Naked Science - Intelligence

Copyright Michael Curtis 2020

All rights reserved

                    

Part 3

Mistakes in Mathematics

 

Introduction                                                      

There are recurring mistakes and problems that happen in everyday mathematics or academic mathematics - like figuring out what the date was 3 days ago if today is10th June where you could count backwards : 10th, 9th, 8th and conclude that the date 3 days ago was the 8th June OR you could say “10 – 3 is 7. So the date 3 days ago was the 7th June”. Once you know a correct method for solving a particular problem then the mistake may not ever be made again (or rarely at least!). This chapter looks at common mistakes involving mathematics.

Note: Maybe “three days ago” is capable of two interpretations – and the intelligent person caches in his/her memory the ‘trick question’ aspect of counting back a number of days.

Note: It is useful in some paragraphs, below, to be familiar with equations like

x + 2 = 5.

It just means ‘When 2 is added to a number, the sum is 5’. The number you start off with must be 3 for this to be true. So, x is 3.

 

Counting Between Numbers

Today is Friday December 31st. So, what was the date on Monday? You might think that Monday comes 4 days before Friday and, because of that you calculate 31-4=27. So the date was the 27th December. That is correct. Compare this with counting back 4 days and starting with the 31st:

31st, 30th, 29th, 28th - the 28th is the wrong answer.

When you count between numbers, do not start with a number like 31st when you count backwards - start with one less: the 30th:

30th, 29th, 28th, 27th - the right answer.

Is June to December 6 months? - or is it 7?

Again, it is this consideration of whether the borders of the numbers to use include or exclude the points mentioned in the question. 

 

Scheduled Numbers

If an activity is taking place from December 3rd to December 7th, it is not (7-3=4) 4 days long but one more day than that: 3rd, 4th, 5th, 6th, 7th. So, you should add a 1 in any situation like this.

Note: The above assumes that 3rd-7th means ‘the 7th is inclusive’.

 

Inclusive or Exclusive

If it is 4am on Monday and I say that I will see you in 2 days time, I might mean count today as the first day and count tomorrow as the second day on which we meet. Or, I might mean that Tuesday is the first day and Wednesday is the day on which we will meet.

 

Symmetrical Obvious Fractions

If I have 2 halves of an apple, there is a tendency to assume that I mean 2 parts after cutting down the middle along the line of the stalk. However, I could have mashed the apple before separating it into 2 halves, or cut it into 4 quarters by cutting it downwards and then cutting it across and arrange 2 halves out of those pieces. Fractions need not be symmetrically obtained and they need not be the simplistic idea of what the fractions should be.

 

Averages

Two men are in a room. One is five feet tall, the other is six feet tall. Their average height is five and a half feet. If you were not told their heights but were told that the average height of some men is five and a half feet then you should keep an open mind as to whether or not any of the men is truly five and a half feet tall.

So, averages can mislead.

 

Numbers Have Many Representations

8 is :

8 + 0 or

5 + 3 or

eight eights divided by eight.

There are many representations.

Once, I saw a maths question where x/x was used to express 1 because typically a number divided by itself equals 1.

8 times ‘x divided by x’ is typically 8 as well!

 

 

Choice Stores

 

Overview

A big problem in our thinking is the way we assume that there is only a limited choice for us to believe in. e.g. ‘John drove in his car today. He only has a Honda Accord and so he must have driven the Honda Accord today.’ - this logic relies on the person having only one car. Perhaps you have in your memory the knowledge that John has three cars but the bad habit of accepting the first idea to leap into your mind ( i.e. only the Honda Accord rather than the other two cars) can stop you from using your knowledge.

I recommend that you develop the habit of replacing the idea of ‘Only’ with the idea of ‘At least’. e.g. "John’s Honda Accord" is not seen as being "John’s only car" but as "At least one of John’s cars". If John really has only one car and you know that then you have lost nothing by considering if he has more than one car. If John has more than one car then you have gained a true perspective by thinking, "At least".

A way to think really well is to have deliberate stores of information which you deliberately build up and deliberately look at regularly when thinking. e.g. A list of the cars that John has is a useful store of information when considering which car he drove. This chapter describes useful types of information store.

 

Clutching at straws

Maybe we are just grateful when an idea pops into our heads as to what is going on. Maybe we trust that a ‘static explanation of what is going on’ is correct because we lack the brain power to put that suggestion on hold and dig around for alternatives.

