Percents...Enough Said...Not Learned
What is the hardest mathematics topic to teach secondary students...if the goal is deep understanding of the concepts involved and the ability to apply that understanding meaningfully in relevant situations.
Many topics would get votes...but I can almost guarantee (with 110% confidence) that percentages would be a leading candidate. And by percentages, I view it as a part of proportional reasoning, plus the associated symbolisms and nuances.
Consider these two problems:
 Which is greater 37% of 92 or 92% of 37?
 True or False: An item whose price increased by 10% then decreased by 10% is still selling at its original price.
Both problems frustrate students because they "play" with their intuitions. And, research documents that few students are proficient proportional reasoners, with most students mired in memorizing rules to determine one of A, B, or C, given the other two in A% of B = C. The luckier ones may have advanced to using the mneumonic "is over of equals percent over 100" to produce C/B = A/100....but neither approach equates (let alone leads) to understand percentages, or to even utilizing the formulaic ideas correctly.
Some relevant research about students (and teachers?):
 Many ignore the percentage sign, treating 1/2 and 1/2% equally.
 When explaining the meaning of 25%, many produce the relationship "1 part out of 4", yet do not connect this to "25 parts out of 100."
 Many merely place a decimal before a number and then remove the percent sign, obtaining 0.55 and 0.110 for 55% and 110% respectfully.
 Many do not connect percents and rate/proportion, let alone making proper connections to fractions and decimals
 Many cannot use visual models to make productive connections, especially when performing a "percent" algorithm.
So what is a teacher to do? A good place to start is to review what is known about student (and adult) understanding of all aspects of percents, as well as productive techniques for teaching/learning the same.
Thought there are many great resources, three good places to start are:
 Lembke, L. & Reys, B. (1994). "The Development of, and interaction between, intuitive and schooltaught ideas about percents." Journal for Research in Mathematics Education. 25: 237259.
 Parker, M. & Leinhardt, G. (1995). "Percent: A Privileged Proportion." Review of Educational Research. 65: 421481.
 Lamon, S. (2011). Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers. Routledge.
Speaking from my own personal experience, probably the worst place to start is to build a pedagogical approach on one's own approach to doing percent problems, let alone one's personal ability to reason proportionally. That is, after 42 years of teaching mathematics (i.e. relearning mathematics), I am getting close to being able to help students understand percents.
Note: If you have other candidates for the hardest math topic to teach, send them to me...with a justification.
