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Visible algebra II

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Wolfram website. The code for the demos is in the Mathematica notebook algebra2.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

More about visible algebra

I have written about visible algebra in previous posts (see References). My ideas about the interface are constantly changing. Some new ideas are described here.

In the first place I want to make it clear that what I am showing in these posts is a simulation of a possible visual algebra system.  I have not constructed any part of the system; these posts only show something about what the interface will look like.  My practice in the last few years is to throw out ideas, not construct completed documents or programs.  (I am not saying how long I will continue to do this.)  All these posts, Mathematica programs and abstractmath.org are available to reuse under a Creative Commons license.

Commutative and associative operations

Times and Plus are commutative and associative operations.  They are usually defined as binary operations.  A binary operation $*$ is said to be commutative if for all $x$ and $y$ in the underlying set of the operation, $x*y=y*x$, and it is associative if for all $x$,$y$ and $z$ in the underlying set of the operation, $(x*y)*z=x*(y*z)$. 

It is far better to define a commutative and associative operation $*$ on some underlying set $S$ as an operation on any multiset of elements of $S$.  A multiset is like a set, in particular elements can be rearranged in any way, but it is not like a set in that elements can be repeated and a different number of repetitions of an element makes a different multiset.  So for any particular multiset, the number of repetitions of each element is fixed.  Thus $\{a,a,b,b,c\} = \{c,b,a,b,a\}$ but $\{a,a,b,b,c\}\neq\{c,b,a,b,c\}$. This means that the function (operation) Plus, for example, is defined on any multiset of numbers, and \[\mathbf{Plus}\{a,a,b,b,c\}=\mathbf{Plus} \{c,b,a,b,a\}\] but $\mathbf{Plus}\{a,a,b,b,c\}$ might not be equal to $\mathbf{Plus} \{c,b,a,b,c\}$.

This way of defining (any) associative and commutative operation comes from the theory of monads.  An operation defined on all the multisets drawn from a particular set is necessarily commutative and associative if it satisfies some basic monad identities, the main one being it commutes with union of multisets (which is defined in the way you would expect, and if this irritates you, read the Wikipedia article on multisets.). You don't have to impose any conditions specifically referring to commutativity or associativity.  I expect to write further about monads in a later post. 

The input process for a visible algebra system should allow the full strength of this fact. You can attach as many inputs as you want to Times or Plus and you can move them around.  For example, you can click on any input and move it to a different place in the following demo.

Other input notations might be suitable for different purposes.  The example below shows how the inputs can be placed randomly in two dimensions (but preserving multiplicity).  I experimented with making it show the variables slowly moving around inside the circle the way the fish do in that screensaver (which mesmerizes small children, by the way — never mind what it does to me), but I haven't yet made it work.

A visible algebra system might well allow directly input tables to be added up (or multiplied), like the one below. Spreadsheets have such an operation In particular, the spreadsheet operation does not insist that you apply it only as a binary operation to columns with two entries.  By far the most natural way to define addition of numbers is as an operation on multisets of numbers.

Other operations

Operations that are associative but not commutative, such as matrix multiplication, can be defined the monad way as operations on finite lists (or tuples or vectors) of numbers.  The operation is automatically associative if you require it to preserve concatenation of lists and some other monad requirements.

Some binary operations are neither commutative nor associative.  Two such operations on numbers are Subtract and Power.  Such operations are truly binary operations; there is no obvious way to apply them to other structures.  They are only binary because the two inputs have different roles.  This suggests that the inputs be given names, as in the examples below.

Later, I will write more about simplifying trees, solving the max area problem for rectangles surmounted by semicircles, and other things concerning this system of doing algebra.

References

Previous posts about visible algebra

Other references

 

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Visible algebra I supplement

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook algebra1.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Active calculation of area

In my previous post Visible algebra I constructed a computation tree for calculating the area of a window consisting of a rectangle surmounted by a semicircle. The visual algebra system described there constructs a computation by selecting operations and attaching them to a tree, which can then be used to calculate the area of the window. 

I promised to produce a live computation tree later; it is below.

Press the buttons from left to right to simulate the computation that would take place in a genuine algebra system.  Note that if you skip button 2 you get the effect of parallel computation (the only place in the calculation that can be parallelized).

In Visual Algebra I the tree was put together step by step by reasoning out how you would calculate the area of the window: (1) the area is the sum of the areas of the semidisk and the rectangle, (2) the rectangle is width times height, (3) the semidisk has half the area of a disk of radius half the width of the rectangle, and so on.  So the resulting tree is a transparent construction that lets you see the reasoning that created it.  

