What to blame for misconceptions in mathematics?
Oliver Knill, December 19, 2015
detects that half of a test group
of college students can not decide correctly whether 1/5 is larger than 1/8.
Psychologists attribute this to memorization techniques used by the students.
A different interpretation is that students fail to give the
right answer simply because they lack any pre-wired techniques to solve such basic
problems reliably. It is possible that it is not the memorization which is the problem, but
that it is the lack of mastering of firm algorithmic structures and the presence of
gaps in knowledge foundations, which is the root of the problem.
One of the students in my Math
E320 Extension school class mentioned this
The interpretation of that "psychology today" article is that students have problems to compare 1/5 with 1/8 because
"they have memorized the algorithm of cross multiplication."
I believe that the source of the problem that many students do not dare to cross multiply because they
have been instructed to hold off standard techniques. The generic technique of
cross multiplication would give and prove the right answer. While also simple ``common sense" thinking:
[if you divide a cake into 5 pieces or 8 pieces, in which case are the pieces larger ]
gives the solution, common sense will fail for more complex problems.
[December 25 added: here is a beautiful example from here: you take 60 stairs when going up an escalator and 90 when returning down the
same stairs, how many stairs would you take if the stairs would not move?]. It is not only remarkable about this example appears
in a popular journal in Germany; it should also be noted also that the problem can be solved by using pre-wired techniques
which one learns as a mathematician. If you think that shortcuts work, then no, the obvious answer 75 is totally wrong!
You have to learn techniques to think in order to solve such problems reliably. And this needs memorization as it requires
to know the technique!
Mathematics gives us the power to solve problems which go beyond simple intuition and which also allows us
to make sure that the thought process is correct and the statement is true and not just a lucky guess.
Intuition and argumentation is important, but in more abstract concepts already present in basic
geometry or probability, common sense can fool us (Monty Hall, Geometric proofs that all triangles are equilateral etc).
Mathematics is not politics, where fuzzy thinking, sloppy definitions or rhetorical tricks rule.
Having solid thinking and procedural problems skills is one of the main goals in
math education. Why is it true that 1/5 is larger than 1/8? It is because a simple
cross multiplication (or producing a common denominator) renders this equivalent to 8 is larger than 5.
The algorithm of cross multiplication
is not the devil. It not only solves this particular problem, it also proves the statement.
Here is a maybe provocative claim:
The demonisation of learning "procedures" and acquiring "knowledge" done by some
psychologists the last few decades is one of the main reasons for poor testing outcomes. Indeed, there is
a cultural war on "Knowledge" and "Algorithmic thinking". It can be summarized in
the statement ``you shal not memorize" or ``apply prewired algorithms".
When we work with students on calculus problems, it is staggering to observe that often,
basic arithmetic skills like cross multiplication working with fractions
are not solidified. It is as if some inner voice tells the students, not to do anything automatic.
They feel helpless and insecure.
Maybe the following recommendation is more reasonable:
Building solid knowledge and algorithmic skills, which at first are used
blindly by applying patterns gives the confidence and hunger for a deeper understanding.
It is the soil,* on which creativity can flourish later.
* yes, sometimes the soil is dirty and unattractive.
Here is the study and the citations of the
The Richland et al. (2012) article reports a string of shocking
findings gleaned from two other recent articles (Givvin et al., 2011;
Stigler et al., 2010). Two of the questions assessed whether or not
students understand what a fraction is.
Students were shown a number line from -2 to 2 and asked to draw a
line marking the approximate location of two numbers: -0.7 and 13/8.
Percentage who answered correctly: 21%.
Students were asked "If a is a positive whole number, which is
greater: a/5 or a/8?" Fifty percent would answer correctly if they
just guessed. Percentage who answered correctly: 53%.
If you've been assuming high school graduates fully understand how
fractions work, these results say otherwise. Some fell back on procedural
knowledge, probably because that's the only knowledge they had about
fractions. For example, seeing two fractions near each other apparently
triggered an urge in some students to use the cross-multiplication
procedure they had memorized.
Hiebert, J. C., & Grouws, D. A. (2007). The effects of classroom
mathematics teaching on students learning. In F. K. Lester, Jr. (Ed.),
Second handbook of research on mathematics teaching and learning
(Vol. 1, pp. 371-404). New York, NY: Information Age. Givvin, K. B.,
Stigler, J. W., & Thompson, B. J. (2011). What community college
developmental mathematics students understand about mathematics, Part
II: The interviews. The MathAMATYC Educator, 2(3), 4-18. Richland,
L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual
structure of mathematics. Educational Psychologist, 47(3), 189-203.
Stigler, J. W., Givvin, K. B., & Thompson, B. (2010). What community
college developmental mathematics students understand about mathematics.
The MathAMATYC Educator, 10, 4-16
- December 20, put first online, December 28: added the escalator riddle