Introduction

This is the first of a series of articles filling out the theory behind the invention of the LinearAbacus\(^{TM}\)¹. The principles that inform the use of the linear abacus in teaching and learning are:

1. All teaching and learning is framed by Natural Language Communication. Or, as Halliday (1993) put it: English is a metalanguage for all learning.

2. Embodiment is prior to and founds emerging concepts.

3. Meaning is made by cultures and societies, sense is made by individuals.

4. From the point of view of “learning”, concepts are better thought of as “conceiving things”(dynamic) rather than as “things conceived” (static). That is to say process is more important than endpoints.

5. Making sense with arithmetic happens through producing and comprehending arithmetic expressions.

These principles are operationalised in a scheme for constructing and sequencing teaching and learning experiences in the classroom as communicative events. This scheme is called a Pedagogical Image of a Concept or more briefly, a PIC.

For those who have the eyes to see, the broader argument being proposed here is: English teaching and Mathematics teaching are natural companions in the quest to produce clear thinking individuals.

Pedagogical Images of Concepts

The following diagram sets out the key elements of the system of teaching and learning using the PIC model.

A PIC models sense making with Arithmetic expressions, it answers the question, what concept is expressed by the Simple Number Sentence \(5\div2\frac{1}{2}= 2\)? That is when and where can it be used to express a thought about the world? The double arrow and circle symbol marking each side of this triangle (\(\leftarrow \circ \rightarrow\))represents a communicative event . It is where speaking, writing, drawing, calculating, manipulating is talked about between Teacher and Students, Student and Teacher, Student and Student. That is to say all teaching and learning is conceived of as made up of communicative events: Someone saying something about something to someone. The deep thought here is that for a communication to be coordinated across words, models, and calculations there has to be a coordination of sense making with words, sense making with Arithmetic, and sense making with manipulations. The possibility of communicating a sense in any of words, arithmetic, or manipulations depends on coordinating a syntax and semantics between modes of thinking. In a PIC,

1. the Material Model is about gestures, through the linear abacus hands and eyes coordinate to embody the sense of numbers and calculations. The very act of counting on the linear abacus uses gestures to embody one-to-one hand eye coordination as a basis for building concepts of ordinality and cardinality.

2. Calculations are thinking with arithmetic. Getting an answer to \(5\div2=?\) isn’t sufficient, the concept is conceived in the calculation as argument, so for the concept of a mixed number : \[5\div2=(2+2+1)\div2 = \frac{2}{2}+\frac{2}{2}+\frac{1}{2}=1+1+\frac{1}{2}=2\frac{1}{2},\] where “\(2\frac{1}{2}\)” is understood as an additive construction.

3. Solving word problems stand in for understanding concepts in the world. The classroom is not the real world hence why a PIC is an “image” of a concept. So making sense of the meaning of “\(\div\)” is to pay attention to the use of numerals, for example compare these three uses of the mixed number “\(2\frac{1}{2}\)”:

   1. If Jack has 5 marbles and Jill has 2 marbles then Jack has \(2\frac{1}{2}\) times more marbles than Jill,because \(5\div2=2\frac{1}{2}\) .

   2. If Jack has 5 marbles and this is \(2\frac{1}{2}\) times as many as Jill then Jill has two marbles because \(5\div2\frac{1}{2}=2\).

   3. If Jack has 6 marbles and this is \(2\frac{1}{2}\) times as many as Jill then Jill has \(2 \frac{2}{5}\)marbles because \(6\div2\frac{1}{2}= 2 \frac{2}{5}\).

The examples given to illustrate the contribution of wordproblems to making sense of arithmetic concepts is most valuable because it introduces one of the most important innovations of the PIC model as an instructional system. 3a and 3b are about a syntax of arithmetic and 3c is about a semantics of arithmetic: 3c doesn’t make sense in respect of marbles being marbles. That arithmetic can be described as having a syntax and a semantics allows it to be systematically aligned with making sense in English as well making sense with material models. A PIC is a framework for building sequences of sensible communications about arithmetic concepts that scaffold the goal of learning with understanding.

In this brief article the focus will be building the classroom level machinery for teachers and students to communicate about number sentences and wordproblems- \(wp\leftarrow \circ \rightarrow calc\) and \(SNS\leftarrow \rightarrow WP\) . Other articles will explore how communicative events in the classroom can be informed by the other dialogical relations in a PIC.

