Is this exam question really too difficult?

[sk1105]

A lot of school students in the UK are making quite a fuss at the moment about a question that appeared on a recent GCSE maths exam (GCSE exams are taken in the UK by 15/16-year-old students). The discussion has since spawned its own Twitter hashtag, a variety of memes and has been covered by national newspapers. Only, I don't really see what the problem is. The question is as follows:

Hannah has a bag containing $n$ sweets. Six of these are orange and the rest are yellow. She takes a sweet from the bag at random and eats it, before choosing at random another sweet. The probability that she takes 2 orange sweets is 1/3. Show that $n^2-n-90=0$.

Note that there is no mention of finding the actual value of $n$.

I was able to draw a tree diagram and solve it in about 3 lines of algebra, although being an undergraduate I realize that this might not be a fair comparison. However, I remember being taught at GCSE level that the sort of set-up given above strongly lends itself towards drawing a tree diagram; after doing that the required quadratic falls out easily.

Of course there may be those who were perhaps intimidated by the seemingly unconnected statements in the question and this distracted them from just applying the techniques they were taught, but aren't questions like this specifically intended to distinguish the more gifted students? I get the feeling from the reaction that a lot of students almost felt cheated that such a challenging outside-the-box question had been included. Without wanting to open too large a can of worms that perhaps belongs in its own thread, could this be indicative of the idea being put forward by some (this interview is an example) that children are feeling more and more entitled to quick success without putting in the hours?

 

[pasmith]

A lot of school students in the UK are making quite a fuss at the moment about a question that appeared on a recent GCSE maths exam (GCSE exams are taken in the UK by 15/16-year-old students). The discussion has since spawned its own Twitter hashtag, a variety of memes and has been covered by national newspapers. Only, I don't really see what the problem is. The question is as follows:

Hannah has a bag containing $n$ sweets. Six of these are orange and the rest are yellow. She takes a sweet from the bag at random and eats it, before choosing at random another sweet. The probability that she takes 2 orange sweets is 1/3. Show that $n^2-n-90=0$.

Note that there is no mention of finding the actual value of $n$.

Here is a picture of the actual question. Note that "show n^2 - n - 90 = 0" is merely part (a). For all we know, part (b) is "Hence find n." This is the sort of hand-holding GCSE questions provide; at a higher level the question might be simply

Hannah has a bag containing $n$ sweets. Six of these are orange and the rest are yellow. She takes a sweet from the bag at random and eats it, before choosing at random another sweet. The probability that she takes 2 orange sweets is 1/3. Find n.

 

[Vanadium 50]

Where is it supposed to be hard?

P = \frac{6}{n} \frac{5}{n-1} = \frac{1}{3}

simplify

P = \frac{30}{n(n-1)} = \frac{30}{n^2-n} = \frac{1}{3}

Cross-multiply

90 = n^2- n

and rearrange

n^2- n - 90 = 0

 

[Hepth]

There was a math prize thing that I won our school in in Michigan in high school that had problems much harder than this. Heres a random example:

http://math.hope.edu/mmpc/graphics/exams/1999.II.problems.pdf

Honestly I don't know how I even got some of them right looking back.

edit: The domino one I still don't get how to "prove"...

 

[ShayanJ]

Where is it supposed to be hard?

P = \frac{6}{n} \frac{5}{n-1} = \frac{1}{3}

simplify

P = \frac{30}{n(n-1)} = \frac{30}{n^2-n} = \frac{1}{3}

Cross-multiply

90 = n^2- n

and rearrange

n^2- n - 90 = 0

The problem doesn't lie in the solution you presented, but in the path that the students should undertake to reach the conclusion that this is the way to solve it. I don't know enough about UK educational system, but the fact that such an issue made its way to the media, persuades me that I can safely assume what I'm going to say is correct. The point is, the students are always given clear questions when no interpretation is needed. Its always clear what to do. As the OP mentioned:

Note that there is no mention of finding the actual value of n.

The problems the students were given were always as such. "Find n", "Find the probability of such and so", "Find the value of the expression". Always clear questions where a clear algorithm for solving them was taught to them. But this question suddenly, with no preparations, asks the students to first interpret the question correctly, then find the correct algorithm for solving it. This is the problematic part of the question which made it so controversial. Of course there is nothing wrong with the question itself, its just that the educational system didn't prepare students for such a question. But I don't want to talk about whose fault it is.

