I decided to mix things up from my 100 Days of Productivity posts to write an advice post instead! I’ve been reading the book “How to Solve It” by George Polya (which I would definitely recommend!) and this is a post summarising some of the techniques given in there on how to solve problems. I’ve also added some advice that I have found helpful in the past. Hope you enjoy!
The basic idea is that problem solving can be split into 4 stages:
Understanding
Creating a plan
Executing the plan
Reviewing the solution
In this post I’m going to go through how to approach each stage and then give an example of the technique being used on an actual problem.
Understanding
Before you can even start thinking about solving a problem you need to make sure you fully understand what it is asking.
Some questions to ask yourself at this stage are:
What data is given?
What are the unknowns?
What is the condition?
It might also help to try rephrasing the problem in your own words.
If it’s appropriate for the given problem, I cannot emphasis enough how helpful it is to DRAW A DIAGRAM.
If you don’t understand all the terms or symbols in the question, it’s really important that you look them up at this stage or there is no way you’ll be able to progress with the problem!
Creating a plan
So this is definitely the most difficult stage and most likely the part you’ve been stuck on in the past.
Some questions to ask yourself at this stage are:
Do you know of a related problem?
Can you think of a related problem having the same or similar unknown?
If you cannot solve the proposed problem can you think of a simpler related problem?
Did you use all the data?
Did you use the whole condition?
Do you know an analogous problem?
Hopefully these questions will help you make progress. However, if they don’t, the best thing to do at this stage is to take a break and work on something else. This allows you to come at the problem afresh and more often than not make a breakthrough with the problem. Ideally you should leave the problem until the next day, but if you’re pushed for time even just going for a 5 minute walk can help you clear your head and stop going round in circles with the problem.
Executing the plan
Once you’ve got a plan, this stage is relatively simple - just make sure you don’t skip any steps and check your work frequently to avoid errors!
Reviewing the solution
This stage is often overlooked since it’s much more interesting to move onto the next problem. However, it is really important for consolidating learning and improving your problem solving skills.
Firstly, you should check your answer. The best way to do this is a “sanity check” i.e. does the answer seem reasonable? For example, if you get an answer that indicates a car is travelling at 2000 km/h you have clearly gone wrong somewhere. If you are using variables, try inputting some values into your solution and see if it seems reasonable. It may also be useful to work through each step again paying particular attention to any steps you found difficult.
Secondly, you should consider the implications of the result - this stage will help you when solving similar problems in the future. You might want to consider some real world applications of the results or similar problems that could be solved in the same way to reinforce what you have just learnt.
Some questions to ask yourself at this stage are:
Can you check the result?
Can you check the argument?
Can you see the answer at a glance?
Could you solve the problem differently?
Did you use all the data?
Can you use the result for some other problem?
Let’s look at an example of the technique in practice!
22.1.16 || 22/100 Days of Productivity Somehow managed to drag myself out of the house to the library today. Attempting to get to grips with compiler construction - although having talked to the other people on my course at least I'm not the only one struggling. Just got to keep at it 💪.
20.1.16 || 20/100 Days of Productivity So nice to get a change of scenery for studying 😊. Looking forward to going to a formal this evening but need to make sure I get all my work done before then.
I decided to mix things up from my 100 Days of Productivity posts to write an advice post instead! I’ve been reading the book “How to Solve It” by George Polya (which I would definitely recommend!) and this is a post summarising some of the techniques given in there on how to solve problems. I’ve also added some advice that I have found helpful in the past. Hope you enjoy!
The basic idea is that problem solving can be split into 4 stages:
Understanding
Creating a plan
Executing the plan
Reviewing the solution
In this post I’m going to go through how to approach each stage and then give an example of the technique being used on an actual problem.
Understanding
Before you can even start thinking about solving a problem you need to make sure you fully understand what it is asking.
Some questions to ask yourself at this stage are:
What data is given?
What are the unknowns?
What is the condition?
It might also help to try rephrasing the problem in your own words.