The resulting tree could obviously be simplified.  But if you were designing a few such windows, why should you simplify it?  You certainly don't need to simplify it to speed up the computation.  On the other hand, if you are going on to solve the problem of finding the maximum area you can get if the perimeter is fixed, you will have to do some algebraic manipulation and so you do want a simplified expression.    

Later, I will write more about simplifying trees, solving the max area problem, and other things concerning this system of doing algebra.

Remark

What I am showing in these posts is a simulation of a possible visible algebra system.  I have not constructed any part of the system; these posts only show something about what the interface will look like.  My practice in the last few years is to throw out ideas, not construct completed documents or programs.  (I am not saying how long I will continue to do this.)  All these posts, Mathematica programs and abstractmath.org are available to reuse under a Creative Commons license.

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1.000… and 0.999…

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Note: This post uses MathJax. If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
 
Recently Julian Wilson sent me this letter:
It is well known that students often have trouble accepting that $0.999\ldots$ is the same number as $1.000\ldots$.  However, there is at least one context in which these could be regarded as in some sense as being distinct. In a discrete dynamical system where the next iterate is formed by multiplying the current value by 10 and dropping the leading digit, and where you make a note at each iteration of the first digit after the decimal point, then 0.9999… generates a sequence of 9s, whereas 1.0000… generates a sequence of 0s. The imagery is of a stretching a circle, wrapping it ten times around itself and recording in which sector (labeled 0 to 9) you end up.
 
From the dynamical systems perspective, being in state 9 (and remaining there after each iteration) is different from being in state 0.
The $0.9999\ldots =1.0000\ldots$ equation is associated with several conceptual difficulties that math students have, which I will describe here.

The decimal representation is not the number

Another way of describing the equation is to say that "$0.999\ldots$" and "$1.000\ldots$" are distinct decimal representations of the same number, namely $1$. Julian's proposal provides a different interpretation of the notation, in which "$0.999\ldots$" and "$1.000\ldots$" are strings of symbols generated by two different machines.  Of course, that is correct.  But they are both correct decimal notation that correspond to the same number.

Mathematical writing will sometimes use notation to mean the abstract mathematical object it refers to, and at other times the text is referring to the notation itself.  For example,

$x^2+1$ is always positive.

refers to the value of $x^2+1$, but

If you substitute $5$ for $x$ in $x^2+1$ you get $26$.

refers to the expression "$x^2+1$".  Careful authors would write,

If you substitute $5$ for $x$ in "$x^2+1$" you get $26$.

This ambiguity in using mathematical notation is an example of what philosophers call the "use-mention" distinction, but they apply the phrase to many other situations as well.  Mathematicians have an operational knowledge of this distinction but many of them are not consciously aware of it.

Definitions

A decimal representation of a number by definition represents the number that a certain power series converges to. The two power series corresponding to 1.000… and to 0.999… both converge to 1:

\[1+\sum_{i=1}^{\infty}\frac{0}{10^n}=1\]

and

\[0+\sum_{i=1}^{\infty}\frac{9}{10^n}=1\] 

They are different power series (mention) but converge (use) to the same number.

Most students new to abstract math are not aware of the importance of definition in math. As they learn more, they may still hold on to the idea that you have to discover or reason out what a math word or expression means.  In purple prose, THE DEFINITION IS A DICTATOR. 

This does not mean that you can understand the concept merely by reading the definition.  The definition usually does not mention most of the important things about the concept.

Completed Infinity

A common remark by newbies about $0.999\ldots$ is that it gets closer and closer to $1$ but does not get there. So it can't be equal to $1$.  This shows a lack of understanding of completed infinity.  The point is that the notation "$0.999\ldots$" refers to a string beginning with "$0.$" and followed by an infinite sequence of $9$'s.  Now "$s$ is an infinite sequence of $9$'s" means precisely that $s$ has an entry $s_n$ for every positive integer $n$, and that $s_n$ is $9$ for every positive integer $n$. 

  • The expression is gradually unrolling over time, and does not ever "get there". 
  • All the nines are already there.

Both the preceding sentences are metaphorical.  They are about how you should think about "$0.999\ldots$".  The first metaphor is bad, the second metaphor is good.  Neither statement is a formal mathematical statement.  Neither statement says anything about what the sequence really is.  They are not statements about reality at all, they are about how you should think about the sequence if you are going to understand what mathematicians say about it. 

Metaphors are crucial to understanding math.  Too many students use the wrong metaphors, but too often no one tells them about it.