 

[PietKuip]

The problems the students were given were always as such. "Find n", "Find the probability of such and so", "Find the value of the expression". Always clear questions where a clear algorithm for solving them was taught to them. But this question suddenly, with no preparations, asks the students to first interpret the question correctly, then find the correct algorithm for solving it. This is the problematic part of the question which made it so controversial. Of course there is nothing wrong with the problem itself, its just that the educational system didn't prepare students for such a question. But I don't want to talk about whose fault it is.

But school should prepare for the unexpected. Students should be able to interpret a short question like this.

 

[ShayanJ]

But school should prepare for the unexpected. Students should be able to interpret a short question like this.

Of course students should be able to solve such a problem without seeing it before. The problem is, some part of their learning process didn't work well to provide them such an insight.

 

[Intrastellar]

I know someone who did this exam. He was very surprised to know that this question went viral because, according to him, it was not the most difficult question in the exam.
The people who were complaining about this question were mainly complaining about the wording: "Hannah ate some sweets, prove this equation" as can be seen from this link https://twitter.com/hashtag/EdexcelMaths?src=hash
I could be wrong, but I highly doubt that those who were complaining have successfully answered all the other questions in this exam. The wording of this question gave the weaker students something to complain about.

The second part of the question asks the students to solve this equation to find the value of n.

 

[Vanadium 50]

If the position is "I only do cookbook problems, and I haven't see this before", I don't have much sympathy. Understanding implies being able to use information in new ways, and a student who can do this problem should get more points than someone who can't. If the complaint was that they didn't understand the English, I have some sympathy, since English and Math (sorry, Maths) should be decoupled. But only some. "My score was low because I am illiterate" is not the most powerful defense.

Given that there is exactly one piece of information given, and exactly one way to express it in a formula, this doesn't appear to me to be unfairly difficult. Yes, you need to know some probability to set up the equation, but I don't think it's unreasonable to ask a probability question.

I don't see why this is particularly difficult.

 

[sk1105]

For all we know, part (b) is "Hence find n."

Yes it turns out that was part (b), however since part (a) is a "show that..." question, its solution is not necessary for the second part, and the complaints have been directed towards part (a). I entirely see your point about hand-holding though, and I agree your suggested wording of the question would represent a higher level.

Of course there is nothing wrong with the question itself, its just that the educational system didn't prepare students for such a question.

I sort of agree with you, but I think a lot of the complaints came from a sense of injustice that students had been taught the skills relevant to probability and algebra, but had not been prepared explicitly for questions of this type. It takes a certain degree of academic maturity to take a step back and think "OK, this is a new context but still a probability question. What techniques do I know that can be useful here?". It seems that a lot of 16-year-olds were not able to do this (although selection bias is likely to be an issue), but they should bear in mind that if there were no questions beyond the reach of most students then everyone would receive the highest grades and the exam would be pointless.

Mind you, it could be argued that schools themselves are relying too much on the formulaic hand-holding type questions, when really they should be teaching students to recognise when to apply certain techniques. Essentially I think it comes down to a mixture of the framework of GCSE exams, the way the syllabus is taught, and the students' attitudes, but having quoted Shyan's post I will comply with their request by not trying to apportion levels of blame to each factor!

 

[ShayanJ]

Yes it turns out that was part (b), however since part (a) is a "show that..." question, its solution is not necessary for the second part, and the complaints have been directed towards part (a). I entirely see your point about hand-holding though, and I agree your suggested wording of the question would represent a higher level.
I sort of agree with you, but I think a lot of the complaints came from a sense of injustice that students had been taught the skills relevant to probability and algebra, but had not been prepared explicitly for questions of this type. It takes a certain degree of academic maturity to take a step back and think "OK, this is a new context but still a probability question. What techniques do I know that can be useful here?". It seems that a lot of 16-year-olds were not able to do this (although selection bias is likely to be an issue), but they should bear in mind that if there were no questions beyond the reach of most students then everyone would receive the highest grades and the exam would be pointless.

Mind you, it could be argued that schools themselves are relying too much on the formulaic hand-holding type questions, when really they should be teaching students to recognise when to apply certain techniques. Essentially I think it comes down to a mixture of the framework of GCSE exams, the way the syllabus is taught, and the students' attitudes, but having quoted Shyan's post I will comply with their request by not trying to apportion levels of blame to each factor!

I'm not sure how my post was received so I should explain that I wasn't defending the complaining students. I was just explaining the reason behind such complaints. Its true that schools don't seem to do a good job in teaching kids to think properly instead of just injecting some maths, physics, etc. in their heads. But it doesn't mean students can say "OK, its schools' fault so we have the right to complain in such situations".
And I completely agree with the fact that in such a situation, these kind of questions can decide between average/good students and students who actually know what they're learning and so these question should be there so that the latter students get higher grades.