If it’s appropriate for the given problem, I cannot emphasis enough how helpful it is to DRAW A DIAGRAM.
If you don’t understand all the terms or symbols in the question, it’s really important that you look them up at this stage or there is no way you’ll be able to progress with the problem!
Creating a plan
So this is definitely the most difficult stage and most likely the part you’ve been stuck on in the past.
Some questions to ask yourself at this stage are:
Do you know of a related problem?
Can you think of a related problem having the same or similar unknown?
If you cannot solve the proposed problem can you think of a simpler related problem?
Did you use all the data?
Did you use the whole condition?
Do you know an analogous problem?
Hopefully these questions will help you make progress. However, if they don’t, the best thing to do at this stage is to take a break and work on something else. This allows you to come at the problem afresh and more often than not make a breakthrough with the problem. Ideally you should leave the problem until the next day, but if you’re pushed for time even just going for a 5 minute walk can help you clear your head and stop going round in circles with the problem.
Executing the plan
Once you’ve got a plan, this stage is relatively simple - just make sure you don’t skip any steps and check your work frequently to avoid errors!
Reviewing the solution
This stage is often overlooked since it’s much more interesting to move onto the next problem. However, it is really important for consolidating learning and improving your problem solving skills.
Firstly, you should check your answer. The best way to do this is a “sanity check” i.e. does the answer seem reasonable? For example, if you get an answer that indicates a car is travelling at 2000 km/h you have clearly gone wrong somewhere. If you are using variables, try inputting some values into your solution and see if it seems reasonable. It may also be useful to work through each step again paying particular attention to any steps you found difficult.
Secondly, you should consider the implications of the result - this stage will help you when solving similar problems in the future. You might want to consider some real world applications of the results or similar problems that could be solved in the same way to reinforce what you have just learnt.
Some questions to ask yourself at this stage are:
Can you check the result?
Can you check the argument?
Can you see the answer at a glance?
Could you solve the problem differently?
Did you use all the data?
Can you use the result for some other problem?
Let’s look at an example of the technique in practice!
Our problem is this: Find the diagonal of a cuboid given its height, width and length.
Understanding
This problem clearly calls for a diagram:
We are given the height, width and length which are labelled as w, h, l on the diagram and we are trying to find the diagonal labelled d.
The diagram now makes the problem much clearer and maybe even gives us a hint as to how to solve the problem.
Creating a plan
So now what?
First we ask ourselves: do we know of any similar problems? After a few minutes of thinking we decide we haven’t seen any problems like this before.
Okay, could we solve a simpler problem? We think for a few more minutes and are still stuck so decide to go and make a cup of tea. When we get back we suddenly have the idea that it would be fairly easy to find the diagonal of one of the faces using Pythagoras’ theorem!
c²=√(a²+b²)
We can then work out that the length of the diagonal along the bottom face from front right to back left is √(w²+l²) which we write on the diagram. Suddenly we see the solution to the original problem - we can apply Pythagoras’ theorem again with this intermediate result!
Executing the plan
It would be easy to get carried away and skip steps now but make sure to include all your working!
We have: d² = h² + (√(w²+l²))²
= h² + w² + l²
So d = √(h² + w² + l²)
Reviewing the solution
Okay lets do a quick sanity check of the result. If we let w=3cm, h=4cm and l=5cm roughly what value would you expect for d? So from the diagram it looks like it should be longer than l but not by too much - therefore we probably expect d to be between about 5cm and 10cm. When we type this into a calculator we get d roughly equal to 7.1cm which is within our estimate - therefore it seems reasonably likely we got the correct answer!
How might we apply this result? One obvious real-world application of the result would be just to figure out the length of the diagonal of the room we’re sitting in. If we take the result further we could figure out the length of the diagonal from one corner of a cube to its centre. Now consider erecting a flag pole in the centre of a rectangular roof with ropes joining the top of the pole to each corner of the roof. We now have the ability to calculate the length of the ropes given the size of the roof by considering the top of the flagpole as the centre of a cuboid!
Okay well this has been a long post but that’s about it! Hopefully this has helped someone - please let me know if it has!