We need a math ed text for teachers

I am thinking of precalculus through typical college math major courses.  The issues I have discussed in this post are occasionally written about in the math ed literature but I have had difficulty finding many articles (on the web and on JStor) about these specific ideas.  Anyway, articles are not what we need.  We need a modest paperback book specifically aimed at teachers, covering the kinds of cognitive difficulties math students have when faced with abstraction. 

What I have written in abstractmath.org and in the Handbook are examples of what I mean, but they don't cover all the problems and they suffer from lack of focus.  (Note that the material in abstractmath.org and in posts on this blog can be used freely under a Creative Commons license — click on "Permissions" in the blue banner at the top of this page). 

Among math ed researchers, I have learned a lot from papers by Anna Sfard and David Tall. 

References

 
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Generating a Collatz tree

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I have written a short Mathematica program that generates the function tree of the Collatz function.  The code is in the document collatz.nb on the abmath website.

Examples

Here are some examples.  The first one is generated by the integers between 1 and 26.  (27 is to be avoided because it makes a shoot that is 111 nodes high.)  The primes from 1 to 26 generates the same tree. 

This one is generated by the odd numbers from 1 to 26: 

This is generated by the even numbers from 1 to 50:

 

Remark

This program is not of great import, but it was fun doing it and I learned more Mathematica. In particular, I learned that you cannot assign to a parameter in a function definition. For example, I had to write fv[gl_List, n_Integer] := (nn := n; ggl := {nn}; (Sow[1]; While[! MemberQ[ggl, cf[nn]], (Sow[nn]; Sow[cf[nn]]); nn = cf[nn]]) // Reap) instead of fv[gl_List, n_Integer] := (Sow[1]; While[! MemberQ[gl, cf[n]], (Sow[n]; Sow[cf[n]]); n = cf[n]]) // Reap) (where n:=cf[n] wouldn't have worked either).

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Algebra is a difficult foreign language

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Note: This post uses MathJax.  If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

Algebra

In a previous post, I said that the symbolic language of mathematics is difficult to learn and that we don't teach it well. (The symbolic language includes as a subset the notation used in high school algebra, precalculus, and calculus.) I gave some examples in that post but now I want to go into more detail.  This discussion is an incomplete sketch of some aspects of the syntax of the symbolic language.  I will write one or more posts about the semantics later.

The languages of math

First, let's distinguish between mathematical English and the symbolic language of math. 

  • Mathematical English is a special register or jargon of English. It has not only its special vocabulary, like any jargon, but also used ordinary English words such as "If…then", "definition" and "let" in special ways. 
  • The symbolic language of math is a distinct, special-purpose written language which is not a dialect of the English language and can in fact be read by mathematicians with little knowledge of English.
    • It has its own symbols and rules that are quite different from spoken languages. 
    • Simple expressions can be pronounced, but complicated expressions may only be pointed to or referred to.
  • A mathematical article or book is typically written using mathematical English interspersed with expressions in the symbolic language of math.

Symbolic expressions

A symbolic noun (logicians call it a term) is an expression in the symbolic language that names a number or other mathematical object, and may carry other information as well.

  • "3" is a noun denoting the number 3.
  • "$\text{Sym}_3$" is a noun denoting the symmetric group of order 3.
  • "$2+1$" is a noun denoting the number 3.  But it contains more information than that: it describes a way of calculating 3 as a sum.
  • "$\sin^2\frac{\pi}{4}$" is a noun denoting the number $\frac{1}{2}$, and it also describes a computation that yields the number $\frac{1}{2}$.  If you understand the symbolic language and know that $\sin$ is a numerical function, you can recognize "$\sin^2\frac{\pi}{4}$" as a symbolic noun representing a number even if you don't know how to calculate it.
  • "$2+1$" and "$\sin^2\frac{\pi}{4}$" are said to be encapsulated computations.
    • The word "encapsulated" refers to the fact that to understand what the expressions mean, you must think of the computation not as a process but as an object.
    • Note that a computer program is also an object, not a process.
  • "$a+1$" and "$\sin^2\frac{\pi x}{4}$" are encapsulated computations containing variables that represent numbers. In these cases you can calculate the value of these computations if you give values to the variables.  

A symbolic statement is a symbolic expression that represents a statement that is either true or false or free, meaning that it contains variables and is true or false depending on the values assigned to the variables.

