## Patient problem solving

I recently read a piece by D Pearcy called “Reflections on Patient Problem Solving”, from Mathematics Teaching 247. It was an interesting article that looks at how teachers need to allow time for students to try their own ideas out while problem solving, rather than just coax them along in a “this is how I would do it” kind of way.

Pearcy’s definition of problem solving is looking at something you have never encountered before that is difficult and frustrating at times, takes a reasonable amount of time, can be solved more than one way and can be altered or extended upon easily. He then goes on to ask whether this is actually happening in classes or if teachers are just walking students through problems, rather than allowing them to problem solve.

He quotes Lockhart (2009) – “A good problem is one you don’t know how to solve” and states that it follows that if you give hints then it defeats the point of setting problems. He goes on to say that maths advocates talk of the importance of maths as a tool to problem solving – but that this isn’t actually happening if students are not being allowed to get frustrated and struggle through to a solution.

He explains how he finds it difficult not to give hints when students are struggling, both because it is in most teachers nature to help, and because of the external pressure to get through the syllabus quickly. This is something I too have encountered and something I have become increasingly aware of as I try to allow time for struggle. Other factors at play are maintaining interest, and increasing confidence. If we let students struggle too much they may lose interest and confidence in their ability – thus it is important to strike a balance between allowing the struggle but not letting it go too far. This is certainly something I keep in mind during lessons, and I feel it is something that we all should be aware of when planning and teaching.

This is an interesting article that looks at a specific problem and allowing students time to struggle and persist. This importance of this is paramount, in my view, and this is also the view expressed by the author of the article. I find it very hard to not offer hints and guidance when students are struggling. One way I manage to combat this at times is by setting problems I haven’t solved yet, thus leaving me a task to complete at the same time. This can work well, particularly at A Level and Further Maths level as then I can take part in the discussion with the students almost as a peer. This is a technique I have used often with my post 16 classes this year.

I have been reading a lot about problem solving recently, and a recurring theme is that teachers can often stifle the problem solving they are hoping to encourage by not allowing it to take place. This is something we need to be aware of, we need to have the patience to allow students the time to try out their ideas and to come up with solutions or fall into misconceptions that can then be addressed.J

.

*Have you read this article? If so, what are your thoughts on it? Have you read anything else on problem solving recently? I’d love you to send be the links if you have and also send me your thoughts. Also, what does problem solving look like in your classroom? Do you find it a struggle not to help? I’d love to hear in the comments or via social media.*

**Further Reading on this topic from Cavmaths:**

Dialogic teaching and problem solving

**References:**

Pearcy.D. (2015). Reflections on patient problem solving. *Mathematics Teaching. ***247** pp 39-40

Lockhart, P. (2009). *A Mathematician’s Lament*. Retrieved from: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

## Dialogic teaching and problem solving

I recently read an article entitled “Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency” by K. von Duyke and E Matusov. It was an interesting article that looked at some student teacher interaction in a lesson where students were asked to solve a mathematical problem dividing one dollar between three people. They had found an interesting exchange between the teacher (“John”) and a student who had approached the problem topologically and has a correct solution using physical coins but hadn’t calculated the amount each person had. This has irked and perplexed the teacher – seemingly because she hadn’t come up to the solution he had in mind. This is an interesting revelation and one that we, as maths teachers sometimes fall into. There can often be many ways to solve a problem in mathematics and all are equally valid. My view is that we need to be looking at solutions presented to us by our students with an open mind before telling them they are wrong. In this case the student had come up with her own approach and had the correct solution – an outcome that feel should be celebrated.

The authors use this as a starting point for a discussion on various pedagogies, suggesting that to really allow this sort of maths to thrive in the classroom teachers need to take a dialogic approach – to discuss with the students where their thinking has come from and help them refine their models. They also suggest that the reason John was keen to dismiss this valid reasoning in this case was due to his favouring of a more rigid pedagogical structure. I tend to agree with the researchers. We are there to help students make their own meaning, their own links, in mathematics. Obviously we need to pass on the relevant subject content, but in an open ended task like this it is important to ensure all solutions are explored and refined.