Not all arithmetics are equal.

The PIC system recognises three Arithmetics²: Naive Arithmetic, Operational Arithmetic, and Algebraic Arithmetic. We can take the “\(+\)” sign as a case study to distinguish these arithmetics.

1. Naive Arithmetic is mostly about a naive understanding of “\(+\)” when interpreted as “and”. So ,“if Jack has 2 cats and 3 birds and 6 dogs and 2 gold fish how many pets does he have?” can be represented as \(2+3+6+2=13\). The “\(+\)” in the expression is very much like the SUM function in a spreadsheet, it simply means accumulate. This “\(+\)” is not a binary operation, which it is in both Operational and Algebraic Arithmetic ³.

2. Operational Arithmetic respects the binary nature of the signs “\(+\)” and “\(-\)”. This leads to well formed Simple Number Sentences like: \(5+3=8\), \(8-6=2\), etc. If these simple number sentences are extended then they imply a bracketing, for example \(((8-6)+5)=7\) vs \((8-(6+5))=-3\)Operational Arithmetic is the arithmetic of primary education, it is the frame for introducing numerals like \(\frac{2}{3}\) ,\(^-5\), \(3.75\).Operational Arithmetic is a semiotic system that uses four signs for“operators” (\(+,-,\times,\div\)).

3. Algebraic Arithmetic, like Operational Arithmetic, respects the binary nature of the operator signs. It differs from Operational Arithmetic in that it only uses the signs “\(+\)” and “\(\times\)”.

It is argued that these are different arithmetics because the signs used in expressions in each arithmetic mean different things - if the signs for expressing are different then what is being expressed is different. Signs themselves are part of cultural meaning, unlike our use of them which may have many senses. These senses arise from conflating semiotic systems. It is part of the deep theory of a PIC that it models thought as an interaction of signs through a transcoding of interpretations. This follows on from the work of Charles Saunders Pierce :

“To say, therefore, that thought cannot happen in an instant, but requires a time, is but another way of saying that every thought must be interpreted in another, or that all thought is in signs.” (CP 5.253)

At the level of the classroom the work of this transcoding can be illustrated by what we call Context Framing Sentences (CFS), for instance:

1. CFS: If Jack has 5 marbles and Jill has 2 marbles then Jack has \(2\frac{1}{2}\) times more marbles than Jill, because \(5\div2=2\frac{1}{2}\) , because \(5\times\frac{1}{2}=2\frac{1}{2}\) .

The CFS shows a path to the development of arithmetic concepts from Natural Language through Operational Arithmetic to Algebraic Arithmetic, which is to say a CFS shows a path (one that isn’t yet clearly mapped) from thinking with language to thinking algebraically.

Operational Arithmetic has a syntax.

Operational arithmetic is the arithmetic of the everyday world and it is the arithmetic that should be taught as the precursor to algebraic thinking. Operational arithmetic is a process oriented arithmetic. To understand this compare these two English translations of the simple number sentence \(2+6=8\) :

1. Two plus six makes eight, and

2. Two plus six is eight.

Translating “\(=\)” as “makes” gives the sentence a constructive sense, whilst translating it as “is” gives a truth-value sense⁴. The “truth-value” interpretation of arithmetic statements, which is the basis of algebraic statements, is a very sophisticated abstraction and one that needs to wait for some years after primary school. Numerals are the clothes that numbers wear when they are used in communications between people. It is numerals that are used in writing simple number sentences in operational arithmetic, and so making sense with numerals is at the heart of teaching and learning of numeracy. Numerals in simple number sentences (\(5\div2=2\frac{1}{2}\)) are like words in simple sentences (“Jack jumped the candle stick”) in that they have roles to play as parts of speech. The following table sets out a syntax for the use of numerals for expressing Additive, Multiplicative, and Exponentative thinking in Operational Arithmetic.

Thinking with exponents has been included for completeness but for the primary curriculum only additive and multiplicative thinking are considered. This syntax gives a new set of “facts” - the triple of additive number sentences and the triple of multiplicative sentences - that establish the deep connection between the three core concepts in each of Additive and Multiplicative thinking. If I am given one of the number sentences from triple of number sentences then I know the other two. This is a very powerful insight when it comes to conceiving of numbers. For example if I know \[5\div2=\textcolor{blue}{2\frac{1}{2}}\] then I also know \[2 \times \textcolor{blue}{2\frac{1}{2}}=5\] and \[5\div\textcolor{blue}{2\frac{1}{2}}=2\] Just like if I know “Jack jumped over the candle stick” then I know “the candle stick was jumped over by Jack”⁵ .

