## Group Work Issues

Recently I wrote this post (2017) that highlights various ways that I can see group work being of benefit to students study in mathematics. In the post I allude to there being many issues around group work that can have a detrimental effect on the learning of the students and I intend to explore them a little further here.

The benefits of group work can be vast, and are often tied to the discussion around the mathematics involved in a way consistent with the writings of Hodgen and Marshall (2005), Mortimer and Scott (2003), Piaget (1970), Simmons (1993), Skemp (1987) and Vygotsky (1962) amongst others. These perceived benefits give the students a chance to try things, make mistakes, bounce ideas around and then find their way through together. Seeing the links between the things they know and its application within new contexts or the links between different areas of maths.

So what are the down sides?

Good et al. (1992) warn that group work can reinforce and perpetuate misconceptions. This is an idea that is also expressed by von Duyke and Matsov (2015) who feel that the teacher should be able to step in and correct any misconceptions that the students express, although this would be difficult in a classroom where a number of groups are working simultaneously and it also goes against the feelings expressed by some researchers, such as Pearcy (2015), that students should be allowed to get stuck and not receive hints. This is a tricky one to balance. As teachers we clearly do not want misconceptions becoming embedded within the minds of our students, but we do want to allow them time to struggle and to really get to grips with the maths. I try to circulate and address misconceptions when they arise but in a manner that allows students to see why they are wrong, but not give them the correct answer.

Another potential pitfall of group work is related to student confidence. Some students worry about being wrong and as such will not speak up. This is an issue that transcends group work and that we need to be aware of in all our lessons and is discussed at length in “inside the black box” (Black and Wiliam, 1998). It is part of our jobs as teachers to create an environment where students do not fear this, and are comfortable with talking without fear of being laughed at. I try to create a culture where students know it’s better to try and be wrong than not to try at all. This classroom culture is discussed by Hattie (2002) as an “optimal classroom climate” and it is certainly a good aim for all classrooms.

The other main downside to group work is behaviour related (Good et al., 1992). Group work can be more difficult to police, and it can become difficult to check that everyone is involved if you have a large class that is split into many groups. This can give rise to the phenomenon known as “Social Loafing”, which is where some members of the group will opt out in order to have an easy ride as they feel other group members will take on their work as well (Karau and Williams, 1993). This is something that teachers need to consider and be wary of. The risk of these issues having a negative impact on learning can vary wildly from class to class and from teacher to teacher. I would advise that any teacher who is considering group work needs to seriously consider the potential for poor behaviour and social loafing to negatively impact the lesson and to think about how they ensure it doesn’t. Different things work for different people. Some people assign roles etc. to groups. Some set up a structure where students can “buy” help from the teacher or other groups. Often a competitive element is introduced. All of these can be effect or not, again depending on the class and on the teacher so it is something we need to work on individually. I’ve written before about one method I’ve had some success with here (2013).

So there are some of the worries around group work and thoughts on what needs to be considered when embarking on it. As mentioned in my previous post, I feel that group work is an inefficient way to introduce new concepts and new learning, but I do see it as something that can be very effective when building problems solving skills and looking at linking areas of mathematics together.

*What are your thoughts on group work? And what are your thoughts on the issues mentioned in the article? I’d love to hear them via the comments or on social media.*

**Reference list / Further reading:**

Black, P. and Wiliam, D. 1998. *Inside the black box: Raising standards through classroom assessment*. London: School of Education, King’s College London.

Cavadino, S.R. 2013. Effective Group Work. 5^{th} July. *Cavmaths.* [online] accessed 14^{th} July 2017. Available: https://cavmaths.wordpress.com/2013/07/05/effective-group-work/

Cavadino, S.R. 2017. Student led learning in maths. 13^{th} July. *Cavmaths.* [online] accessed 14^{th} July 2017. Available: https://cavmaths.wordpress.com/2017/07/13/student-led-learning-in-maths/

Good, T.L., McCaslin, M. and Reys, B.J. 1993. Investigating work groups to promote problem-solving in mathematics. In: Brophy, J. ed. *Advances in research on teaching: Planning and managing learning tasks and activities*. United Kingdom: JAI Press.

