"and" and "or" in Set Theory and English

Relating the English words "and" and "or" to concepts in set theory and logic can be done by starting with basic examples and reasoning by analogy to more complex cases. It must also be remembered that human language is generally less precise than formal logic. In natural language, we recognize a range of meaning for a word and so we must look not for one definition, but a few definitions. Here, below, I attempt to give a listing of the use-cases of the words "and" and "or".

and:

1. between two objects, P and Q means the set \(\{P, Q\}\).
2. between two sets, P and Q means the set \(P\cup Q\).
3. between two propositions, P and Q means that \(\{P, Q\}\) is contained in the set of all true propositions.

A possible objection to use-case 2 for "and" is that the conjunction "and" is used in the definition of intersection and so P and Q on sets should refer to set intersection, not union. However, a set or a list or the designation of a category refer to objects (conceptual ones or otherwise). Our best analogy is with with use-case 1, where we have both items included in the set. This conforms to customary usage. For example, in the designation The Department of Math and Computer Science, customary usage indicates this means "the department which contains math courses and computer science courses", or, more expansively, "the department which contains math courses and contains Computer Science courses". We recognize this rendering as use-case 3, which serves as the argument for use-case 2. If we called it The Department of Math or Computer Science or The Department of Math and/or Computer Science, this would imply that the course name would be valid if it contained math courses only, computer science courses only, or both types of courses. The name of such a department would not inspire confidence in the applicant interested in one of these areas specifically, that the department offered the kind of courses he/she was interested in.

or:

1. between two objects, exclusively, either P or Q means exactly one element of the set \(\{P, Q\}\).
2. between two object, inclusively, P or Q means any element of the set \(\{P, Q\}\).
3. between two sets, exclusively, x is in either P or Q means \(x\in P\cup Q - P\cap Q\).
4. between two sets, inclusively, x is in P or Q means \(x\in P\cup Q\).
5. between propositions, exclusively, either P or Q means \(|\{P, Q\}\cap T| = 1\), where \(T\) is all true statements.
6. between propositions, inclusively, P or Q means \(\{P, Q\}\cap T\ne \emptyset\).

At least some of the above inclusive cases are sometimes expressed in English using "and/or" in place of "or" as a means of indicating the inclusive nature of the conjunction intended. The exclusive uses are sometimes emphasized by using the word "either", as above.

We can take our example of The Department of Math and Computer Science and express it in a way that uses set intersection. Let M be the set of departments in a university offering the majority of mathematics courses and let C be the set of departments offering the majority of Computer Science courses. Then \(M\cap C\) will either be a one element set containing The Department(s) of Math and Computer Science or the empty set, meaning there is no such department at the given university. However, this is forcing the issue and use case 2 for "and" is the more natural expression.

We can also see set intersection in the exclusive use-cases of "or". Use-case 3, in particular, can be expressed as \(P\cup Q - P\cap Q\). Depending on the context, this might appear redundant. For example, we might say that a washroom is made for either males or females. We take M as washrooms intended for male occupancy and F as washrooms intended for female occupancy. This can be seen as use-case 3, but if the context of the conversation is public, multiple-occupancy washrooms the intersection is empty. In fact, the nature of the statement may be to emphasize the fact that the intersection is empty and the speaker is really saying \(M\cup F = M\cup F - M\cap F\), that is, "all of the washrooms I have in mind are for single sex occupancy, none of the washrooms I have in mind are accepting of both male and female". There may be other washrooms where the intersection is non-empty (e.g., single occupancy), but not the washrooms the speaker has in mind.

Shortcuts in Teaching Mathematics Lead to Quicker Dead Ends

There are lots of times in life where shortcuts lead to efficiency. Efficiency is great, provided it is actually effective at achieving your goals (or the goals you should have). On the other hand, you can sometimes efficiently achieve a short term goal only to find yourself at a dead end later on.

If you're in a maze, not every step closer to straight-line proximity with the cheese is necessarily actually getting you closer to eating the cheese. In a maze, you can be right beside the cheese with just a single wall between you and the cheese and you might be as far away as possible from being able to eat the cheese. Sometimes you need to head in a direction that seems to take you further from your goal, in order to be closer to achieving it. Do you want to have a really strong smell of cheese or do you want to actually eat cheese?

Take weightlifting as an example. Improving your technique often takes you back to lighter weight. Your goal is to lift lots, right? Well, lighter weight seems like the wrong direction if you're thinking naively. But improved technique will take you further along a path that can actually lead to "having cheese" rather than just "smelling cheese", both because you will be less prone to injuries which will set you back and because you will be training your muscles to work along a more efficient path. So, suck it up!—and reduce the weight if you have to.