  • $\pi\gt0$ is a symbolic assertion that is true.
  • $\pi\lt0$ is a symbolic assertion that it is false.  The fact that it is false does not stop it from being a symbolic assertion.
  • $x^2-5x+4\gt0$ is an assertion that is true for $x=5$ and false for $x=1$.
  • $x^2-5x+4=0$ is an assertion that is true for $x=1$ and $x=4$ and false for all other numbers $x$.
  • $x^2+2x+1=(x+1)^2$ is an assertion that is true for all numbers $x$. 

Properties of the symbolic language

The constituents of a symbolic expression are symbols for numbers, variables and other mathematical objects. In a particular expression, the symbols are arranged according to conventions that must be understood by the reader. These conventions form the syntax or grammar of symbolic expressions. 

The symbolic language has been invented piecemeal by mathematicians over the past several centuries. It is thus a natural language and like all natural languages it has irregularities and often results in ambiguous expressions. It is therefore difficult to learn and requires much practice to learn to use it well. Students learn the grammar in school and are often expected to understand it by osmosis instead of by being taught specifically.  However, it is not as difficult to learn well as a foreign language is.

In the basic symbolic language, expressions are written as strings of symbols.

  • The symbolic language gives (sometimes ambiguous) meaning to symbols placed above or below the line of symbols, so the strings are in some sense more than one dimensional but less than two-dimensional.
  • Integral notation, limit notation, and others, are two-dimensional enough to have two or three levels of symbols. 
  • Matrices are fully two-dimensional symbols, and so are commutative diagrams.
  • I will not consider graphs (in both senses) and geometric drawings in this post because I am not sure what I want to write about them.

Syntax of the language

One of the basic methods of the symbolic language is the use of constructors.  These can usually be analyzed as functions or operators, but I am thinking of "constructor" as a linguistic device for producing an expression denoting a mathematical object or assertion. Ordinary languages have constructors, too; for example "-ness" makes a noun out of a verb ("good" to "goodness") and "and" forms a grouping ("men and women").

Special symbols

The language uses special symbols both as names of specific objects and as constructors.

  • The digits "0", "1", "2" are named by special symbols.  So are some other objects: "$\emptyset$", "$\infty$".
  • Certain verbs are represented by special symbols: "$=$", "$\lt$", "$\in$", "$\subseteq$".
  • Some constructors are infixes: "$2+3$" denotes the sum of 2 and 3 and "$2-3$" denotes the difference between them.
  • Others are placed before, after, above or even below the name of an object.  Examples: $a'$, which can mean the derivative of $a$ or the name of another variable; $n!$ denotes $n$ factorial; $a^\star$ is the dual of $a$ in some contexts; $\vec{v}$ constructs a vector whose name is "$v$".
  • Letters from other alphabets may be used as names of objects, either defined in the context of a particular article, or with more nearly global meaning such as "$\pi$" (but "$\pi$" can denote a projection, too).

This is a lot of stuff for students to learn. Each symbol has its own rules of use (where you put it, which sort of expression you may it with, etc.)  And the meaning is often determined by context. For example $\pi x$ usually means $\pi$ multiplied by $x$, but in some books it can mean the function $\pi$ evaluated at $x$. (But this is a remark about semantics — more in another post.)

"Systematic" notation

  • The form "$f(x)$" is systematically used to denote the value of a function $f$ at the input $x$.  But this usage has variations that confuse beginning students:
    • "$\sin\,x$" is more common than "$\sin(x)$".
    • When the function has just been named as a letter, "$f(x)$" is more common that "$fx$" but many authors do use the latter.
  • Raising a symbol after another symbol commonly denotes exponentiation: "$x^2$" denotes $x$ times $x$.  But it is used in a different meaning in the case of tensors (and elsewhere).
  • Lowering a symbol after another symbol, as in "$x_i$"  may denote an item in a sequence.  But "$f_x$" is more likely to denote a partial derivative.
  • The integral notation is quite complicated.  The expression \[\int_a^b f(x)\,dx\] has three parameters, $a$, $b$ and $f$, and a bound variable $x$ that specifies the variable used in the formula for $f$.  Students gradually learn the significance of these facts as they work with integrals. 

Variables

Variables have deep problems concerned with their meaning (semantics). But substitution for variables causes syntactic problems that students have difficulty with as well.

  • Substituting $4$ for $x$ in the expression $3+x$ results in $3+4$. 
  • Substituting $4$ for $x$ in the expression $3x$ results in $12$, not $34$. 
  • Substituting "$y+z$" in the expression $3x$ results in $3(y+z)$, not $3y+z$.  Some of my calculus students in preforming this substitution would write $3\,\,y+z$, using a space to separate.  The rules don't allow that, but I think it is a perfectly natural mistake. 