 

[JorisL]

[...]
It takes a certain degree of academic maturity to take a step back and think "OK, this is a new context but still a probability question. What techniques do I know that can be useful here?".

This is in part a problem with teaching methods in my opinion.
A couple weeks ago my old high school teacher made the news, apparently he's making videos (for his students) since 4 years for two reasons.

• One reason is that since he teaches students with 8 hours of maths a week (38 hours a week) so they get quite far ( basic calculus I without the rigour ), a lot of parents aren't able to help/discuss their homework so he perceived it as a vacuum. Getting the explanation again certainly helped.
• The second reason is that this allowed him to introduce a flipped class system. The students would study appropriate parts of the book by themself/in groups. And prepare a little project. In my time he did this for volumes of revolution. Here he would introduce the topic through a video and start with some exercises prior to assigning the project.

The second part is what I think we should aim at even in high school. (this is my opinion)
It is certainly possible with the wealth of information at ones fingertips (thanks to Google and forums like this) we have on the internet.
It teaches one self-study but also how to apply the knowledge.

About the commotion; people always look for excuses. I say this is one of those cases.
Secondly life would be boring if we wouldn't complain often enough(national sport in Belgium)

 

[StevieTNZ]

Have any Maths teachers said anything about the question? All I've heard are reports of some students being dis-satisfied with the question.

 

[sk1105]

Those are the only reports I've seen as well. I think the media have catastrophically over-hyped this, with headlines such as "The 'impossible' maths question solved". Indeed, the fact that the reports I've seen all focused mainly on the students' Twitter posts and did not even mention teachers or Ofqual (England's exams regulator) suggests that this has been sensationalised to create a story where there isn't/shouldn't be one. After all, I don't know about other countries but the UK media do love to whip up a scandal.

 

[russ_watters]

I'm really not following -- do the students complain about something specific that they find unfair about the question or is it just vague "it's too hard" complaints?

 

[Vanadium 50]

According to the BBC it was so hard that it made them want to cry. At least one claimed to be traumatized.

In the US, this might be called a "microaggression".

 

[sk1105]

I'm really not following -- do the students complain about something specific that they find unfair about the question or is it just vague "it's too hard" complaints?

I think the motivation behind the initial complaints was specific - "Hannah has some sweets...show this equation" seemed like too large a leap in reasoning for a lot of people. But it's also fairly probable that a large number of students just couldn't do the question but haven't thought about precisely why, so they made the generic "it's too hard" complaints. I can perhaps understand the specific complaints, notwithstanding the fact mentioned above that there must be some questions beyond the reach of all but the top achievers, but the vague complaints I think reflect the sense of wanting things presented on a plate that has been discussed above.

According to the BBC it was so hard that it made them want to cry. At least one claimed to be traumatized.

In the US, this might be called a "microaggression".

Not being from the US, I had to Google 'microaggression'. I can completely see that those students who couldn't do the question might feel persecuted by the exam board. It's a widely known stereotype that teenagers will exaggerate the consequences of unfortunate events with particular emphasis on perceived oppression and "me vs. the world" (not long out of the teenage years myself I can certainly remember being guilty of this!). Perhaps that is a factor here, especially given the use of the word 'traumatised'.

 

[f95toli]

But school should prepare for the unexpected. Students should be able to interpret a short question like this.

If the position is "I only do cookbook problems, and I haven't see this before", I don't have much sympathy. Understanding implies being able to use information in new ways, and a student who can do this problem should get more points than someone who can't.

But -as has already been pointed out above- most schools are teaching in a way that is extremely focused on doing well on the exam. The goal is to get an A* on the exam,. not to actually learn anything (and in some subjects -such as Latin- this is more or less explicitly stated by the teachers)
Hence, there is a LOT of focus on rote learning and doing problems from previous exams, meaning even good student (i.e. someone who who is well prepared) will not expect to see anything unfamiliar in an exam. My step-son is in year nine and is very good at maths (best in his year in a good school) and we have therefore repeatedly asked his teacher go give him more challenging work. However, her response is that he should only focus on what will be included in the exam, i.e. he should just keep doing the same type of problem over and over again until his exam (which will be in year 11)...

