# Students understand during course but can't solve exam

I am teaching a math class where the students, most of them, tell me that they can understand the materials given by me during the course. I test them during the course too and they seem to get it. However, when it is time to face quiz or exam problems, they don't know what to do.

At first, I thought that my exam is too hard. However, I usually, definitely, give problems that need the understanding of materials I give in the class to solve, but add some "spices" to them. By spices, I mean I always assume that in an exam, there should be a little bit of challenge, not just repeating exercises. It is a test after all, right? I will not repeat problems I gave previously during the course. Sometimes, even when I intentionally give the same problems, they still don't know what to do, most likely because they do not repeat until they understand after class.

For example of the problem, instead of just asking how many ways we can arrange a group of 4 letters, I ask how many ways we can arrange a group of n letters. The point is that I won't repeat and will give a small added challenge during exam.

I thought I should give them homeworks and also tutorial sessions, but the math program is new (just two batches now), so we have no tutor..nor grader. I do it myself almost always. We have no money for tutors either. Thus, I can't do anything else (like finishing paper, doing administrative task, etc) other than helping my students. Indeed, it's gonna be so tiring.

I genuinely want them to succeed and this is just my second year of formal lecturing. So happy knowing they understand me when I teach, but no idea yet how to make them exercising more often, other than reducing the covered materials maybe? Doesn't seem wise. Or should I make the exam..a repetition of problems, where only the information about the numbers given is different? It will kinda help but I am just a little bit afraid they will only be able to do specific problems with no small added restriction, resulting in less flexibility of problem solving.

ADDED: It could also be due to my experience as an undergrad student in one of the leading universities in my country. I am still so fresh to remember my days in college where I was given problems in exams (most of them) that was served more like challenges rather than competency tests. Hence my mindset was like "Exam is supposed to challenge you with extra work". Now I am teaching in a not so well-known university where I should've known the standard set would be much lower, which is in the form of fewer materials given. Hence, I am still in the mindset that while university is a place to learn, it is also a place for those who can flexibly apply what they learn, now matter how high or low the standards are.

Any suggestion on how do I demonstrate flexibility to students other than by having them learn the basics and giving nonrepeated problems (in exam) to test them?

Some of my students do fine with the extra work exams, but most of them, predicted from quizzes, still need more work and guidance before midterm.

• I suspect that the jump from $4$ to $n$ would seem like a big jump to some students. Another comment is that I think exam questions should usually be easier than the homework questions, so whatever twist is added on the exams probably shouldn't increase the difficulty as compared to the homework. – eternalGoldenBraid Oct 4 '18 at 20:34
• "instead of just asking how many ways we can arrange a group of 4 letters, I ask how many ways we can arrange a group of n letters." - if they do not know the generic way of finding it, if they only can count up the cases, then they won't be able to solve it. Do you expect them to come up with a generic formula during a test? I don't mind giving more complex problems, but one that sort of a combination of two-three simpler known problems, where one has to do an extra step, but it should not be like deriving a completely new formula. – Rusty Core Oct 4 '18 at 22:16
• You are most of the problem. Exams are a high pressure situation. Look at gymnastics...do you ask someone to do a full twisting back flip for the first time in a competition (when he has been practicing regular back flips?) If anything it is the opposite and your practice problems should be slightly harder than the test problems (and SLIGHTLY, not super spicy). You are making the classic error of doing something that is interesting to YOU. Who has more sophistication than your students. This is not about spice. It is about training. – guest Oct 4 '18 at 22:24
• I gave them the same problem with big numbers but not n. However, the comments are really helpful. About extra steps problem, I make sure I include at least one. Sometimes, instead of counting donuts, I just rewrite the problem to counting candies but with bigger numbers given. I want to make sure they use combinatorial argument, not just by listing them. – bms Oct 5 '18 at 12:52
• I agree with DRC about outside work and classwork being harder (potentially much harder...) than timed, in-class tests. – Benjamin Dickman Oct 8 '18 at 17:12

Do NOT give exam questions that are intentionally more challenging than homework or in-class problems. I would recommend precisely the opposite.

The point of the exam is really a spot-check that students know the basics and aren't just faking their way all through the class. If there is a time-limit, then that is already a lot more pressure/high-stakes than open-ended homework problems. Clearly the hardest problems confronted should be in the time-free context, with available open resources and study partners, etc. -- not with the simultaneous additional overhead of a time limitation; that's geometrically increasing the difficulty, and arguably really is unfair. Use the exam situation as a spot-verification that students have the expected individual "automaticity" with basic skills.

I think I can sympathize in that I made the similar mistake of writing "interesting" (i.e., with-"spices") test problems in my first year of teaching. That simply never works. My rule of thumb now is that each test question needs to be completely transparent to me on first sight. Writing the answer sheet should take at most one-quarter of the time available to students. If I look at a problem and think that it's "interesting", then that's a red flag that it needs to be cut from the test.

If you're new to teaching (or just unsure of what direction to go in), then I recommend reading Steven G. Krantz, How to Teach Mathematics, published by the American Mathematical Society.

