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Students often misuse the equals sign to indicate "I've done this operation" rather than the proper use indicating numerical equivalence.

Eg. Tax is paid using the rule: \$3 572 plus 32.5c per \$1 over $37 000. How much tax is paid on \$78 540?

Answer 1: 78 540-37 000 = 41 540 x 32.5% = 13 500.50 + 3572 = $17 072.50

Answer 2: 3 572+(78 540-37 000)x 32.5% = 3 572+41 540 x 32.5% = 3 572+13 500.50 = $17 072.50

Both methods produce the correct answer. But Answer 1 misuses "=" to mean "now we do this", perhaps it also shows a lack of understanding of order of operations and algebra, but fundamentally it is a good method.

How could this method be written to not misuse the equals sign?

How could this method be used to introduce the correct use of the equals sign?

Observation: What they are doing is writing the expression (a+(b-c) x d) using a form of postfix notation: bc-dxa+ which is quite a rational thing to do.

pdmclean
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    I recommend you to correct students when they do this. Make them rewrite the problem properly without misuse of the = sign. You are teaching them skills. Organized processes will help them with future word problems. – guest Sep 06 '18 at 02:40
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    Have you given the students any other mathematical 'grammar' they can use? Most students are unaware that you can use words in maths (having essentially been trained never to do so), and are unlikely to write much more than they think they need to. – Jessica B Sep 06 '18 at 08:15
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    @pdmclean "but fundamentally it is a good method" - no, it is a sloppy and incorrect notation. Just plain wrong. No matter what answer they get, this is a problem NOT DONE in my book. They obtain this notation from inattentive elementary school teachers and from calculators, where "=" indeed means "and now calculate". – Rusty Core Sep 06 '18 at 19:37
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    @pdmclean The notation in Answer 1 is infix, not postfix. If you replace all " = n" by a pair of paren's enclosing everything to their left then you get a valid infix expression - as in my answer. It is written as it would be input to an infix calculator, where '=' does evaluation. Note: it may confuse readers that your edit uses $a,b,c,d$ differently than in my answer, but that's easy to fix (if you so desire). – Bill Dubuque Sep 06 '18 at 20:05
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    Possible duplicate of Issues with "equals", where does this come from and how do I combat it? – Daniel R. Collins Sep 07 '18 at 01:01
  • @RustyCore You might consider thinking about skills and ideas that students have achieved, rather than only criticising those they haven't. There's a difference between having high standards and being harsh. Feelings play a big part in the learning process. – Jessica B Sep 07 '18 at 06:42
  • I feel like the number of downvotes on this question (question and answers) is surprisingly high. – Jessica B Sep 07 '18 at 06:42
  • Maybe students should be allowed some flexibility on how to present solutions on homework and tests at this level. It's hard to argue that they did something wrong if they get the right answers. The teacher should avoid such presentations, however. They should always use Jessica B's method. Later, when several steps must be done in parallel, it will become obvious to students that their usual string-of-equalities approach won't work, e.g. when totaling up an invoice with various unit costs. – Dan Christensen Sep 07 '18 at 15:53
  • @Jessica B I am not criticizing the students, I am criticizing their elementary teachers first and foremost. – Rusty Core Sep 07 '18 at 16:02
  • @DanChristensen I don't think the level of the students has been specified. This could be anywhere between late primary school and undergraduate (and to some extent I think that affects the approach to dealing with the problem). – Jessica B Sep 07 '18 at 16:19
  • @JessicaB I think a purely numerical calculation such as Answer 1 should be accepted on an test or homework assignment at the primary and secondary school level. Teachers shouldn't get away with it, though. – Dan Christensen Sep 07 '18 at 16:38
  • Dan, I think your solution is practical. Unfortunately, the math oriented brain sometimes wants to think that every problem can be nipped in the bud, rather than thinking about how people learn and develop. Consider language. We learn native language grammar from imitation and practice (almost osmosis) rather than explanation. I don't think math is the same as language...still there are times when we need to learn different things at different times. – guest Sep 10 '18 at 18:36

3 Answers3

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I would write this as:

Earnings over $\$37,000$: $\$78,540-\$37,000 = \$41,540$

Base tax: $\$3,572$

Additional tax: $\$41,540\times 0.325 = \$13,500.50$

Total tax: $\$3,572+\$13,500.50 = \$17 072.50$

Jessica B
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  • A good answer, though I'd be careful with having : follow a number, as it is hard to read properly, for me. Also, : is sometimes used as a sign for division (at least in Finland), so I would try to not place it between two numerical quantities. – Tommi Sep 06 '18 at 08:23
  • @TommiBrander A colon not used for division in the UK, although it is used for ratios. Maybe you can think of a better symbol, but I massively prefer it to an equals, which is what UK students would probably default to. A new line could be used instead. I think if they don't get equals signs, this would be a big step in the right direction, even if it does come up with some non-ideal notation. – Jessica B Sep 06 '18 at 08:29
  • I think colon is fine, as long as is not between two numbers, as on the first row with numbers. Similar to how in mathematical writing one should not start a sentence with a symbol, and how one should try to add words between symbols between expressions so that they are easier to distinguish from each other. So rather than "for all $x$, $x=y$" write "for all $x$ we have $x=y$". E.g. Earnings over 37000 $ are: .... – Tommi Sep 06 '18 at 09:13
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    There is a point here. E.g.: My statistics class allows use of the formula card that comes with the text. One of the formulas is presented, "Mean of the variable $\bar x$: $\mu_{\bar x} = \mu$". Every semester there are students who copy the equation as: "$\bar x$: $\mu_{\bar x} = \mu$" (i.e., anything from a math symbol on must be part of the equation, including the colon symbol). – Daniel R. Collins Sep 06 '18 at 22:11
  • Not only such a breakdown is a valid notation, it also shows the thinking of a student, and the sequence of operations. When I was in school, we were encouraged to break down larger expressions into separate steps, and explaining each step. Just getting an answer without proper explanation did not count. – Rusty Core Sep 06 '18 at 22:54
  • I'll accept this because it focuses on understanding and making the reasoning clear. It builds on student knowledge. It uses the equals sign correctly while not being insistent on the "correct" mathematical convention. Thanks! – pdmclean Sep 06 '18 at 23:15
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Update. It appears that the intent of this answer was not clear to some readers, so I will elaborate. The most common way to fix the student's computations is simply to break out the subexpressions onto separate lines (as in the other answers). There are however other noteworthy approaches that offer some advantages. Below we briefly present one such approach.

