Problem Sets
TIPS FOR PROBLEM SETS & math (Cal Newport)
General tips for math courses
Take notes on sample problems, answers and explanations. Take notes on the teacher’s explanations – the big picture (put it in your own words so you are sure you understand it). Engage with the material and try to understand it while still in class.
If you don’t understand something after looking at your notes and textbook GOOGLE it (Math World & Wikipedia have clear explanations for most problems)
When reviewing the material, try to re-create the steps, answers and especially the underlying concepts recorded in your notes OUT LOUD as if you were teaching a class.
A Four Step Process for Solving Hard Problem Sets
The motivating idea behind this strategy is simple: your brain can only work productively on a hard problem for 1 -3 hours before needing to reboot. To reboot your brain, so more productive work can be accomplished, requires a significant break. Preferably overnight.
Here’s a four step strategy built around this idea.
Step 1: Pick Off the Simple, Prime the Hard
Your first block of work should occur early in the week. Set aside 2 – 3 hours, in the morning. Make this the first thing you do that day (when your energy is at its highest). Your goal is two-fold. First, you want to solve easy problems. Your strong focus will help you avoid stupid mistakes. Second, you want to tackle at least two hard problems. You probably won’t solve them. This is why they are hard. But you can do something almost as important: prime them.
To prime a hard problem is to discover exactly why you can’t solve it. Pick an obvious approach — even if you suspect it won’t work — and start working through the problem until you get stuck. Identify why you are stuck. Ask what you need to figure out to make progress. What is it that makes this hard? Then take a break…
Step 2: Think in the Shower
For the next 2 – 3 days, think about how to get around the obstacles you discovered while priming. Don’t do this formally, in the library, with books around you. Instead, do this while walking around campus. While waiting for class to start. In the shower. I used to solve my Algorithms take home exam problems, for example, while jogging.
This is when breakthroughs occur. If you end up with a great insight, take 20 minutes, next time you can spare it, to sit down and write it down formally. If needed, prime a new hard problem so you can keep making progress as your wander campus throughout the week.
If you encounter ambiguities in the problem description that are giving you trouble, send concise questions to your TA requesting clarification. You don’t want these details to slow down progress any longer than they need to. (You might end up e-mailing your TA many times early in the week. This is okay so long as the questions are specific and concise. Don’t wait until office hours. By then, it’s too late.)
Step 3: Meet with your Problem Partner
A team effort is crucial for problem sets. But it has to be the right effort. Don’t meet with a large group. These are rarely efficient. Most of the time is spent griping about the class. Usually, there is one kid in the group who actually did the work, and, in the end, everyone copies off of him. Avoid this. The “smart kid” is often wrong, and likes the group because it boosts his self-esteem. Not to mention that your lack of understanding will come back to tag you on the exam.
The other extreme is to work alone. I see this a lot at MIT. Too many movies like Good Will Hunting got people thinking that to be smart at math means you should be able to stare at a problem for 5 – 10 seconds and then instantly solve it. Sorry. Doesn’t work that way. I walk past real geniuses every day — people, for example, who are my age and are also tenured professors — and guess what: it takes them a long time to solve hard problems; and they work with other people. The ideal configuration for a problem set is a single partner who is at roughly your ability and is willing to meet earlier in the week.
Meet with this partner for 2 – 3 hours to discuss progress made so far. Check your answers on the easy problems. Trade insights on the hard problems. Make new, collaborative attacks on those that still resist solving.
Step 4: Finalize the Problem Set at Office Hours
Show up early to office hours. Arrive understanding exactly why you are stuck on the small number of problems (hopefully) on which you are still stuck. Translate this into a small number of highly specific questions. Ask the TA these questions right after he or she arrives. The key here — and I base this on my own TA experience — is to avoid simply saying: “I don’t know how to do this problem, help!” That’s frustrating. Instead, you need targeted information that shows the effort you’ve expended. For example: “I’ve been trying approach XX, it’s promising, but I keep getting stuck with YY, can you point me in the right direction?”
Bring your laptop to office hours and work on finalizing these problems right there. If small questions or ambiguities pop up as you make progress, the TA can be asked on the spot. Aim to leave office hours with a completed problem set. Notice, this is much different from most students who arrive at office hours with very little done. You are arriving with most of the work done, and are just filling in the details.