Hattie, J. 2012. *Visible learning for teachers: Maximizing impact on learning*. Abingdon: Routledge.

Hodgen, J. and Marshall, B. 2005. Assessment for learning in English and mathematics: A comparison. *Curriculum Journal*. **16**(2), pp.153–176.

Karau, S.J. and Williams, K.D. 1993. Social loafing: A meta-analytic review and theoretical integration. *Journal of Personality and Social Psychology*. **65**(4), pp.681–706.

Mortimer, E. and Scott, P. 2003. *Meaning making in secondary science classrooms*. Maidenhead: Open University Press.

Pearcy, D. 2015. Reflections on patient problem solving. *Mathematics Teaching*. **247**, pp.39–40.

Piaget, J. 1970. *Genetic epistemology*. 2nd ed. New York: New York, Columbia University Press, 1970.

Simmons, M. 1993. *The effective teaching of mathematics*. Harlow: Longman.

Skemp, R.R. 1987. *The psychology of learning mathematics*. United States: Lawrence Erlbaum Associates.

von Duyke, K. and Matusov, E. 2015. Flowery math: A case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency. *Pedagogies: An International Journal*. **11**(1), pp.1–21.

Vygotsky, L.S. 1962. *Thought and language*. Cambridge, MA: M.I.T. Press, Massachusetts Institute of Technology.

** **

## Student Led Learning in Maths

Student led learning seems to be a bit of a hot topic at the moment. I’ve seen these two graphics making the rounds on twitter, I can’t find them now, but one was a slide proclaiming why student led learning was better and one was the same slide but altered to say it was worse. This of course came with great debate from all quarters.

It is also certainly a much talked about issue in the teaching and learning meetings we are having in my school.

This morning we had a great presentation from a food tech colleague who described a fantastic lesson where students had been allowed to lead their own learning on the function of eggs in cooking. A number of different recipes had been provided (as diverse as Egg Custard and Chick Pea Burgers) and students were given the choice as to what to cook and asked to investigate.

This sounded interesting, so I began to think about the applications this may have in a maths classroom. While studying for my Masters I read a lot about group work and other pedagogical approaches to the teaching of mathematics. I found that there was a lot of evidence to suggest that, on the teaching of new content, “whole class teaching”, i.e. direct instruction, was the most effective method (Reynolds and Mujis, 1999). However, this approach can often lead to students being proficient in algorithmically following a process to achieve and answer – ie they can have an instrumental understand of the topic but not a deeper understanding of the underlying concepts. This can lead to issues when students encounter a question that is phrased in a different way or that requires a variety of mathematical topics to solve. (e.g Avital and Shettleworth, 1968, Davis, 1984 and Skemp, 1976)

This was an area that interested me and my dissertation focus was using group work and other problem solving ideas to deepen conceptual understanding at A Level maths. I found that with my cohort explicitly teaching problem solving approaches and then setting problems that required a variety of approaches to be solved in groups to be effective. Some real success was had when I used problems I had not encountered and as such was able to act like a member of the group while bouncing ideas around.

My findings backed up the work of others who had suggested problem solving as a good tool to deepen conceptual understanding. (e.g. Avital and Shettleworth, 1968, English and Halford, 1995, Hembree, 1992, Karp, 2004, Silver and Marshall, 1999, and Zeitz, 2006)

In the new maths GCSE we are seeing questions that are focussed on testing a deeper understanding using problems that require thinking about and often require a number of mathematical techniques to solve. This is a move away from predictable questions and as such, teaching methods aimed at giving algorithms to students to solve types of questions will no longer work.

One simple example is questions based on ratio. Previously ratio questions usually took one of two forms, use a ratio to scale up a recipe or split this amount into this ration. Both are easily solvable by an algorithm and I’ve seen this taught this was and correct answers given by students who don’t really know what a ratio is. Now we are seeing ratio questions that include other areas of maths, such as densities, as well as questions where the language is quite important and a better understanding of what is going on is required.

i.e. A student who is taught, “When you see a ratio you add, divide then multiply”, will get full marks on a question asking “Sana and Jo split £110 pounds in the ratio 6:5, how much does Sana get?” but may get nothing if the question asks: “Fred and Nigel split some money in the ratio 6:5, Fred gets £10 more than Nigel. How much does Nigel get?” Even though there is a comparable level of mathematics used.