This leads me back into a discussion I had recently regarding the purposes of assessment in mathematics which came about from this blog that I wrote on a question with multiple solutions. John R Walkup (@jwalkup) said that we should be assessing all methods to ensure that students can do it. I think that to an extent he has a point. We do need to test that our students can complete the content, and we should be doing this with low order questions where they are directed to practice and recreate skills. However, maths is about making links, making your own links, and solving problems that are unfamiliar – trying the methods you know to see if you can find a solution to a problem, you have never seen.

It is the latter that is increasingly being tested in our terminal external exams in the UK as we move to the new specification GCSE and A Level tests, so we need to be preparing our students to be successful in this type of question. I think that the dialogic approach mentioned here is an extremely powerful tool in this quest. It allows us to help students explore their thinking and create their own links. I heard a colleague recently explain to a student that maths was about “finding shortcuts, and finding tricks” this worried me a little at first but then he continued “we all have hundreds of tricks and shortcuts that we have developed over years of doing maths. If Mr Cavadino and I were to teach you our tricks they wouldn’t make sense to you and it would overwhelm you.” I can understand this point – if a student notices that d = s x t can be rearranged simply in a triangle because they understand how to rearrange that equation then they will save themselves time. If they learn the technique without understanding what is taking place they open themselves up to the possibility of more errors.

In the article the authors use the term heterodiscoursia, which means legitimate simultaneous diverse discourse. The suggestion is that as part of the dialogic teaching teachers should be allowing discussions and methods to abound and thrive in the classroom. They suggest that this mix of discourses allows students to bounce ideas, allows the teacher to correct any misconceptions and helps build meaning making and engagement. Their suggestions are certainly in line with my observations from my own lessons that have allowed these types of discussion to develop and I think that it would be beneficial to explore how this can be allowed to grow with my other classes.

The authors have some practical suggestions for us maths teachers. They suggest that we need to be familiar with the fact that there are often different solutions and be able to develop them. We need to allow the students to frame the question into a context that works for them then use that context to find a solution which is salient, and we need to be able to question our preconceived notion of the solution. This sounds like sound thinking, I feel that these are things we should all do while we are trying to build the problem solving capacity of our students.

The authors go on to discuss dialogic pedagogy and how this can in effect allow teachers to discuss the problem almost as a peer – I find that this is a great tool when working with A Level students. If i can find problems that I haven’t done before then I can share my thinking with them and we can work through them together. This has been very successful when developing problem solving strategies.

The authors go on to discuss dialogic pedagogy and how this can in effect allow teachers to discuss the problem almost as a peer – I find that this is a great tool when working with A Level students. If i can find problems that I haven’t done before then I can share my thinking with them and we can work through them together. This has been very successful when developing problem solving strategies.

**Reference:**

Von Duyke, K. and Matasov, 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. [accessed 23/5/2016] http://dx.doi.org/10.1080/1554480X.2015.1090904

## Forming and Solving Equations

While checking the work of a year 11 student on Friday I came across a question that could have been a great one for the higher GCSE students to practice their skills together and also their selection of which mathematics to use.

The question was to find the area of this triangle:

A great question. One that to you or I is straightforward but that would take GCSE level students and below a bit of thinking and let’s them hone their skills.

The way to tackle it is to use Pythagoras’s Theorem to form an equation, solve for x then find the area. I feel is beneficial as it combines Pythagoras’s Theorem with a decent amount of algebra then includes the find the area bit at the end.

In this case though, that wasn’t the question. There was more information on offer and the question was:

Which is still a fairly nice form and solve an equation problem.

*3x + 1 + 3x + x – 1 = 56*

*7x = 56*

*x = 8*

*A = 0.5×7×24 = 84*

There is a niceness to this question that goes beyond the question itself. It shows us a great way of differentiating within lessons. Just be leaving out a tiny portion of the information, in this case the perimeter, we can make the question much harder. This idea is something I’ve been working on in various places. M1 questions can be made much easier by providing a diagram, for example.