The triple of number sentences can be given a signature, which we call a synset⁶ , so \(+<8_s,2_o,\textcolor{green}{6_v}>\) and \(\times<8_s,2_o,\textcolor{blue}{4_v}>\). To understand why the numerals “\(6\)” and “\(4\)” are colour coded as different to the other numerals is to understand how Operational Arithmetic is a process arithmetic. “\(\textcolor{green}{6_v}\)” and “\(\textcolor{blue}{4_v}\)” are names for an additive comparison of two numbers (DIFFERENCE) and a multiplicative comparison of two numbers (QUOTIENT or RATIO) respectively. The name of a relation between numbers is not the same as the name of a number, that is to say the words "numerals" and "numbers" are not synonyms. Some examples might help, consider these CFS:

1. If Jack is filling a \(5lt\) bucket using a \(2lt\) jug, then he will have to empty two and a half jugs full into the bucket, because \(5\div2=\textcolor{blue}{2\frac{1}{2}}\)

2. If Jack has a jug that holds \(2lt\) and he empties two and a half jug fulls into a bucket to fill it, then the bucket must hold \(5lt\),because \(2\times \textcolor{blue}{2\frac{1}{2}} =5\).

3. If Jack has filled a bucket with a \(2lt\) jug by emptying it two and a half times into the bucket then the bucket must hold \(5lt\), because \(2\div\textcolor{blue}{2\frac{1}{2}}=5\).

In the English use of the numerals the “\(5\)” and “\(2\)” refer to amounts of stuff, but the numeral “\(\textcolor{blue}{2\frac{1}{2}}\)” refers to the act of emptying the jug into the bucket. The numerals in this triple “\(\times<5,2,\textcolor{blue}{2\frac{1}{2}}>\)” are not all talking about the same sort of thing, that is to say the numerals have a different meaning depending on “the part of speech” they have in the number sentence. This description of additive and multiplicative synsets establishes a “grammar” of operational arithmetic. The deep thought here is: Operational Arithmetic supports two syntactic patterns for expressing an arithmetic thought.

1. The comparative syntax: \[((\_)\;op\;(\_))=(\_)\] and,

2. The constructive syntax: \[((\_)\;op\;\_)=(\_)\] .

Whereas Algebraic arithmetic supports only one syntactic form: -\[((\_)\;op\;(\_))=(\_)\] This simplification of forms is made possible because of the invention of Integers and rational numbers which take over the syntactic work performed by the symbols “\(-\)” and “\(\div\)” in operational arithmetic. Scaffolding this transition in the machinery of thinking is the task of educators.

Operational Arithmetic has a semantics

Semantics arise when a syntax is used to talk about something in the world. In a PIC Operational Arithmetic gathers its semantics form two communicative events: the dialogue with Wordprolems and the dialogue with the linear abacus. The linear abacus is designed to be both a tally machine (modelling counts) and a measuring device (modelling a ruler). These two models give significance to numerals in two different ways as follows:

1. A Count treats numerals as names of a place in an order. On the linear abacus counting is performed by moving beads from right to left toward the knot which signifies a “0”⁷. The deep thought here is that the linear abacus models a count number as both ordinal and cardinal at the same time, and it is the ordinality of numbers that founds sense making in operational arithmetic. The \(\textcolor{red}{10}\) in the above diagram represents the \(10^{th}\) thing counted and ipso facto a collection of \(10\) things.

2. A Ruler treats numerals as scalars (lengths are multiples of a common measure) which names a span⁸. On the linear abacus, keeping the one hundred beads as contiguous, the string of beads acts as a metre rule with \(1cm\) as the common measure. Of course the 100 beads can be partitioned in any number of other ways to model informal units of measure.

Distinguishing between numerals that name Countable things or Measurable stuff is to respect the distinction between discrete and continuous phenomena in the world. The tension between these two experience of the tangible inform both the semantics of Natural Languages and Operational Arithmetic. For instance, reflect on why one of these sentences makes sense and the other not:

1. I have a bag of marbles.

2. I have a bag of waters.

“Marble” can take a plural but “water”, in this sentence, can’t. In English “marble” and “water” are parts of speech called count and non-count (or measure) nouns respectively. The very syntax of English distinguishes experiencing countable things and experiencing measurable stuff, and the same deep opposition between these semantics that structures English syntax influences sense making with arithmetic.

