Can you afford not to be a numbers person?

I must go back and see if I can get some help with the weighted mean. I’m sure I could do this in a different context.

I wonder how well we would score if this was not a multiple choice test, providing the correct answer plus a few distractors. Real life situations tend to be less helpful.

Multiple choice answers no doubt make life much easier for the marker, whether human or electronic, but I don’t approve of them. They can of course lead you to guessing the answer – law of odds says you’ll get some right – or giving you a hint that if yopur answer is not their you should see where you might have made a mistake.

I think the best questions are where you show your workings, or logic, that lead up yto the answer you’ve given. Even if you have made a mistake at some point, the examiner will see whether you’ve followed the right procedure and just made a careless slip.

“Weighted Mean
Also called Weighted Average
A mean where some values contribute more than others.

Mean
When we do a simple mean (or average), we give equal weight to each number.
Here is the mean of 1, 2, 3 and 4:
weighted average equal
Add up the numbers, divide by how many numbers:
Mean = 1 + 2 + 3 + 4 = 10 = 2.5

Weights
We could think that each of those numbers has a “weight” of ¼ (because there are 4 numbers):
Mean = ¼ × 1 + ¼ × 2 + ¼ × 3 + ¼ × 4
= 0.25 + 0.5 + 0.75 + 1 = 2.5
Same answer.

Now let’s change the weight of 3 to 0.7, and the weights of the other numbers to 0.1 so the total of the weights is still 1:
weighted average more weight
Mean = 0.1 × 1 + 0.1 × 2 + 0.7 × 3 + 0.1 × 4
= 0.1 + 0.2 + 2.1 + 0.4 = 2.8
This weighted mean is now a little higher (“pulled” there by the weight of 3).”

The weights in the examples I did were the quantities attached to different prices, so the average price achieved was influenced by how many of each were sold.

Multiple choice questions (MCQ) and the multiple answer variants with more than one correct answer are widely used in education and some of the advantages might not be obvious. I have found them very useful in formative assessment, where users can assess their own strengths and weaknesses, as in the example we have been using. They are well liked by users as long as negative-marking is avoided. Increasing the number of distractors decreases the success of guesswork and with numbers as answers there is no need to struggle to think up plausible wrong answers.

Computer-based tests allow valuable extra features, depending on the software and which features are enabled. For example it is useful to look at the stats for each question and see which ones most users get right and most get wrong. The latter can be down to poor or misleading wording such as double-negatives in the question or other presentation issues. I used to make formative tests available after a lecture or workshop, have a look at what the students found difficult and if necessary go over something again at the next class. A selection of questions from a pool can be presented at random. The advantage of redoing a calculation, for example, provides valuable practice and new questions encourage repeated use. It can be useful of students can compare their performance in tests or individual questions with their peers. Anonymous MCQ tests mean that only the user can see their marks. Obviously this would not apply in summative assessment where the marks are to be used in measuring performance.

I expect that Rachel is using software that offers a variety of features that make it easier to understand where help is most needed.

Worked answers very much have a place but don’t lend themselves well to self-assessment.

Thanks Malcolm. I understand the principle and uses of weighted averages but haven’t ever used them in a financial context.

MCQ are ideal for computer based learning (CBL) because even a computer can be trusted to get the marking right.

I don’t know hat other kinds of questions, if any, can be used reliably in CBL.

Perhaps the next step up would be missing word or phrase questions, e.g.

“Name the three most common causes of house fires?”

where the computer may have some change of testing against a set of correct answers.

I expect that short answer or long answer questions will usually need to be marked by a human.

I thought the explanation I found might be useful to others who wondered what was meant by weighted average, wavechange.

Derek – A couple of the systems I used did offer the option of putting in missing words/phrases but there are possible pitfalls. If a correct answer was tumble dryer, would the software allow tumble drier, tumble-dryer or clothes dryer (US name for such things). I avoid this option but sometimes had questions that allowed some answers to be typed in. As you say, the disadvantage is the need to mark these questions manually before the system would calculate an overall mark. Nevertheless, if you are going to use computer-based tests for assessment rather than just self-evaluation, I think it is very valuable to use this type of questions rather than relying on MCQ.

