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Last night at 8pm I was sat at home nervously awaiting the start of the first #mathsjournalclub discussion. I really wasn’t sure if anyone would take part – especially after I had found the article relatively hard going! The article didn’t turn out to be what I had expected, but once I had got through the first few pages I found it very interesting and it gave me lots to think about. 

Thankfully, lots of people took part and I really enjoyed hosting the discussion (even if it was surprisingly tiring!). There were some great points made and lots of excellent discussion- see below for some of these. 

If you didn’t take part this time please consider it next time. I will be releasing the poll to choose the article next week so if you have a suggestion send it to me or to the @mathjournalclub account. 

To try and distill the discussion I created the following storify… Please don’t be mad if all of your tweets didn’t make it – there were so many of them!

On the 24th August between 8pm and 9pm the first #mathsjournalclub discussion took place. We were talking about the article “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, Hughes and Ely. Discussion was frantic – this is an attempt to provide some kind of record.

On the 24th August between 8pm and 9pm the first #mathsjournalclub discussion took place. We were talking about the article “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, Hughes and Ely. Discussion was frantic – this is an attempt to provide some kind of record.

On the 24th August between 8pm and 9pm the first #mathsjournalclub discussion took place. We were talking about the article “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, Hughes and Ely. Discussion was frantic – this is an attempt to provide some kind of record.

https://storify.com/tajbennison/getting-started

Very short post today to just promote the first #mathsjournalclub discussion tonight between 8pm and 9pm. We are going to be talking about “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, J, Hughes, G and Ely, R which is available here if you haven’t read it yet.

Last week I posted some possible themes for the discussion, for convenience I have reproduced them below:

• What did you think were the key points of the article?
• Do you agree with the four ways given that “students typically represent functional relationships: graphs, tables, verbal descriptions and equations” ?
• Has this article impacted on how you introduce and teach the understanding of functions?
• What are the limitations of the article/research?

The more people who can take part the better the discussion will be so I really would love it if you could make it tonight.. Don’t forget to use the hashtag #mathsjournalclub in all your tweets so that I and others can follow the discussion.

After writing my original Matlab program to plot Barnsley’s fern I thought I would look into it in a bit more detail and read his original paper here and have had a play around with the parameters in the program. I have been trying to generate a fern that looks a bit like one that I saw at Bradgate Park in Leicestershire.

  After a while of adjusting the parameters and getting ferns that I wasn’t completely happy with I decided to look online and see if anyone had come up with some parameter sets already. It turns out they had, and using the given parameters for the Cyclosorus fern  I can come up with something pretty good.

In the notation of the previous post, the affine transformations are given by

\( \begin{align} f_1(x,y) &= \begin{pmatrix} 0.000 & 0.000 \\ 0.000 & 0.250 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 0.000 \\ -0.400 \end{pmatrix} \\ f_2(x,y) &= \begin{pmatrix} 0.950 & 0.005 \\ -0.005 & 0.930 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -0.002 \\ 0.500 \end{pmatrix} \\ f_3(x,y) &= \begin{pmatrix} 0.035 & -0.200 \\ 0.160 & 0.040 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} -0.090 \\ 0.020 \end{pmatrix} \\ f_4(x,y) &= \begin{pmatrix} -0.040 & 0.200 \\ 0.160 & 0.040 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} 0.083 \\ 0.12 \end{pmatrix} \end{align} \)

with associated probabilities \(0.02, 0.84, 0.07, 0.07\). These give the following plot after a million iterations.

Have play with the parameters and see what you can come up with. I have also found a very nice bit of html5 and javascript code that generates Barnsley’s ferns, this is here – take a look at it.

When I was teaching mean from a frequency table I remember looking on MyMaths and seeing the questions they had (I don’t use MyMaths that much to be honest), and they had this….

Not holding back now…. I think this is an awful question. This is mainly because of the third column that is titled \(fx\). At no point in the question is either \(f\) of \(x\) defined!! Maybe I am being a bit pedantic, but to me because of this having a column titled \(fx\) is meaningless and sloppy. Ok, I know that \(fx\) seems to be often used in this context, but I really don’t like.