In Operational Arithmetic the concepts of countable and measurable are distinguished by distinguishing the meaning of numerals used to name them.

1. In Operational Arithmetic totals of countable things are named using the syntax \[0+\textcolor{green}{10}=\textcolor{red}{10}\] and,

2. In Operational Arithmetic amounts of measurable stuff are named using the syntax \[\textcolor{blue}{10}\times 1=\textcolor{violet}{10}\] (The \(\textcolor{blue}{10}\) here is put at the start to reflect the common practice for denoting measurements , eg \(\textcolor{violet}{10}cm\) which means \(\textcolor{blue}{10}\times 1cm=\textcolor{violet}{10cm}\))

As a count \(10\) can be thought of as a Difference from \(0\) and as a measure it can be thought of as a Multiple of \(1\). It can’t be overemphasised just how important the understanding of “\(0\)” and “\(1\)” are for establishing algebraic thinking and hence the importance of carefully scaffolding the use of these numerals as the foundation of arithmetic reasoning- whether as index (additive) or proportion (multiplicative). The use of this colour coding for the semantics of numerals allows us to explore the expressive range of operational arithmetic as it is met in the classroom⁹.

These operational arithmetic descriptions of a number as a process in respect of either “\(0\)” or “\(1\)” are the basis for understanding more complex arithmetic expression in an operational mode. A complex number expression like \[((3\times 5)+7)\] is evaluated operationally as follows: \[((3\times 5)+7)\rightarrow (((1\times \textcolor{blue}{3})\times\textcolor{blue}{5})+ 7)=15+7 \rightarrow((0+\textcolor{green}{15})+\textcolor{green}{7})=22\]
The static phrase \(((3\times 5)+7)\) becomes operationally dynamic by the gestalt of perceiving numerals as relative to “\(0\)” or relative to “\(1\)”¹⁰. These observations are part of the exploration of the PIC dialogues \(wp\leftarrow \circ \rightarrow calc\) and \(LA\leftarrow \circ \rightarrow calc\) , which will be taken up in other articles.

The meaning potential of Operational Arithmetic

Keeping in mind that a synset is a “signature” for a triple of simple number sentences that are related as syntactic forms for conceiving of the comparison of two numbers in respect of their increasing or decreasing. The semantic dimension of these syntactic forms is, what in the world they can be used to talk about. Keeping in mind that our colour coding scheme for numerals refers back to a theory of “grammar” for operational arithmetic (that is how to produce and comprehend simple number sentences when talking about the world). Here is the legend for the colour coding:

The use of purple for coding measure numerals leverages a subtle point in the semiotics of colour - purple is a mixture of red and blue. This is intentional because it points to an important point in the semiotics of operational arithmetic. Consider these sentences:

1. I have a twenty-five centimetre long block of wood.

2. It is twenty-five centimetres long.

When discussing English usage the distinction between singular and plural forms (centimetre/centimetres) is not too strict, but in a mathematical context the distinction carries conceptual weight. “twenty-five centimetres” suggests an additive construction whilst “twenty-five centimetre” suggests the scaling construction. The official SI unit is \(cm\) and this is never pluralised. Most children approach measures as additive constructions and coming to understand them as scaling a unit is important for establishing multiplicative thinking. So our colour analogy is, Purple is an extension of red by mixing with blue.

Colour parsing wordproblems

The colour coding machinery is a metalanguage for students over the interpretation of wordproblems. Let’s take a wordproblem and analyse what thinking with this metalanguage promotes. Consider:

• Jack has 12 marbles and this is 3 more than Jill, how many marbles does Jill have?

To parse this as a simple number sentence we note the following:

1. Marbles are countable hence \(\Rightarrow \textcolor{red}{12}\).

2. “how many marbles..” indicates the unknown answer numeral is a count \(\Rightarrow \textcolor{red}{?}\).

3. “3 more” marks an additive relation, hence \(\Rightarrow \textcolor{green}{3}\).

4. “this is 3 more” also indicates that the unknown number is less than \(\textcolor{red}{12}\).