From the NC – Key stages 1 – 4:

5. Numeracy and mathematics
5.1
Teachers should use every relevant subject to develop pupils mathematical fluency.
Confidence in numeracy and other mathematical skills is a precondition of success
across the national curriculum.
5.2
Teachers should develop pupils’ numeracy and mathematical reasoning in all
subjects so that they understand and appreciate the importance of mathematics.
Pupils should be taught to apply arithmetic fluently to problems, understand and use
measures, make estimates and sense check their work. Pupils should apply their
geometric and algebraic understanding, and relate their understanding of probability
to the notions of risk and uncertainty. They should also understand the cycle of collecting, presenting and analysing data. They should be taught to apply their mathematics to both routine and non-routine problems, including breaking down more complex problems into a series of simpler steps.”

For more fun facts, click here (pdf)

The NC was notable in that it prescribed what shuld be taught, but not how it should be delivered, apart from the mandatory cross-curricular elements, referenced by the jolly phrase Teachers should use every relevant subject to develop pupils mathematical fluency.

The shift away from ‘rote’ learning of tables started in the ’60s, as educationalists pored over newly emerging theories of learning and knowledge acquisition and has never returned, despite evidence emerging later that suggested learning by rote often has useful benefits, particularly later in life.

If you’re aged over 65 then you probably know your tables by heart. And I’d be willing to bet you use them on a regular basis.

The irony is that virtually all Primary school teachers were themselves never subjected to rote learning techniques and – as a consequence – were found in the ’80s to have a somewhat slender grasp on numeracy and mathematics (I’m being generous…).

It’s always baffled me as to why it fell out of favour; as a Musician rote learning is part and parcel of the process. There’s good reason why a concerto soloist doeosn’t have the MSS on the piano.

Thanks Ian. I’ll have a look at this later.

Worryingly in Maths teaching the UK is ranked 27th among developed countries, slipping down a place from three years ago, the lowest since it began participating in the Pisa tests in 2000.

There’s a lot of research out there, and this study focussed on teacher behaviour. The summary was that using multilevel modelling techniques it was found that teacher behaviours were able to explain between 60% and 100% of pupils’ progress on the Numeracy tests.

IN practical terms, In Years 1 and 2 (4 -6), a great deal of the maths work children do is still very practical and related to everyday experiences.  
Around the classroom will be number lines, number tracks, number grids, 100 squares, examples of number bonds, examples of multiples and mathematical vocabulary. There will also be a variety of resources, such as coins, dice, dominoes, playing cards, beads and plastic bricks for counting. 
Pupils are encouraged to use correct mathematical language, such as ‘greater than’, ‘sum’, ‘difference’, and they will talk about and explain what they have been doing. For example: “We did this because…”, “I worked out my answer by…”, “It can’t be because…”

Teachers usually prepare lessons that include a mental starter, a whole-class introduction to a new piece of work, group activities and then a plenary at the end to review their work. 
Lessons involve investigating, finding out, working together and talking about what they have done. Children learn that making mistakes is part of learning.

Year 1

• In Year 2, children continue working on numbers. They order numbers to 100, use odds and evens, rehearse and use addition and subtraction facts regularly in their mental work. They also use number lines, tracks and 100 squares.
• They learn that subtraction is the inverse of addition (eg 12 + 7 = 19 , 19 – 7 = 12, 19 – 7 = 12) and continue to develop their mathematical language (eg half, quarter, sum, digit, fraction)
• The × and ÷ symbols will be introduced.
• Some children work on numbers to 1,000, finding missing numbers in a sequence (eg 750-650).
• In multiplication, they double and halve numbers. Some children start to learn their 2, 5 and 10 times tables.
• Pupils are asked to solve number problems and investigate the properties of numbers.
• Teachers will be aware of each child’s ability and will provide work to challenge them. They will be given opportunities to practise and apply their skills, ensuring they develop confidence and competency.

