What Is Numeracy?

Numeracy and the Science Teacher: Subject Knowledge Revisited.

Allan Soares and Stephanie Prestage

School of Education, University of Birmingham

Paper presented at the British Educational Research Association Conference, Cardiff University, 7–10 September 2000

One of the requirements for student science teachers is to develop pupils’ understanding of numeracy through science (DfEE, 4/98). This research, with a group of ten secondary postgraduate science student teachers at one higher education institution, and two teachers with different lengths of teaching experience set out to determine the extent to which teachers’ knowledge base is functional and sufficient for the demands of developing pupils’ understanding of graphical representation. Discussions with the science student teachers on the their use of graphical representation in their science lessons and interviews with two of the student teachers and the more experienced teachers suggest that previous experiences with graphical representations are, on their own, inadequate for developing understanding in other learners; neither is the length of teaching experience on its own. Special attention needs to be given to developing these students’ learner-knowledge and teacher-knowledge so that they may engage critically with the use of graphical representation in science. A professional discourse beginning at the training stage needs to be maintained through on-going professional development.

Introduction

Most of the student science teachers entering the secondary postgraduate course at this higher education have specialised in a limited area of science, usually with physics, chemistry or biology related degrees, and have had considerable experiences with data handling. They can expect to have to teach some aspects of science outside the area of their particular specialism during the training year, and also once they qualify as teachers. Many are usually capable of identifying broad areas where their subject knowledge and understanding are inadequate. This is usually biology for the physical scientists and physics for the biological scientists. There is therefore a need for the trainee science teacher to develop the subject knowledge and understanding in the areas identified as inadequate before teaching; at the very least at the level appropriate for ‘telling’ accurately. The adequacy of these teachers’ grasp of subject knowledge and understanding areas within their own specialisms is often assumed without question. Only when teaching do uncertainties arise. Thus subject knowledge and understanding are hugely important issues for secondary science teachers and the demands placed on trainee science teachers would appear to be quite considerable (Finlayson, Lock et al).

To raise standards of numeracy OfSTED (1998), in its review of secondary schools in England (1993-97), recommended the development of numeracy across the curriculum. It:

... has, therefore, to be a whole school matter to establish a clearer vision of what numeracy is, and how to develop numeracy skills both in mathematics and other subjects. ¤

...The need to reduce inconsistencies in approach between teachers and across different subjects will be vital if progression in numeracy skills is to be established and implemented consistently throughout Key Stages 3 and 4. ¤

The Initial Teacher Training Science National Curriculum (DfEE 1998) for secondary science requires trainees to know how pupils’ numeracy and other mathematical abilities can be developed as an integral part of science teaching. Personal experience of working with secondary Science: Chemistry student teachers and schools over the years has suggested that:

• there is an assumption that developing numeracy through science is straightforward, with the quantitative aspect of science ‘naturally’ allowing for this development. The past experiences of the students, through the courses they have followed in secondary and higher education are seen as adequate preparation for teachers to develop pupils’ numeracy skills. If there is a problem it is because of the pupils’ inability to perform the basic numerical operations.
• student teachers uncritically adopt the practice seen on teaching placements.

This paper considers existing literature on the differing perceptions of mathematics in the science curriculum as well as roles and responsibilities of the different curriculum groups for developing mathematical skills and concepts. It then goes on to explore issues concerned with graphical representationamong a group of science teachers, ten postgraduate trainees and two more experienced science teachers including understanding of subject knowledge. The research seeks to inform aspects of professional development and to define areas of subject knowledge that will support student science teachers in developing a critical and questioning approach to professional choices in their planning.

Context

Concerns about poor communication between teachers of mathematics and teachers of science in schools have been raised for some time now (DES 1979, Cockcroft 1981, Cornelius and Gott 1988) and certainly before the introduction of the National Curriculum. The development of the original National Curriculum in England (DfEE 1998) and its first two revisions (DfEE 1991, DfEE 1995) did not help in establishing better communications between the two subject areas, with each discipline creating its school curriculum without reference to the other. Blame is placed variously. The apparent failure to provide pupils with the basic mathematical skills needed in science courses continues to frustrate science teachers (Gill 1995, Dodd 1995). Whilst mathematics teachers claim over expectations by science teachers. In a study of mathematics and science teachers ratio, percentages, transposition of formulae and graphical work in science were identified, among others, as the specific skills causing the greatest concern among science teachers. Mathematics teachers in the study considered all these areas to have been covered but acknowledged that there might be a mismatch in timing of when it is taught in mathematics and when it is required in science (Dodd and Bone 1995, Benson and Selinger 1998). A dislocation in the levels demanded, the timing and organisation of teaching, together with the lack of liaison between departments were also perceived as contributing to the inability of pupils to successfully use these skills in science. Despite continuing calls for greater liaison and agreement in the use of mathematical terms to minimise pupils’ confusion and difficulty, cries of ‘what do they teach them in maths?’ are not still not uncommon in secondary school science departments.

