This is the first of a set of notes I am writing to help me develop my thoughts about how particular topics in my book Abstracting algebra should be organized. This article describes my plan for the book in some detail. The present post has some thoughts about presenting binary operations.
Before binary operations are introduced
Traditionally, an abstract algebra book assumes that the student is familiar with high school algebra and will then proceed with an observation that such operations as $+$ and $\times$ can be thought of as functions of two variables that take a number to another number. So the first abstract idea is typically the concept of binary operation, although in another post I will consider whether that really should be the first abstract concept.
The Abstracting Algebra book will have a chapter that presents concrete examples of algebraic operations and expressions on numbers as in elementary school and as in high school algebra. This section of the post outlines what should be presented there. Each subsection needs to be expanded with lots of examples.
In elementary school
In elementary school you see expressions such as
 $3+4$
 $3\times 4$
 $34$
The student invariably thinks of these expressions as commands to calculate the value given by the expression.
They will also see expressions such as
\[\begin{equation}
\begin{array}[b]{r}
23\\
355\\
+ 96\\
\hline
\end{array}
\end{equation}\]
which they will take as a command to calculate the sum of the whole list:
\[\begin{equation}
\begin{array}[b]{r}
23\\
355\\
+ 96\\
\hline
474
\end{array}
\end{equation}\]
That uses the fact that addition is associative, and the format suggests using the standard school algorithm for adding up lists. You don’t usually see the same format with more than two numbers for multiplication, even though it is associative as well. In some elementary schools in recent years students are learning other ways of doing arithmetic and in particular are encouraged to figure out short cuts for problems that allow them. But the context is always “do it”, not “this represents a number”.
Algebra
In algebra you start using letters for numbers. In algebra, “$a\times b$” and “$a+b$” are expressions in the symbolic language of math, which means they are like noun phrases in English such as “My friend” and “The car I bought last week and immediately totaled” in that both are used semantically as names of objects. English and the symbolic language are both languages, but the symbolic language is not a natural language, nor is it a formal language.
Example
In beginning algebra, we say “$3+5=8$”, which is a (true) statement.
Basic facts about this equation:
The expressions “$3+5$” and “$8$”
 are not the same expression
 but in the standard semantics of algebra they have the same meaning
 and therefore the equation communicates information that neither “$3+5$” nor “$8$” communicate.
Another example is “$3+5=6+2$”.
Facts like this example need to be communicated explicitly before binary operations are introduced formally. The students in a college abstract algebra class probably know the meaning of an equation operationally (subconsciously) but they have never seen it made explicit. See Algebra is a difficult foreign language.
Note
The equation “$3+5=6+2$” is an expression just as much as “$3+5$” and “$6+2$” are. It denotes an object of type “equation”, which is a mathematical object in the same way as numbers are. Most mathematicians do not talk this way, but they should.
Binary operations
Early examples
Consciousnessexpanding examples should appear early and often after binary operations are introduced.
Common operations
 The GCD is a binary operation on the natural numbers. This disturbs some students because it is not written in infix form. It is associative. The GCD can be defined conceptually, but for computation purposes needs (Euclid’s) algorithm. This gives you an early example of conceptual definitions and algorithms.
 The maximum function is another example of this sort. This is a good place to point out that a binary operation with the “same” definition cen be defined on different sets. The max function on the natural numbers does not have quite the same conceptual definition as the max on the integers.
Extensional definitions
In order to emphasize the arbitrariness of definitions, some random operations on a small finite sets should be given by a multiplication table, on sets of numbers and sets represented by letters of the alphabet. This will elicit the common reaction, “What operation is it?” Hidden behind this question is the fact that you are giving an extensional definition instead of a formula — an algorithm or a combination of familiar operations.
Properties
The associative and commutative properties should be introduced early just for consciousnessraising. Subtraction is not associative or commutative. Rock paper scissors is commutative but not associative. Groups of symmetries are associative but not commutative.
