Are we really Numerate? How numbers lead us up the garden path!
Our Politicians and Business Leaders talk about the need for us all to be both literate and numerate when we leave school and as such literacy and numeracy are key subject components of the National Curriculum and beyond.
But are we infact teaching the literacy and numeracy required for success in the real world?
I think not and I will be posting my thoughts on this over the next few weeks on this blog
Here I make a start, looking at Numeracy.
I call the numeracy that we learn at school and in our universities “Math 1.0″ (And in general this is the only form of numeracy we are taught, so most if not all of our leaders are only numerate to the level of Math 1.0)
This is a Math that is useful, but only in the very simple domain of counting and manipulating pure numbers. This domain is what Donald Wheeler (one of the few statisticians in the world who seems to understand this stuff) calls “Math World” a strange world that has very little bearing on everyday reality. It is very misleading in fact when, in the Real World, we use Math 1.0 to manipulate, interpret and compare measurements
It was Walter Shewhart (the man who has been called the father of quality) who said “Data is meaningless outside of its context”. Using my language here, he could have rephrased this as “Data is meaningless unless processed using Math 2.0″ (Math 2.0 is a way of working with numbers that keeps the important context ‘in view’)
Math 1.0 is the Math of the Counters. Math 1.0 works for the abstract Math-World. Math 2.0 however is needed for Real-World Problems.
Many of our “number experts” (mathematicians and statisticians for example) base their life-long working knowledge on this Math 1.0, so they are then part of the problem. Math 1.0 is entrenched in academia and science.
It is now one of those implicit unquestioned assumptions (like water is to fish and air to birds) that Math 1.0 is numeracy and that Math 1.0 describes the sole reality of numbers. There will certainly be a few people in very high powerful places who know about Math 2.0 but are happy for the rest of us to just learn Math 1.0. When it comes to comparing things, Math 1.0 does not clarify issues, instead it clouds them.
All this means so few people know or understand the limitations of numbers, and therefore that numbers can be used to keep us all in the dark (ages?) about most things. We will never really know whether our Health Service or Schools are getting better or worse using Math 1.0. What is certain is that using Math 1.0 we get into endless debate about the trivia from the data (we can call this “noise”) and we will nearly always be missing the important understandings (we can call this the “signal”). Without Math 2.0 the useless information (noise) is drowning out the important information (signal).
So although it seems ludicrous that some, if not most, of our main ‘experts’ in Maths and Statistics use a Math that was devised for the special case of pure numbers and counting and that is strictly NOT applicable to numbers as MEASUREMENT.
But it is such experts that write a numeracy curriculum for our schools, universities and accountants that is based on a special case with numbers (the Math of simple counting – Math 1.0).
In the real world most of the important numbers we deal with on a day to day basis are to do with measurement, or involve counts that are being used as measures, and so we need to apply “Math 2.0″ in order to interpret these situations.
When we use simple Math 1.0 for interpreting data measurements we create problems and misunderstanding. Because we have come to rely on numbers in every facet of life and business (we found we could no longer trust the word of leaders, doctors, scientists etc so we needed their numbers) numbers now heavily impinge on our emotions.
We can get very angry when we see numbers we don’t like. The problem is often there is no valid reason to get angry with the numbers, it is Math 1.0 we should be getting angry with. We should be getting angry that we are not taught the ‘numeracy of measures’ at all.
Everyday we are all making decisions with sometimes life-threatening or very severe unintended consequences because of a lack of real-world numeracy because we don’t have the skills of Math 2.0
No-one is excluded. Politicians, scientists and business leaders all continually make poor decisions when they apply Math 1.0 thinking to real-world Math 2.0 situations, making us depend on numbers in a way that is totally irrelevant, abstract, misleading, artificial, and distorting.