This situation has arisen partly because of the emphasis on "accountability" (i.e. schools and teachers have to do well, or face serious trouble) which is to a large extent based on exam results, but also because of grade inflation: students now need very good grades to get into an OK university, and they are repeatedly told from an early age how important it is get to into a Russell group university if they want to pursue any form of "serious" career.
This has also resulted in a situation where many the students who do get into these universities are woefully unprepared despite having very good grades. I have friends who are admissions tutors at "mid-level" universities and they are very unhappy with the quality of students that are coming through the current system.

 

[ShayanJ]

But -as has already been pointed out above- most schools are teaching in a way that is extremely focused on doing well on the exam. The goal is to get an A* on the exam,. not to actually learn anything (and in some subjects -such as Latin- this is more or less explicitly stated by the teachers)
Hence, there is a LOT of focus on rote learning and doing problems from previous exams, meaning even good student (i.e. someone who who is well prepared) will not expect to see anything unfamiliar in an exam. My step-son is in year nine and is very good at maths (best in his year in a good school) and we have therefore repeatedly asked his teacher go give him more challenging work. However, her response is that he should only focus on what will be included in the exam, i.e. he should just keep doing the same type of problem over and over again until his exam (which will be in year 11)...

This situation has arisen partly because of the emphasis on "accountability" (i.e. schools and teachers have to do well, or face serious trouble) which is to a large extent based on exam results, but also because of grade inflation: students now need very good grades to get into an OK university, and they are repeatedly told from an early age how important it is get to into a Russell group university if they want to pursue any form of "serious" career.
This has also resulted in a situation where many the students who do get into these universities are woefully unprepared despite having very good grades. I have friends who are admissions tutors at "mid-level" universities and they are very unhappy with the quality of students that are coming through the current system.

I'm curious about the situation in other countries. Now because most of the people here are from US(I don't mean in terms of the fraction of all members, but in terms of people who usually contribute to such threads), is there a similar problem in US too?(To anyone living in US who knows the educational system from early years)

 

[mesa]

The n^2-n-90=0 actually threw me off. The question simply should have read, "..., how many candies were there?".

 

[Orodruin]

My step-son is in year nine and is very good at maths (best in his year in a good school) and we have therefore repeatedly asked his teacher go give him more challenging work. However, her response is that he should only focus on what will be included in the exam, i.e. he should just keep doing the same type of problem over and over again until his exam (which will be in year 11)

This honestly makes me want to cry. If he is already done with a certain material early, it means it was likely very simple for him. Forcing him to do the same simple thing over and over is awful for development. If he would be allowed to develop, the easier things would still be easier but he would be stimulated and learn things on a deeper level. That is how you encourage people to learn, not by boring them to death.

I'm curious about the situation in other countries. Now because most of the people here are from US(I don't mean in terms of the fraction of all members, but in terms of people who usually contribute to such threads), is there a similar problem in US too?(To anyone living in US who knows the educational system from early years)

There are problems like this all over the world. A colleague told me some time back that on a Swedish standardised test, it is not enough to provide a counter example when asked to prove something false to get the top grade on the problem ...

 

[Silicon Waffle]

Have any Maths teachers said anything about the question? All I've heard are reports of some students being dis-satisfied with the question.

Then the teacher can hold a pool party at which all the dissatisfied kids are invited to enjoy themselves. This approach works best as sort of a teamwork or team building activity in adult world to lessen conflicts among individuals. Kids will forget soon and learn that bad grades or unfairness means nothing in this world.

 

[ShayanJ]

This honestly makes me want to cry. If he is already done with a certain material early, it means it was likely very simple for him. Forcing him to do the same simple thing over and over is awful for development. If he would be allowed to develop, the easier things would still be easier but he would be stimulated and learn things on a deeper level. That is how you encourage people to learn, not by boring them to death.There are problems like this all over the world. A colleague told me some time back that on a Swedish standardised test, it is not enough to provide a counter example when asked to prove something false to get the top grade on the problem ...

I think the problem is that, actual scientists and researchers (should I blame you too?) don't bother themselves about making their country's educational system better. Or maybe some of them bother, but authorities don't listen! Anyway, it seems a bad idea that the books written to educate kids and the tests to evaluate their learning, are composed by people who don't have much to do with science.
It seems like many other(almost all) problems of the today's world. We need more responsibility and care from people who can do something.

 

[JorisL]

I think the problem is that, actual scientists and researchers (should I blame you too?) don't bother themselves about making their country's educational system better. Or maybe some of them bother, but authorities don't listen! Anyway, it seems a bad idea that the books written to educate kids and the tests to evaluate their learning, are composed by people who don't have much to do with science.
It seems like many other(almost all) problems of the today's world We need more responsibility and care from people who can do something.

