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Why no modelling?

3/30/2016

Introduction

Following this post on some thoughts about teaching maths, @geoffwake1 asked me this question:

I was surprised to see no reference to developing modelling as a mathematical practice? Why is that?

We agreed to have an email conversation about modelling in the (my) maths classroom - this post contains an edited version of this conversation.

Time

Me: I think modelling is an incredibly powerful teaching approach, particularly for the applied (statistics and mechanics) A-level modules which I currently teach - for which it is the whole point! I worked as a financial 'modeller' for a number of years (and have an MSc in Applied Statistics), so I do understand how mathematics can be applied in the 'real-world'..

Geoff: I am a little surprised that you have a background in modelling yourself. My experience is that a lack of any recognition of modelling is often precipitated by a teacher being a 'pure' mathematician who finds engagement with the puzzles that school mathematics can provide satisfying enough - and I guess a lack of experience of modelling themselves. So good news that you understand the importance of modelling and applications.

Me: Yes, I definitely do, but… there are important constraints that are stopping me from creating a 'modelling classroom’, namely the time needed to (1) create, and (2) explore, authentic modelling situations.

Although I have a reasonable amount of experience-of and motivation-towards modelling, I still lack the time, and perhaps the expertise, required to create and present (units of) modelling lessons whilst ensuring the students come out the other end having learned what needs to be learned.

As a teacher with limited resources I make decisions on how I can meet conflicting demands, and I often adopt a 'simple' approach that circumvents 'real-life'. This of course creates a conflict; I know that teaching statistics and mechanics (and to a lesser extent pure) only really makes sense in context, that mathematics should be meaningful and relevant to students, and I admire those very few (if any) full-time classroom teachers that manage to pull it off!

Geoff: I note your honesty in suggesting that you don't have the experience and expertise to develop modelling lessons that would give you confidence that you are both covering content as well as providing students with insight into an important mathematical practice. But it is this deficit for students that underpins my concern.

If we look at the data of student progression from A Level to degree programmes we can get a sense of the scale of the problem. In any one cohort approximately some 9 thousand students go on to study mathematics degrees and something approaching 52 thousand study some form of engineering or physical science degree. These engineering and physical science students will all engage with mathematical models and applications at university with many having to engage in mathematical modelling themselves. This means that they will have little or no experience of what this entails. And of course these students are just a very obvious tip of an iceberg - many other undergraduates will also engage with models and applications. From this perspective alone I would argue that we need to include some experience of models and applications.

Me: Yes, I agree, the modelling process is fundamentally important, although I also enjoy, and think there is value in, solving 'pure' mathematics problems...

Geoff: I’m in the same boat with this. I particularly like working with algebra, functions, graphs and geometry. But my experience is that we have plenty of supporters of mathematical practices associated with maths in these areas so I’m concerned to support modelling and applications above all.

Meaningful modelling

Me: I totally agree. My entire reason for teaching (maths) is to try to give students opportunities to experience wider mathematical and social elements, but as I say: it is difficult to create truly meaningful modelling situations.

As an example, I spent quite a long time last weekend trying to find some interesting/relevant datasets that exhibited exponential growth, but was largely unsuccessful. I could have explored some bacteria data, but this is not the sort of modelling I have in mind.

If I am going to spend the time planning and exploring modelling situations in my classroom, I really want to model something *important*, in the sense that it *matters*, perhaps politically, or socially, perhaps having a *local* relevance for the students in the classroom - this paper has been an inspiration (one of the authors, @LaurieRubel also suggested citydigits.org, which looks exciting). There are also some interesting ideas in this book, and this new ATM book (written by @PeteWrightIOE) looks really interesting. The idea that mathematics can be used to explore social justice, making decisions about important issues, is the kind of thing I am interested in bringing to my students.

In the past, I have created some resources of my own (for KS3/4), such as this on the UK 2010 election, but this quality of modelling experience takes time to plan and explore, and is also difficult to align with the curriculum.

Some of the other online modelling resources, such as Dan Meyer's blog, don't really fit what I would class as meaningful modelling, and can feel contrived, or of little relevance/import to students' lives; if I am going to spend the time bringing modelling situations to my classroom, I want to address matters of importance, socially or politically.

Geoff: Yes, I’m interested see how Dan Meyer promotes a sort of pseudo-modelling that seems to be quite popular among certain teachers. I think one aspect that appeals is that he suggests a narrative that is immediately accessible. On the other hand some of the questions are not particularly meaningfully tackled using mathematics seriously. The same is true of some of the Fermi problems that are promoted by the likes of Tim Gowers – although I entirely agree with his recent comments on the lack of imagination in maths teaching.