• +1 for "If I look at a problem and think that it's "interesting", then that's a red flag that it needs to be cut from the test." I too have succumbed to this, especially in my earlier years of teaching, and one way I found to get around my desire to "show interesting stuff" was through extra credit problems, supplementary handouts, and the like. Also, with the rise of internet math forums in the late 1990s (then mostly at Math Forum), I found another way to satisfy my desire to "show interesting stuff", and now that I'm no longer teaching (since 2005), that's been pretty much my only outlet. – Dave L Renfro Oct 5 '18 at 12:46
• This is indeed a very helpful advice. I usually, as worded in the comments of the post, add what I consider as extra works. Sometimes this extra can mean interesting which can harm them. I need to figure out ways to avoid interesting problem, but here we (not just me) usually will include extra work problem to make sure they use the methods, not just by using brute force. The type of exam you mention is usually served as quizzes here. For lecturers who don't hold quizzes often, adding extra work problem is not recommended I guess. – bms Oct 5 '18 at 12:59
• A particular issue when you teach k12, have a full period dedicated to a test, and some students finish early: We cannot just let them leave and roam. So, I include "Bonus" problems that are hard to ensure no one finishes and to get an idea of how much further some students are ready to stretch. But, I give no extra credit for these problems: I like having problems even in a test setting to reenforce that grades are not the ultimate purpose and, crucially, I believe extra credit points for bonus problems is a way to reward further fast workers, which adds to time pressure. – Benjamin Dickman Oct 8 '18 at 17:08

I always make homework (from the textbook and online in WeBWorK) and written assignments MORE difficult than exam questions. I tell my students this, with the reason being “if you can run 10 miles in training running 5 miles on race day is easy”.

Keep the course learning goals in mind. Your exam questions should be a chance for your students to demonstrate competency.

• +1. However, the course standard set by the department says also that flexibility of problem solving is one of the goals. I agree with you totally. I make my exams not much harder than assignments and quizzes too. Any idea on how we challenge flexibility other than repeating problems? – bms Oct 5 '18 at 13:32

If you ask me in how many ways I can arrange 4 letters or digits, this is what I will do:

1234
1243
1324
1342
1423
1432
2134
2143
2314
2341
2413
2431
3124
3142
3214
3241
3412
3421
4123
4132
4213
4231
4312
4321


Then, I'll count them and answer your question: "Sir, I know, it's 24!!!".

If you ask me how many there are for a general n letters, I don't have the slightest idea, and honestly, when you think about it, it's indeed extremely difficult.

• The problem was not quite the same, but I've shown them the general formula, helped them demonstrate how it is deduced? and let them work on it with big numbers. The point is to indeed make them use the method without brute force. Avoiding the "listing all solutions" process (for some materials only) is one of the subgoals listed. – bms Oct 5 '18 at 13:28
• "...shown them the general formula, helped them demonstrate how it is deduced..." How did you check for understanding and retention when you helped them demonstrate how the formula was deduced? i.e. How do you know they were thinking the right way as opposed to just speaking the right words and writing the right symbols? – WeCanLearnAnything Oct 16 '18 at 15:15

At first, I thought that my exam is too hard. However, I usually, definitely, give problems that need the understanding of materials I give in the class to solve, but add some "spices" to them.

Part of the problem is the "extra spice" you're giving them during the test. Why are you surprising them with extra spice for the first time during the test? That's literally the worst possible time to introduce new material!

I am teaching a math class where the students, most of them, tell me that they can understand the materials given by me during the course. I test them during the course too and they seem to get it. However, when it is time to face quiz or exam problems, they don't know what to do.

This suggests a standard blocking vs interleaving illusion. Many textbooks are organized by concept and procedures rather than by difficulty, so students:

1. Get all the fraction addition and subtraction questions right in the adding and subtraction fractions section of the textbook.
2. Get all the multiplying fractions questions right in the multiplying fractions section of the textbook,
3. Then they get all the dividing fractions questions right in the dividing fractions section of the textbook,

Students and teachers conclude "Holy cow, look at all these right answers, this is amazing, we're really getting it!"

Then students arrive at the test and face:

'There are 5 and 1/4 equal pieces of cake and 6 people to share them. How much should each person get?'

And various students respond:

$$5\frac{1}{4}\times6=31\frac{1}{2}$$

$$5\frac{1}{4}\times6=\frac{21}{4}\times6=\frac{126}{4}$$

$$5\frac{1}{4}+6=11\frac{1}{4}$$

$$5\frac{1}{4}-6=?$$

$$6\div5\frac{1}{4} = 6 \times 5\frac{4}{1}$$ ... "When we divide, the big number goes first, then you flip and multiply..."

i.e. The students used section titles of textbook to select the operation correctly instead of the meaning of the word problems. This is an illusion of understanding. When those section titles are not there on the exam, those non-meaningful habits continue and students are forced to guess somewhat randomly, which, of course, is terrible in a math class.

Try interleaving! It drastically reduces speed and accuracy in the short run but forces meaningful thinking and retention in the longer run, and effect sizes are huge. [Google it.]

• About your first point, I was given these "extra spices" during my undergraduate years. There, exam was supposed to test, not repeat. That is kinda built in my view. However, I am well aware this university where I teach is not the same as mine, but the materials and standards are reduced already. How much more should them be reduced? Also, in discrete math course, students have to recognize the objects and the methods, not to remember only certain examples of objects given in exercises. How do you think? As for your second point, I appreciate it and will soon google it. – bms Oct 15 '18 at 14:10
• The problem with "extra spices" is that they often seem like a small leap to the teacher while they are, in fact, a gigantic conceptual leap to the students. Why are you not introducing "extra spices" during the course? Why wait until the exam to introduce "extra spices" for the first time? Have you tried introducing these variations earlier in the course, to, say, one student and have them think aloud as they work? If you do, prepare for a shock, as this type of "user-testing" often shows drastic holes in knowledge, understanding, and work habits. – WeCanLearnAnything Oct 16 '18 at 15:19