Probably this happened and as a result scientists don't bother any more (except if there's tv shows and popularisation involved).
Try explaining to a government system with 98+% of the people in charge that the current system is seriously flawed and fixing it will cost xx amount of money.
They'll ignore it, I guarantee it.

 

[dragoneyes001]

how many questions involve situations? train A leaves ###### @ 3 pm going 100 km/h while train B...etc... how would Hanna eating sweets be any different?

any student who found the question difficult beyond not knowing the math needed to solve the question are academic failures. I won't reduce the language because its true the entire point of education is to use what you learn in situations you will encounter. put simply learning to count means you can understand the change you receive at a store.

 

[collinsmark]

The n^2-n-90=0 actually threw me off. The question simply should have read, "..., how many candies were there?".

I just have to ask then. How would you go about solving for n such that you don't end up with n^2 - n - 90 = 0, or something very similar to it, as an intermediate part of the solution?

I suppose you could just start trying out different numbers and see if they work, and maybe you'll eventually get lucky and pick the number n = 10 to show the probability is 1/3. But what deterministic way would you use to solve for n in such a way that n^2 - n - 90 = 0, or something very similar to it, doesn't come up in the process?

[Edit: I ask because when I solved for n, the n^2 - n - 90 = 0 equation naturally popped out. Is there some other straightforward way to solve this problem where it doesn't? (I suppose one might use a geometric approach with a compass and straightedge. But that's probably not expected on an exam [or maybe not even allowed], and I doubt it would be very straightforward).

 

[Vanadium 50]

maybe you'll eventually get lucky and pick the number n = 10 to show the probability is 1/3. But what deterministic way would you use to solve for n in such a way that n2−n−90=0 n^2 - n - 90 = 0 , or something very similar to it, doesn't come up in the process?

Well, "kids today" (and now I sound like an old man) greatly prefer the "guess and check" method, the idea being they can test several potential solutions in less time than it takes to actually solve the problem. A quasi-reasonable approach might be "the situation has changed for the second sweet, but not by very much. Therefore,

\left( \frac{6}{n} \right)^2 \approx \frac{1}{3}

Or n \approx 6\sqrt{3}, which is 10.4. So 10 is not crazy. A good student might notice that this is an upper bound, and a very good student might notice that either n or n-1 must contain two powers of 3: one to give you the 1/3 and another to cancel the 6. So there really are only two plausible solutions, n = 9 and n = 10, and n = 10 looks better.

 

[Intrastellar]

Well, "kids today" (and now I sound like an old man) greatly prefer the "guess and check" method, the idea being they can test several potential solutions in less time than it takes to actually solve the problem. A quasi-reasonable approach might be "the situation has changed for the second sweet, but not by very much. Therefore,

\left( \frac{6}{n} \right)^2 \approx \frac{1}{3}

Or n \approx 6\sqrt{3}, which is 10.4. So 10 is not crazy. A good student might notice that this is an upper bound, and a very good student might notice that either n or n-1 must contain two powers of 3: one to give you the 1/3 and another to cancel the 6. So there really are only two plausible solutions, n = 9 and n = 10, and n = 10 looks better.

A very good student might also notice that n and n -1 must swap values when going from n = 10 to 9, and that is only possible if 9 is actually -9. That makes n = 10 the only possible solution. But is this not a bit too much to expect from a high school student ? I think @collinsmark was looking for something a reasonable high school student can figure out which would lead them to avoid the quadratic. If a high school student can reason "the situation has changed for the second sweet, but not by very much." then this student is already very good in my book. Keep in mind that these students still have two years of high school left before they graduate.

 

[J.J.T.]

I'm not really sure this is that difficult, but I would not have approached in with the "calc-II" style proof method that the first guy on this thread solved it with. I probably would have done this backward and just solved the Quadratic first to get n=10 or n=-9 of which only n=10 is a possible solution then worked from there to prove that the probability and numbers fit i.e. (6/10)*(5/9)=30/90 simplified is 1/3. I would probably have an issue if they marked that as an incorrect response. Mostly because it's inappropriate to ask for students to do a non-cookbook problem in a cookbook manner. You can't have it both ways, either give cookbook problems and expect cookbook correct responses, or give critical thinking problems and expect correct responses to be based on critical thinking.
Obviously this answer would only work in a problem based on "real-world" type math, as if you were solving a conceptual math problem you would have to include the extraneous solution of (-9) in your calculation.

Damn... just realized this is a month old lol... I guess there's not that much activity I didnt go back that far...

 

[Silicon Waffle]

Very well said! J.J.T, I totally agree!