Binary operation as function
The first definition of binary operation should be as a function. For example, “$+$” is a function that takes pairs of numbers to numbers. In other words, $+:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a function.
We then abstract from that example and others like it from specific operations to arbitrary functions $\Delta:S\times S\to S$ for arbitrary sets $S$.
This is abstraction twice.
 First we replace the example operations by an arbitrary operation. such as multiplication, subtraction, GCD and MAX on $\mathbb{Z}$, or something complicated such as \[(x,y)\mapsto 3(xy1)^2(x^2+xy^3)^3\].

Then we replace sets of numbers by arbitrary sets. An example would be the random multiplication on the set $\{1,2,5\}$ given by the table
\[
\begin{array}{cccc}
\Delta& 1&2&5\\
\hline
1&2&2&1\\
2&5&2&1\\
5&2&1&5
\end{array}
\]
This defines a function $\Delta:\{1,2,5\}\times\{1,2,5\}\to\{1,2,5\}$ for which for example $\Delta(2,1)=5$, or $2\Delta 1=5$. This example uses numbers as elements of the set and is good for eliciting the “What operation is it?” question.  I will use examples where the elements are letters of the alphabet, as well. That sort of example makes the students think the letters are variables they can substitute for, another confusion to be banished by the wise professor who know the right thing to say to make it clear. (Don’t ask me; I taught algebra for 35 years and I still don’t know the right thing to say.)
It is important to define prefix notation and infix notation right away and to use both of them in examples.
Other representations of binary operations.
The main way of representing binary operations in Abstracting Algebra will be as trees, which I will cover in later posts. Those posts will be much more interesting than this one.
Binary operations in high school and college algebra
 Some binops are represented in infix notation: “$a+b$”, “$ab$”, and “$a\times b$”.
 “$a\times b$” is usually written “$ab$” for letters and with the “$\times$” symbol for numbers.
 Some binops have idiosyncratic representation: “$a^b$”, “${a}\choose{b}$”.
 A lot of binops such as GCD and MAX are given as functions of two variables (prefix notation) and their status as binary operations usually goes unmentioned. (That is not necessarily wrong.)
 The symbol “$(a,b)$” is used to denote the GCD (a binop) and is also used to denote a point in the plane or an open interval, both of which are not strictly binops. They are binary operations in a multisorted algebra (a concept I expect to introduce later in the book.)
 Some apparent binops are in infix notation but have flaws: In “$a/b$”, the second entry can’t be $0$, and the expression when $a$ and $b$ are integers is often treated as having good forms ($3/4$) and bad forms ($6/8$).
Trees
The chaotic nature of algebraic notation I just described is a stumbling block, but not the primary reason high school algebra is a stumbling block for many students. The big reason it is hard is that the notation requires students to create and hold complicated abstract structures in their head.
Example
This example is a teaser for future posts on using trees to represent binary operations. The tree below shows much more of the structure of a calculation of the area of a rectangle surmounted by a semicircle than the expression
\[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\]
does.
The tree explicitly embodies the thought process that leads to the formula:
 You need to add the area of the rectangle and the area of the semicircle.
 The area of the rectangle is width times height.
 The area of the semicircle is $\frac{1}{2}(\pi r^2)$.
 In this case, $r=\frac{1}{2}w$.
Any mathematician will extract the same abstract structure from the formula\[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\] This is difficult for students beginning algebra.
References
 Abstracting algebra. An outline of the proposed book.
 Abstraction in abstractmath.org.
 Algebra is a difficult foreign language. G&G post.
 Formal language in Wikipedia.
 Monads for high school III. G&G post.
 Monads in Wikipedia
 Natural language in Wikipedia.
 Noun phrases in Wikipedia.
 Symbolic language of math in abstractmath.org.
 Visible Algebra I.G&G post.
 Visible algebra I supplement. G&G post.
 Visible Algebra II. G&G post.
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