Me: Yes, I am inclined to agree with him. But it comes back to the constraints, and the ability, to bring meaningful modelling situations into the classroom. The lottery article I mentioned above is a brilliant example: it is authentic, full of local and social relevance, but was created by people with a vast amount of expertise and time. This kind of meaningful project is not easy to produce, or re-produce, in the classroom. Are there any resources that you might recommend that fit these criteria, what I would call meaningful modelling, that are readily available?

Geoff: Interestingly there was a time when modelling was an essential part of the A Level curriculum and there are resources that were developed at that time. An important set of resources that did this really well was the SMP 16-19 course materials.

My own career move to work in maths education in a university was precipitated by my interest in modelling in mechanics and I joined the Mechanics in Action Project at the University of Manchester. Our group contributed to the SMP 16-19 materials and also produced a wide range of other resources that supported the teaching of mechanics more generally. As part of that work we produced a number of practical kits to support practical work and teacher-led demonstrations in the classroom. These helped teachers promote mathematical models and modelling at different scales whilst also 'covering' content. I'm going to suggest at this point you take a look at the Mechanics in Action resources and let me know what you think about the suggested classroom activities. Would these provide you with a starting point to try something in your classroom?

Me: Thanks for this. Yes, they are simple enough to be do-able, although I would have to talk to the Physics department (or perhaps the premises staff!) to get the equipment; I will definitely consider using them when teaching mechanics next year.

But again, I come back to my concern about coverage of content: it's all well and good experimenting with pulleys on strings and so on, but is this really a much more valuable learning experience than solving a few mechanics problems?

Also, I might ask: are these activities really meaningful, or motivational, for students? Perhaps they will give students a better 'feel' for the kinds of problems they will ultimately face, but are they really motivated by a few pulleys, or a car rolling down a ramp? Perhaps they might be, but if I am to devote the time to modelling, I want it to have real meaning for the students, to transform the way they view the world...

Geoff: Well, I’m with you all the way on this. I want to see modelling tasks that really do focus on meaningful applications and I also want to work on developing tasks and lessons in which reality and mathematics are mutually supportive, indeed preferably where understanding the ‘reality’ being modelled leads to understanding of the mathematics. I’ve done some writing about this recently in a journal article - you can find the final version that is not copyrighted here - but first of all try out this Four Card Problem. Maybe this gives some insight into how I believe mathematical thinking can be facilitated by context.

Core maths seems the ideal space in which modelling and applications might be developed – although I hope it’s not too late by then and we may have already lost some of students. The qualifications that the awarding bodies have developed all try to address the issue in different ways. Are you likely to get involved in teaching in this post-16 area?

Me: I have no plans to, but will definitely consider it. Thanks for this conversation, it has made me think very carefully about how I can introduce meaningful modelling into my teaching next year.

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It's not about the posters

3/26/2016

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This article in the Evening Standard, taken from this TES article written by @tombennett71, has created a lot of comment.

Here is the headline and image:

I exchanged a number of Twitter comments with Tom about this article, as did many other people. This post contains some thoughts about my exchanges with Tom and others.

I'll start with this comment from @ProfDanielMuijs:

If the article is 'perfectly sensible', why has there been such a strong reaction?

The tone of the Evening Standard article

I came to the TES article via the Evening Standard. I was caught by the headline: the 'Government's education/behaviour czar' appears to be 'warning' us about what is and isn't 'proper' teaching, and that students who do "group work" are not being taught properly.

Clearly this headline is designed to get a reaction. Perhaps some people did not read beyond it? There is plenty to get worked up about in the title - but it does not bear much resemblance to the body of the original text of the TES article.

(Incidentally, Tom did not see the Evening Standard article in advance of it being published.)

A lot of group work is "not teaching"

In the body of the TES article, Tom suggests that some activities such as DVDs and posters are not a good use of lesson time. I personally do not use posters or DVDs in my teaching either, as I have not found them particularly useful.

However, there are some parts of the article that I do have issue with, such as this comment about group work:

To be honest, a lot of group work is “not teaching”. It just looks like it.

I think it would be difficult to argue that developing students ability to work with others is not an important aspect of education. Of course, some group work may not be effective, but this is the case with all approaches to teaching; why single out group work if this is not a negative evaluation of group work in general?