*Have you used questions in a similar way? If so I’d love to see them, please do get in touch.*

*Cross-posted to Betterqs here.*

## An interesting area problem

Here’s an interesting little question for you:

*Have you worked it out? How long did it take you to see it?*

It took me a few seconds at least, I had screenshotted the picture and was reaching for the pencil when the penny dropped, and that’s why I thought it was an interesting question.

The answer is, of course, 100pi. This follows easily from the information you have as the diagonal of the rectangle is clearly a radius – the top left is on the circumference and the bottom right is on the centre.

*So why didn’t I spot it immediately? *

I think the reason for me not spotting it instantly might be the misdirection in the question, the needless info that the height of the rectangle had me thinking about 6, 8, 10 triangles before I had even discovered what the question was.

I see this in students quite often at exam time, they can get confused about what they’re doing and it links to this piece I wrote earlier about analogy mistakes. The difference is I wasn’t constrained by my first instinct but all too often students can be and they can worry that it must be solved in the manner they first thought of.

Earlier today a student was working on an FP1 paper and he was struggling with a parabola question, he had done exactly this, he had assumed one thing which wasn’t the right way and got hung up in it. When he showed me the problem my instinct was the same as his, but when I hit the same dead end he had I stepped back and said “what else do we know”, then saw the right answer. I’m hoping that by seeing me do this he will realise that first instincts aren’t always correct.

I’m going to try this puzzle on all my classes tomorrow and Friday and see if they can manage it!

*How quickly did you see the answer? Do you experience this sort of thinking from your students? I’d love to hear any similar experiences.*

Cross-posted to Betterqs here.

## Passivity in the maths classroom

Today I managed to find a few minutes to browse the latest issue of Maths Teaching, the ATM journal. One article that caught my eye was the “from the archive” section, where Danny Brown (@dannytybrown) introduced an article that was first published in 1957. The article was written by Ruben Schramm and is entitled “The student’s passive attitude towards mathematics and his activities.”

The article discusses mathematics teaching, particularly the nature of students who often, for whatever reason try to find an algorithmic method to follow to solve a problem, looking to recognise the problem and answer it in a similar way to how they have answered questions before. This is a problem that was obviously prevalent in the 1950s, as evidenced in the paper, but it is still prevalent now, and I feel the nature of our exam system must at least hold a portion of the blame. The questions on exams tend to be very similar and students will learn methods to answer them whether the teachers like it or not. This is one issue I hope will be dampened a little with the upcoming changes to the exams.

Schaum suggests that this passivity in maths, this tendency to look for algorithms, is in part down to how students see mathematics. He suggests that when they see teachers solve problems on the board by delivering a slick, scripted solution they can get a feeling that it is via “witchcraft” and see the whole process of uncoordinated steps, rather than a series of interconnected mathematical ideas. The latter would encourage the students to drive the mathematics from their internal ideas, and this would lead to them being more able to apply their knowledge in new contexts. If we can develop this at all levels then I feel we really would be educating mathematicians – ie giving students the skills to be able to apply their knowledge in new contexts, rather than teaching them to follow a recipe to answer a question.

Schaum goes on to discuss authority, the infallible authority that students see in their teachers and in the mathematical theorems and formulae. It is suggested that students see these theorems as infallible, and as such they reach out for them in their memories and try to apply them to problems. This can mean that the problem they are applying them too is only vaguely similar to the problem the theorem or method is actually there to solve. Schaum calls these “analogy mistakes”, and suggests that it is down to how comfortable with the content students feel that mean they revert to them. I feel that this is true in part, but that also the pressure of exams can lead students to confuse things in their head if they have opted to learn algorithms rather than looking to develop a deeper understanding.

I’ve had a couple of examples of these “analogy mistakes” in lessons and exams recently. A year 12 student came to an afterschool elective as she was trying to solve some coordinate geometry problems involving tangents. She had gotten herself really confused because in her notes she had written tangent gradient is perpendicular (when discussing circles) but she didn’t think it should be perpendicular because a tangent at a point should have the same gradient as the curve. I spend a little time discussing where her misconception had come from (her notes should have said “perpendicular to the radius”) and discussed how she could remember this more easily if she has thought about the graphs and sketched them.

