I never learned to think of anything mathematical as a problem that could be coherently solved.  I never "got" numerical thinking, and I was never doing anything more than applying formulae or tricks to the questions.  I never really understood why anything was right beyond the most basic arithmatic questions.

When I took college algebra, the pinncle of my mathematical accomplishments, there was a particular type of problem that was just beyond my ability to comprehend.  I spent hours and hours fighting with those problems, and I remember on three separate occasions "getting it," and finally understanding how to do the thing.  I then forgot the next day, and had to go through the same process again.  I needed to pass the course, and didn't want to take it again, so I pushed through, but when it came to the test, I STILL didn't understand the problem type.  I had some time on the final, so I did my best, and without any understanding at all did a bunch of stuff to the problem that I didn't really understand, but seemed reasonable at the time, and eventually stopped, because I couldn't see anything more to do.  I'd actually gotten it right, to my shock and amazement, but I couldn't understand why.

Later, I took three quarters of statistics.  That was easy, really.  I never came anywhere close to understanding what on earth the formulas that produced correlations or standard deviations and whatnot were, but I could see quite clearly how to plug data sets into the equations, and could understand how to use the information they produced.  But the maths behind all that?  Pure and incomprehensible gibberish.  I thought about the people who had developed those formulae with a bit of awe, because the relations between the data going in and the information coming out was completely and totally mysterious to me.

On the other hand, I had a great intuitive sense for probabilities.

I've since gotten better at basic arithmetic, and can sometimes remember or intuit basic multiplication.  When pressed, I think i remember how to do long division.

However, I find intepreting statistical data to be quite easy, and think in percentages and fractions all the time.  However, that doesn't have any connection to math in my mind.

I wonder how much of that is differential brain development - some people get that portion much earlier than others, and some people get a lot more of it than others.

While I don't deny that poor pedagogy can sour people on math, I also don't think its deniable that some people are just better at math and numerate things than others.  

Yet, interestingly, those traits don't seem fixed, as observed above.

I'm rubbish at maths. Although algebra ... I am ok with.

My personal list of what people should know. This is specifically aimed at people who do not use any advanced mathematics in their normal lives, but at the basic things even the most innumerate people should get a hang of.

1. Add, subtract, multiply, division.
Sure. Including tables. And lots of practice with everything, especially with simple calculations, to give 'the feel'. People should be completely clear that if they can do 4 * 8, they can do 40 * 800. Truth is, many people are not fully aware of this.

I guess people should learn the multiplication and long division algorithms, but it is important to realize that these algorithms were 'developed' for efficiency, not for robustness. The multiplication algorithm starts at the back of the numbers and works forward, meaning that the largest chance of mistakes lies at the front, where the most significant numbers are. In the same vein, standard long division gives you only the number of digits, and therefore the order of magnitude of the answer when you are completely finished. If you are disturbed in the calculation, you can easily make a factor 10 mistake

Second downside of these algorithms: hardly anyone ever uses them anymore for more than simple calculations, while they are designed for very long calculations, by people trained to work meticulously. In any situation where the answer matters, you will use a calculator.
This automatically brings us to 2:

2. Estimation.
If people can do quick calculations giving the correct order of magnitude, and the first digit give or take one, they can verify answers of their calculators.  If can they can do the first 2 digits correct, they will be within a few percent of the answer, and that is good enough for most things.

There are many estimation methods, and I think they should be taught stronger than the standard algorithms. In modern society, estimation is the more important skill.  First and for all, people have to be able to find orders of magnitude. You can't read newspapers if can't `feel `the order of magnitude of 3 million euros for 1500 people.
Also important is rounding. When people see 7847, they should not think `complicated number' but `a bit less than 8000'.  
On a side note, I sometimes wonder whether we shouldn't simply learn floating point notation to children. 1.3 e5 is definitely not more difficult to learn to understand than 130.000. In practice we usually round to powers of 1000. Think of how money is reported: 682 million ( 3 digits accuracy); 68 million (2 digits, but sometimes 68.2 million); 6.8 million, but rarely 6.82 million,  682.000 ( 3 digits, but different notation), 68.000 ( hardly ever 68.200).  Because most people do not know a decent notation for large numbers, they have to rely on words, leading to the trouble with American and European billions, and the messing around with prefixes. Do you know how much smaller a nanometer is than a millimeter?

Further estimation techniques: tables. Everyone should know both the multiplication table and the addition table by heart, and `know' the answer in the same way they `know' the meaning of a word when they see it, without thinking. We learn children not to spell words when reading, this is exactly the same.  
On top of the tables, we should teach some extra, useful sums. 2x25, 3x25,4x25 are standard calculations in our society. It is also good to let them practice the tables of 11, 12, 15 and 20. If only to calculate +10%, +20%, etc. Think what a paradise the world would be if people were good at calculating the effect of budget increases and price reductions!
If people know these, they can do `first digit' calculating: round all numbers to 1 digit, know the answer by heart, think about order of magnitude (number of zeros), round to 1 digit ( more is fake accuracy). I think this is the only level of arithmetic that people can really learn to do with the same ease as reading.