I used to chat with a maths teacher but have lost contact after moving home. I remain convinced that a major factor in poor numeracy is the fact that many don’t need to process numbers very often.

Learning to use a slide rule was a great help to me because it’s necessary to do a very rough calculation of answers in order to know where to put the decimal point. Calculator users often have no feeling about whether their answers are reasonable or whether their is a mistake. I like to check answers by doing a calculation in a different way, if possible. To give a simple example, when adding up a column of numbers I would start at the top and work downwards and then start at the bottom and work upwards.

That was a very real fear of Mathematics teachers in the ’80s/90s when the ubiquity of mobile calculators (and later ‘phones) made it easy to do calculations, but supplanted the kids’ need to check the results.

I agree that it’s certainly an issue; but I also suspect the real problem comes back to the paucity of serious differentiation in the early stages. When kids arrive at a reception class, they’re all assumed to know nothing. My wife is the family mathematician and we both taught our children to read, become numerate and even an element of critical thinking – something else that’s sadly lacking. But we were both told not to do this by the local infants’ school – since “We find it easier if we start them all on the same page…”.

So given the sheer size of reception classes, combined with this approach it’s hardly surprising that the children with unsupportive backgrounds flounder whilst those with highly motivated parents are held back.

Couple that with the fact that at later stages maths is decidedly illogical it’s little wonder we – as a nation – deliver maths so poorly.

I had the opportunity to meet students from many countries and at one time I noticed that Greek students were much better at doing calculations than UK students. One said that he had not been allowed to use a calculator at school. OK, it was a small sample size and an unconfirmed statement, but it seemed plausible.

I agree with what you say, and if I had had kids I would have done my best to provide support with their education. However, I can identify with what you were told by the primary school because anyone who is ahead of their peers can disengage if ‘taught’ what they have already mastered. That’s certainly the case with undergrads, and it seems to depend on the individual. I liked to put in some material that would at least challenge the better students even if it was lost on the weaker ones.

My father held numeracy in high regard and I will not forget his reaction when he was told of a school where the times tables were posted on the classroom wall.

There is little point in looking back at a world without calculators and computers, but I wonder if there are lessons to be learned from teaching in other countries.

Your point about disengagement is certainly valid in later years but schools are expected to differentiate the work in classrooms for precisely that reason. Children are also, I suspect, far more adaptable and if the work is insufficiently challenging they will tend to work down to that level.

As an aside in the early ’50s when I was sent to school for the first time I could already read fluently and can remember still (and quite vividly) the way the class was taught the alphabet. By the second class (now year 2) it was decided I was so far ahead I had to be moved up a year – something that would not only now never be countenanced but which, at the time, did me absolutely no good whatsoever. It took me two years to ‘recover’ but by year 6 I had done so, passed the 11 plus. and started Grammar school at the age of 10. I was far too immature and it again took me a couple of years to recover from that.

I know little about school teaching, Ian, but have a tremendous respect for teachers and have done my best to encourage postgrads and undergrads to look as teaching as a choice rather than as a fall back position. I suspect you are right about children being more adaptable and in small group teaching it’s far easier to keep an eye on who is not engaged.

I can relate to your challenge of moving up a year to some extent. I was doing very well in a very ordinary village school when my parents moved home half way through the year. The teacher wanted to put me back a year, especially since I would have been on of the youngest pupils but my father thought that I could cope. It was a real struggle to start with, but I think the challenge did me good. I don’t recall enjoying doing calculations in these days but it became fun when I had settled in to being at a Grammar school some with rather old fashioned values.

I might have another look at Rachel’s site. From the comments it looks as if the questions may be from a pool and there will be ones I’ve not tried.

Can we count on you standing in for Ian if he needs a break from dispensing morning humour. As an early bird you would be a good candidate.

That’s a nice thought. The trouble is I’m only an early bird on days when I’ve got to swiftly fly the nest and head off to work.