Another example was in a recent exam one of my students had answered part of a question on alternative from incorrectly, she had done the alternative form bit well and the answer was 25 Sin(x + a), but it then asked her for the maximum she had written -25. When I questioned her about this after it seems she had fallen victim to an “analogy mistake”, she had remembered that “maximum is positive” when discussing second derivatives and in the pressure of the exam this memory had taken over, rather than the rational thought process that should have flagged up that the maximum or the function would be 25, which is definitely bigger than -25.

In his preface Danny Brown suggested that one way to counteract this would be by questioning and discussion, if we remove the authority from the discussion and don’t validate the answers by issuing statements saying they are correct or incorrect, but rather open them as conjecture to the class who then can discuss this, then we can allow students to develop their own mathematical ideas. Lampert (2001) also discussed this idea and suggests that as teachers we need to be striking the right balance between allowing students to discuss and conjecture and ensuring they understand what is important and aren’t making mistakes. This is something I strive for in my own classroom, and something I am currently working on trying to improve.

*This post was cross posted to Betterqs here.*

## 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

## Understanding students’ ideas

I read a really interesting article today entitled “Teachers’ evolving understanding of their students’ mathematical ideas during and after classroom problem solving” by L.B. Warner and R.Y. Schorr. It is a great report that looks at three teacher’s responses to their students’ solutions to a problem, and it discusses in detail how the teachers reflected on them together. It is well worth a read for all maths teachers.

The teachers were middle school maths teacher and they were presented with a problem to solve by the researchers they then presented their classes with the problem and debriefed afterwards. It was clear that the teachers didn’t have the thorough subject knowledge of a high school maths specialist and this lead to them failing to pick up some misconceptions and not allowing students to explore their own methods if they didn’t understand it, rather moving them on to a method that was more familiar to the teacher. The reflections of the teachers are interesting, they all appear to become frustrated with themselves when analysing their responses and are able to reflect on this by offering alternatives. It does show that deeper subject knowledge is important to allow that exploration to take place. The study showed that in this context when the teachers just told students how to fix their mistakes, rather than question students as to why they had made them, this led to student confusion. This suggests that we should be striving to understand our students thinking whenever possible and using that to combat their misconceptions so they don’t fall into similar traps again. This will also allow students to see why they are coming up with these misconceptions.

There are many teachers who, at times, fail to understand the lines of mathematical thinking taken by their students when solving problems. This can lead to not giving the proper amount of credit to valid ideas and it can lead to teachers failing to spot misconceptions. Some students may have a perfectly valid method but as the teacher may not see where they are going they can sometimes block this route off. This has deep links to “Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency” by K. von Duyke and E Matusov , which I read recently (you can read my reflections here). I feel that it shows that deep subject knowledge is important, as is allowing students time and space to work through the problem on their own. Rather than saying, “No, do it this way” we should, be encouraging students to follow their nose, as it were, and see if they can get anywhere with it. It is always possible to show the students the more concise method when they have arrived at the answer to bui8ld their skill set.

Warner and Schorr believe that subject content, as well as pedagogical content is vitally important to teachers to enable than to know how to proceed when a student is attempting a problem. They look at relevant literature on this and quote Jacobs, Philip and Lamb (2010) who suggest that this is something that can be achieved over time and Schoenfield (2011) who says that teachers tend to be more focussed on students being engaged in mathematics and replicating the solutions of the teacher rather than allowing students to meander their own way through so the teacher scan identify their understanding and misconceptions. The latter would, in my opinion, be a much better way of developing, and I agree with JPL that this is a skill one can develop over time.

ReferencesJacobs, V. R., Lamb, L. L. C., and Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking.

Journal for Research in Mathematics Education,41, pp 169–202.Schoenfeld, A. H. (2011). Toward Professional Development for Teachers Grounded in a Theory of Teachers’ Decision Making.

ZDM, The International Journal of MathematicsEducation,43pp 457–469.Von Duyke, K. and Matasov, 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 1-21Warner, L.B. and Schorr, R.Y. (2014). Teachers’ evolving understanding of their students’ mathematical ideas during and after classroom problem solvin.

Proceedings of the 7th InternationalConference of Education, Research and Innovation,Seville, Spain, pp 669-677.## Share this via:

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