The negative evaluation of group work brings to my mind concerns about the individualisation of learning, as described by @gbiesta in his paper Interrupting the Politics of Learning:

To highlight these aspects of the politics of learning – that is, the political work which is being achieved through the notion and language and discourse of learning – is not to deny that there may be some good aspects to learning (although I am becoming less and less optimistic about this, precisely because of the problems outlined above), but to be aware that the language of learning, which fundamentally is individualistic, individualising and process-oriented rather than substantive, is not an innocent language, but actually a language which exerts a powerful influence on what we can be and how we can be – one that tends to domesticate rather than to emancipate.

This quotation leads to other underlying issues I have with Tom's article. Perhaps I am reading too much into the text, but the tenor of the article to me is a language of learning that is process-oriented, not substantive. It suggests (to my reading) that the only aim of teaching should be the transmission of content, where every second counts, where student engagement - the 'holy grail of zany education' - necessarily requires the 'sacrifice of content'.

Of course we should not waste students time with ill-thought out activities, and of course knowledge matters, but I also think it is important that we aim to provide children with a meaningful education, rather than aiming to fill them with as much content as we can in the time alloted.

Politics not posters

Some of this might explain why there was a number of negative responses to the article; teachers agree and disagree about this kind of thing all the time. However, I would argue that the particularly strong reaction to this article may have been a political reaction.

This week the government has proposed some highly prescriptive measures on those working in education. Perhaps this article was perceived as yet another prescription on a profession that feels as though it is being stripped of its autonomy and creativity. I think this might have been why I was initially motivated to respond to the Evening Standard headline: Here is the government's education/behaviour tsar/czar, espousing what is and what is not proper teaching.

However, I think I was naive to have this reaction. I suggested that Tom could be construed to be making further prescriptions on behalf of the government, to which he replied:

I then (also wrongly) suggested that being appointed by the government meant that Tom was speaking for the government:

My disagreement with Tom's views on teaching was wrapped inside my political distrust of anyone who would choose to work with this government:

I am encouraged by Tom's response to my 'political' concerns, and hope that he is able to change education for the better.

Conclusion

So why did people have such a strong reaction to this article? Was it the inflammatory Evening Standard headline? Was it disagreement with the content? Was it political? Or do people just love making posters?

I think @BodilUK sums up the discussion well with this comment:

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Marking

3/20/2016

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I’m realising I did it for me, not for the pupils.

I'm writing this post as part of the conversation about marking, and how useful it is. In this post by @jo_facer, Jo talks about how she doesn't mark students' work, but rather makes notes as she reads their work, and then gives feedback to the whole class on the main issues, which they act on.

If we can give feedback to students in this way, why do we mark students work? Do we do it for us, as teachers, instead of the students, as Jo suggests?

I would agree we can get as good a picture of what the students need help on without putting our marks on their work, and I can see that it might indeed be beneficial not to do so. I also agree that marking is time consuming, especially if we write the same comment on many pieces of work.

So why are we putting our pen to their paper?

I thought about this as I marked some AS Maths work today - exam problems on C2 radians, circles and triangles. As I marked it, I too made notes, like Jo. There were those who had not completed it, those who I would need to speak to in person as (it appeared) they had not really understood the underlying concepts, and those who might be able to help others as they appeared to have a more secure understanding. There were also common errors that a number of students had made, which I too decided to discuss with the whole class.

However, there were also a multitude of individual and specific errors/issues that I wanted to highlight to each student. I could make notes about each of these and go round to each student individually to talk about them, but this would not be feasible. Alternatively, I could ignore these issues, and only focus on the main ones, but this would be doing a disservice to my students - the detail matters, especially at A-level.

So, surely the most effective (only?) way to share this information with each student is to make a note on their work and ask them to revise their solution? I use the approach of putting a star next to the issue, perhaps with a comment/hint if required; students then spend the start of the next lesson working together to revise their solutions, creating a second version, improving on the issues that are specific to them.

How could I do this effectively without marking their work?

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Watching video

3/20/2016

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The process

We have watched a couple of videos of teaching/learning in our department meetings so far: see this post and then this one. We are developing a process for watching the videos (once a week), which is currently:

First viewing, during which each member of the team makes notes on what stands out for them, without evaluation.
First conversation, to identify a basic-level action as a focus for the second viewing.
Second viewing, viewed through the lens of the agreed focus.
Second conversation, to discuss the basic-level action and the (possible) impact on learning.
Conclusion: How might we use this basic-level action in our teaching?