Using expressions and writing about them

  • If I write "If $x$ is an odd integer, then $3+x$ is odd", then I am using $3+x$ in a sentence. It is a noun denoting an unspecified number which can be constructed in a specified way.
  • When I mention substituting $4$ for $x$ in "$3+x$", I am talking about the expression $3+x$.  I am not writing about a number, I am writing about a string of symbols.  This distinction causes students major difficulties and teacher hardly ever talk about it.
  • In the section on variables, I wrote "the expression $3+x$", which shows more explicitly that I am talking about it as an expression.
    • Note that quotes in novels don't mean you are talking about the expression inside the quotes, it means you are describing the act of a person saying something.
  • It is very common to write something like, "If I substitute $4$ for $x$ in $3x$ I get $3 \times 4=12$".  This is called a parenthetic assertion, and it is literally nonsense (it says I get an equation).
  • If I pronounce the sentence "We know that $x\gt0$" we pronounce "$x\gt0$" as "$x$ is greater than zero",  If I pronounce the sentence "For any $x\gt0$ there is $y\gt0$ for which $x\gt y$", then I pronounce the expression "$x\gt0$" as "$x$ greater than zero$",  This is an example of context-sensitive pronunciation
  • There is a lot more about parenthetic assertions and context-sensitive pronunciation in More about the languages of math.

Conclusion

I have described some aspects of the syntax of the symbolic language of math. Learning that syntax is difficult and requires a lot of practice. Students who manage to learn the syntax and semantics can go on to learn further math, but students who don't are forever blocked from many rewarding careers. I heard someone say at the MathFest in Madison that about 25% of all high school students never really understand algebra.  I have only taught college students, but some students (maybe 5%) who get into freshman calculus in college are weak enough in algebra that they cannot continue. 

I am not proposing that all aspects of the syntax (or semantics) be taught explicitly.  A lot must be learned by doing algebra, where they pick up the syntax subconsciously just as they pick up lots of other behavior-information in and out of school. But teachers should explicitly understand the structure of algebra at least in some basic way so that they can be aware of the source of many of the students' problems. 

It is likely that the widespread use of computers will allow some parts of the symbolic language of math to be replaced by other methods such as using Excel or some visual manipulation of operations as suggested in my post Mathematical and linguistic ability.  It is also likely that the symbolic language will gradually be improved to get rid of ambiguities and irregularities.  But a deliberate top-down effort to simplify notation will not succeed. Such things rarely succeed.

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Mathematical and linguistic ability

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This post uses MathJax.  If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen.  Sometimes you have to do it two or three times.

Some personal history

When I was young, I was your typical nerdy geek.  (Never mind what I am now that I am old.)

In high school, I was fascinated by languages, primarily by their structure.  I would have wanted to become a linguist if I had known there was such a thing.  I was good at grasping the structure of a language and read grammars for fun. I was only pretty good at picking up vocabulary. I studied four different languages in high school and college and Turkish when I was in the military.  I know a lot about their structure but am not fluent in any of them (possibly including English).

After college, I decided to go to math grad school.  This was soon after Sputnik and jobs for PhD's were temporarily easy to get.

I always found algebra easy.  When I had to learn other symbolic languages, for example set theory, first order logic, and early programming languages, I found them easy too.  I had enough geometric insight that I did well in all my math courses, but my real strength was in learning languages. 

When I got a job at (what is now) Case Western Reserve University, I began learning category theory and a bit of cohomology of groups. I wrote a paper about group automorphisms that got into Transactions of the AMS.  (Full disclosure: I am bragging). 

The way Saunders Mac Lane did cohomology, he used "$+$" as a noncommutative operation.  No problem with that, I did lots of calculations in his notation.  In reading category theory I learned how to reason using commutative diagrams.  That is radically different from other math — it isn't strings of symbols — but I caught on. I read Beck's thesis in detail.  Beck wrote functions on the right (unlike Mac Lane) which I adapted to with no problem.  In fact my automorphisms paper and many others in those days was written with functions on the right. 

Later on in my career, I learned to program in Forth reasonably well. It is a reverse Polish language. Then (by virtue of summer grants in the 1990's) to use Mathematica, which I now use a lot:  I am an "experienced" user but not an "expert".

Learning foreign languages in studying math

I taught mostly engineering students during my 35 years at CWRU (especially computer engineering). When I used a text (including my own discrete math class notes) some students pleaded with me not to use $P\wedge Q$ and $P \vee Q$ but let them use $PQ$ and $P+Q$ like they did in their CS courses.  Likewise $1$ and $0$ instead of T and F.  Many of them simply could not switch easily between different codes.  Similar problems occurred in classes in first order logic. 