What follows is a series of common math shortcuts and suggestions for avoiding the pitfalls of the shortcuts (like, avoiding the shortcuts 😜). Some of these are statements that arise from students who hear a teacher express a proper truth the first time. But, when asked to recall the statement, the student expresses an abbreviated version of the statement that differs in a way that makes it not true anymore. Sometimes the student really did understand, but was experiencing a "verbally non-fluent moment" or just didn't want to expend the energy to explain. The teacher, trying to be positive, accepts this as a token of the student paying attention and then gets lazy himself and doesn't add a constructive clarification.

In any event, the quest for simplicity and clarity has pitfalls. Merely making something appear simple is not a sure path to understanding. Go deeper. However, I don't promise that I will never be guilty of any of these, because, explaining is hard and takes time and we don't always want to.

Two Negatives Make a Positive

Why it's bad: It is a false statement. If I stub my toe and later hit my thumb with a hammer it is not a good thing, it is cumulatively bad (two negative displacements 🤔). Talking about multiplication as if it were magic, may not always produce a desire to understand (it does for some of us), but instead a desire to check out from the understanding business and maybe sigh and say that math just isn't your thing. #badlearningoutcome

How it happens: Quick summary (not intended to be fully accurate) and humor. Going back to the basic full statement and providing examples might seem exasperating and so it is tempting to call it magic tell students not to worry: it's magic that works and you're allowed to use it on the test. Unfortunately, the quick summary could leave students flustered and believing that math isn't something they can ever become good at because some of it gets "explained" as if it was magic. 

Better: A negative number multiplied by a negative number, produces a positive number.

Expansion: Negative numbers are a way of expressing a change in direction, as in, (sometimes figuratively) a 180° change in direction. $10 is something you want to see coming your way. -$10 can go to someone else.

With this in mind you can explain that subtraction is equivalent to adding a negative number. A negative number is equivalent to multiplying the corresponding positive number by -1. The negative on the front of the number means you move to the "left" on the number line (a useful metaphor). If you pick a random number on the number line and multiply it by -1 it is like mirroring it about zero (0). Also, multiplying by -1 is equivalent to taking its difference with zero. That is,
$$0-z=-1\cdot z = -z,$$ which are more or less a part of the definition and development of negative numbers in a technical sense.

Displacement is one of the easiest illustrations to use. Suppose I am standing on a track that has chalk lines marked from -50 m to 50 m. I am standing at the 0 m mark facing the end with positive numbers. Instructions show up on a board which are in a few different forms:

1. Advance 10 m. 
2. Go back 10 m.
3. Move 10 m.
4. Move -10 m.

It is easy enough to see that 1 is equivalent to 3 and 2 to 4. Following a list of such instructions will result in a predictable finishing position which can be worked out one step at a time in order or can be put into a single mathematical expression. Commutativity and associativity can be explored by comparing differences in order applied to the mathematical expression as well as the list of expressions. I can reorder the sequence of instructions and produce a new expression that gives the same result or I can tweak the expression and spit out revised instructions that parallel the revised expression and produce the same result. Arithmetic is intended to express very practical things and so if something you do with your math expression causes a disconnect with the real life example, you are guilty of bad math. The problem will not be with commutativity or associativity, but with your failure in implementation. It is worth investing a good deal of time on this, but it will probably have to be brought up again and again when you move on to other things because students sometimes get slack on their efforts at understanding when something appears too easy.

The next step is to understand that reversing direction twice puts you back on your original direction. We can see how this works on the track by working with the displacement formula:
$$d = p_f - p_i,$$ where \(p_f\) is final position and \(p_i\) is initial position. It's easy to illustrate on a chalk/white board that if I start at the -20 m mark and travel to the 50 m mark I will have traveled 70 m, not 30 m. Using the formula to get there will require combining the negative signs in a multiplication sense.

I would love to have a simple example that involves two negative non-unity numbers, but while I've done this arithmetic step countless times in the solving of physical and geometrical problems, I have trouble isolating a step like this from a detailed discussion of some particular problem and still retaining something that will lead to clearer understanding than the displacement example.

Heat Rises

Why it's bad: It is a false statement.

How it happens: Unreasonable laziness.

Better: Hot air rises (due to buoyancy effects).

