# Is this exam question really too difficult?

1. Jun 5, 2015

### 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 realise 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?

2. Jun 5, 2015

### pasmith

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

3. Jun 5, 2015

Staff Emeritus
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

$$n^2- n - 90 = 0$$

4. Jun 6, 2015

### 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"...

5. Jun 6, 2015

### ShayanJ

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:
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.

Last edited: Jun 6, 2015
6. Jun 6, 2015

### PietKuip

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

7. Jun 6, 2015

### ShayanJ

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.

8. Jun 6, 2015

### 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.
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.

9. Jun 6, 2015

Staff Emeritus
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.

10. Jun 6, 2015

### sk1105

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!

11. Jun 6, 2015

### ShayanJ

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.

12. Jun 6, 2015

### JorisL

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)

13. Jun 6, 2015

### 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.

14. Jun 7, 2015

### 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.

15. Jun 7, 2015

### Staff: Mentor

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

16. Jun 7, 2015

Staff Emeritus
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".

17. Jun 7, 2015

### sk1105

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.

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'.

18. Jun 9, 2015

### f95toli

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.

19. Jun 9, 2015

### ShayanJ

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)

20. Jun 9, 2015

### mesa

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