How can we help students understand graphs and overcome some of the conceptual difficulties they have?
In this rigorous study the researcher investigated whether mathematics instruction based on questioning that targeted students’ knowledge about their mathematical thinking (metacognitive questioning) would promote students’ learning and enhance their understanding of graphs. (Examples of metacognitive questions are: ‘Does this make sense? Why is s/he doing it that way? What did I/we do last time we had a similar problem?). Students used such questioning approaches within small group learning situations in which they worked together to solve the same problem (co-operative learning). The research focused on 196 (Year 9) students and six teachers in two schools. The study found that the students significantly improved their understanding about how to interpret and construct graphs. Research evidence also showed that co-operative small group working structured through metacognitive questioning engaged more students and produced more elaborate verbal contributions from the participants when compared to the control groups.
Keywords:
Israel; Key Stage 3; Secondary schools; Numeracy; Pupil grouping; Speaking and listening; Metacognition; Learning processes
As background to the study the author described research into students' understanding of graphs, constructivist and cooperative learning, metacognitive instruction and students' "alternative conceptions" (beliefs and understandings that diverge from generally accepted models) of graphs. Some of the conceptual difficulties that students showed when interpreting or constructing graphs related to:
The researcher suggests that these alternative conceptions (or misconceptions) limit students' ability to develop what she refers to as 'graph sense'. She quotes research that suggests that 'graph sense' means students looking at the entire graph, or the relevant part of it, and understanding that it expresses the overall way in which two variables change in relation to each other.
The findings showed that students who had been instructed in metacognitive questioning:
than did those students who had not received the intervention. There was no difference between the two samples in relation to conceptual understanding before the study began.
Specifically the results demonstrated that students in the intervention group improved their understanding relative to the control sample in two ways:
On the other hand there was little change in either sample in the tendency of some students (about 25% in each case) to assume graphs always showed positive relationships. In addition, the number of students treating graphs as 'icons' fell equally for both samples by about the same amount - from about 20% of each sample to about 10%.
The research findings suggested that when compared with the control group, students in the intervention class:
Control group students tended to spend more time working individually, (even though they were organised in co-operative groups) and asked more questions of a technical nature. However, off task behaviour was rarely an issue in either group. In addition, students in both groups continued to provide or receive similar numbers of answers without explanation.
The intervention was based on a 10 lesson, two week "Linear Graph Unit" intended to develop students' understanding of key points about graph work, such as: graph slope, intersection point, rate of change and other aspects of graph interpretation. The students were not given graph construction work in any class, as the researcher planned to use activities based on graph construction as a measure of how well the students could transfer ideas. All students used the same textbook and tackled the same problems/tasks. The teachers taught each class using a similar lesson structure consisting of: introduction (10 mins), group work (30 mins) and whole class review (5 mins).
Both sets of three classes used the same cooperative approach in their mathematics lessons. The teacher organised the students into groups of four each made up of one high achieving, one low achieving and two middle achieving students. Each student in turn read a problem and tried to solve it. While doing so s/he explained the task and proposed an approach for solving it to the other members of the group. If the team of four disagreed, they discussed the issue until all agreed upon a solution which they then wrote down. During the discussion they presented their own viewpoints and worked together to arrive at the best option for proceeding. When none of the team knew how to solve a task, they asked the teacher for help.
What was different was that the group which had been taught how to use metacognitive questioning used a series of questions in order to prompt comprehension, help decide choice of strategy, forge connections with previous experience and reflect on both the solutions reached and the process by which they got there. These are described further on the next page.
The metacognitive questioning approach was aimed at developing the students' thinking individually and collaboratively in the group context. The researcher had already been involved in the development of a teaching approach called IMPROVE which was designed to encourage metacognitive discourse throughout the curriculum. This entailed students working in small groups and asking probing or prompting questions of themselves and each other. The questions used in the IMPROVE programme around which discussion was developed focused on four key areas:
To help students further in understanding the graph tasks the students were guided by the acronym DATA which represented the four stages:
The researcher believed that by engaging in open questioning and encouraging responses using mathematics terms teachers could change students' behaviours in mathematics.
The study drew on a sample of 196 eighth grade (Year 9 in the UK) students (96 boys, 100 girls) from six classes in two different schools. The schools were similar in terms of size and socioeconomic range of pupils. Within each school, three classes were randomly chosen from a pool of seven classes in which mathematics was taught in mixed ability classrooms. The students in each class were similar in terms of academic ability and prior mathematical knowledge.
The teachers were all female, with a degree in mathematics and at least five years experience in teaching mathematics. All the teachers received two days of training on pedagogical issues related to the teaching of the mathematics unit on linear graphs. Except for practising the use of questions intended to prompt metacognitive discussion on the part of the students, the teachers' training was identical. The students were given the same test before and after the intervention.
The results of the pre and post intervention tests were analysed for the two samples and compared statistically. Two separate tests, one of graph interpretation and one of graph construction, which was intended as a measure of knowledge transfer, were used. Each aimed to assess the "alternative conceptions" held by students and to see whether they were replaced by the generally accepted ones as a result of the intervention. The students' test responses were judged by two experts in mathematics education.
In order to compare the discussion behaviours of the two samples, a research student blind to the purpose of the study made observations of the interactions between students in 24 small groups (four groups selected randomly from each of the six classrooms). The research student was prepared for the coding of the behaviours by prior watching and analysing videos of cooperative learning settings. The observer recorded observations of each student in the group four times for 1 1/2 minutes each time and also coded the interaction level of the entire group. The total observation time for each group of four in one lesson was almost ten minutes. Observations were made twice a week so the total observation time for each group was forty minutes.
Whilst writing this digest the author became aware of a number of implications for teachers:
An implication for headteachers and others involved in raising achievement is:
A number of interactive research summaries on the GTC Research of the Month website provide significant detail about dialogue in the context of collaborative learning see:
Effective Teachers of Numeracy at: http://www.gtce.org.uk/shared/medialibs
/31435/93134/numeracy.pdf
Improving learning through cognitive intervention: Do you want to know more about how teachers have used cognitive intervention to develop pupils' thinking skills? at:
http://www.gtce.org.uk/shared/medialibs/31435/
93128/103260/raisestudy.pdf
Researching Effective Pedagogy in the Early Years at: http://www.gtce.org.uk/shared/medialibs/
31435/93128/103260/earlyyears.pdfSocial interaction as a means of constructing learning: the impact of Lev Vygotsky's ideas on teaching and learning at: http://www.gtce.org.uk/shared/medialibs/
31435/93134/VygotskyROM.pdf
A short digest on Encouraging Student/Student Interaction in Science lessons:
http://www.educ.sfu.ca/narstsite/publications/research/
encourage2.htm
National Education Research Forum bulletin editions 1 and 2 contain a number of articles about research evidence related to classroom dialogue and co-operative groups, see:
www.nerf-uk/bulletin
National Numeracy Strategy: Learning is more effective when common misconceptions are addressed, exposed and discussed in teaching, at: http://www.standards.dfes.gov.uk/numeracy/prof_dev/self_study/
effectiveteaching/14699/?leaf=4
TTA website, for an article about building on pupils' misconceptions in mathematics: From Nottingham to Prague: Building on best practice internationally
http://www.tta.gov.uk/php/read.php?resourceid=1926