And you can make progress with a wrong interpretation of what is going on. Pretty soon, observed facts will contradict your explanation. “Oh! He turned into his driveway. So he is not jogging along the whole street!”; but the concept of a man jogging the whole street has caused you to be cautious not to knock into him, perhaps. The wrong view of what is going on has still served a purpose.

What is really undesirable is where your wrong interpretation of what is going on goes unchallenged. Eg. Computer code that should be adding a given number to itself  but you mistyped and the number gets multiplied by itself. You run a test on your code and give it a 2. It multiplies 2 by itself to give you 4. Unfortunately, if you add 2 + 2 you will also get 4. So you think that your test demonstrates that your code is correct.

When an assumption goes un-contradicted, you might commit to more risky behaviour because the assumption seems to hold true.

Having some memorised possible interpretations of a scenario can be helpful: ‘stores’. So your mind has more than a vague “Oh, remember to question if an assumption is true!” heuristic. Your mind benefits from a memory of common interpretations of what you are seeing.

I suppose a downside of having that list is an over-reliance on it: a scenario might really be novel.

With the jogger, you know that he is either going to continue jogging in a straight line or do something else; but if you memorised a ‘store’ of useful information about the jogger then you also have the option to consider that he could turn into the next driveway since you saw him do it before. But if you assume he will do exactly that then you are limiting your thought avenues.

 

Switching off the Filter

Sometimes, the facts which we perceive or the system which we first encounter is only a partial picture. A modern example of that is the computer spreadsheet with a filter on it. I search for a number that should be in the spreadsheet but fail to find it – until the filter is turned off; but what if I am not asking myself, “Is a filter on?”; then I might declare to people that a number is categorically not on the spreadsheet.

A presenter of a lesson might make a bad choice of graphic to display such that only part of the big picture is shown on a slide of information. Someone assuming that the slide is sufficient or who assumes that the slide is emphasising significant facts (when it is not) can become lost; whereas someone else is thinking, “Is this enough of a picture for me to see the essentials of the system?”

So, it is a habit which needs developing: a habit of questioning if the minimal necessary information for comprehension of a system is available to you.

Another example is a 2 section phonebook with business phone numbers in one section and residential numbers in the other section.

You might flip open the directory and seek a residential number but you are browsing the business numbers section.

An awareness of syllabus or framework and a habit of doubting the information immediately on display is helpful here.

 

 

 

Goals Store

Early in this book, it was shown how people set themselves goals to achieve. We prioritise goals and prefer to achieve the goal of eating over the goal of watching television when we have not eaten for a long time.

By focusing intently on a particular goal, you might ignore that your focused-upon goal is not the only goal that you want to achieve. In the food example, a person’s body makes him aware that he is hungry and so the television will not captivate all of his attention. Compare that with a warehouse worker with 2 tasks: shift boxes to the far wall and remove boxes from the cupboard by the far wall. One task can get in the way of the other: if the cupboard is blocked by the moved boxes then the boxes will need to be moved away so that the cupboard can be emptied and then moved back. So it makes sense to consider all of one’s goals and to arrange their order of execution based on efficiency.

This is why a goals store is a good topic to consider regularly in your thoughts.

People say that cats, no matter where they go, think about their exit strategy were they to be attacked. While I know at least one cat that isn’t too bothered, the concept is interesting: there is the task at hand but it is of little consequence if the cat finds itself cornered. It is thinking about a ‘bigger picture’.

 

Memorised List Store

 

In the example of a person with three cars, a person thinks that there is only one car rather than at least one car. What would help a person in this situation is a deliberate effort to memorise lists like the list of cars.

Similarly, if dogs have different breeds then it is good to have a mental list of this diversity of breeds. In that way, when the word ‘dog’ is mentioned, the idea of breed variety rather than a static image is likely to spring to mind.

 

Auto-Pilot Store

 

There is a tendency for people to go into an auto-pilot mode when they are doing something repetitive. For example, a driver could be so used to the idea of following the road he is on that he misses his exit road; or a person so used to meeting his friend at a dining room at lunchtime does not stop to recall that his friend told him recently that he would not be turning up today.

As a bit of a daydreamer, I have had to behave against my natural inclination in order to improve at avoiding auto-pilot errors. Eg. Missing my train stop because I am thinking so hard about an idea.

And isn’t that kind of focused thinking something to be applauded? Not if it gets my dinner burned in the other room!

At work, I know that I am ‘on the clock’ and so I just can’t go into daydream mode. Maybe my work environment is part of that mental calibration to frequently step back from what I am doing ‘in the now’ and to think about a whole framework of tasks which need doing.