If you combine the above techniques, you get what I call the `good enough algorithm for multiplication' Many people use some version of it for quick calculations.

 I assume everyone knows the standard algorithm ( in 432x67: 7x2=14, write down 4, carry 1. 7x3= 21, add carry gives 22, write down 2, carry 2, etc. When finished with 7x432, write down a zero below this, and do 6x432 in front of it, you know the drill)

The `good enough algorithm' is this:
Round both numbers to 2 digits. I.e. 430 and 67
Calculate first digit times first digit, including order of magnitude.
So : 400*60 = 24000. Write down ( preferably, compare the order of magnitude with the `first-digit method, 400*70=28000)
Do second digit of number times first digit of number two.
30*60 = 1800. Write down
 Second digit of number2 times first digit of number1.
7*400=2400
Add all numbers, round to 2 digits. DO NOT calculate second digit times second digit, it is too small to be significant.
Result is 28000. The correct answer is 28944. Accidentally, the answer is the same now as the first digit method, but that is a bit of coincidence. This method is guaranteed to be good to within a few percent, the first-digit method is not guaranteed to be good.

Upsides of this algorithm: it is fast, especially on paper, doesn't require error-prone carries, and you start with the most important numbers.
Downsides: it is not exact. If you want exact, use a calculator ( or the standard algorithm). Also, you have to be aware of the order of magnitudes during your calculation. But that is in my opinion not so bad, spend the time gain on thinking about magnitudes, and the change of errors is small)

OK, enough about estimation. There are loads of other techniques, especially for division.

3. Types of numbers.
Already mentioned, I think. Start with integers, including zero. Let them learn the table of zero. Treat like all other numbers. Add negative numbers.

In my opinion, decimal numbers should come before fractions. People understand and `feel' decimal numbers in a way they don't `feel' fractions. Besides, decimal numbers are a special case of fractions. Once they are introduced to fractions, they will see infinitely repeating fractions like 1/3=0.333... . That might be enough in the direction of real numbers for all normal purposes. ( real numbers are `numbers with infinite decimals', like points on a line)

Focus on the meaning of numbers: you can't add apples to pears; 1000 apples is an integer, that is a counting number. But 1000 meters is a measured value, and will be 'imprecise' (once again, a hint at real numbers). Values usually need units. 1000 meters is 1 kilometer. When you multiply or divide, units change too. 50 kilometers in 2 hours is 25 `kilometers per hour'.  10 pizzas for 7 people is ten-seventh `pizzas per person' and is approximately 1.4 `pizzas per person'

In the case of fractions, my experience is that not-mathematically inclined people have enormous trouble with them. I suspect the far majority of people will never calculate 1/8 + 1/7 in a situation were transforming to (approximate) decimal numbers is not sufficient. Besides, people who have trouble with adding fractions will have no idea what to think of 15/56. Is it more than a quarter? Let them use decimals for complicated fractional calculations.

What is important in my view: Converting to decimals; Understanding that 2/4 is 1/2; 1/8 is smaller than 1/7; The equivalence between fractions and division. Especially this last part is often not fully understood by people, many people really have to think about `2 times 5 divided by 3 is the same as two-thirds times 5'.

4. Basic geometry. What are curves, areas, volumes. Area of rectangle, volume of rectangular beam. What is an angle. Sloped side of a straight-angled triangle is shorter than its straight sides combined. Pythagoras is nice, but doubt if people really -need- it.

I guess I am forgetting important things here, but guiding principle should be: do people really need it? If not, it is perhaps fun and interesting, but for people with little mathematical feeling, it is not fun, not interesting and only distracting from the stuff that is important. We should be careful not to throw the important stuff and the fun stuff in one basket, so that people call it all `complicated math' and forget about it.

5. Selected advanced topics. Outside of arithmetic, there are some areas that are important.

Basic, very basic algebra. Numbers represent values, and those values can also be represented by symbols. People understand this part easily. It's a miracle, but everyone understands that the letters GDP stands for a number that represents a monetary value ( a lot less people realize it is represent a flow, money per time, but that is another story).  The hard part is showing that you can manipulate the symbols, and use that to get  answers about the values. There is a lot of algebra people understand if you fill in the numbers, but not in pure symbols.
Still, people should understand `if a=bc, then b=a/c', and `if a= b+c, then b=a-c'. Preferably they understand that you divide both sides by c. But still, it would already be great progress if no university-level educated person got scared from simple formulas.