In the early days of calculators when most of them were reverse Polish, some students never mastered their use. 

These days, a common complaint about Mathematica is that it is a difficult language to learn; at the MAA meeting in Madison (where I am as I write this) they didn't even staff a booth.  Apparently too many of the professors can't handle Mathematica.

I gave up writing papers with functions on the right because several professional mathematicians complained that they found them too hard to read. I guess not all professional mathematicians can switch code easily, either. 

There are many great mathematicians whose main strength is geometric understanding, not linguistic understanding.  Nevertheless, to become a mathematician you have to have enough linguistic ability to learn…

Algebra

The big elephant in the room is ordinary symbolic algebra as is used in high school algebra and precalculus.  This of course causes difficulty among first year calculus students, too, but college profs are spared the problem that high school teachers have with a large percentage of the students never really grasping how algebra works.  We don't see those students in STEM courses.

It is surely the case that algebra is a difficult and unintuitive foreign language.  I have carried on about this in my stuff about the languages of math in my abstractmath site. 

Some students already in college don't really understand expressions such as $x^2$.  You still get some who sporadically think it means $2x$.  (They don't always think that, but it happens when they are off guard.)  Lots of them don't understand the difference between $x^2$ and $2^x$.

In complicated situations, students don't grasp the difference between an expression such as $x^2+2x+1$ and a statement like $x^2+2x+1=0$.  Not to mention the difference between the way $x^2+2x+1=0$ and $x^2+2x+1=(x+1)^2$ are different kinds of statements even though the difference is not indicated in the syntax.

There are many irregularities and ambiguities (just like any natural language — the symbolic language of math is a natural language!): consider $\sin xy$, $\sin x + y$, $\sin x/y$.  (Don't squawk to me about order of operators.  That's as bad as aus, außer, bei, mit, zu.  German can't help it, but mathematical notation could.)

One monstrous ambiguity is $(x,y)$, which could be an ordered pair, the GCD, or an open interval.  I found an example of two of those in the same sentence in the Handbook of Mathematical Discourse, and today in a lecture I saw someone use it with two meanings about three inches apart on a transparency.

Anyway, the symbolic language of math is difficult and we don't teach it well.

Structuring calculations

There are other ways to structure calculations that are much more transparent.  Most of them use two or three dimensions.

  • Spreadsheets: It is easy to approximate the zeros of a function using a spreadsheet and changing the input till you get the value near zero. Why can't middle school students be taught that?
  • Bret Victor has made suggestions for easy ways to calculate things.
  • My post Visible Algebra I suggest a two-dimensional approach to putting together calculations.  (There are several more posts coming about that idea.)
  • Mathematica interactive demos could maybe be provided in a way that would allow them to be joined together to make a complicated calculation. (Modules such as an inverse image constructor.)  I have not tried to do this.

A lot of these alternatives work better because they make full use of two dimensions.  Toolkits could be made for elementary school students (there are some already but I am not familiar with them).  

It is impractical to expect that every high school student master basic algebraic notation.  It is difficult and we don't know how to teach it to everyone. With the right toolkits, we could provide everyone, not just students, to put together usable calculations on their computer and experiment with them.  This includes working out the effect of different payment periods on loans, how much paint you need for a room, and many other things.

STEM students will still have to learn algebraic notation as we use it now.  It should be taught as a foreign language with explicit instruction in its syntax (sentences and terms, scope of an operator, and so on), ambiguities and peculiarities.

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exposition, math, representations, Uncategorized, understanding math

Visible Algebra I

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This is the first in a series of articles about how algebra could be implemented without using the standard language of algebra that so many people find difficult. The code for the graphs are in the Mathematica notebook Algebra1.nb.

An algebra problem

Suppose you are designing a window that is in the shape of a rectangle surmounted by a semicircle, shown above for the window with width 2 and rectangle height 3. 

This example occurs in a tiresomely familiar calculus problem where you put a constraint on the perimeter of the window, thus turning it into a one-variable problem, then finding the values of the width and height that give the maximum area.  In this post, I am not going to get that far.  All I will do is come up with a calculation for the area.  I will describe a way you might do it on a laptop five or ten years from now. 

You have an algebra application that shows a screen with some operations that you may select to paste into your calculation.  The ones we use are called plus, times, power, value and input. You choose a function called value, and label it "Area of window". You recognize that the answer is the sum of the areas of the rectangle and the area of the semicircle, so you choose plus and attach to it two inputs which you label "area of rectangle" and "area of semicircle", like this:

 

The notational metaphor is that the computation starts at the bottom and goes upward, performing the operations indicated.