Good schools do, but one major issue in UK state schooling is class size. Imagine attempting to teach 36 four-year-olds at the same time…

It’s not looking good for the older ones either: http://www.bbc.co.uk/news/uk-england-38506305

So many issues covered in that piece.

It possibly helps to know how schools manage class sizes (apologies if I’m telling you things you already know). Since that piece focusses on the Secondary sector, and since a mother with a dyslexic child was an interviewee, expressing her concerns about how rising class sizes would affect her offspring, it’ll help if we understand how secondary schools address these issues.

Most Secondary schools that do well in OFSTED assessments attract sufficient numbers to meet their capacity. That’s an average of 30 children per class. Now, you could be forgiven for assuming that means each class the child attends will have 30 kids. But that’s not how the system works.

Special needs departments exist in almost all secondary schools; some schools even specialise solely in educating children with specific needs, but I’ll stick to mainstream secondary.

Within each school the children with special needs (Dyslexia, Dyspraxia, Autistic tendencies, visual impairments, auditory impairments and a whole host of other conditions) are sometimes integrated into the main classes, accompanied by a special needs teacher, or taught in very small groups – such as 6 or 7. That serves to reduce overall class size and give extra help to those who most need it.

Then there’s a selection process. Generally, Secondary school intake management don’t place much credence in statistical attainment reports from feeder Primaries. There’s a whole host of reasons for that, but one is that Primaries – usually being smaller – tend to mirror community expectations, so a child classed as ‘very bright’ from one school might well be far less able that a child classed as ‘very bright’ from another.

The result of all this is that initially most children are taught for the first couple of terms, at least, in large classes. Around the third term the schools will normally seek to enact streaming: classes in Maths specifically and in Science become segregated by perceived ability. Thus, if we take a typical, four-form entry, then at the third term there will be five streamed groups for Maths and Science created. The most able will be taught in a large group – say, 36, while those experiencing real difficulty will be taught in classes of >20.

For many years we’ve known that children thrive in smaller classes, but the Government has always stated “there’s no evidence of that”, usually while sending their own children to Private schools with far smaller classes. It’s true that there’s no hard evidence, however; because there have never been any serious large scale studies done that focussed solely on class size.

As an aside classes in Africa and in the Far East are far larger, but there’s an entirely different culture there, one in which the Teacher is held in great respect.

Straight algebra says:

S + L = 49 (1)

S – L = 36 (2)

So 36 <= S <= 49

Solving (1) and (2) as simultaneous linear equations gives:

2S = 85

S = 42.5

But that does not make sense, because you can't have half a dog!

If, perhaps, there are also some medium (or other) sized dogs, then:

S + M/2 = 42.5 or S = 42.5 – M/2, so
if M = 1, S = 42 or
if M = 3, S = 41, or
if M = 5, S = 40, and so on down to…
if M = 13, S = 36 and L = 0. (so no large dogs at all.)

I would want to know if there are any medium dogs before attempting the calculation. We are not always provided with all the information needed to solve a problem.

The simple calculation is easy enough to do – the fact that it gives a silly answer reveals that we might need more information (or might have been given incorrect information).

If we had been told that 48 dogs were competing, I’m sure we’d all agree that 42 of them were small and not stop to wonder if there were dogs of any other size.

When we use maths in engineering or science (or anything else), it is always good to formally state the assumptions and limitations of our work.

A further possibly it that we are dealing with some peculiar local region, where they do not use base 10 arithmetic.

For example, if base 11 were used, in base 10, the reported total number of dogs is 44 + 9 = 53 and the difference is 33 + 6 = 39. That gives a solution with 7 large dogs and 46 small ones, or 42 small dogs, if you need to report the answer in base 11.

Perhaps those fond of quoting statistics on Which? Convo might like to follow the advice in your last sentence, Derek.

[Edit – I was referring to the post about stating assumptions and limitations.]

I suppose that we must assume that the ‘dogs’ in the competition are not the kind with mustard, but that would be a way of accounting for the half dog.