Poor Use of Mathematical Language (PUML)
A prerequisite skill to writing good mathematics is the ability
to write well in one's native tongue. People who cannot write
a complete English sentence should take remediation in English
composition before reading on.
What may surprise some students is that good writing using
mathematical symbols (even in the write-up of homework problems)
consists of using complete sentences, setting up one's ideas
clearly and then following through on the details, much as
one expects from a good English essay. The language and symbols
of mathematics are used just like regular English words and phrases
to express ideas, albeit ideas which one would often struggle to
use any other means of expressing.
Nobody studies mathematical writing as a subject. Your mathematics
professor(s) got to be good writers of mathematics, if good they be,
by reading papers and books by other mathematicians,
not by reading a treatise such as this one. If a book on good
mathematical writing does exist (and there are probably a
number of such books), they will say much more than I say here.
I will only describe the most common example of poor mathematical writing
I see when grading students' work: Using Equals as a Conjunction
The word ``equals" has a very specific meaning. It requires two
objects, and it asserts that these two objects are the same. In
mathematics, the two objects are usually quantities, like the
mathematical expression , or the number 7. Even within
this tight definition, mathematical equations, as I mentioned earlier,
come in two varieties: identities and conditional equations.
A conditional equation is one such as the equation
which is only for particular values of (in this case one
particular value). In algebra courses
one often sees conditional equalities in homework problems accompanied
by the instruction ``Solve the equation". There are some quantities
that are the same regardless of the value of the variable. A familiar
example is the identity
which is true no matter what real value takes. These two types
of equations encompass the two most common (and only?) valid ways to
use an `equals' () sign.
Consider the typical calculus problem of evaluating a limit like
What we are given here is not an equation, but an expression. If
we begin writing a series of equalities to simplify/evaluate this
expression, we will want them to be identities, as in
The original expression, along with each of the ensuing expressions,
as it turns out, are all equal to the number .
In contrast, suppose we begin with a (conditional) equation like
which we are asked to solve. If a student who understands very well
the discussion of UAS and UMD (found earlier in this
piece) makes a mistake, she is most likely to do so writing
Such a string of equalities asserts three things:
(i) and (ii) are conditional equations in their own right, but it
should be clear that they do not have the same solutions as the
original equation (nor does (i) have the same solution
as (ii)). And (iii) has no solution at all, for it is never true.
What I am really saying is that the string of equations
- that ,
- that ,
- and that .
is really three equations, and there is no common solution between
them (and, even if there had been, such a solution would have no
relevance to the original problem, that of solving ).
The student most likely never intended to assert these three
equations in place of the original; she simply began writing
out her ideas, and used an equals sign to join them together whenever
it seemed some sort of conjunction was required.
The student who writes (1) actually appears to have
some facility in the techniques for solving linear equations,
but lacks the ability to put her ideas onto paper in a meaningful
fashion. One good way to express the solution of the previous
The symbol can be translated here as ``which implies".
Yes, (2) is more writing than (1), much in
the same way the complete sentence ``I am taking the train to Chicago
this weekend" requires more writing than the three words ``weekend,
Chicago, train". A more favorable comparison is between
(2) and the same ideas expressed in
If the sum of three times and five is seven, then subtracting
five from both sides and dividing by three yields the value of
two-thirds for ,
or, perhaps more literally,
Assuming that the sum of three times and five is seven,
this implies that three times is two, and that is
Top Algebra Errors Made by Calculus Students
Full List of Grading Codes
Thomas L. Scofield