 

Big Picture Principles Store

 

The big picture is like the contents page of a book. It is the framework of headings and sub-headings under which the passages of a book are written. The big picture is also like a map of a town in that a room might be inside a house which is on a street which is in a town. The room is framed by containers in a similar way to how written passages are contained by sub-headings which are contained by headings and all of those headings are contained by a book with a title. The big picture is also like the list of dog breeds which come under the heading ‘Dog’. The big picture is the container which contains somewhere within it the system which you are examining. e.g. ‘Dog’ contains ‘Highland terrier breed’.

And maybe a chapter in a textbook only means something when you consider where the chapter sits within the overall chapter and section framework of the book. In Biology, you might have studied all about photosynthesis and be able to answer questions about it but you might not index it in your mind as part of a broader topic of ‘energy sources of living things’.

And when you walk past a green tree on the way home, you might not ‘join the dots’ that the academic subject meets up with the real world right in front of you.

The big picture is a reminder to you not to limit your area of thinking. If you wait at a train platform for a train and think that it is the only platform on which your train can arrive then you will not notice your train arriving on the platform opposite you. By regularly thinking about the big picture, you can notice that there is a variety of platforms contained by the railway station. You are more likely to consider the other platforms from that time onwards.

You want a discipline of stepping back from the immediate thought and considering the syllabus or big picture framework which it sits in. Otherwise, you might complete a task perfectly but fail at the bigger task.

Also, problem solving often involves ‘joining the dots’ between systems so that they are applied in a useful way. So, in maths, a chapter to do with trigonometry and a chapter to do with quadratic equations might both need to be harnessed to answer a maths question involving both concepts.

On a more mundane level, maybe the shoe shop is closed but you need new shoes. You do not know any other shoe shop but, at an abstract level, you know that a telephone directory is a way to discover any type of shop. So the fresh problem is resolved by ‘joining the dots’ of the information which you have to play with.

Caches of information come together. So it is a good thing that your mind throws at you the idea of a telephone directory when you ask it, “How can I find a shoe shop?”. Somehow, you have previously indexed the answer to the more abstract question, “How can I find a shop?” but your mind still throws it into your thought space when you are thinking specifically about shoes.

So does that mean that the practice of indexing solutions is very important? Yes. When a tough maths question appears, during homework, you revisit all topics of your textbook if the solution approach has stumped you. But, in an exam without textbooks, you need a cache of your syllabus in your memory.

And on top of that principle, probably a habit of deliberately revising the existence of solutions which could greatly benefit you one day. Eg. A particular search engine that is specialising in a particular type of search is going to be preferable perhaps to a one-size-fits-all search engine; but if you forgot that you once encountered such a search engine then its existence won’t be able to help you solve a search problem.

A passion for indexing solutions!

And a thinking habit for developing a big picture awareness is simply a habit of asking yourself: “Where does this information fit within a bigger picture?”

 

Useful Methods’ Store

 

Note: This overlaps with some of what I said in the Big Picture Store section above.

There is information worth remembering because it is useful for problem solving. For instance, it is good to know how to convert miles into kilometres. Unlike the memorised list store, the useful methods store is specifically concerned with listing ways of doing things (methods).

In mathematics, there are problem solving methods which can be applied time and time again to different problems. It is good to have memorised such frequently used methods so that you can draw them from memory at a time when they may prove helpful to you.

On a more down-to-earth level, it is good to know the phone number of a doctor in case you need to ring one in a hurry. It is good to know a bus timetable or which months end on the 30th. If you do not take the time to learn manipulatable information or methods then you limit the number of conclusions you can come to.

 

Label Interpretations’ Store

Communication involves symbols (writing, speech, etc.) which a person sends out and receives. The interpretation of those symbols or labels can take time when a word or phrase is new to you. e.g. ‘Promissory estoppel’. It is to your advantage to make a note of new language and to regularly revise it. In this way, when you next encounter the language, you can quickly interpret its meaning.

Note: A weakness of using labels is that it can be easy to hear a term without interpreting what it means. This is a block on how much information is available to us. There needs to be a habit of seeking the meaning behind a label in order to be able to work with the wider system of information.

 

Best Evidence’ Store

There is a difference between knowing something to be true and being virtually certain that something is true. We seek the evidence and best evidence that exists for the truth of a statement (e.g. The sky is blue and so I doubt it will rain in the next four hours). Sometimes we rely on the best evidence to decide our opinion. e.g. "I am so sure that the person in the distance is my office colleague that I will wave."