The next week, we start by talking about how the previous week's discussion/focus/action has affected our practice. How did we employ the action? What evidence do we have that it improved learning?

This is much like this protocol written by @wimbs [for instructional practices/routines, we talk about basic-level actions]:

Aims

The immediate aim is to identify pedagogical actions that might improve students' learning, to be able to name these actions, to develop a language with which we can talk about good practice.

We must not forget that the underlying aim is to improve learning. I think these questions from this article on Improvement Science by @DavidPriceOBE are useful to keep in mind:

Perhaps the biggest challenge is #2 - trying to identify to what extent the basic-level actions improve learning.

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Department meeting 180316

3/18/2016

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Yesterday I posted two videos of students working in my lessons. In our department meeting today, we discussed the second video.

First viewing - initial observations

We started the meeting by solving the problem ourselves, then we watched the video.

In this bit of audio, we have just watched the video for the first time, have made a few notes, and are talking about our first impressions. The aim of this part of the meeting is to identify something what we might want to focus on in a second viewing.

Here are some initial observations:

Katy [00:37] "...I wrote fluid down, I just wrote words down to start off with... they were constantly changing and re=drafting things... there was a lot of comparing between people. It was very clear for us to see what they were doing, and we saw variety of techniques about how they were going about trying to solve the problem... re-drafting, comparing, referencing, ... "

Rich [01:36] "The girls didn't want to... they covered it all up... they were all standing there really close..."

Christian [01:45] "...the thing that I wrote the most for was this little subtitle called mistakes... there were lots of mistakes happening in that video, I think, and for me the most interesting thing for me was how Danny dealt this the mistakes... there was different ways - he ignored a couple of them, he gave a hint at the beginning... he corrected one of them..."

Danny [02:58] "I noticed that Yeasin [middle student of the left-most group] was on the right track for ages and they ignored him... someone else was looking in the book because they didn't really believe him, and he had the right answer all the time but they didn't really do anything about it... "

Identifying a focus for the second viewing

This is only the second time we have done this, and we are still developing how we want the process to operate.

In this next clip we spend a few minutes discussing the best way of conducting this video-watching process, and what the aims are - to identify and name basic-level actions that will improve our teaching.

In this clip I am aiming to pull some of the most interesting ideas together, to give examples of what might be fruitful, to focus the group. At [02:23] I ask again what we would like to focus on, saying "I like all of them, so I'll let you decide."

At this point, a crucial event occurs - Katy suggests we focus on one of our basic-level actions, namely:

'Student validation': students evaluate/validate each others responses, reaching a consensus of opinion rather than students always relying on the teacher, thereby distributing authority across the community, allowing them the space to make sense of responses .

This important event gives the session the required focus.

Following this, Rich makes the interesting comment [02:55]: "Mmm, although I'm not sure they reach a consensus, I think normally the dominant person just.. does it."

We then discuss how this might be connected to student validation: Who has the authority/status in this classroom? How did the students reach a consensus of opinion (if indeed they did)?

We then watched the video a second time with the action student validation as our focus.

Discussion following the second viewing

This discussion starts with some interesting comments from Christian [00:20]: "All of the groups made some errors... why did those errors happen? They then righted themselves... and then I suppose what's interesting is: have all of them learned from that, or would they make the same mistakes again? It would be nice to say that the conversations helped them, but.. how do we know it helped them? I think I would want my students to be having those conversations, but then I would want to know - are they now going to be able to do that type of question?"

Pooja agrees [02:50], suggesting some of the students might have been "passive", and that some students were waiting for the correct answer to appear somewhere else: "what I typically see is that when they see the solution they understand it, but will they be able to deliver it on their own afterwards?"

Katy then talks about the 'split group' [04:48]: "There was a lot of talk at the start about who the authority was... they were unsure about what they were doing, and unsure who they should rely on... other groups seemed they could validate what they were writing better, they had a stronger conviction... but why? I wonder if they had stronger prior knowledge, perhaps they were just more confident... that was the most interesting group because they relied on the most authority things... maybe for them it was less about the sitting trying to work something out mathematically, and it was more of a social exercise, like who do we believe?"

We then ran out of tie. We wrapped it up by deciding to focus on the action student validation in our teaching over the next week.

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Conversation with Aysha

3/18/2016

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Yesterday I posted two videos of students working in my lessons. In the first video, one of my students, Aysha, had decided to teach some other students who had come for some extra help.