Expansion: Fluids (liquids or gases) circulate in part due to differences in density of the fluid(s) and  temperature affect that density. More dense fluid goes down and less dense fluid moves up. This only applies to one heat transfer mechanism which we might call "mass flow". It is not applicable to heat conduction which 

So, why does hot air rise? The cause is density differences. Denser gasses will fall below less dense gasses and "buoy up" or displace the less dense gasses. Density is affected by both atomic weight and how far apart the molecules are. The "average distance" between molecules is affected by temperature. The warmer the temperature the faster the particles move and bang into each other and tend to maintain a sparser configuration making them less dense (collectively). Higher density means higher pressure. Buoyancy is the result of the difference in pressure from the bottom (high density, high pressure) to the top (low density, low pressure) of a fluid. (Pressure is like a push spread out over an area.)

You can either take the perspective of the less dense fluid and think about positive buoyancy (with less dense stuff, the buoyant force overcomes gravitational pull) or the perspective of the more dense fluid and think about negative buoyancy (with more dense stuff, the gravitational pull overcomes the buoyant force).

If you want a shortcut, "hot air rises" is better than "heat rises". It's a small difference, but it is still the difference between true and false.

Cross Multiply and Divide

Why it's bad: It does not require the student to understand what they are doing. For some students, this is all they remember about how to solve equations. Any equation. And they don't do it correctly because they don't have a robust understanding of what the statement is intended to convey.

How it happens: Proportions are one the critical components of a mathematics education that are invaluable in any interesting career from baking, to agriculture, to trades, to finance, to engineering, to medicine (or how about grocery shopping). Teachers are rightly concerned to turn out students who can at least function at this. It breaks down when it is disconnected from a broader understanding of equations. Students may look at the easy case of proportions as a possible key to unlocking other equations, which has a degree of truth to it. However, it is important to emphasize the reason why cross multiply and divide works (below). Without understanding, there is little reason to expect the simple case to spillover to help in the hard cases.

Better: Always do the same thing to both sides of an equation. If both sides are equal, then they are equal to the same number (for a given set of input values for the variables). If I do the same thing to the same number (expressed differently on each side), I should get the same result. The result might be expressed differently, but it should still be equivalent.

For simple equations, often the best operation to do (to both sides, as always) is the "opposite" of one of the operations shown on one or both sides of the equation. Cross multiply and divide is a specialization of this principle that is only applicable in certain equations which must be recognized. The ability to recognize them comes from having good training on the handling of mathematical expressions (see remarks on BDMAS).

Expansion: Provided "do" means "perform an operation", the above is pretty valid. The other type of thing you can "do" is rearrange or re-express one or both sides of an equation such that the sides are still equivalent expressions. Rearrangement does not have to be done to both sides of the equation because it does not change its "value". When operations are performed, they must be applied to each side in its entirety as if each entire side was a single number (or thing). Sometimes parentheses are used around each side of the equation so that you can convey that distributivity applies across the "=" sign, but from there, distributivity (or lack thereof) needs to be determined based on the contents of each side of the equation.

Most Acronyms (but Especially FOIL and BDMAS)

Why they're bad: My objection is qualified here (but not for FOIL). An acronym can sometimes summarize effectively, but it is not an explanation and does not lead to understanding. In rare cases, understanding may not be critical for long term proficiency, maybe. But an acronym is a shoddy foundation to build on. If you're trying to make good robots, use acronyms exclusively.

How it happens: Acronyms can make early work go easier and faster. This makes the initial teaching appear successful—like a fresh coat of paint on rotten wood. Teacher and student are happy until sometime later when the paint starts to peel. Sometimes after the student has sufficient understanding they may continue to use certain acronyms because of an efficiency gain they get from it, which may lead to perpetuating an emphasis on acronyms.

Better: Teach students to understand first. Give the student the acronym as a way for them test if they are on the right track when you're not around. Very sparingly use as a means of prompting them to work a problem out for themselves. (My ideal would be never, but realistically, they need to be reminded of their back up strategy when they get stuck.) Never, ever take the risk of appearing to "prove" the validity of operations you or others have performed by an appeal to an acronym (unless it is a postulate or theorem reference)—that's not just bad math, it is illogical.

Expansion: Certain acronyms, if you stoop to use them, can possibly be viewed as training wheels. Maybe BDMAS qualifies. But is there a strategy for losing the training wheels or are the students who use the acronym doomed to a life of having nothing else but training wheels to keep from falling over?

