know exactly what all the steps are going to be. $\endgroup$ – Henry Towsner May 15 '18 at 17:39
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    $\begingroup$ I disagree with the claim that "the student seems entirely incapable of even conceiving of doing these as separate steps". If I challenge a student that, e.g., $3 + 4 = ... = 14$ is false, the most common response is, "But that's not what I meant!". I would say that this is purely a grammatical-training issue, not a conceptual one. $\endgroup$ – Daniel R. Collins May 26 '18 at 10:16
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I've read that this issue goes all the way back to elementary school where a teacher will write on the board, "3 x 4 = " and ask students to fill in the blank. This creates a subtle impression that an = sign is a procedural thing rather than a statement of equivalence, i.e. that it means something like "the next step is". This leads to statements like yours where each of the steps is separated by an = sign.

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  • $\begingroup$ I wish I had a superpower that let me appear in every one of those classrooms and write $3 \times 4 + 0$ and so forth, and let them think about that. $\endgroup$ – Daniel R. Collins May 27 '18 at 23:44
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    $\begingroup$ @DanielR.Collins A very good kind of question to ask young children is "4+2=__+3". $\endgroup$ – Steven Gubkin Oct 12 at 0:55
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    $\begingroup$ @StevenGubkin: I actually have that as a question on my 1st-day diagnostic quizzes for introductory community college courses. For remedial algebra it gets about an 85% success rate (N = 137). $\endgroup$ – Daniel R. Collins Oct 12 at 2:23
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My experience with this type of written work is because students use "=" in a different way than we mathematicians use it.

Sometimes it is used to mean “the result of this calculation is”, like the above student did, and then they continue with the next calculation, just like a calculator would do. And students' understanding of equality or the equal sign affects their success in mathematics in general.

For example, without a true understanding of equality, students struggle with solving equations in a general way, such as in solving a differential equation or solving an equation with the variables being polynomials or vectors.

I don't remember the first paper I read about this but a quick Google search came up with the following news article about related research which might help: https://www.sciencedaily.com/releases/2010/08/100810122200.htm

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The problem stems from the proliferation of worksheets, so entrenched into the American school culture. Get rid of the worksheets and teach students how to record a problem and a solution on a clean sheet of paper, preferably in a notebook, so that the record stays for at least the duration of a school year.

In short, use GFSA (Given-Find-Solution-Answer) or GFAS (Given-Find-Assumptions-Solution) process, which is used in technical colleges, but is completely absent from schools.

Given: write down what is given without repeating the problem verbatim; instead introduce constants and variables that later will be used in the solution; draw a picture if needed, use introduced symbols in the picture. The problem must be completely recognizable by reading the Given section without referring to the original problem in a textbook.

Find: clearly state what should be found.

Assumptions: this may be part of a Given part, for example, when solving geometrical problems involving vehicles or walking we may assume that the Earth's surface is flat, or when solving an equation we may assume that the roots are real numbers.

Solution: a step-by-step process, which usually involve some textual explanations, but not too much. There should be enough text to understand what numbers mean. Solve symbolically first, then plug in the numbers.

Answer: clearly state the answer, some professors require to underline it for clarity.

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The above problem can be solved as follows:

Given: rectangle (draw a picture) with sides (mark sides with letters, say "a" for the longer side, and "b"). On the picture mark segments of the longer side as a1 and a2, for example. Then write a1 = 4m; a2 = 3m; b = 2m

Find: Area of the rectangle (introduce a letter, say "s"), write s - ?

Solution:

1) Generic formula: s = a * b = (a1 + a2) * b;

2) Plug in numbers: s = (3 + 4) * 2 = 7 * 2 = 14;

alternatively, s = (3 + 4) * 2 = 3 * 2 + 4 * 2 = 6 + 8 = 14;

Each separate calculation is terminated with semicolon. It must be beaten into the subconcious that equal sign means that stuff on the left equals to the stuff on the right, so 3+4=7×2=14 is incorrect solution, and the problem is marked as not done, plain and simple.

Students must be taught to split task into clearly marked steps, like:

s = s1 + s2, where s1 = a1 * b, s2 = a2 * b;

Then students would write:

Area of the left part: s1 = 3 * 2 = 6 (m2) Area of the right part: s2 = 4 * 2 = 8 (m2)

Notice (m2) in parentheses. There are two approaches to dimensions: one is to write dimensions everywhere throughout the expression, like: s = 4m * 2m = 8m2 (no parentheses), this is useful for dimensional analysis, like meters times meters equals meters squared, but can be clunky for school math. Another option is to specify dimension of the resulting value only, in this case put it in parentheses.

s = 6 + 8 = 14 (m2)

Answer: the area of the rectangle is 14 m2. (underline "14 m2").

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Obviously, this particular problem does not require such an elaborate solution, but the students must be taught the principle, the scaffolding of the generic solution. The above problem can be solved simply as:

1) s = a * b; 2) s = (3 + 4) * 2 = 7 * 2 = 14; Answer: 14 (m2)

Introduce symbolic solution as early as possible, this improves conceptual understanding, and will pay back when they start algebra.

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  • $\begingroup$ This answer to "How should we first teach variables?" and this answer to "How can you be perfect in maths?" have some similar notes about how to solve word problems. $\endgroup$ – Jasper May 25 '18 at 22:39
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In my experience use notebooks where you can easy check their work and give instructions on how to correct it. To my students I always say that writing mathematics is like writing a small meaningful phrase. It has to be aesthetically appealing, must be correct, good grammar etc. So they have to write it as beautiful as they can and respecting rules and meanings. A lot of them complain that I call mathematics, or any expression beautiful. However with this they (after 2 or 3 months) start to get it, and apply it constantly. At the beginning of this school year I had easily 35 out of 40 students doing this kind of abuse to the notation, right now is around 15/40. And the other are slowly getting it. I want to point out that a lot of this mistakes are imprinted by private tutors, peers or sometimes a teacher that allows this kind of notation use.

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