Remember that the three numerals in the question and answer to a simple number sentence word problem represent a synset triple of possible number sentences. Given the numerals in this word problem We are looking at this synset form of the problem: \(+<\textcolor{red}{12},\textcolor{red}{?},\textcolor{green}{3}>\)

Now the wordproblem can be paired with a number sentence interpretation:

• Jack has \(\textcolor{red}{12\; marbles}\) and this is \(\textcolor{green}{3\;more}\) than Jill, \(\textcolor{red}{how\; many}\) marbles does Jill have? \(\rightarrow \textcolor{red}{12}-\textcolor{green}{3}=\textcolor{red}{9}\)

Understanding operational arithmetic as structured by synset triples opens the communication to a far wider engagement with the concepts than is the case in present teaching and learning practice. This richness is amplified when the number sentence “\(\rightarrow \textcolor{red}{12}-\textcolor{green}{3}=\textcolor{red}{9}\)” is explored coordinated with the other two number sentences in the synset:

1. \(\textcolor{red}{12}-\textcolor{red}{9}=\textcolor{green}{3}\rightarrow\)Jack has \(\textcolor{red}{12\; marbles}\) and Jill has \(\textcolor{red}{9\; marbles}\),\(\textcolor{green}{how\; many \;more}\) marbles does Jack have than Jill?

2. \(\textcolor{red}{9}+\textcolor{green}{3}=\textcolor{red}{12}\rightarrow\)Jill has \(\textcolor{red}{9\; marbles}\) and Jack has \(\textcolor{green}{3\;more}\) than her, \(\textcolor{red}{how\;many\;marbles}\) does jack have?

The richness goes up another step when these synsets are embedded in a PIC and the concepts coordinated and consolidated by the connections: \(wp\leftarrow \circ \rightarrow LA\) , \(wp\leftarrow \circ \rightarrow Calc\) , and \(Calc\leftarrow \circ \rightarrow LA\) . These are matters for other articles.

Wordproblems\(\leftrightarrow\) SNS

The PIC dialogue between wordproblems and number sentences (\(SNS\leftarrow \rightarrow WP\))can now go both ways. Consider this practice exercise:

• Exercise: Using the same numerals and the same theme as in the example above, write a comparison wordproblem. The synset form to use for this task is \(+<\textcolor{red}{12},\textcolor{red}{9}, \textcolor{green}{?}>\)

• Answer: Jack has \(\textcolor{red}{12\;marbles}\) and Jill has \(\textcolor{red}{9\;marbles}\) , how \(\textcolor{green}{many \;more}\) does Jack have \(\textcolor{green}{than}\) Jill?

The PIC, through the syntax of operational arithmetic, brings both numeracy and literacy together to bear on the task of building concepts and on the tasks of producing and comprehending in two different but now coordinated semiotic systems, English and Mathematics. Here are some more base examples:

1. At the start of the week a plant was \(\textcolor{violet}{9cm\; tall}\) and by the end of the week it was \(\textcolor{violet}{12cm\; tall}\) ,\(\textcolor{green}{how\; much}\) had it grown? \[\leftrightarrow\textcolor{violet}{12}-\textcolor{violet}{9}=\textcolor{green}{3}\Rightarrow +<\textcolor{violet}{12},\textcolor{violet}{9},\textcolor{green}{3}>\]

2. At the start of the week a bamboo shoot was \(\textcolor{violet}{9cm\; tall}\), by the end of the week it had \(\textcolor{blue}{tripled \;in\;height}\), \(\textcolor{violet}{how\; tall}\) was it at the end of the week?\[\leftrightarrow\textcolor{violet}{9}\times\textcolor{blue}{3}=\textcolor{violet}{27}\Rightarrow \times<\textcolor{violet}{27},\textcolor{violet}{9} \textcolor{blue}{3}>\]

3. Jill had bag of \(\textcolor{red}{9\; marbles}\) and Jack had a bag with \(\textcolor{blue}{2 \frac{1}{3}\;times\;as\;many}\) ,\(\textcolor{red}{how\;many\;marbles}\) does Jack have?\[\leftrightarrow\textcolor{red}{9}\times\textcolor{blue}{2 \frac{1}{3}}=\textcolor{red}{21}\Rightarrow \times<\textcolor{red}{21},\textcolor{red}{9},\textcolor{blue}{2 \frac{1}{3}}>\]