You know (or are told by the system) that the area of a rectangle is the product of its width and height, so you replace the value called "area of rectangle" with a times button and attach two values called $w$ and $h$:

 

You also determine that the area under the semicircle is half the area of a circle of radius $r$ (where $r$ must be calculated).

 

You have a function for the area of a circle of radius $r$, so you attach that:

Finally, you use the fact that you know that the semicircle has a radius which is half the width of the rectangle.

Now, to make the calculation operational, you attach two inputs named "width" and "height" and feed them into the values $w$ and $h$.  When you type numbers into these buttons, the calculation will proceed upward and finally show the area of the window at the top.

In a later post I will produce a live version of this diagram.  (Added 2012-09-08: the live version is here.) Right now I want to get this post out before I leave for MathFest.  (I might even produce the live version at MathFest, depending on how boring the talks are.) 

You can see an example of a live calculation resembling this in my post A visualization of a computation in tree form.

Remarks

Who

  • This calculation might be a typical exercise for a student part way along learning basic algebra. 
  • College students and scientists and engineers would have a system with a lot more built-in functions, including some they built themselves.

Syntax

  • Once you have grasped the idea that the calculation proceed upward from the inputs, carrying out the operations shown, this picture is completely self-explanatory.
    • Well, you have to know what the operations do.
    • The syntax for standard algebra is much more difficult to learn (more later about this).
  • The syntax actually used in later years may not look like mine.
    • For one thing, the flow might run top down or left to right instead of bottom up. 
    • Or something very different might be used. What works best will be discovered by using different approaches.
  • The syntax is fully two-dimensional, which makes it simple to understand (because it uses the most powerful tool our brain has: the visual system).
    • The usual algebraic code was developed because people used pencil and paper. 
    • I would guess that the usual code has fractional dimension about 1.2. 
    • The tree syntax would require too much writing with pencil and paper.  That is alleviated on a computer by using menus.
    • Once you construct the computation and input some data it evaluates automatically.
  • It may be worthwhile to use 3D syntax.  I have an experiment with this in my post Showing categorical diagrams in 3D.

Later posts will cover related topics:

  • The difficulties with standard algebraic notation.  They are not trivial.
  • Solving equations in tree form.
  • Using properties such as associativity and commutativity in tree form.
  • Using this syntax with calculus.
  • The deep connection with Lawvere theories and sketches.

References

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exposition, language of math, math, representations, Uncategorized, understanding math

The mystery of the prime numbers: a review

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The Mystery of the Prime Numbers: Secrets of Creation v. 1, Matthew Watkins (Author), Matt Tweed (Illustrator). Inamorata Press, 2010.

The author and illustrator describe some of the mysteries of the distribution of primes, ending with Riemann's harmonic decomposition of the distribution. (If you don't understand what all that means, you will if you read the book and concentrate a lot). It is the first part of a trilogy which the author promises will culminate in a connection with quantum theory.  I can't wait.

The most remarkable thing about this book is the presentation.  You do not have to understand symbolic math.  The ideas are communicated using many, many pictures and metaphors. For example, you have a number of apples and you can't arrange them into a rectangle (a row doesn't count as a rectangle) then the number is a prime. Technical terminology is avoided, especially when the terminology consists of ordinary English words with new meanings.  So a prime factorization of a number is a "cluster" — and of course it really is since the order the primes occur in is irrelevant. On other occasions he will use the technical term (for example "distribution of primes" but warn you against your understanding being contaminated by the everyday meaning such as "distribute two pencils to each student").

The author is not afraid of saying the same thing several times, using different metaphors and rewording.  He will notice that some ideas will make you uncomfortable, such as the prime number theorem which "ought" to tell you the exact number of primes less than a number instead of merely estimate it.  (How many of your teachers ever admitted that an idea may make you uncomfortable and this is why it does…)  His explanation of functions and graphs is pictorial (using the endograph — arrows from one place to another on the real line) and kinetic (a ball rising from the x axis, hitting the graph, and being knocked horizontally to the y axis).

This makes me believe a reader who finds algebraic equations hard to understand and the nomenclature baffling can still get a reasonable mental picture of what primes are and what the Prime Number Theorem and Riemann's theorem on the distribution are actually saying.  It won't be easy: such a reader will have to concentrate and stop and think a lot (and I hope doodle pictures) and at times it will be slow going.  But this book makes it possible for someone who has no self-confidence in their ability to understand math to understand some deep stuff that many mathematicians find as astonishing as anything in math.

