At all school levels, it is clearly important that students are able to distinguish elements of mathematical theory which are just results of *decisions made* by people, such as terminology, notation and so forth, from results which have been *discovered*. Technically, the first domain consists of mathematical *definitions*, while the second consists of mathematical *theorems*, or mathematical *results*. In the first domain, we find definitions of mathematical *concepts* such as rectangles, triangles, prime numbers, functions and so forth. We also find *conventions* regarding notation, terminology, rules for expressing things, and so on. In the second domain we find the Pythagorean theorem, algebraic laws such as the distributive and commutative laws, and other elements of mathematical theory which require an *explanation* for why they are true, or formally speaking a *proof*. For elements in the first domain, students must essentially learn to accept decisions made by the community. If asked “why” it is true that

$$a^{3} = a \cdot a \cdot a$$

for all numbers *a*, the teacher can do little but essentially respond that this is so because mathematicians have decided so: Mathematicians have *agreed* that the notation

is to be shorthand for *a* · *a* · *a*. Similarly, if asked why a full circle is 360°, again the teacher can do little but explain that this is something mathematicians have agreed about. One could *motivate* the choice by going into the details of why this choice was made historically, but students still must accept that this is just a choice.

However, students should learn to relate in a completely different manner to elements from the second domain, namely the “theorem” domain. For these, students may rightfully ask for a justification, or a proof, for why the result is true. Typically, the term “proof” must be interpreted in the school context, that is, we are referring to explanations which can give students a meaningful understanding at the grade level in question. And even if understanding an (intuitive) proof of the result is beyond the reach of the student at the present time, it is clearly an advantage if the student understands that there *is* an explanation. Then the student understands the nature of the mathematical situation, and avoids feeling stupid for not understanding “why” the result is true. This is important at all school levels. In particular, it is crucial for the long-term building up of mathematical understanding.

The division of mathematical theory into “definitions” and “theorems” may be considered as a distinction between mathematical *language* and *content*, respectively. For this reason, we refer to it as the *LC distinction*. The language domain represents definitions, while the content domain represents theorems. The formal distinction itself is described in the field of mathematical logic. See e.g. (Shoenfield 1967). Note, however, that while in mathematical logic one would typically take the word ‘language’ to mean an underlying (formal) language in which both theorems and definitions are expressed, we use the word language in a different sense. Here, we consider new definitions as *extensions* of the mathematical language, and thus as becoming a part of it. This mechanism corresponds to *extensions by definitions* in a first order logical language, see section 4.6 in (Shoenfield 1967). In mathematics textbooks at the university level, the distinction between theorems and definitions is usually very clear and explicit. This is often not the case in school mathematics books. There is a deep divide between textbook traditions at different levels here.

It should be emphasized that the term ‘language’ is used in many different ways across the field of mathematics education research. Our use of it here is *mathematics theory oriented*, as opposed to, for instance, a *learning theory oriented* use. Speaking in terms of mathematics teaching, it is clearly impossible to teach content (in our sense) without also teaching language; the LC distinction is not related to mathematical teaching or learning as *processes*. The distinction concerns mathematical theory as a *body of knowledge*.

When starting in school, children will typically meet a lot of L mathematics in the beginning stages. They will learn that numbers are written using some particular (chosen) symbols, that the symbol “+” is used for adding numbers, they will be informed about what the word ‘rectangle’ means, and so forth. However, the C category also quickly comes onto the scene. When children find that 2 + 3 = 5 by counting first 2 and then 3 objects, they are discovering the mathematical theorem “2 + 3 = 5”. This is a mathematical result, so formally it belongs in the C category. However, it is normally not *used* like this in school mathematics. For this reason, we did not count results of arithmetical calculations as theorems (C mathematics) in our framework for categorizing test items, see “Methods”.

In spite of its importance at all school levels, the distinction between L and C is rarely discussed in mathematics education literature. See e.g., (Clements et al. 2013; English & Bussi 2008; Niss 2007). It does not fit into well-known frameworks for mathematical competencies (see, for instance, Kilpatrick et al. 2001; Niss 2015; Niss & Jensen 2002), due to the fact that neither L nor C can naturally be described as corresponding to “competencies’’ when taken separately. On the other hand, the LC distinction is closely related to research on the role of definitions in school mathematics, for instance research concerning the difference between *concept image* and *concept definition* (Niss 1999). While research on the role of definitions in education is related to the L side, research on the role of *proof* in mathematics education (Hanna 2000; Pedemonte 2007; Tall 2014) is related to the C side. However, none of the two research traditions mentioned here put their emphasis on the *distinction* between the two sides. Further, the LC distinction is very different from the process/object duality described in (Sfard 1991) and subsequent developments. Also, most of the well-known theories of *concept learning* in mathematics can essentially be viewed as adaptations of general subject-independent learning theories. As such, they fail to pick up the LC distinction, which is more or less particular to mathematics.

It should be remarked that the LC distinction is meaningful to speak of only relative to a *specific way of building up* mathematical theory. While in many countries today it is customary to define 3 · 2 to be 2 + 2 + 2, it is certainly possible to define it as 3 + 3 as well, thus switching the role of the factors. With the latter choice of definition, the result 3 · 2 = 2 + 2 + 2 is formally in the C category; it is a result which can be proved. Thus there is a certain aspect of subjectivity to the division of mathematical theory into L and C. However, this does not substantially affect the situation we are considering.