Today I asked Aysha why she decided to help the other students, how it made her feel, and what she learned from the experience - here is the conversation.

I have transcribed some of parts of the conversation that I found interesting:

[00:58] Me: "What did it feel like when you were doing it..."

[01:27] Aysha: "When they were actually asking me for help.. that's when I felt confident, it gave me a bit of a boost..."

[01:46] Me: "And how did that make you feel?"

[01:49] Aysha: "Good, because it made me feel a bit smart!"

[01:59] Me: "What did you find difficult about it?"

[02:05] Aysha: "It was alright but... when I was walking around, I didn't know if I was too close or too far... I thought, what if they feel intimidated... I feel like that sometimes when someone tries to help me... they were working, they would ask for help, then they would go back to their work, and I was like, shall I still stand here or not."

[02:49] Me: "What did you learn from doing this?"

[03:05] Aysha: "When I was at the board writing [before this video], it was hard, because they're working, I'm explaining, and so many things are happening. When I'm looking at the board, I don't know what they're doing, and when I look back they're all looking, and then they'll ask me to go back over things, and I'll go back over it, and they'll do little bits on their paper... I think it's better to do it how I did it there [in the video] close and in pairs than to do it on the board.. because they responded better, because they asked me to do it like that instead of doing it on the board."

[03:57] Me: "And did you learn anything about yourself?"

[04:01] Aysha: "Yeah, that pair work is actually helpful, because I always though I would like to do my work independently, but then other people's opinions and work does actually help, because they have knowledge that you don't..."

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Two videos

3/17/2016

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This post contains two videos of students working in my classrooms.

What/how are the students learning in these videos, and from whom?

What pedagogical methods are being employed?

Video #1: Aysha

This video is of a completely unplanned session with a few students who wanted some extra help. Aysha (stood up) had done a bit more work on this topic and organically took the role of 'expert'.

As we join the lesson, they have been working on various problems for around 15 minutes (without my help), and are currently working on this:

What/how are the students learning in this video, and from whom?

Video #2: Whiteboards

In this video, students are working on whiteboards around the room in groups of 2 or 3. They are working on this problem:

What/how are the students learning in this video, and from whom?

I would love to hear your comments below...

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A student conversation while solving a maths problem

3/5/2016

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The problem

This post contains an excerpt from a conversation between two of my students. This is Mohammed's second lesson on geometric series; the previous lesson we explored what a geometric sequence was through a few basic problems. Daria missed the previous lesson and is trying to catch up.

Mohammed and Daria are working together to solve this problem:

The clips

Now play each clip as you read the summary: how is learning taking place?

What is significant about each clip for you? What are your thoughts after listening to all the clips?

They have just started the problem.

Mohammed: "Mmm, this is weird... we are not given anything in-between [the 2nd and 5th terms]."

In this clip, Mohammed compares this to something he has done in a previous module, and proposes a method of solution.

Mohammed and Daria are looking through the textbook to find more information (Mohammed: "It should be here.")

As they flick through the book, Mohammed is surprised to find logs in the chapter on series:

Mohammed: "What is this? This log... Look, there's log..."

Mohammed then finds what he was looking for, but it appears that his method might not work for this problem.

Mohammed: "It wouldn't work, we need two values."
Daria: "But we do have two numbers, don't we?"
Mohammed: "Yes, but they're not close to each other... Wait a minute..."

In this clip, Mohammed proposes that the common ratio is "a fraction" as the "numbers are going down."

Daria: "Do you want me to try a half?"

She tries a common ratio of 1/2 to see if it works (it doesn't), at which point Mohammed notices that the common ratio must also be negative, and explains why.

Mohammed turns to another group for help.

Mohammed: "Have you guys done part 2... it's weird..."

Mohammed: "There has to be some way of working it out, let me just look in the textbook... This is very weird... Wait a minute, what if we do simultaneous equations, we could just do simultaneous equations!"
Daria: "Oh yeah!"

Mohammed guides Daria through the method: "It will be better to put the other one on top... the ar^4... it will cancel easier."

After solving the problem, Mohammed and Daria discuss how they might have solved the problem more efficiently.

Mohammed: "I don't know why I didn't do that at the beginning... it's weird... I don't know why I didn't see it!"
Daria: "It's kind of interesting that we used so many different things to find... I'm pretty sure there's one, sort of, formula to find this."

Mohammed: "I really like this chapter because you can check all your answers... if it doesn't make sense, if you don't get the next term, then you're wrong."

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