4. A recipe calls for \(\textcolor{red}{2\;cups}\) of flour. Jack wants to \(\textcolor{blue}{double}\) the recipe, \(\textcolor{red}{how\;many\;cups}\) of flour will he need\[\leftrightarrow\textcolor{red}{2}\times\textcolor{blue}{2 }=\textcolor{red}{4}\Rightarrow \times<\textcolor{red}{4},\textcolor{red}{2},\textcolor{blue}{2}>\]

5. A recipe calls for \(\textcolor{violet}{2 \frac{1}{2}\;cups}\) of flour. Jack only needs \(\textcolor{blue}{\frac{1}{2}}\) the recipe, \(\textcolor{violet}{how\; much}\) flour does he need?\[\leftrightarrow\textcolor{violet}{2 \frac{1}{2}}\times\textcolor{blue}{\frac{1}{2}}=\textcolor{violet}{1\frac{1}{4}}\Rightarrow \times<\textcolor{violet}{1\frac{1}{4}},\textcolor{violet}{2 \frac{1}{2}},\textcolor{blue}{\frac{1}{2}}>\]

6. One bottle holds \(\textcolor{violet}{2 \frac{1}{2}\;cups}\) whilst a smaller bottle only holds \(\textcolor{violet}{1 \frac{1}{4}\;cups}\), \(\textcolor{blue}{how\;much\;bigger}\)/\(\textcolor{green}{how\;much\;bigger}\) is the first bottle? (compare bigger to smaller)\[\leftrightarrow\textcolor{violet}{2 \frac{1}{2}}\div\textcolor{violet}{1\frac{1}{4}}=\textcolor{blue}{2}\Rightarrow \times<\textcolor{violet}{2 \frac{1}{2}},\textcolor{violet}{1 \frac{1}{4}},\textcolor{blue}{2}>\] \[\leftrightarrow\textcolor{violet}{2 \frac{1}{2}}-\textcolor{violet}{1\frac{1}{4}}=\textcolor{green}{1\frac{1}{4}}\Rightarrow +<\textcolor{violet}{2 \frac{1}{2}},\textcolor{violet}{1 \frac{1}{4}},\textcolor{green}{1\frac{1}{4}}>\]

7. One bottle holds \(\textcolor{violet}{2 \frac{1}{2}\;cups}\) whilst a smaller bottle only holds \(\textcolor{violet}{1 \frac{1}{4}\;cups}\), \(\textcolor{blue}{how\;much\;smaller}\)/\(\textcolor{green}{how\;much\;smaller}\) is the second bottle? (compare smaller to bigger \[\leftrightarrow\textcolor{violet}{1 \frac{1}{4}}\div\textcolor{violet}{2\frac{1}{2}}=\textcolor{blue}{\frac{1}{2}}\Rightarrow \times<\textcolor{violet}{1 \frac{1}{4}},\textcolor{violet}{2 \frac{1}{2}},\textcolor{blue}{\frac{1}{2}}>\] \[\leftrightarrow\textcolor{violet}{2 \frac{1}{2}}-\textcolor{violet}{1\frac{1}{4}}=\textcolor{green}{1\frac{1}{4}}\Rightarrow +<\textcolor{violet}{2 \frac{1}{2}},\textcolor{violet}{1 \frac{1}{4}},\textcolor{green}{1\frac{1}{4}}>\]

All this may feel overwhelming at first sight but understand, analysing these wordproblems creates a communicative site in the classroom. These aren’t simple exercise to be rushed through they are opportunities to thoughtfully explore and coordinate concepts expressed in Natural language and how those same concepts appear in arithmetic. The colour coding coordinates sense making in both English and Maths. Each wordproblem in the above set of examples is one of a triple of wordproblems that go with the given synset. The richness of the PIC approach lies in exploring arithmetic with the three coordinated simple number sentence forms simultaneously. It is a single concept that holds these three simple number sentences together as a synset triple, each number sentence in the triple is a tautology in respect of the other two. The full richness of the approach happens when the full PIC is brought to bear in forming understandings: \[WP\leftarrow \circ \rightarrow Calc\] \[WP\leftarrow \circ \rightarrow LA\] \[LA\leftarrow \circ \rightarrow Calc\] The deep thinking here is that conceiving a concept is always an unending story. A concept is never completely captured by any expression in any semiotic system. Where primary arithmetic has six concepts to deal with (both in language and arithmetic), algebraic arithmetic condenses these to a single additive and a single multiplicative concept (overseen by the concepts of “\(0\)” and “\(1\)”) which sum up the synset triples. That the synset triple of simple number sentences are tautologies for each other establishes the unity of the new concepts. This summation is a mathematical concept that doesn’t have an expression in natural language and is part of the toolkit of mathematical thinking.