Alternative views exist which propose an integrated view of the disciplines. Attempts have been made to show how these two core subjects could reinforce each other with science providing the relevance or the resource for mathematics and the latter providing a vital tool for scientific analysis (Dodd 1991, McKenna , DfEE Nat Num Proj). Brown and Wall (1976) created a model with a continuum of possible interactions between the disciplines (Figure 1)

maths for maths maths science science for

math’s with and with science’s

sake science science maths sake

Figure 1.

The ‘calls’ for greater liaison seem to be in line with points two and four on the continuum rather than the centre which would require teaching in concert to develop conceptual understanding of mathematical and scientific concepts, laws or principles; in other words ‘integration’ (Roebuck and Warden 1998). Other literature appears to suggest such ‘integration’ increases understanding, motivation and interest in mathematics and science and learning in general but this approach may be too radical to be accepted with any degree of confidence. (Meir et al 1998, Westbrook 1998).

The focus for this project is the use of graphical representation in science, an issue that had been raised by the science students during their training year. Evidence from the AKSIS Project ( ) which looked at the use of graphs in science investigations suggests that ‘most pupils regarded graphs as an end in themselves’ and that over 75% were constructed incorrectly; additionally there was little teaching about the construction and use of graphs. St John Jesson (1983) suggested that the problems pupils might have in using mathematical skills could be due to teachers overestimating children’s mathematical ability or underestimating the difficulty of the mathematics required. Merely being able to do the mathematics, having the learner-knowledge in mathematics (Prestage & Perks 1999, 2000), being able to answer the question correctly, may be insufficient knowledge for the student teachers to develop aspects of pupils’ numeracy through science.

Methods

The research was undertaken by the tutor to the science trainee teachers and a mathematics education tutor at the higher education institution during the trainees’ one year postgraduate course.

Three data sets were collected from:

• assessment and discussion of student teaching files - during their placements in school the student science teachers are required to keep files containing lesson plans; using these files activities involving data handling and graphs covered in lessons were identified and subject knowledge issues uncovered; examples from the content analysis and teaching files were then used in group discussions with the trainee science teachers to explore issues concerned with data handling and graphs
• content analysis of a popular Key Stage 3 science scheme (nelson 1887) – the students reported that many of their teaching decisions were based upon a published scheme; a content analysis of a science scheme was carried out to identify and categorise the specific examples of data handling and graphical exercises included which would support the development of mathematics.
• four interviews were carried out with two student teachers (ST1 and ST2) as well as two experienced teachers (ET1 and ET2). One of the more experienced teachers had two years teaching experience (ET2) had previously undertaken the postgraduate course and was considered by colleagues to be an excellent and thoughtful teacher, while the other (ET1) with more than twenty years experience, worked closely with the institution on its postgraduate course.

The two trainees were selected for the interviews because they had included data handling and graphs in their science lessons the most and had not considered the development of pupils’ abilities in these areas through science lessons to be particularly problematic. All participants were assured that the purpose was to explore issues around the use of data handling and graphs in science lessons and not to assess their competence in helping pupils develop their skills. The mathematics tutor carried out the interviews as the science tutor might have compromised their responses.

Three examples of data (Appendix 1) one from the science scheme analysed, and the other two modified questions used by APU ( ) and AKSIS ( ) were used to initiate discussion around the areas of mathematics and science subject knowledge. The prompts in Table 1 were used through the interviews.

Science subject knowledge:

What aspect of science would pupils be working on with such data? Have you done any activity where you collected such data with a class or seen it in a text?

This is to help focus on the data and make them comfortable.

Mathematics learner knowledge

What graph would you draw to represent this data? Would you draw it and talk about its construction while you are doing this?

With prompts to encourage discussion about: axes, scale, origin, discrete/continuous data, alternative representations, interpolate, extrapolate

Mathematics teacher knowledge

What help might a pupil need to construct this graph? What difficulties might they encounter? What are the mathematics teaching points for this type of graph?

Why might the data be represented in graphical form? What is gained by drawing the graph? What questions does the graph help to answer?

Table 1: Prompts for the interviews

The interviews, each lasting 30-45 minutes, were recorded and transcribed.