In Conclusion
Repeated fresh attacks are how hard problems are solved in the real world. Problem sets teach you this skill. The issue, however, is that professors often forget to convey this strategy to their students, many of whom still believe that the high school style, big push tactic for finishing work is still applicable. So keep this advice in mind. Until you’ve approached a problem fresh, 3 – 4 times, you haven’t really yet tried to solve it.
Taking a Math Test
• First look over the entire test. You'll get a sense of its length. Try to identify those
problems you definitely know how to do right away, and those you expect to have
to think about.
• Do the problems in the order that suits you! Start with the problems that you know for
sure you can do. This builds confidence and means you don't miss any sure points just
because you run out of time. Then try the problems you think you can figure out; then
finally try the ones you are least sure about.
• Time is of the essence - work as quickly and continuously as you can while still
writing legibly and showing all your work. If you get stuck on a problem, move
on to another one - you can come back later.
• Work by the clock. On a 50 minute, 100 point test, you have about 5 minutes for a
10 point question. Starting with the easy questions will probably put you ahead
of the clock. When you work on a harder problem, spend the allotted time (e.g., 5 minutes)
on that question, and if you have not almost finished it, go on to another problem.
Do not spend 20 minutes on a problem which will yield few or no points when there are
other problems still to try.
• Show all your work: make it as easy as possible for the Instructor to see how much
you do know. Try to write a well-reasoned solution. If your answer is incorrect, the
Instructor will assign partial credit based on the work you show.
• Never waste time erasing! Just draw a line through the work you want ignored and
move on. Not only does erasing waste precious time, but you may discover later that
you erased something useful
(and/or maybe worth partial credit if you cannot complete the problem). You are
(usually) not required to fit your answer in the space provided - you can put your
answer on another sheet to avoid needing to erase.
• In a multiple-step problem outline the steps before actually working the problem.
• Don't give up on a several-part problem just because you can't do the first part.
Attempt the other part(s) - if the actual solution depends on the first part, at least
explain how you would do it.
• Make sure you read the questions carefully, and do all parts of each problem.
• Verify your answers - does each answer make sense given the context of the problem?
• If you finish early, check every problem (that means rework everything from scratch).
General tips for math courses
Take notes on sample problems, answers and explanations. Take notes on the teacher’s explanations – the big picture (put it in your own words so you are sure you understand it). Engage with the material and try to understand it while still in class.
If you don’t understand something after looking at your notes and textbook GOOGLE it (Math World & Wikipedia have clear explanations for most problems)
When reviewing the material, try to re-create the steps, answers and especially the underlying concepts recorded in your notes OUT LOUD as if you were teaching a class.
A Four Step Process for Solving Hard Problem Sets
The motivating idea behind this strategy is simple: your brain can only work productively on a hard problem for 1 -3 hours before needing to reboot. To reboot your brain, so more productive work can be accomplished, requires a significant break. Preferably overnight.
Here’s a four step strategy built around this idea.
Step 1: Pick Off the Simple, Prime the Hard
Your first block of work should occur early in the week. Set aside 2 – 3 hours, in the morning. Make this the first thing you do that day (when your energy is at its highest). Your goal is two-fold. First, you want to solve easy problems. Your strong focus will help you avoid stupid mistakes. Second, you want to tackle at least two hard problems. You probably won’t solve them. This is why they are hard. But you can do something almost as important: prime them.
To prime a hard problem is to discover exactly why you can’t solve it. Pick an obvious approach — even if you suspect it won’t work — and start working through the problem until you get stuck. Identify why you are stuck. Ask what you need to figure out to make progress. What is it that makes this hard? Then take a break…
Step 2: Think in the Shower
For the next 2 – 3 days, think about how to get around the obstacles you discovered while priming. Don’t do this formally, in the library, with books around you. Instead, do this while walking around campus. While waiting for class to start. In the shower. I used to solve my Algorithms take home exam problems, for example, while jogging.
This is when breakthroughs occur. If you end up with a great insight, take 20 minutes, next time you can spare it, to sit down and write it down formally. If needed, prime a new hard problem so you can keep making progress as your wander campus throughout the week.