Operational Arithmetic models additive and multiplicative thinking.

Modern arithmetic (the one that includes “\(0\)” in its vocabulary) is as wondrous as language. Operational arithmetic isn’t the end of the journey it is just a sensible way to start with young minds. The operational arithmetic described here is a bridge between Natural language thinking and Algebraic thinking. In fact more than being a bridge it gives the tools to build the bridge. The colour coded synset description of operational arithmetic is a way of exploring additive and multiplicative thinking about the world - remember Peirce: “all thought is in signs”. Just as a Natural Language encodes a world of categories so Operational Arithmetic encodes a world of quantities. A colour coded synset is a way of interrogating what can be expressed about the world using the four standard operations as content. To explore the full range of semantics expressible in the syntax of operational arithmetic a slight extension to the colour coding is required. Comparisons between numbers are not always between either amounts of the same stuff, or counts of the same things. We need a way to distinguish synsets that describe the comparison of two measures, for example \[\times<120km,2hr,60kph>\] or the comparison of two counts, \[\times<120_{zebra},2_{lions},60_{zebra/lion}>\] which of course extends the arithmetic to rates and ratios. But just as importantly it allows fruitful discussions of nonsense: \[_+<\textcolor{violet}{120km_1},\textcolor{violet}{2hr_2},\textcolor{green}{???}>\]

In conclusion

The thinking behind the work presented here has been heavily influenced by the writings of M.K.A.Halliday (M A K Halliday and Christian, 2013). His pioneering work on Systemic Functional Grammar (SFG) see words and the syntax of sentences that contain them as making meaning on a continuum. He coins the phrase “lexicogrammar” to describe the formal description of language he produced. As with all linguists with whom I am aware, his treatment of numerals isn’t sufficient. Most grammarians shuffle numerals off into a category of their own, the opening paragraph of the wikipedia article on “numerals” gives an adequate summary:

>In linguistics, a numeral in the broadest sense is a word or phrase that describes a numerical quantity. Some theories of grammar use the word “numeral” to refer to cardinal number that act as a determiner that specify the quantity of a noun, for example the “two” in “two hats”. Some theories of grammar do not include determiners as a part of speech and consider “two” in this example to be an adjective. Some theories consider “numeral” to be a synonym for “number” and assign all number (including like the compound word “seventy-fifth”) to a part of speech called “numerals”.  Numerals in the broad sense can also be analysed as a noun (“three is a small number”), as a pronoun(“the two went to town”), or for a small number of words as an adverb (“I rode the slide twice”).(https://en.wikipedia.org/wiki/Numeral_linguistics #:\textasciitilde:text=Some%20theories%20of%20grammar%20use,example%20to%20be%20an%20adjective.)

The description of an operational arithmetic, that sits behind this article, is an attempt to give a “numerogrammar” of operational arithmetic: a description of how numerals function in language, in arithmetic, and between them. In fact “Operational Arithmetic” can act as a probe for a deeper understanding of a lexicogrammar.

In SFG the ideation mode of language describes sentences in terms of participants (nouns and noun phrase), processes (verbs and verb phrases), and circumstances (the how, where, and when) of the action between Participants. In operational arithmetic Numerals (noun like) that refer to numbers functioning as Participants, Numerals that name relations (verb like) function as Processes (in simple number sentences that use “\(=\)” with the sense of “make”), and Circumstances¹¹ are the framing of numbers as referring to counts or measures. Numbers for Measures have a further degree of Circumstance in respect of attributes (SI units).

But the description of the “numerogrammar” of operational arithmetic is not dependent on SFG. Beyond being a “theory” of numeral use, it is primarily a tool for engineering change in the classroom. In the classroom it is the communicative events that sit between the nodes of a PIC that give a 360 degree view of a conceiving of the world using arithmetic. The deep thought here is that when numerals are said, heard, read, written in the classroom they have senses.

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