# What does math education research know about difficulty vs. effectiveness?

I've asked basically the same question previously on on math.SE, then cogsci.SE without much response, surely here is the place to ask this.

As anecdotal evidence is plentiful, but unfortunately for whatever conclusion you wish to draw, I'm hoping for answers that are backed by actual education or cognitive psychology research.

There seems to be a prevalent view among mathematicians and math educators that forcing students to struggle is an effective way to improve retention and understanding.

Since I first asked the question, several points have become clearer:

• There's sufficient (for me) evidence that required math courses are sometimes used to "weed out" weaker students as a form of "population control". I'm ignoring those.
• There are researchers who teach (historically, surely) who consider teaching to be a distraction from research and so have an incentive to embrace any teaching philosophy that minimizes their efforts, such as "let them struggle with it and prevail". Putting aside ethical and/or utilitarian arguments on the wisdom of such an approach, here too the nature of the course is not motivated by a belief that difficulty increases the benefit for students. I'm ignoring these too.
• There are various "non-traditional" approaches such as Problem-based learning, peer-learning, constructivist approaches, the moore method (and variants), lecture-skepticism (my term) and others which claim, and can often demonstrate quantitatively, improved performance by students (yet often, interestingly, decreased confidence). They share a de-emphasis on the traditional lecture in favor of students taking a more active role in the learning process (So in a sense, working harder). They often simply require a larger investment of time by the students which might itself account for the difference.
• Research suggests that humans have a natural tendency to "skim" when a suitable opportunity presents itself, so that increasing the difficulty involved in processing the material (even extremely artificially, as in using a less readable font) forces more engagement from the reader/student and improves retention (see papers). Similarly, when students have a strong preexisting intuition on the subject, such as when studying newtonian physics (which we all have much experience with), students appear to resist the learning of new concepts that replace their existing intuitions unless actively confronted with their shortcomings (see papers). This also can be interpreted as evidence that learning is improved when students are forced to break out of a tendency to use "cruise-control".
• The theory of "cognitive load" has been pointed out as relevant on cogsci.SE, but from what I've skimmed (naturally) so far, the idea is that under increased cognitive load learning breaks down. I've not seen yet results demonstrating that increasing cognitive load improves learning other then, as just mentioned, by forcing engagement.
• Vygotsky's theory on the "Zone of proximal development" is intriguing in it's claim that learning is most effective when students tackle problems which they "cannot solve unaided, but can solve with some guidance". The discussion is centered on children's education, it's unclear to me how widely applicable it is to a collegiate math setting.
• The related idea of a teacher as providing "scaffolding" also matches up with research on human memory: we remember new facts better when we connect them to existing memories and concepts. It seems plausible to claim that a student who derives a result on his own has built up a more substantial structure around the discovered fact and is therefore more likely to remember it.

That's an outline of 3 different senses of "difficulty" in learning:

• Challenge ("He sets impossible standards, so you work really hard or crash. or both.")
• Engagement ("I really have to pay attention to make sense of it, it's exhausting!")
• Time spent ("It took me 3 days to do that assignment, it was really difficult!").

Engagement seems specifically to be an issue of growing importance in recent decades, vis the debate on the effects of TV/video games and the increased diagnosis rate for attention/hyperactivity disorders.

I'm looking for answers that shed more light on the issue, point out research results and generally make the issue clearer. Though I am not an educator (math or otherwise), a good guideline are answers indicating research that would help someone with teaching duties, in math, at the college level, form an informed policy on the difficulty level of a course intended (one would hope) to help students achieve their best.

The original question on Math.SE was sparked by a bunch of quotes I've accumulated through readings on math. I've included them here:

The first is by Lebesgue:

When I was a rather disrespectful student at the Ecole Normale we used to say that 'If Professor Jordan has four quantities which play exactly the same role in an argument he writes them as $u$, $A''$, $\lambda$ and $e_{3}'$ Our criticism went a little too far but, nonetheless, we felt clearly how little Professor Jordan cared for the commonplace pedagogical precautions which we could not do without, spoiled as we were by our secondary schools. <...> Professor Jordan's only object is to make us understand the facts of mathematics and their interrelations. If he can do this by simplifying the standard proofs, he does so; <...> But he never goes out of his way to reduce the reader's trouble or compensate for the reader's lack of attention.