If you encounter ambiguities in the problem description that are giving you trouble, send concise questions to your TA requesting clarification. You don’t want these details to slow down progress any longer than they need to. (You might end up e-mailing your TA many times early in the week. This is okay so long as the questions are specific and concise. Don’t wait until office hours. By then, it’s too late.)
Step 3: Meet with your Problem Partner
A team effort is crucial for problem sets. But it has to be the right effort. Don’t meet with a large group. These are rarely efficient. Most of the time is spent griping about the class. Usually, there is one kid in the group who actually did the work, and, in the end, everyone copies off of him. Avoid this. The “smart kid” is often wrong, and likes the group because it boosts his self-esteem. Not to mention that your lack of understanding will come back to tag you on the exam.
The other extreme is to work alone. I see this a lot at MIT. Too many movies like Good Will Hunting got people thinking that to be smart at math means you should be able to stare at a problem for 5 – 10 seconds and then instantly solve it. Sorry. Doesn’t work that way. I walk past real geniuses every day — people, for example, who are my age and are also tenured professors — and guess what: it takes them a long time to solve hard problems; and they work with other people. The ideal configuration for a problem set is a single partner who is at roughly your ability and is willing to meet earlier in the week.
Meet with this partner for 2 – 3 hours to discuss progress made so far. Check your answers on the easy problems. Trade insights on the hard problems. Make new, collaborative attacks on those that still resist solving.
Step 4: Finalize the Problem Set at Office Hours
Show up early to office hours. Arrive understanding exactly why you are stuck on the small number of problems (hopefully) on which you are still stuck. Translate this into a small number of highly specific questions. Ask the TA these questions right after he or she arrives. The key here — and I base this on my own TA experience — is to avoid simply saying: “I don’t know how to do this problem, help!” That’s frustrating. Instead, you need targeted information that shows the effort you’ve expended. For example: “I’ve been trying approach XX, it’s promising, but I keep getting stuck with YY, can you point me in the right direction?”
Bring your laptop to office hours and work on finalizing these problems right there. If small questions or ambiguities pop up as you make progress, the TA can be asked on the spot. Aim to leave office hours with a completed problem set. Notice, this is much different from most students who arrive at office hours with very little done. You are arriving with most of the work done, and are just filling in the details.
In Conclusion
Repeated fresh attacks are how hard problems are solved in the real world. Problem sets teach you this skill. The issue, however, is that professors often forget to convey this strategy to their students, many of whom still believe that the high school style, big push tactic for finishing work is still applicable. So keep this advice in mind. Until you’ve approached a problem fresh, 3 – 4 times, you haven’t really yet tried to solve it.
Taking a Math Test
• First look over the entire test. You'll get a sense of its length. Try to identify those
problems you definitely know how to do right away, and those you expect to have
to think about.
• Do the problems in the order that suits you! Start with the problems that you know for
sure you can do. This builds confidence and means you don't miss any sure points just
because you run out of time. Then try the problems you think you can figure out; then
finally try the ones you are least sure about.
• Time is of the essence - work as quickly and continuously as you can while still
writing legibly and showing all your work. If you get stuck on a problem, move
on to another one - you can come back later.
• Work by the clock. On a 50 minute, 100 point test, you have about 5 minutes for a
10 point question. Starting with the easy questions will probably put you ahead
of the clock. When you work on a harder problem, spend the allotted time (e.g., 5 minutes)
on that question, and if you have not almost finished it, go on to another problem.
Do not spend 20 minutes on a problem which will yield few or no points when there are
other problems still to try.
• Show all your work: make it as easy as possible for the Instructor to see how much
you do know. Try to write a well-reasoned solution. If your answer is incorrect, the
Instructor will assign partial credit based on the work you show.
• Never waste time erasing! Just draw a line through the work you want ignored and
move on. Not only does erasing waste precious time, but you may discover later that
you erased something useful
(and/or maybe worth partial credit if you cannot complete the problem). You are
(usually) not required to fit your answer in the space provided - you can put your
answer on another sheet to avoid needing to erase.
• In a multiple-step problem outline the steps before actually working the problem.
• Don't give up on a several-part problem just because you can't do the first part.
Attempt the other part(s) - if the actual solution depends on the first part, at least
explain how you would do it.
• Make sure you read the questions carefully, and do all parts of each problem.
• Verify your answers - does each answer make sense given the context of the problem?
• If you finish early, check every problem (that means rework everything from scratch).