It is important to bear in mind that the person in the distance is not known to be the person’s colleague. However, the person chooses to act upon the virtual certainty that the person in the distance is the office colleague. It is vital not to forget that the status of the person’s belief is ‘based on best evidence’ rather than ‘based on known fact’.

A good thinker needs the habit of questioning his belief in a statement or idea. He needs to know if a belief is based on fact, virtual certainty, or lesser proof - even after he has decided to act as if the proposed idea is true.

Balanced against that are the challenges of ‘real time’ reactions to the world around us. Maybe my reactions are more knee-jerk when I am out and about walking than when my work environment encourages me to be reserved in my decision making. But I am sure that many people are naturally going to take their time over reacting whether they be in the work place or in a recreational environment.

Through the exercise of challenging what we see, we can improve our sense of what is ‘first pass’ information and what is confirmed information.

                         

Updated Information’ Store

We modify our beliefs when fresh information arrives. We also modify our view of the world when we see things changing. It can be difficult to remember that things have changed. This was seen in the auto-pilot example where a person is so used to going to a dining room that he does not recall that his schedule has been updated and that the friend will not be turning up today.

So, it is a good habit to ask if our picture of the world has been updated: "Is there a fresher version of the facts for me to look at?". This overlaps with the idea of asking for the best evidence as to the truth of a belief - as well as overlapping with the idea of the auto-pilot.

 

Observation’ Store

This simply involves remembering to observe. Some people observe the world out of habit. Other people do not observe the world in an active way. If a car is speeding towards a person then his reflex action is to get out of the way - he responds to the observation. Compare that with a situation where reflexes are not involved. e.g. Noticing that a plug is not in the wall socket but still wondering why the television is not working. Also, compare that with someone who is aware that the television is not working but has not taken the time to observe the wall socket.

So, observation is a great tool to have at your disposal; and it is even better when you ask, "Does what I am observing have any effect upon how efficiently I can achieve my goal?". There is a sea of information available to us; much of it is not helpful; some of it is. We need to see the sign which warns us that a shop is closing early today. We need to pick up on the sigs which guide our thinking.

enefit of good observation is that you can notice things and learn about them despite not having directly encountered those things. e.g. You might learn the address of a company by reading its stationery despite reading that stationery to get at other information written on it.

Observation is something that can be improved by deliberately taking time to observe things.

 

Recency’ Store

Recent events are often relevant in the present moment. For instance, a person says that his car is being serviced. A minute later, you ask the person for a lift to your home. By not focussing on recent information, your world view is limited. This is a specific case of the updated information store. You can progress from limited knowledge to an informed view if you pay attention to the update which is the fact that the person’s car is not available.

 

It’ Store

Communication tends to involve ‘reading between the lines’. If I start talking about someone called Jane then I might later refer to ‘she’ and it is implied that I mean Jane. The use of the word ‘it’ is like this too. The labels store was concerned with the need to see beyond a label and look at its meaning. The ‘it’ problem is a specific case of the labels store problem: there is a need to see beyond words like ‘he’, ‘she’and ‘it’. Somewhere in your mind, you need to be keeping a note of what specifically ‘it’ is representing; and whom ‘he’ or ‘she’ is representing.

As an aside, often, regulations can be stated without giving you the specifics of the actors. Eg. “No ball games” probably means that nobody can play ball games but is there ever an exception? Surely the person who owns the estate can do what he likes with a football!

No parking”.. but then someone parks and has a valid reason why.

 

Mistakes and Efficiency’ Store

Some activities have mistakes associated with them. For instance, in algebra or computer code writing, there may be T1 and T2 (labels for two specific numbers) which might be confused with one another if you are trying to work quickly. In coding, this sometimes happens when I copy and paste code for re-use elsewhere but now T1 needs to be T2 in its spelling or else the logic will not work in its new context in the computer program.

In the real world, you might make tea using time inefficiently by putting teabags in a pot before you put the kettle on. If done the other way round, you can make a cup of tea in less time by starting the kettle and putting the teabags in while water is boiling. That is not necessarily a mistake but it is an inefficiency.

If we memorise our mistakes and inefficiencies then hopefully we can repeat them less often in future.

Since frequently performed activities have their own mistakes (or inefficiencies) attached to them which you might regularly fall into the trap of making, it makes sense to memorise the mistakes which you make in each situation and to memorise the inefficient things you do in each situation.

So, a person sitting down to a game of chess might spend time recalling the typical mistakes he makes and thus reminds himself what not to do. A driver who regularly gets into the wrong driving lane can be prepared for the mistake which he does out of habit - and thus avoid making the mistake.