Findings and Data Analysis

Data set 1: Data handling activities from inspection of lesson plans in trainees school placement files

The following two examples were typical of the data handling activities that arose from practical work in the trainees’ science lessons.

Example 1

In a Year 7 science lesson on fruit batteries pupils were asked to investigate which fruit would make the best battery. By measuring the voltages across two metal strips inserted into the fruit pupils presented the following results in their books.

FruitVoltage

Apple0.76

Orange0.74

Grapefruit0.78

Pupils were then expected to draw a bar chart before drawing the conclusion that the grapefruit battery was the best one.

Example 2

To investigate the effect of concentration of reactant A on the rate of reaction pupils were asked to measure the time it takes for a reaction to reach a certain point when using different concentrations of the reactant. The following results are obtained.

Concentration of reactant A (mol/dm3)Time of reaction (s)

1.05

0.815

0.630

0.450

0.280

0.1120

Pupils were expected to plot a graph of time against concentration before drawing the conclusion that the rate of reaction increases as the concentration increases.

Example 1 is typical of the questions used by the students. From the group discussions it was clear that there was an expectation that bar charts or line graphs would be drawn or plotted even when the pattern or conclusion could readily be deduced from the table of results.

The plotting of time on the y-axis, in example 2, raised some confusion. ‘It’s because the textbook asks you to do that but I have always plotted time on the x-axis’ was a majority response. The terms ‘dependent’ and ‘independent’ variables were explored by the science tutor with the group but the majority thought that ‘time was always the independent variable’. The group indicated that subject knowledge issues around data handling graphical activities had not been discussed with or raised by mentors other than being told that ‘pupils always have difficulties with scales and plotting graphs’. No explanation was given as to why a bar-chart, pie-chart or line graph might be appropriate.

Data set 2: Content analysis of a popular Key Stage 3 scheme of work

A summary of the data handling and graphical activities encountered in the Key Stage 3 scheme is shown in Table 1. The purpose of the activities where only the drawing of graphs and charts was required was seen by the trainees as ‘ a useful means of presenting data so that patterns could be more easily spotted’ even though no such demands were being made by the activity. Why a bar/pie chart or line graph might be the most suitable for particular data was never considered by the students and consequently raising this with pupils was not seen to be important.

Table 2: Data handling and graphical activities in the Key Stage 3 scheme of work

The following example is common in this scheme and begins to explain the planning of the student teachers.

Things to do

You know that crude oil is a mixture of substances. The percentage of each substance is shown in this table.

Name of substance in crude oil% of substance in crude oil

fuel gas2

petrol6

naphtha10

kerosine13

diesel oil19

fuel and bitumen50

a) Draw a bar chart to show this information.

b) Choose 4 of the substances found in crude oil. Draw pictures to show a use for each of themJohnson et al 1993, p.27

From the data provided, the pupils are asked to draw a chart. No questions are asked about the chart, there seems no mathematical reason for drawing the chart since the data are clear in the table. The learned behaviour ( professional traditions) of the student teachers is clarified. There are numerous such examples, nearly half of those involving data handling activities, in the published scheme used in this study. Discussions with this group of trainees revealed that the their justification for the use of such activities is that many staff in their placement schools ‘tend to follow such schemes closely and … feel obliged to stick to these schemes’ quite rigidly, a finding consistent with OfSTED’s view of science teaching at Key Stage 3 (OfSTED 1998). Sentiments such as the ‘authors of such schemes will have considered the best ways of approaching the teaching through selecting and organising the content and there seems no point in changing or questioning the content and tasks’ were not uncommon within this group of trainee science teachers. The authors have to an extent made the decisions for teachers on the tasks set and the approaches to teaching.

Data set 3: The interviews

This data is reported under the three headings of the prompts for the interviews.

Science subject knowledge

None of the interviewees talked much about science. It might be that the examples that were chosen were not ‘good’ examples for discussing science but in the data there is little that relates directly to science. For example with the ‘shoe survey’ one of the student teachers described the event:

It would be useful to collect such data from the whole class…for doing things like graphs, getting pupils to make such measurements and making it relevant to pupils (ST1)

But ET2 also did this for ‘falling spinners’:

Science behind the data? Well by adding paper clips you are adding weight so it is falling quicker (ET2)

However, variation was mentioned by both of the experienced teachers at the beginning of the ‘shoe survey’: with the most experienced teacher explaining the science purpose of this activity as:

OK this would be something where they would be looking at variables and I guess they would be looking at discrete variables as opposed to continuous variables and they would probably expect the kids to plot a bar chart for this (ET1)