Surely this just sets a bad example for students… especially once they get to A-Level and beyond. Clarity of mathematical exposition is important, and it should make logical sense. It’s like if a student writes \(E(X)\) without first defining the random variable \(X\) – makes me want to cry a little!

If you follow me on Twitter then you probably know that I am pretty passionate about engaging with academia, both in terms of bringing current mathematical research into the classroom (if possible) and by engaging with the mathematical education community.

I feel pretty lucky in that I am based near Nottingham with the University of Nottingham on my doorstep. Not only has it got a beautiful campus and a great maths department but is also the home of The Centre for Research in Mathematics Education (CRME) convened by Malcolm Swan

Among others, Jeremy Hodgen (@JeremyHodgen), Colin Foster (@colinfoster77), Peter Gates @petergates3), Geoff Wake (@geoffwake1), Diane Dalby, Andy Noyes and Mark Simmons (@simmo1363) are all based at the CRME.

They hold an excellent series of seminars during term time, which I try to go to – I think I went to most of them this year – and are well worth the time.

If you have something similar near to you I would recommend that you get involved, and if you are near the CRME come to some of the seminars next year.

Back in May, there seemed to be a craze of posting “shelfies” of our book collections; as mine are spread out over numerous shelves I collated all of mine into a blog post.

Unfortunately I can’t remember who started this (if it was you please come forward!) but many of us on Twitter shared photos of books and briefly mentioned some of your favourite ones. I said about how it would be good to have a central list of books that have either been recommended or reviewed by us on Twitter, maybe listing some of the things that were good about the books.

After this, Stuart Price (@sxpmaths) set off and created website where we could submit books and links to reviews etc to do exactly this. I couldn’t really believe how quickly he got this website up and running.

Unfortunately until now, I haven’t gotten around to submitting any reviews or mentions, but this academic year I plan to change this. I am aiming to give myself some reading time every day (even if it is just 15 minutes) so that I can start to work through my large backlog of books, post reviews on my blog and then send to Stuart to update his site.

I’d really like to encourage other people to submit books and reviews to Stuart too, I think it could become a valuable reference if collaboratively we build it up so that it is fairly comprehensive.

It is well known that for some people mathematics causes great anxiety and an article from the Telegraph by Javier Espinoza (@JaviereTMG) last Tuesday (11th August) reported on research linking parental maths anxiety to lower attainment in mathematics by their children. Have a read…

Aside from the fairly awful headline (ability and attainment are not necessarily the same), like many reporting of educational research there was

• No link to the original research paper
• Claims made that are not well explained.

The above things really do irritate me – how hard is it to include a link to a research paper? Luckily the paper is actually pretty easy to find online through the webpage of Professor Sian Beilock, one of the authors of the study who is quoted in the Telegraph article. In fact you may recognise the name as she is the author of the fairly popular book “Choke: What the Secrets of the Brain Reveals About Getting it Right When you Have To”. The paper is available at this link on her research lab’s homepage and was published in Psychological Science OnlineFirst, an official journal of the Association of Psychological Science.

Overall I feel this is a very interesting piece of research, and their methods are explained very well. The study looks at a group of 438 children (from 90 separate classes) across 29 different schools in the Midwest states of the US. These children were selected from a larger group of 868, from whom children were removed for various reasons, including the need for parental data for each child. As the authors admit, the children whose parents chose to participate came from higher socio-economic status (SES) schools than those that did not – unfortunately this seems common with this kind of study. Despite this the students were still drawn from schools where the percentage of children receiving free or reduced price lunches (a common proxy for socio-economic status, though we all know this is not without its faults) ranged from 0% to 97%.

Teacher’s math anxiety and knowledge were also assessed, and due to missing data from the teachers the number of children who could be used reduced to 379 (211 girls, 168 boys) across 27 different schools with 76 teachers (73 female and 3 male).

