The skilled maths teacher sets the pace

Abstract

Skilled mathematics teachers do things that most of us who teach mathematics do.  The trick is that they do not just keep pace with the practice, they also set the pace, writes Joakim Samuelsson.

Despite the criticism that Swedish schools and these mathematics teachers often have to endure, there are in our schools mathematics teachers whom school administrators, colleagues, pupils, former pupils and parents perceive as good maths teachers.

A larger study (Samuelsson, 2012) describes and analyses how an experienced and, as perceived by those around them, skilled maths teacher at a senior level school conducts their mathematics lessons. This article, which is that part of the larger study, shows how the skilled maths teacher works with language in their teaching.

Background

School is one of the largest workplaces we have in the country.  There, teachers and pupils work along with many other staff groups.  Apart from all these people who are found on a daily basis in schools, virtually all adults in Sweden have attended school.  They have thus acquired their own specific picture of what a school is and how it and the people who work in the school should act.

The fact that so many people work in schools and have spent so much time in schools results is many people having opinions about school.  Thus, there are very many who feel they know what a good school is, as well as a good head teacher, a good teacher and a good pupil etc.
Perhaps because so many people have personally experienced it, schools are subject to critical viewpoints from many different directions.  In studying the teaching of mathematics, it can be observed that pupils' mathematical knowledge is a perennial topic of discussion in society in general and in schools in particular (Swedish Government Official Report (SOU), 2004).

The teacher plays a crucial role in determining which maths skills are developed in a school and society (SOU 2004, Hattie, 2009).  It is the teacher's decisions in the teaching situation that determine what pupils in the school have the opportunity to learn (Samuelsson, 2003).

An issue often raised in relation to school mathematics in Sweden is the large number of teachers who teach mathematics but who lack the qualifications.  If one lacks training or knowledge in an area, it is often much more difficult to know what to look and listen for in the classroom.  Without knowledge within a specific field, one does not know for example the appropriate way in which to represent the mathematics for the pupil group in question, or how pupils in this age group think about different mathematical phenomena, the misconceptions the pupils have, difficulties the pupils usually have with the content or what the critical points in the teaching can be, etc.

Despite the criticism that Swedish schools and these mathematics teachers often have to endure, there are in our schools mathematics teachers whom school administrators, colleagues, pupils, former pupils and parents perceive as very good maths teachers. The purpose of the following study is to describe and analyse how an experienced and, as perceived by those around them, skilled maths teacher at a senior level school conducts their mathematics lessons.

Knowledge of their teaching methods could provide a different and probably more positive picture of how it is to work as a teacher than those fateful descriptions we often encounter in the press.  To gain an insight into what characterises an experienced skilled maths teacher, I followed a senior-level teacher in his mathematics lessons during his final term before retirement.

The maths teacher is perceived by school administrators, colleagues, pupils, former pupils and parents to be a very good maths teacher.  The results of the entire study are presented in a larger report (Samuelsson, 2012).  This article provides only a brief example of how the teacher works with language in his teaching of mathematics.

Finally, there is also a discussion on what it is that characterises the skilled maths teacher's teaching of mathematics on a more general level.
The skilled maths teacher's use of language

At the first lesson during my field studies, I felt that it was a lesson with a clear mathematical focus.  The feeling was that, as a participant, one was drawn in to what could be described as the mathematical discourse.  During the mathematics lessons, the teacher and the pupils worked with mathematics, they thought mathematics and they utilised the mathematical concepts.

The teacher expected the pupils would be familiar with terms and concepts and that they would also use them since the concepts have a meaning.  As part of the instruction, it is emphasised that the words/concepts used mean different things, the symbols used also have a meaning, while at the same time mathematical reasoning is frequently applied in mathematics teaching.

The meaning of words

In mathematics teaching, a variety of words and expressions are used that are not commonly found in pupils' everyday lives.  The linguistic expression denotes a signification or a meaning. Essential elements of what we might call knowledge are found in the relationship between an expression and that which the expression refers to.  In the subject teacher's teaching, this form of language use is very clear.  It is about, for example, what different prefixes stand for; milli, cent, hecto, kilo, etc.

Through the teacher also expecting the pupils to master these prefixes, he returns constantly to them in his teaching when the pupils, for example, discuss mathematical problems.  An example of this phenomenon is when the 'percentage' concept is to be introduced. The teacher then connects this to centimetres.  He says:

"If we know what the cent stands for in centimetre, maybe we can figure out how much a percentage of something is."

That the prefixes have a meaning and are important in the teacher's teaching is clear from the note that sits next to the board in the teacher's classroom.  On the note are all prefixes that are relevant to the pupils and their learning in mathematics, listed and described.
On several occasions, the teacher stresses that it is called kilogram and not kilo, for the purpose of getting the pupils to realise that kilo means a thousand of something, in this case grams.

Another example of when the prefixes are used is when the pupils are working with scale. When asked, "How long is a centimetre on the map in reality?".  The map's scale is 1:4 000 000, as is 1 centimetre is 4 000 000 centimetres in reality.  To simplify this, the teacher makes use of the prefixes.  Since cent means one hundredth, the following line of reasoning can be applied. We draw a vertical line between the second and third zero from the end of 4 000 000.  This means that we get 1 cm representing 40,000 m.

                                 m
1 cm thus corresponds to 4 000 0|00 cm

Since kilo means 1 000, the following line of reasoning can also be applied. We draw a new vertical line so that three zeros from 4 000 0 "disappear". One centimetre on the map thus corresponds to 40 km in reality.

