# Researching the relation between language and (mathematics) teaching

The mathematical language of the teacher, the vocabulary she uses while speaking about learning and the learner, and the language she employs while speaking about her practice are three relatively independent teaching-related themes that constitute a fertile soil for LiME research.

## Theme 7: Teacher’s mathematical language

How should the teacher present students with an algebraic task? Should she say “Solve x2 > 4 for x” or rather “Find the numbers that, when substituted for x, turn x2 > 4 into a true proposition”? While solving the inequality, should she opt for “Lets transpose the 4” or for “Let us subtract 4 from the numbers [functions] on both sides of the inequality”?12 Does it even matter? The same seemingly obvious answer imposes itself in both cases: given the equivalency of the two expressions, the first proposition is preferable to the second because of its simplicity. But is it generally so? Is the simpler always the better? The answer I wish to promote is that opting for simplicity as a rule may be a mistake, because simplicity does have its price: it comes on the expense of proper opportunities for learning. Thus, in both pairs above, the first expressions are about symbols rather than mathematical objects. The teacher who uses the first type exclusively, and thus appears to be always talking about the visible entities that do not stand for anything but themselves, never offers her students incentives for constructing mathematical objects. And yet, without those abstract entities, thus without seeing x2 > 4 and x2-4 > 0 as “representing” the same sets of numbers, the learners may be unable to make sense of her manipulations on the symbols. Incapable of telling what is being done in the form of a sensible story, the different parts of which can be deduced one from another or from other things they know, they can only learn by memorizing. Investigating teachers’ mathematical language and its possible impact on students’ learning is an important topic for LiME research. Looking for lexico-grammatical forms that would balance the tension between the needs for simplicity and the needs for the fidelity to mathematical precision is another.

## Theme 8: The language in which the teacher speaks about learners and learning

In the classroom, the teacher tells stories not only about mathematical objects, but also about her students. These two types of classroom storytelling can be called mathematizing and subjectifying (Sfard, 2008). The stories of this latter type are not necessarily explicit: they are often merely deducible from other things the teacher says or does. Thus, if the boy by name Gur is one among many students in the classroom who managed to arrive at a correct solution to a problem, but the teacher greets his result with the particularly cheerful exclamation “How wonderful, Gur!”, her exaggerated praise, and especially the use of the superlative “wonderful”, lets us understand that this student’s success came as a surprise. The teacher’s implicit storytelling becomes an indirect act of building Gur’s identity as the “weak” learner of mathematics.

Indeed, whether explicit or tacit, the teacher’s subjectifying stories are powerful means of shaping students’ identities. They are at their most effective when properties of the students’ actions are translated into characteristics of the student himself. For instance, the reifying story “Gur is a weak student”, while seemingly equivalent to saying, “so far, Gur has been unsuccessful in most of his mathematical assignments”, implies a much greater pessimism about the possibility of future change. The pessimism aggravates when, in addition, Gur’s story hints at an independent reason for his being a “weak student”. This is the case, for instance, in statements such as “Gur has a learning disability” or “Gur sutfers from dyscalculia”.Yet, as long as there are no independent ways for finding inner factors, if any, behind students’ learning difficulties, dyscalculia remains a discursive construct, not unlike abstract mathematical objects. Yes, objectification, as desirable as it may be in the teacher’s mathematical language, becomes risky and avoidable in her language of subjectifying. Here, speaking in terms of as-if permanent properties may impact the choice of opportunities for learning offered to the students, and ultimately, it may determine these students’ life trajectories.

Whereas all kinds of subjectifying may be harmful to student’s emerging identities, its covert versions are most risky. Implicitly told stories, because of their being invisible to the storyteller, remain practically uncontrollable. In result, the teacher’s language may do things that go against her own better judgement. This is where LiME researchers’ ability to see what remains invisible to the teacher herself becomes invaluable. Flagging the hidden linguistic traps is this researcher’s first task. In this endeavor, she should pay particular attention to first-person stories of those who went through the experience of being identified by their teachers, parents or friends in unhelpful, harmful ways.The researcher’s other assignment is to guide the teacher to a language in which the most harmful things are simply unsayable - a language in which semantic voids open in those places where her language, so far, carried the greatest risks.

Precautions must be taken by the teacher also when she speaks about her students’ learning. Here, she may be prone to talk, thus think, in terms of deficit - of what the learners “do not yet do” rather than of what they are actually doing.Thus, the teacher who says “Gur did not succeed in comparing two sets with regard to number” implies that Gur was, indeed, trying a comparison, but, unfortunately, used improper procedure or failed to perform correctly the proper one. And yet research has shown that only too often, very young children, when asked “Where is there more?” perform potentially useful tasks that, however, have nothing to do with quantitative comparisons (the task may be that of choosing one of two sets, choosing from them or naming their contents - see Lavie & Sfard, 2019). In this and similar situations, the teacher remains convinced that her students failed in something that they did not try to do. The language of deficit deprives her of opportunities to see where the children are coming from - in this case, she fails to see the inner workings of their rich, but numberless worlds. In result, it also makes her ignorant of what they need in order to get from where they are now to where she wants them to be. The advice given to the learner in one of the last paragraphs may now be offered also to the teacher: to make learning-teaching interaction more effective, also teachers must be capable of dialogic engagement. More specifically, the teacher must remain alert to the possibility that her words lead the student to tasks different from those she has in mind. In identifying those places where she may be most likely to fall in the trap of deficit discourse, the teacher can benefit, as before, from LiME researcher’s help.

## Theme 9: The language in which the teacher speaks about her professional practices

All that has been said above about learners and learning becomes applicable to the teacher within the context of her own life-long professional learning. There are at least two phenomena that research on teacher learning has been showing with striking consistency: teachers’ practices are highly resistant to change and their stories about their own practices are often quite different from those told by external observers (Spillane, 2004). Both these findings may now be accounted for, at least partially, by the existence of unacknowledged disparities between the language used by the teachers and those employed by their mentors and observers. Heyd- Metzuyanim and Shabtay (2019), who noted considerable differences between the first-person and third-person stories about practices of one teacher, hypothesized that such disparities may result from an unrecognized incommensurability between the discourses of different storytellers rather than from the teachers’ failure “to see herself correctly”. In general, however, not much thinking has been invested so far into the role of language in shaping teachers’ practices. This is a rich and timely theme to be yet studied in a systematic way by LiME researchers.