The "Moore Method" (and variants thereof) is a well known approach to math education. Here's an excerpt from P.R. Halmos' autobiography "I want to be a mathematician":

Can the mathematician of today be of any use to the budding mathematician of tomorrow? Yes. We can point a student in the right direction, put challenging problems before him, and thus make it possible for him to "remember" the solutions. Once the solutions start being produced, we can comment on them, we can connect them with others, and we can encourage their generalizations. Almost the worst we can do is to give polished lectures crammed full of the latest news from fat and expensive scholarly journals and books—that is, I am convinced, a waste of time. You recognize, I am sure, that I am once more advocating something like the Moore method. Challenge is the best teaching tool there is, for arithmetic as well as for functional analysis, for high-school algebra as well as for graduate-school topology.

Lastly, here's a quote from the preface to "Mathematics Made Difficult":

there is no doubt that an absolute ignoramus (not a mere qualified ignoramus, like the author) may become slightly confused on reading this book. Is this bad? On the contrary, it is highly desirable. <..misleading redaction...> it is hoped that this book may help to confuse some uninitiated reader and so put him on the road to enlightenment, limping along to mathematical satori. If confusion is the first principle here, beside it and ancillary to it is a second: pain. For too long, educators have followed blindly the pleasure principle. This over-simplified approach is rejected here. Pleasure, we take it, if for the initiated; for the ignoramus, if not precisely pain, then at least a kind of generalized Schmerz

## Previous questions elsewhere

• math.stackexchange.com/questions/702044
• cogsci.stackexchange.com/questions/5921

## Related Papers

• Designing Effective Multimedia for Physics Education - www.physics.usyd.edu.au/super/theses/PhD(Muller).pdf by Derek Muller TED talk: www.youtube.com/watch?v=RQaW2bFieo8
• Fortune favors the bold - web.princeton.edu/sites/opplab/papers/Diemand-Yauman_Oppenheimer_2010.pdf by Diemand-Yauman & Oppenheimer
• Is this teaching maths for math’s sake or maths to students that will use it for someone useful like engineering?
– Ian
Mar 26 '14 at 21:24
• Yes it is. Feel free to explain why you think why those cases should be considered seperately.
– user370
Mar 26 '14 at 21:42
• I suspect many posters here over-value struggle (and even worse at math stack exchange). The goal should be productive use of time. Most learning with least time spent. Time is valuable and people have other uses for it (other subjects, work, recreation, etc.) Yes, there is a value in struggle but not in wasteful struggle. Sep 5 '18 at 18:51
• I want to offer a bounty on this question but I don't think it's eligible because the start a bounty link doesn't appear. I might ask a question on Meta Stack Exchange about whether all questions should be eligible for a bounty when I get my next chance to. Mar 18 '20 at 2:59

Perhaps the key-word needed here is not just struggle but productive struggle.

Hiebert and Grouws (2007) discuss two key features of mathematical teaching/instruction for "promoting conceptual understanding" (p. 383). Their paper can be found here:

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. Second handbook of research on mathematics teaching and learning, 1, 371-404. Link.

To the point, their second feature is called Students Struggle with Important Mathematics.

The authors trace this line of thought back at least to John Dewey (p. 388):

The article continues with references to work by cognitive theorists such as Lev Vygotsky (whose Zone of Proximal Development is mentioned in the OP), the mathematician George Pólya (whose "How to solve it" effectively laid the groundwork for the study of mathematical problem solving), and the developmental psychologist Jean Piaget.

The authors continue with a discussion of some material on work done outside of mathematics before returning to studies specific to mathematics and mathematics education. You remark that your "question is in natural peril of being too broad," but hopefully this source (and the references contained therein) can get you started on tackling it.

• Totally agree with the "productive" struggle concept. Apr 10 '14 at 17:00

Here is an NPR article that discusses how teachers' efforts to engage learners in productive struggle (or not) may be culturally situated. (Of note, Benjamin cites Jim Hiebert above, who has written The Teaching Gap with Jim Stigler, interviewed in this article.) Jim and Jim have conducted research on how instructional approaches differ culturally between nations.