This, I feel, is where group work / “student led learning” could be very effective in maths teaching. Once content has been taught students need to practice that content in new setting and to mix it up with other content that has been learned. Tasks need to be set and students need to be given adequate time to get stuck and struggle. This will build resilience and problem solving skills as well as allowing students to see where various strands of maths can be applied.

This ties in with something I read recently that Kris Boulton (2017) had written about the use of learning objectives. Kris argues that sometimes it is important not to use learning objectives as this tells students exactly what maths they need to be using to solve a problem. This is a big factor in this idea around problem solving and I would go further and say that it’s important not to set problems that involve topics you have taught in the last few lessons as this will have the same effect as having an objective such as “use Pythagoras’s Theorem to solve problems involving areas.”

I hope to write more about this in the coming weeks as I look to further apply the findings of my dissertation to KS3 and 4. My thoughts at the moment are that this “student led” approaches are good for the development of these skills once the core content has already been taught. There are, of course, many draw backs to group work and other student led approaches, but they are for another post for another day.

**Reference List:**

Avital, S.M. and Shettleworth, S.J. 1968. *Objectives for mathematics learning; some ideas for the teacher*. Toronto: Ontario Institute for Studies in Education.

Boulton, K. 2017. Whywe need to get rid of lesson objetives. 17^{th} April. *To the real*. [online] accessed 13^{th} July 2017. Available: https://tothereal.wordpress.com/2017/04/17/why-we-need-to-get-rid-of-lesson-objectives/

Davis, R.B. 1984. *Learning mathematics: The cognitive science approach to mathematics education*. London: Croom Helm.

English, L.D. and Halford, G.S. 1995. *Mathematics education: Models and processes*. New Jersey, United States: Lawrence Erlbaum Associates.

Hembree, R. 1992. Experiments and relational studies in problem solving: a meta analysis. *Journal for research in mathematics education*. **33**(3), pp.242–273.

Karp, A. 2004. Conducting Research and Solving Problems: The Russian Experience of Inservice Training. In: Watanabe, T. and Thompson, D. eds. *The Work of Mathematics Teacher Educators. Exchanging Ideas for Effective Practice*. Raleigh, NC: AMTE, pp.35–48.

Reynolds, D. and Muijs, D. 1999. The effective teaching of mathematics: A review of research. *School Leadership & Management*. **19**(3), pp.273–288

Silver, E.A. and Marshall, S.P. 1990. Mathematical and scientific problem solving: Findings, issues and instructional implications. In: Jones, B.F. and Idol, L. eds. *Dimensions of thinking and cognitive instruction*. Hilsdale, New Jersey, United States: Lawrence Erlbaum Associates, pp.265–290.

Skemp, R.R. 1976. Relational understanding and instrumental understanding. *Mathematics Teaching*. **77**, pp.20–27

Zeitz, P. 2006. *The art and craft of problem solving*. USA: John Wiley.

## Effective Pedagogy

Recently I’ve done a fair bit of reading for my dissertation and two of pieces of literature have had very similar titles, there was The Effective Teaching of Mathematics (Simmons 1993) mentioned here, and then there was “The effective teaching of mathematics: a review of research” (Reynolds and Mujis 1999).

It is the second one which I want to share some thoughts on today. It is an interesting article which is aimed at school leaders and policy makers and looks to a variety of sources to create an idea of effective maths teaching.

The main areas it looks at are pieces of teacher effectiveness research, both from the UK and from the USA, and professional evidence on teacher effectiveness from the UK – namely the three most recent reports on maths teaching from Ofsted (most recent as of 1999).

**Whole class teaching**

This mixture of academic and professional evidence is analysed and brought together and the article finds that all three areas suggest that “whole class teaching” is the most effective way of teaching maths. That isn’t to say they suggest that we all lecture to silent classes for entire lessons, rather they are advocating a form of “active” instruction, which would punctuate the instruction with questioning to assess the learning and to see where the class needs expanding on and opportunities for practice and consolidation.