The children completed measures of achievement – both in reading and mathematics, and in math anxiety at the beginning and end of an academic year. Parents completed a questionnaire designed to grade their math anxiety and find out how often they help their children with their homework. Teacher’s maths knowledge and anxiety was assessed at the middle of the year (why not at the end?..)

In their paper they provide more detail than was given in the article. It seems that parental anxiety had an impact on children’s attainment when the parents were frequently helping with homework (this didn’t carry over to reading attainment). They then hypothesised that this decreased performance throughout the year would also lead to increased maths anxiety in the children. To do this they generated 5000 (better than some, but they could easily have done more) bootstrap samples to test the strength of the indirect path from “parents’ math anxiety through childrens end-of-year math achievement to children’s end-of-year math anxiety at three levels of parents’ homework-helping behaviour”. This showed that when parents frequently helped their children with their math homework, parents’ maths anxiety was related to their children’ end of year math anxiety, though as far as I can tell this has not been established in a causal way.  They also showed that the effect of parental anxiety on children’s attainment (when frequent homework help is given) was still present even when controlling for the maths knowledge of the parent.

Does anyone know of any similar studies in the UK?

I think this is a valuable piece of research as, to me, it highlights the need to engage with parents and try to reduce their maths anxiety as well as the students. Of course, students experience differing levels of help with homework, but where homework is given it is clear that it is given with good intentions. If it is the case that simple statement by parents can negate the positive effect of help, then I am sure many parents would like to know how to tackle this. I think further study is needed to confirm some of these results and explore in detail how parental help could differ as a function of math anxiety, but it certainly can’t do any harm to engage parents more with this.

As an aside I’m very interested in these Woodcock-Johnson tests that were used – does anyone have a copy?

So, this holiday seems to be going pretty quick, and it we are now only a week away from the first #mathsjournalclub discussion! Next Monday (24th August 2015) at 8pm we are going to be discussing “A Glimpse into Secondary Students’ Understanding of Functions” by Brendefur, J, Hughes, G and Ely, R. The paper is here if you haven’t had a chance to read it yet.

I sat down to read it at the weekend with a bacon sandwich and found it an interesting read.  

  We shall start the discussion with a general “What did you think of the article?” and see where it leads us, but here are a few things you may like to think about:

• What did you think were the key points of the article?
• Do you agree with the four ways given that “students typically represent functional relationships: graphs, tables, verbal descriptions and equations” ?
• Has this article impacted on how you introduce and teach the understanding of functions?
• What are the limitations of the article/research?

One of the referenced papers is also worth a read as it is referred to extensively when the authors discuss covariational reasoning. This is the paper “Applying covariational reasoning while modelling dynamic events: A framework and a study” by Carlson, Jacobs, Coe, Larsen and Hsu.

I’m looking forward to hearing everyone’s thoughts next week. Also, please send any articles to me if you want them on the Poll for the next article selection….

I sometimes like to show photos in a lesson and ask either specific maths questions prompted by the photo or a general “What maths can you see?” These are pretty good when what I had planned finishes a bit too early – especially if I can tie the photos into the maths we have been studying. 

Yesterday I was in Whitstable, Kent and took the following couple of photos (among a few others):

• What is the cheapest combination of picking 1 from each section of the menu? 
• How many different combinations of dishes will achieve this cheapest price?
• What about the most expensive combination?
• What do you think the restaurant makes the most money on?

  
The above picture shows a pile of empty oyster shells from an oyster shack. The man in the top left corner of the picture is standing behind the wall and gives a sense of scale. This photo could lead into some good discussions on estimation (and the problems with estimation) such as:

• How many oysters have been de-shelled to create the pile? 
• What assumptions have you used?
• If each oyster costs £2 how much money has been spent on Oysters to create this pile?
• If the shack sells 500 oysters a day, how long did it take for the pile of shells to become this big?

Whilst I’m wary of spending too long on this type of activity all the time, I do find them both fun activities and activities that lead to lots of valuable mathematics. As a bonus, they seem to engage pupils pretty well too.