Next: logic. Venn diagrams should be taught to everyone and their relation to normal, language based arguments. For those who do not know them, imagine a circle representing all dogs, and one representing all animals. The dog circle with be entirely in the animal circle, showing that `all dogs are animals', but `not every animal is a dog'. Draw the circle of `long haired animals'. The overlap between dogs and long-haired animals are `all long haired dogs'. You can now easily talk about `long haired animals that are not dogs', which is the rest of the long-haired circle. Add a few extra overlapping circles, say `objects heavier than 10 kilos' , and the language to describe them would be horrible, while the drawing is still clear.)

Also: graphs and functions. A function is a relation, a graph shows the relation. How axes can be changed to make graphs look different( changing scales, `breaks' in the axes. I would almost add the `rule of difficult graphs' : if you are the intended target of a graph, and you do not understand the graph, the person who made the graph is probably a fraud who wants to lie to you. There is more useful stuff here. With practice and many examples ( graphs!), even the most innumerate person can understand important things about functions, such as inverted function ( swap the axes), one-on-one functions (if you swap the axes, you have a nice function) , one-on-many relations( if you swap the axes, some points have multiple values, many-on one(3-d graphs, people understand these surprisingly well). I would leave many-on-many relations out of the picture, (if only because they are, well, hard to picture)

My last, and tentative suggestion would be `introduction to statistics', a sort of `sex education for new paper readers'. This course would have no math in it, just an explanation what can and cannot be done with statistics, how uncertainty works, an explanation of correlation (without math), that it doesn't tell of the direction of causation, and might be an artifact of the data.  

These are my suggestions for the minimum level of numeracy at 15, to be taught to everyone. Of course, most of it is already in normal curricula. The important part in my view is to find the tings that are NOT critical for everyone. Its my view that a clearer distinction between important-for-everyone math and math-for-people-who-are good-at-math would be useful, to prevent the people who have trouble with it from forgetting the useful stuff together with the things they will never use again.

In short, I have 3 main points:
+Core numeracy should be aimed skills that people will use in normal life.
+Arithmetic should focus on quick grasping of approximate answers and order of magnitudes. This should be so ingrained that people don't think about using their calculator. Also, the meaning of numbers is important
+ higher skills should be aimed more at `passive consumers of math', who are not going to do complicated math themselves, but will read about it in the newspapers.

Maths has to be taught in context (I think). As you know, I am watching the forth series of The Wire in the moment.
And one of the characters is trying to teach maths to the kids. A rather desperate affair, but he manages to teach them fraction, by tapping into their interest in rolling dice and gambling. It turns out to be very successful.  

That's how the basics of maths should be (and probably are - hopefully) being taught.

A pleasure

I therefore claim to show, not how men think in myths, but how myths operate in men's minds without their being aware of the fact. Levi-Strauss, Claude

I can deal with my finances and taxes and I understand ratios and percentages and trig, how to estimate, and how to plot stuff and understand trends and to apply this to every day stuff.  I function fine in real life. I can cut through the crap of manipulated data in the media.

I function well enough to scrape through the maths bit of my physical chemistry PhD but I know it's not good enough.  

I dread being asked in my viva to show how such an such an equation on P and Q forms or scattering constants gives me the size and shape of my micelles.  Someone else has proved it, it is widely accepted and if something isn't working with applying the formulae, I'll ask a mathematician if we need to review it.

However, at what level of student are you aiming? A very significant part the population will never study any philosopher, and I doubt they would study 'abstract logical object that do not depend on real objects'.
Trigonometric functions are another subject that will be hard to teach on any level but 'tan(x) is a button on your calculator'.
I am not sure your real function part would include differential calculus? If so, that is another subject I would not include in 'basic numeracy'. It is important, but you can definitely live without it.

I think your direction is very good, but you are going way beyond the level that would be comparable to 'functionally literate', and more towards the level comparable to 'appreciating high literature'.

The way the brain processes information plays a large part in dyslexia. This is why some forms of dyslexia can be tackled by wearing tinted glasses or using coloured acetate overlays to reduce text contrast but severe dyslexia is linked with the way the brain processes visual information.  

I have a friend who is outstandingly clever and sharp but her dyslexia is very severe even though she is not lazy with trying - she tries to read as much as possible but the bottom line is that it doesn't go in and she relies on audio information.  Her spelling and ability to spots errors is appalling too.

well, you may be poemless, but that was fine prose, imo.

totally describes that feeling of being asked to empty the ocean with a leaky thimble feeling.

christ i'm glad that period of my life is over.

international politics and economics is just as puzzling, but at least i can come to ET and have a laugh while realising how dumb i am!

this diary and thread are vintage ET-

'The history of public debt is full of irony. It rarely follows our ideas of order and justice.' Thomas Piketty