18.1.16 || 18/100 Days of Productivity Finally finished all my work I should have done over Christmas!!! Such a relief - I can actually focus on the new term's work now. Time for a run now and to start looking over this term's lecture notes 😊.
15.1.16 || 15/100 Days of Productivity Did graphics supervision work this morning and notes for my logic and proof module this afternoon (which I am loving so far 😊). Then went for a cute coffee date with the bf before dinner. Need to make some notes on networking this evening and then time for a film with my housemates 😊.
14.1.16 || 14/100 Days of Productivity Making notes on computation theory and networking at the library this morning (even though I'm really hungover 😓). I might make a post on how I take notes if anyone is interested?
13.1.16 || 13/100 Days of Productivity New term = fresh start and an excuse to buy more stationery 😊. Starting my new notebook for this term and making notes on computer networks this morning. Got a reply to one of my internship applications this morning and have been invited for interview! Only problem is that it is in Glasgow (I'm in Cambridge atm for some perspective on distance) during the busiest week of term 😬. Not sure how that's going to work... Anyway hope everyone's having a good start to the new term!
12.1.16 || 12/100 Days of Productivity Went to a coffee shop with a friend this morning and made C/C++ notes. Mock exam this afternoon then I can chill this evening 😊
11.1.16 || 11/100 Days of Productivity Feeling those ~minimalist study vibes~ today. Revising pointers for C/C++ supervision work and then heading to the gym this afternoon 💪
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Hello! I explained this briefly in a masterpost I just did, but I wanted to expand on it a little more! If you’re struggling to find a study method that works for you, or you want to find a way to study for exams, give this a try! Thanks to this method I earned a 95.5% on my biology final/98.8 in the class (and science is a difficult subject for me) but once I discovered this method, the class became so much easier!
First, create a study guide. Do this by writing out the main points and sub-points from your notes/textbook in a few sentences.
So for example, the main point might be “Shakespeare is influential” and then subpoints might be “bc of his use of language” “bc of his play’s societal reflections”, etc!
I recommend doing subpoints in bullet points under the main point!
When you study, this will be a reference, don’t just read the study guide! The guide she be very brief, only a sentence or a cluster of words for each point!
Now you should have a comprehensive list of what you need to know for your exam. If you do this a week at a time, it will be way less daunting than if you have 30 chapters to map out, I promise!
Now, read through your notes, textbook, watch videos, etc! Spend a decent amount of time doing this to refresh it in your mind. This is the point where you should get any questions answered/make sure you understand the material.
Next, return to the study guide you made. Use what you remember (and your notes and other resources, if necessary) to write a paragraph or two expanding on each concept/topic. Explain why you’re correct, and rephrase what you know in different ways. Literally just write as much as you remember! If you’re stumbled, use your class notes or book to help you (is use video lectures to better understand topics).
If you explain something thoroughly in your own words, you can answer questions about it!
When you finish, go back and compare your guide to your class notes to make sure you didn’t leave out anything important.
Put your knowledge to the test! Practice tests are proven to be the MOST effective way to study, so tackle the end-of-chapter questions in your textbook, again, not only giving the answer, but explaining why you’re correct.
Once you get confident, go back to your guide that has just the topics on it, and this time speak the answers. Just like a teacher would teach it, explain it to yourself!
This method takes a decent amount of time, but it’s been really, really effective for me, and it can help prevent you from wasting hours simply rewriting the same notes over and over. Of course, do what works best for you and your class! But if you are looking for something new to try, I believe this method is a good option. Feel free to send me an ask if you have any questions!
My other masterposts:
How to Earn Higher Grades
Setting Study Goals
Top 10 Study Tips
Math Help (part one)
Tips for Finals
Tips for Presentations
Biology Advice
Improving Your Study Habits
9.1.16 || 9/100 Days of Productivity Finished off my C/C++ programming practical this morning and can hopefully get this semantics finished off this afternoon 💪. Might head to the gym after that and then am watching Star Wars with the bf this evening 😊
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