## Thoughts on the understanding paradox and introducing trigonometry

Recently I read a blog entitled “The understanding paradox” (William, 2017) which discussed the idea of maths teaching and put forward the idea that actually, it is better to bypass understanding when first teaching a topic and then fill that understanding in later. This was then applied to the teaching of right angled triangle trigonometry in an example that I found confusing to say the least.

The author, Rufus William, suggested that when teaching trig for the first time we should be solely teaching procedurally using SOHCAHTOA as a mnemonic, but then went on to say we shouldn’t be discussing ratio or similarity and how that links until later on. This confused me as the mnemonic SOHCAHTOA is designed to help you remember the trig ratios. I.e. Sine is the ratio of the opposite side over the hypotenuse. Just by teaching that you ARE teaching the trig ratios and purely by the fact that you are teaching the students that this will work for all right angled triangles you are telling the students that the ratios are the same for any triangle with the same angle no matter what the length of the sides are. THIS IS THE VERY DEFINITION OF SIMILAR TRIANGLES.

This perplexed me a lot and I spent a lot of time thinking about it and asking the author to elaborate on what he meant. The only way I can fathom to teach this without reference to ratio and similarity would be to say: ” “SOHCAHTOA” it gives you 3 triangles. Label the sides circle them to see which triangle you use. Put numbers in, cover the missing one, its either a divide or a times”. To me this seems like a backwards way to go about things. It feels like you are teaching them unnecessary procedures to avoid discussing the underlying concepts of trigonometry, and it doesn’t really make sense to me.

I find that by the time students reach right angled triangle trigonometry they have already met the concept of similarity, I like to use this a way in to discussing the topic and to show that ratio of two sides that are the same in relation to an angle will be the same for all similar triangles. Students will have always encountered simplifying fractions before they meet trig and as such can see why this is. This is when I specifically discuss the sine, cosine and tangent ratios and introduce the procedural manner in which they can solve the problems, although I do avoid the dreaded formula triangles (for many reasons which I have blogged about here). I will show them some common mnemonics, and SOHCAHTOA is one of them. I’m not a fan of mnemonics personally, I’ve never found them that useful except for musical ones, but I know a lot of people do.

Rufus does make some salient points in his post about teachers who refuse to allow students to memorise things and the dangers this will have on learning. Although I’m not entirely sure that they exist, and if they do I certainly don’t think there are many of them. I’ve certainly never met any.

He also suggests that students cannot have a full understanding of the ins and outs of trigonometry when they first meet it. I would very much agree with him in that respect, I know many people who have taught trigonometry for decades and still don’t, but I don’t think that means we have to bypass all information.

**Reference List:**

Cavadino, S.R. 2014. Formula Triangles. 12th October. *Cavmaths. *[online] accessed 14th July 2017. available: https://cavmaths.wordpress.com/2014/10/12/formula-triangles/

Cavadino, S.R. 2016. Catchy Mnemonics. 16th September. *Cavmaths. *[Online] accessed 14th July. Available: https://cavmaths.wordpress.com/2016/09/16/catchy-mnemonics/

William, R. 2017. The understanding paradox. 7th July. *No easy answers. *[online] accessed 14th July 2017. available: https://noeasyanswerseducation.wordpress.com/2017/07/07/the-understanding-paradox/

## 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.

## 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.*

## 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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