BDMAS is a basic grammar summary. But you need to become fluent in the use of the language. A good way to get beyond the acronym is to have clear, practical examples of things you might want to calculate that involve several operations. Calculating how much paint you need is a good way to help convey how orders of operations work. Before you calculate the amount of paint you need, you get the surface area, \(s\). The total surface area is a sum of the surface areas of all surfaces I want to paint. If I have the dimensions of a rectangular room, I can get the area of each wall and add them together. To make the example more interesting, we will omit to paint one of the long walls. Because of order of operations giving precedence to multiplication over addition, I have a simple expression for a simple thing:

$$s = lh + wh + wh = lh + 2wh = h(l + 2w).$$ If you explain how to arrive directly at each expression without using algebra (with reference to simple diagrams), the meaning of each expression can be understood at an intuitive level. Understanding the geometry of the situation gets tied to understanding of the sentences you are making in the language of math. To get the number of cans of paint \(N\), you need coverage, \(c\) in area per can. Then \(N = s/c\). And now, if you didn't already demonstrate how parentheses support the act of substitution in the surface area development, now is a good time, because now you can use substitution for one of the ungainlier expressions for the surface area and get:

$$N=(lh + 2wh)/c.$$ If you also walk through how to do the calculation in multiple simple steps you can draw the parallels with the steps you would take in calculating using the above formula. I realize substitution showed up much later in the curriculum I received than order of operations but I believe this is a mistake. Even if the student is not expected to use substitution in grade 5, why not let them have an early preview so it doesn't seem like it's from outer space when they need it?

Oh, yes, and FOIL. Don't use FOIL outside the kitchen. Better to teach how distributivity applies to multiterm factors which will again be something like a grammar lesson and can incorporate some substitution (e.g., replace \(x + y\) with \(a\) in one of the factors) or "sentence" diagramming, which is beyond the scope of this post.

Using Keywords to Solve Word Problems

Why it's bad: It does not require the student to understand what they are reading which masks long-term learning problems, and leads to long-term frustration for the student.

How it happens: Students normally want something to do to help them get unstuck. Telling them they have to understand what they are reading isn't the most helpful and giving a bunch of examples of similar expressions and finding ones they already understand seems like a lot of work to go through. Keywords are fast and easy to tell students and are often enough to get stronger students started.

Better: Find analogous expressions that are already understandable to the student. If you can find statements that the student already understands at an intuitive level, you may be able to point out the similarity between the statement they are having trouble with and the statements they already can relate to.

Expansion: I am not aware of a standard treatment of this issue that meets my full approbation. We use language everyday and we don't use keywords to figure out what people mean by what they are saying. We shouldn't use language any differently with a word problem. It's the same language!

The words used in grade 6 word problems are all everyday words. What is needed is the ability to understand and use the same words in some new contexts. Providing a lot of examples is probably the way forward with this. Being able to restate facts in other equivalent ways may be a good indication of understanding and accordingly a good exercise.

It's important to recognize that language is complex and takes time to learn. Not everyone will learn it at the same rate and having a breadth and variety of examples with varied complexity is probably necessary for students who struggle more with it. Unfortunately, school doesn't support this kind of custom treatment very well (Cf. "growth" as per Franklin, Real World of Technology).

Conclusion

Explanations, examples, and exercises that lead to genuine understanding are much needed by math students at all levels. I do not believe in the inherent value of making students suffer, figuring everything out for themselves by not giving them the best possible chance of understanding the material with good instruction. But undue opportunities to opt out of understanding are a disservice to them. Training wheels have their place, but we should make every effort to avoid seeming to point to training wheels as any student's long term plan for achieving competency in a subject area.

Mathematics: Principles and Formulae

I gather that when most people think about mathematics, they think about two things, and in this order: numbers and formulas. In doing mathematics, most students are quick to add something else, though not by name: recipes – usually grossly oversimplified recipes. It is this tendency that runs them into so much trouble and many are the able teachers of mathematics that have tried to slap them (figuratively) into using something better, namely, principles.

A famous work of Sir Isaac Newton was titled, in Latin, Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"). But you don't need to be as smart as Sir Isaac Newton to understand the importance of principles as superior to recipes. First of all, what's the difference?

A recipe is a sequence of steps or components with no (particular) reference to reasons. Computers do recipes - algorithms. Students that try to be computers (and nothing more) short-change themselves because there is already a much more efficient tool for that work than themselves. (And thus it is little wonder they complain about the pointlessness of the learning they are engaged in. Often of their own accord, they are not learning the really important lessons from mathematics that they should be.) Principles help me to justify, or at least partly understand, the steps of the recipe and apply it usefully. The ability Principles are one of the key advantages you have over your calculator.