Our lack of real-world numeracy Math 2.0 skills is I believe a big part of the problem why so much today seems to be going wrong. We follow the numbers but we don’t understand the numbers and as a consequence we jump to the wrong conclusions and we take actions misguidedly on the numbers and actually then make matters worse rather than better. (Deming called action based on misguided interpretation of data – tampering and he devised the funnel experiment to help us understand how tampering makes matters worse)
So at this point you may be asking what is this Math 2.0, why isn’t it taught in school and what difference would it make? I will some outline the key differences between Math 1.0 and Math 2.0 next time but here is a taster.
Math 1.0 is an artificial world where lines have no thickness, parallel lines can’t meet and numbers are absolute. When we use Math 1.0 there is only one correct answer and it is not possible to have variation in the answer. (in the real world however variation is always present)
Math 2.0 on the other hand is a real-world Math where lines have thickness, parallel lines can meet and most importantly measured numbers are never absolute. As variation exists in all things Math 2.0 does not ignore its effects (whereas Math 1.0 assumes random variation does not exist)
So here is a little teaser to see if you are working from Math 1.0 or Math 2.0
MATH 1.0
Math 1.0 2 + 2 = 4 YES this is absolute, there is only one answer and that is 4
Math 1.0 implies that this answer is the same whether we are using simple counts or measures.
So 2 inches plus 2 inches will always equal 4 inches
MATH 2.0
Math 2.0 – when simply counting, the results are the same as for Math 1.0
So 2 + 2 = 4 this is absolute, there is only one answer and that is 4
However when adding together measures or comparing measures:
2 + 2 = 4 but only on the average (so each time we take measurements and add them together the answer can vary either side of the number 4 by an amount which Math 2.0 can reliably approximate )
this scenario would be more precisely written as:
2 (v1) + 2 (v2) = 4 (v3)
where v1,v2,v3 is the variation (plus minus 3-standard deviations) that is inherent in each measurement
This brings me on to a further significant difference between Math 1.0 and Math 2.0.
In Math 1.0 there is ALWAYS significance in any change of number and therefore there is value in comparing just two data points. So if something measured 20 last month and 23 this month Math 1.0 says there is a change (an improvement if good stuff, a worsening if the measure is bad stuff. So as Math 1.0 is the math of pure counting if we have 20 apples in one basket and 23 apples in another it is clear that the second basket has (three) more apples in it. Math 2.0 would come to the same conclusion. However if the tree in your garden produced 23 apples this year and 20 last year we actually need Math 2.0 when seeking to make a decision about whether this difference means the tree yield is improving? For we are now not looking at the pure count of the apples we are seeking to use the numbers to give us knowledge about the tree. Now instead of apples and tree performance think of pupil exam success and school performance. And then by way of extrapolation think school success and league tables.
In Math 2.0 we CANNOT KNOW IF THERE IS A DIFFERENCE between 20 and 23 unless we have more data (and then a lot of the time Math 2.0 will show there will be no likely significant change). Math 2.0 tells us that comparing just two data points is ALWAYS meaningless (and of course it can provide the evidence for this). Each time we just compare two data points we are viewing the data outside of its context.
If only journalists were schooled in Math 2.0 we would not have so many meaningless, stupid headlines in our papers. But there again, they probably wouldn’t sell so many newspapers, so you could see that their bosses would be quite happy that their journalists are only numerate to Math 1.0 level.
Using Math 2.0 many headlines in the newspapers would read “Probably no change in the trade figures this month” rather than something that appears very dramatic like “4% fall in trade figures throws UK back into recession”.
Which of these two headlines would make you buy the newspaper – the first one (‘probably no change” so nothing much is happening - a quite likely scenario using Math 2.0) or the second headline derived using the inappropriate use of Math 1.0 ?
Have I grabbed your attention?
If you already know what I mean by Math 2.0 great, please post your own examples here about how Math 1.0 misleads, If you think I am a raving lunatic and it simply can’t be possible that we are being taught the wrong numeracy at school for making sense of the real-world, then please follow, watch and learn. And if you still think I’m being stupid tell me so.
In the next two articles in this series I will be comparing in much more detail some of the differences between Math 1.0 and Math 2.0 and seeking to impress upon the sceptics out there that this is really important stuff.
Next Time: Maths and Science leading us up the (wrong) garden path
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