If we had a hundred authors writing books like this about different parts of math we would (in the long run) have fewer people who hate math or claim it all sounds like gibberish.

The long time reader of Gyre & Gimble will note that I have been saying for years that math should be explained like this.  So naturally I am pleased to recommend this book.  (Even so, all the times I taught primes I never thought of defining a prime number by saying you can't arrange your trinkets into a rectangle.  Oh, the chagrin…)

 

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Metaphors in computing science 2

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In Metaphors in Computer Science 1, I discussed some metaphors used when thinking about various aspects of computing.  This is a continuation of that post.

Metaphor: A program is a list of instructions.

  • I discussed this metaphor in detail in the earlier post.
  • Note particularly that the instructions can be in a natural or a programming language. (Is that a zeugma?)  Many writers would call instructions in a natural language an algorithm.
  • I will continue to use “program” in the broader sense.

Metaphor: A programming language is a language.

  • This metaphor is a specific conceptual blend that associates the strings of symbols that constitute a program in a computer language with text in a natural language.
  • The metaphor is based on some similarities between expressions in a programming language and expressions in a natural language.
    • In both, the expressions have a meaning.
    • Both natural and programming languages have specific rules for constructing well-formed expressions.
  • This way of thinking ignores many deep differences between programming languages and natural languages. In particular, they don’t talk about the same things!
  • The metaphor has been powerful in suggesting ways of thinking about computer programs, for example semantics (below) and ambiguity.

Metaphor: A computer program is a list of statements

  • A consequence of this metaphor is that a computer program is a list of symbols that can be stored in a computer’s memory.
  • This metaphor comes with the assumption that if the program is written in accordance with the language’s rules, a computer can execute the program and perhaps produce an output.
  • This is the profound discovery, probably by Alan Turing, that made the computer revolution possible. (You don’t have to have different physical machines to do different things.)
  • You may want me to say more in the heading above: “A computer program is a list of statements in a programming language that satisfies the well-formedness requirements of the language.”  But the point of the metaphor is only that a program is a list of statements.  The metaphor is not intended to define the concept of “program”.

Metaphor: A program in a computer language has meanings.

A program is intended to mean something to a human reader.

  • Some languages are designed to be easily read by a human reader: Cobol, Basic, SQL.
    • Their instructions look like English.
    • The algorithm can nevertheless be difficult to understand.
  • Some languages are written in a dense symbolic style.
    • In many cases the style is an extension of the style of algebraic formulas: C, Fortran.
    • Other languages are written in a notation not based on algebra:  Lisp, APL, Forth.
  • The boundary between “easily read” and “dense symbolic” is a matter of opinion!

A program is intended to be executed by a computer.

  • The execution always involves translation into intermediate languages. 
    • Most often the execution requires repeated translation into a succession of intermediate languages.
    • Each translation requires the preservation of the intended meaning of the program.
  • The preservation of intended meaning is what is usually called the semanticsof a programming language.
    • In fact, the meaning of the program to a person could be called semantics, too.
    • And the human semantics had better correspond in “meaning” to the machine semantics!
  • The actual execution of the program requires successive changes in the state of the computer.
    • By “state” I mean a list of the form of the electrical charges of each unit of memory in the computer.
    • Or you can restrict it to the relevant units of memory, but spelling that out is horrifying to contemplate.
    • The resulting state of the machine after the program is run is required to preserve the intended meaning as well as all the intermediate translations.
    • Notice that the actual execution is a series of physical events.  You can describe the execution in English or in some notation, but that notation is not the actual execution.

References

Conceptual blend (Wikipedia)

Conceptual metaphors (Wikipedia)

Images and Metaphors (article in abstractmath)

Semantics in computer science (Wikipedia)

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A visualization of a computation in tree form

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Live evaluation of expressions in TreeForm 3, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The code is ad-hoc.  It might be worthwhile for someone to design a package that produces this sort of tree for any expression. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

This demonstration shows the step by step computation of the value of the expression $3x^2+2(1+y)$ shown as a tree.  By moving the first slider from right to left, you go through the six steps of the computation. You may select the values of $x$ and $y$ with the second and third sliders.  If you click on the plus sign next to a slide, a menu opens up that allows you to make the slider move automatically, shows the values, and other things.

Note that subtrees on the same level are evaluate left to right.  Parallel processing would save two steps.

A previous post related to this post is Making visible the abstraction in algebraic notation.

  

 
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