People try to simplify the setting up and solving of proportion problems. They do it this way: "cross-multiply and divide." They don't really know why. It is only a recipe to them. How do you know which numbers to multiply and divide? It increases the number of things to memorize and (I have observed) it distracts people from the real principles. Forget about cross-multiplying and dividing. If I know a/b = c/d, I do what I always do with equations: do "opposites" and do the same thing to both sides (subject to the constraint that the operation must be permissible on the values in the equation). If I want to isolate a, I have a divided by b, so I do the opposite of divide by b, namely, I multiply by b, and I do it to both sides. The end. "Cross-multiple and divide" doesn't bridge well into solving other forms of equations, is unnecessary, gives little clue as to why it is supposed to work, and increases the number of things to memorize. How do you solve an equation?

1. Do "opposites."
2. Do the same thing to both sides.
3. Eliminate any apparent solutions which do not satisfy the original equations or the constraints of the problem (which may be implicit, such as, "negative area is not allowed").

Throw "cross-multiply and divide" and its ilk into the garbage bin.

Here are a few very generic principles to apply to problem solving:

1. Don't ask "What should I do?", ask "What is true?"  You won't make very much ground figuring out what to do if you haven't established a few things that you know about the situation. If you want something to do, then do this: find out more true things and organize them. This often starts with writing down some given information. Perhaps you print a few formulas and consider which variables you know and which you don't. Ask yourself, "Do I know any other equations that involve these unknowns? Are they valid in this situation?" Don't worry too much about whether an equation is going to give you the final answer, but concern yourself with whether it actually holds true in the situation. Thomas Edison didn't invent so many things by only trying things he knew would work. You also will write down formulas and find that they do not help you solve the problem – even though they are true and valid in the situation, they might not be useful to you. It isn't a mistake (yet), to write it down as a tool in the toolbox.
2. Don't ask "Why can't I do this?", ask, "Can this method or statement be proven correct?"  People want to apply rules that are applicable to one type of operation to a completely different operation. They see that they are allowed to make the statement a (b + c) = a b + a c and want to know why they can't say log(b + c) = log(b) + log(c).  This is one or both of two fundamental misunderstandings. Either it is assuming that the parentheses have the same meaning in both cases (which means they don't know what a function is and certainly not what a log is) or they misunderstand the importance of proof. This later misunderstanding in particularly important. Often the student assumes that a similar appearance means a similar treatment is permitted. The fact is, the statement a (b + c) = a b + a c, where a, b, and c are numbers can be proven. Our acceptance of the statement is based on proof from simpler principles. We don't "just know" that that's true (although this example is among the more intuitive that I could have chosen). Similarly, we don't "just know" that the statement  log(b + c) = log(b) + log(c) is true. If we can prove somehow that it is true, then we know we're allowed to make this expansion. On the other hand, if we can find a counter-example, we know it is not true. So here's a counter-example to the above (very, very silly) suggestion: log(10 + 10) = log 20 < 2, but log(10) + log(10) = 1 + 1 = 2; so they are clearly not equal. 
1. This doesn't mean you need to do proofs, but you should be satisfied that a proof exists for what you are doing and that you are not assuming something which cannot be established.
1. (Did you notice yourself proving your own work while you applied that principle? Tricky, huh?)
3. Look for patterns. Patterns have some kind of rule behind them. If you can determine the rule that makes the pattern, it may reduce work and memorization required. The rule will be more generic than the sampling that you noticed the pattern in – otherwise, it is a false pattern, because a counter-example exists. Don't assume the pattern is real. Test it, prove it, disprove it, as may be necessary.
4. Make analogies. The main reason for teaching the principle of proportional triangles is not to teach students how to solve triangle problems – although there's good uses for that. Proportions are so common place that everyone should get comfortable with the archetypical example: proportional triangles. There are analogies between force and momentum, between torque and force, momentum and rotational inertia, fluid pressure and electricity. Some things are more intuitive to us than others and if an analogy exists between something you are familiar with and something you are not, it can help you with a "working understanding" of the unfamiliar concept.

Understanding principles helps you to evaluate proposed solutions or solution methods. Recipes can't do that very effectively. Recipes also don't work well on a "new to you" problem. Knowing principles is a key point of difference between people who know how to implement a given method of solution and someone who can actually develop a method of solution to a problem they have not previously seen. This is true problem solving. Perhaps the most important thing anyone can do to improve their problem solving ability is to prize principles like gold and recipes as mere silver, maybe only bronze.