                             km
1 cm thus corresponds to 4 0|00 0/code>

The above is an example of how the teacher uses words and their meaning for thinking about mathematical problems.  Language is thus given power as a mediating tool, a resource for how one can think in order to create knowledge about the world.  Through the flexible and developable relationship between linguistic expressions and the phenomena they refer to, the pupils can then solve problems that they had not previously encountered. At the same they are being schooled in a way to relate to mathematics.  That is to say, words mean something, and words can help with thought processes.

The meaning of the mathematical symbols

In addition to the words used having a specific meaning, the teacher also emphasises the mathematical symbols' meaning in his teaching. 

When the class, for example, are discussing speed problems, the teacher is very careful to point out that 80 km/h means 80 km per hour, not 80 km an hour.  This figure of thought, that is to say, that it goes 80 km in one hour, is later included in what Samuelsson (2012) presents and is calls, Leif's magical line.  
 
Another example of how the teacher draws attention to the meaning of symbols is when the mathematical operation 'multiplication' is used.  A pupil who writes    when he should calculate what three pastille boxes cost together if one pastille box costs SEK 15, is corrected immediately.  The teacher says that it is 3 times SEK 15, that is  .

Thus, for the teacher, the meaning of the constituent parts is important because they represent a way of thinking.  The group with fifteen crowns is the multiplicand and that is to be multiplied.  How many times we take the multiplicand is indicated by the multiplier (3).  By constantly pointing this out to the pupils, their attention is drawn to the significance of the constituent parts in a multiplication.

The mathematical reasoning

In my first e-mail contact with the teacher, he points out that he is probably just a traditional mathematics teacher and that there is nothing special to come and look at.  After my first week, I feel obliged to remind him of what he said, and also try to explain why I do not think he is a traditional mathematics teacher.

Traditional mathematics teaching has often been associated with teacher run-throughs and individual calculation. In this teacher's classroom, there was constant communication, except when the pupils were to work individually.  Through the classroom practice including these processes, the pupils get an opportunity to follow and evaluate the reasoning of the teacher and other pupils, while they themselves are pushed to devise and implement both formal and informal reasoning.

These processes are generally present in all the different forms of teaching that the teacher uses (see Samuelsson, 2012).  A common conversational structure may look like this:

• The teacher asks a question
• A group of pupils discuss it
• Each group presents their thought process
• The teacher listens and provides feedback and tries to clarify the pupils' train of thought
• The pupils correct the teacher or have comments on other groups' solutions

In the above teaching process, the pupils are at several times given the chance to follow and assess others' reasoning as well as to devise and implement reasoning.  Through continuously engaging in the pupils' thought process, the teacher asks questions that require the pupils to explain themselves, resulting in the development of that their reasoning and communication skills.

The inventory of how the pupils think and feedback are features in all the forms of teaching used by the teacher; whole class teaching at the individual level where new material is presented, whole class teaching at the individual level where problems are discussed, whole class teaching at the group level, individual exercises and during feedback on tests.  This type of inventory has probably greatly helped the teacher to develop his profession knowledge of how pupils think about the mathematics that is taught at senior level school.

Knowledge of the pupils' thought processes has since helped the teacher in his

• planning, because he knows what is difficult, but also in his
• interaction with the pupils, as he can quickly recognise the pupils' problems.

Final reflection

In recent years, there have been significant financial investments in laboratory material throughout our Swedish schools.  Laboratory mathematics has been something that many municipalities have seen as an important part of teachers' professional development. In studying the skilled teacher's teaching, it can be observed that he utilises laboratory models, but only for very short sequences, when something new is to be introduced or when pupils do not understand the abstract model he uses.

In both these cases, the teacher endeavours to get the pupils to assimilate the abstract mathematics and realise that the mathematical terms and concepts that exist provide help in how to think and understand the mathematics.  It is about trying to get the pupils to understand mathematics as the subject is constructed and the relationships between concepts.  My impression is that the teacher is trying to keep his teaching in a mathematical discourse, and only uses concretising material or everyday events to a small degree. However, pictures are frequently used.

In Samuelsson (2012), attention is drawn to the skilled maths teacher, the characteristics of his teaching, and the knowledge used by the teacher in the interaction with the pupils.  Trying to describe what characterises the skilled teachers' teaching is not entirely easy.  There is a risk that what a researcher perceives as a characteristic of the teaching is something that every person, who has devoted themselves to teaching mathematics, is familiar with, and that it therefore does not receive the emphasis it deserves.  We could compare it to a football player. Many of us have played football but there are very few of us who have had the chance to play at the highest level.

Yet we know how to run, shoot and pass, etc.  What probably separates us from the best is that we do the same things but we do it with a totally different quality.  What is difficult to determine in this type of text is when these processes occur, to what extent and in relation to what content, in relation to pupils' questions and what happens in the classroom.  Perhaps it is precisely that which is the finesse of the skilled mathematics teachers.  They do things that most of us who teach mathematics do, but they also do it at the right time.

When, for example, something happens in the class, the skilled maths teachers recognise it and can capture a thought on a maths problem and carry the conversation forward in a fruitful way for the pupils.  Some researchers write that the skilled teachers keep pace with the practice.  I can only agree. The investigated teacher keeps pace with the practice, while at the same time it is he who is setting the pace. In this way, he creates a teaching world where mathematics is front and centre.