Observe that the student uses the equal sign as it functions on an (infix) calculator, i.e. to evaluate the preceding expression. Using variables (and disambiguating parentheses) the student's expression is essentially the first one displayed below, where the equal sign has been (ab)used to annotate (label) subexpressions with their values $\,\color{#0a0}n,\color{#c00}m.\,$

$$\begin{align} ((a\!-\!b =\color{#0a0}n) \times c = \color{#c00}m) + d &\\[.2em] \underbrace{\overbrace{(a\ \ -\ \ b)}^{\Large\color{#0a0} n}\ \ \times\ \ c}_{\Large\color{#c00} m}\ \ +\ \ d\ \ & \end{align}$$

We can greatly improve this by using better ways to annotate the values $\,\color{#0a0}n,\color{#c00}m.\,$ e.g. as above using under/overbraces. Notice how this helps clarify the algebraic structure of the expression, e.g. now we can see with a single glance that it depends linearly on $c$ (interest rate) so we can make inferences about how its value changes with changes in the interest rate. But this innate algebraic structure is greatly obfusctated if instead we dissect the expression into many subexpressions and break them out onto separate lines (into a "straight line computation").

So my point is that we should teach students various ways to present such computations - not only the common straight-line methods. When one encounters more complex expressions the structure-preserving approaches can make a huge difference in reducing complexity. For a less trivial example, below is an excerpt from one of my MSE posts on telescopic induction.. Notice how the over/underline annotations below allow one to comprehend in a single glance the effect of the telescopic cancellations - which would be greatly obfuscated if the intermediate annotated expressions were broken out onto separate lines.

Hint $\: $ First trivially inductively prove the Fundamental Theorem of Difference Calculus

$$\rm\ F(n)\ =\ \sum_{i\: =\: 1}^n\:\ f(i)\ \ \iff\ \ \ \color{#c00}{F(1)=f(1)},\,\ \ \color{#0a0}{F(n) - F(n\!-\!1)\ =\ f(n)}\ \ {\rm for}\ \ n> 1$$

whose proof is simply a rigorous inductive proof of the following telescopic cancellation

$$\rm \underbrace{\overbrace{\color{#c00}{F(1)}}^{\large \color{#c00}{f(1)}}\phantom{-\color{#c00}{F(1)}}}_{\large =\ 0}\!\!\!\!\!\!\!\!\!\!\!\!\!\overbrace{\color{#0a0}{-\,F(1)\!+}\!\phantom{F(2)}}^{\large\color{#0a0}{ f(2)}}\!\!\!\!\!\!\!\!\! \underbrace{\color{#0a0}{F(2)} -F(2)}_{\large =\ 0}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\overbrace{\phantom{-F(2)}+ F(3)}^{\large f(3)}\!\!\!\!\!\!\!\!\!\!\underbrace{\phantom{F(3)}-F(3)}_{\large =\ 0}+\,\underbrace{\cdot\ \cdot\ }_{\large =\,0\,}\overbrace{\cdot\ +F(n)}^{\large f(n)}\ =\ F(n) $$

Bill Dubuque
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    I highly discourage the first notation, because how on earth would a product of equations be defined? – Jasper Sep 06 '18 at 05:33
  • @Jasper (and downvoter). To be sure, the answer is not meant to encourage use of the student's notation (first expression). Rather, it means to explain how to achieve the same goals (preservation of innate structure) using better notation (cf. "much better ways"). Though I would not recommend it, there are in fact ways to use the first notation along with equational operations, e.g. replace uses of the equal sign (for evaluation) by another symbol, e.g. $,\leadsto\quad$ – Bill Dubuque Sep 06 '18 at 13:21
  • I like the annotation idea, it provides a good notational bridge between the student's answer and the correct mathematical answer. – pdmclean Sep 06 '18 at 23:09
  • @pdmclean Glad it was helpful. I edited the answer since it seems the intent was not clear to some readers. – Bill Dubuque Sep 07 '18 at 00:29
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Answer 1,

$$78 540-37 000 = 41 540 \times 32.5\% = 13 500.50 + 3572 = $17 072.50$$

could be rewritten as: \begin{align*} 78540-37000 &=41540\\ 41540\times 32.5\% & =13500.50\\ 13500.50+3572 & =17072.50 \end{align*}

The use of $\leadsto$ has been suggested in the comments below to indicate each next step in the reasoning. At the level where a student makes this kind of mistake, they might not be ready for the introduction of further notation (if they get confused with $=$, they might get even more confused with some new symbol). As "guest" has suggested below, simply writing each step on a different line should be sufficient.

orion2112
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