This idea seems to make a lot of sense to me, the teachers are the experts in the room, and they are best placed to pass on the knowledge. Listening to a well planned presentation and then internalising this and practising to make sense of it seems a good model.

**Group work**

While I was reading this it all seemed very sensible, intuitive and a great way to teach mathematical content, but I started to wonder how the other side of mathematics, the logical thinking and problem solving side, would be catered for in this model. Obviously the writers of the report felt the same as they then moved on to looking at group work and other ways to build problem solving ability into your students.

They looked at the idea of group work, suggesting the opportunity to discuss their mathematical ideas with peers and work out between them how it works would be beneficial. They also feel that scaffolding could enable all students to work within their zone of proximal development, allowing all students a chance to develop. They expressed concerns around social loafing, and the possibility of student misconceptions being reinforced.

Their findings led to many examples of group work being an effective tool in problem solving, but they state that to reap the rewards teachers need to spend a lot of time setting it up. I can see that this may be true, and feel that there could be a place for small group work to tackle these types of problems, especially amongst A level students and others who need to work out how to apply the knowledge learned to solve unfamiliar problems.

The article suggests that group work can be integrated into the active instruction model, taking the place of some of the practice section, and I certainly agree that it could fit. I also feel that modelling a problem solving approach for part of the instruction element of the lesson can give students an insight into how a more experienced mathematician would approach a problem.

**Differentiation**

A rather interesting finding was that poorer, less effective lessons often include overly complex arrangements for individual work. This was a suggestion that those lessons where the teacher has spent all night creating separate worksheets for each student actually had little to no impact, even a negative impact at times. This certainly suggests that this level of time consuming differentiation is unnecessary and that tasks can be differentiated far more easily and effectively by producing a resource that is stepped in difficultly and allowing different start points or moving them on more easily.

*I found that this report was very interesting, it backed up some of the ideas I already had on effective maths teaching and challenged some of the other ones. I am now planning to trial some small group work with some A level students to build problem solving capability. *

**Reference**

Reynolds D & Mujis D, 1999, The Effective Teaching of Mathematics: A Review of the Research, *School Leadership and Management,* Vol. 19 **(3)** pp 273-288 (available online here.)

Simmons M, 1993, *The Effective Teaching of Mathematics*, Longman: Harlow

## Effective Group Work

Last night I attended Teachmeet South Bradford at Appleton Academy and saw some superb presentations. There was one given by Andy Sammons (@amsammons) in which he was discussing independent work. He mentioned that he gives his pupils ten “sammonspounds” per group at the start of a lesson and tells them they can buy ten minutes of his time with it, but that’s all they get. This gives the groups drive to be more independent and to save their time until they are really stuck and have devised good quality questions to ask him. You can read more about the topics Andy spoke about on his blog here) This is a great strategy and got me thinking about the ways I have tried to do group work . It reminded me of one method in particular that I have used a number of times to great success and I wanted to share it with you here.

The first time I used it was with a Y11 class in my NQT year, they all had C’s already in maths and were not very motivated to get B’s. I had taught a topic on Pythagoras and trigonometry and I wanted to do a consolidation/revision lesson on it. I set the room up for grouped tables and assigned them groups on their way into the room. I selected groups so that each of the groups were evenly matched and assigned team captains, envoys, timekeepers and finance managers. It was the captains job to take a deciding vote on any decisions, the envoy was in charge of discussing with other groups, the timekeepers were in charge of ensuring they were not running out of time and the finance manager was in charge of the “money” (in this case counters!). I gave each team a float of 20 counters.

The task itself was an exam paper question based relay, there were some really easy questions, and for each one of those the teams completed they gained 5 counters, they went up in difficulty and there was 10 counter questions, 15 counter questions and 20 counter questions. The teams were told that they could buy my help for 8 counters, or they could buy help from other teams at an agreed fee, but I gave a suggested value of 4 counters. At the end of the lesson the teams cashed up and the winning team received a prize.

This worked well with that first class, they were all shrewd with the questions and only purchased help if they really needed it, it helped with independent thinking. I have now used the set up many times (not always with the same activity) and it does work. It engages them and makes them think for themselves more. And on top of that it is fun for tem and me and some teams get really competitive!