When I tried this problem, I had the useful feeling of befuddlement mixed with the feeling I had seen something similar before. I tried thinking about how much sheep would get eaten in an hour, which is 1/2 + 1/3 + 1/5 = 31/30 sheep, so knew that the answer was a bit less than an hour.
I then started to make the problem easier, by considering what would happen with just the Lion and Wolf. At this point I recognised it as having something to do with harmonic means (here is a nice essay on the harmonic mean from James Tanton) and intuitively and correctly felt as though the answer was the reciprocal of the above number, so 30/31 hours to eat one sheep.
I then justified this answer to myself: each of these fractions is the rate at which each animal eats the sheep, and each animal is eating for the same amount of time, T, so the time taken to eat 1 sheep is given by the equation:
At this point, our approaches differed. Without making the connection to harmonic means and inverse proportion, Sarah decided to continue with her original approach, but this time by finding how much sheep would get eaten in a shorter, 10 minute, time interval, in the hope that she could calculate the number of minutes needed to eat one sheep. On calculating the amount of sheep eaten in 10 minutes (31/180), she then proceeded to divide 180 by 31 to find the number of 10 minute intervals required (5 r 25). Unfortunately she got deterred at this point due to strange numbers involved, concluded that the problem was that we were working with tens, and went on to find the amount of sheep eaten in 6 minutes, but with no better results.
At this point, around 10 minutes later, she was on the way to being stuck. I asked her to explain her feelings at this point. She was:
- Frustrated with her arithmetic ability. I did point out that she had been able to perform the necessary fraction calculations.
- Annoyed at being stuck - she realised this probably wasn't the way to solve the problem, but had no other ideas. This confirmed her belief that she is "not a problem-solver".
This is interesting. Sarah has achieved an A at A level (and gone on to achieve further academic success) and as such must have developed a high level of resilience and problem solving ability. How must students, who have not developed this level of resilience, feel when stuck?
Sarah then returned to the original diagram, and the 'answer' of 31/30 sheep eaten in one hour. At this point she realised that this meant that the time taken to eat one sheep was slightly less than an hour. On changing the 31/30 to 62/60, she then guessed at the answer being about 58 or 59 minutes and then promptly gave up, 15 minutes or so after starting the problem, stating that she "knew this was not the correct answer", but was "close enough" and that she would "probably not to be able to solve it". In short, she was ready to know the answer. I did not resist as I did not want to put her off trying future problems.
Here are my reflections on our attempts to solve this problem:
- We both approached the problem in a similar way, using similar strategies, although Sarah did not make it easier - this would probably have helped her make sense of the problem without having to focus on the tricky fraction additions.
- The importance of knowing a wide range of mathematical techniques can not be understated - strategies will not solve problems on their own. My prior knowledge allowed me to recognise the correct tactic quickly. Sarah continued on the same unfruitful path in the absence of further ideas, and did not reach the end of the problem. The problem does not need knowledge of the harmonic mean but does requires some experience of setting up equations or use of inverse proportion (see below).
- Sarah persevered with the problem for around 15 minutes, finally being dissuaded by the belief she would not reach a solution. Zeitz talks about increasing levels of concentration - that good problem solvers should aim to increase the time they can concentrate for (i.e increase their perseverance). Research shows that perseverance is a huge factor in achievement - but what strategies do we have for increasing perseverance in our students beyond just getting them to work for longer periods? More on this in an upcoming post...
- Sarah had numerous negative feelings about her own mathematical and problem solving ability - I feel as though these might have inhibited her ability to solve the problem. Asking her about these feelings gave me the opportunity to correct her (what I regard incorrect) negative beliefs. Confidence clearly plays a huge part in problem solving - how can we increase our students' confidence?
Post-script: Did this make Sarah a better problem solver?
If we had continued our lives as though this problem never happened, and put it down as a failure, I would suggest not.
After Sarah finished the problem, I revealed a couple of ways of arriving at the solution. The learning comes from reviewing the strategies and tactics involved, not being concerned that a solution had not been reached and learning from the process. That said, it is important to consider how a successful solution might have been reached, and then try to encourage the use of these strategies and tactics in the future.
We discussed how she could have perhaps used the make it easier strategy (solving the problem with just the Lion and Wolf, for example). Although this strategy doesn't won't make her a better problem solver immediately, repeated use of this strategy probably will.
Regarding mathematical techniques (tactics) in general, it is crucial that problems are targeted at the correct students, and that we equip students with a wide range of problem solving tactics. Would Sarah have ever been able to solve this problem with her level of mathematical expertise? I would suggest that she could, although a wider range of tactics (such as familiarity with the harmonic mean) would probably have increased her chances of success.
Regarding this problem in particular, we made connections to other areas of maths, suggesting that the problem is related to speed, time and distance calculations. She worked out the number of sheep eaten per hour, and a simple reciprocal would have given hours per sheep eaten. We then discussed the notoriously difficult idea of inverse proportion. I suggested it was similar to the classic problem: How long does it take 4 women to build a wall if it takes 6 women 3 hours to build the same wall? Admittedly this is a simpler problem, but she immediately answered the problem correctly, suggesting that the tactics learnt here would be useful for future problems.
In conclusion: This was a valuable insight into the problem solving process. To increase our students chances of problem solving success, we must review and practice successful strategies and tactics in equal measure.