## Oblongs

Last week while we were waiting for a swimming lesson to start my daughter told me that one of her teachers had got “higgledy piggledy” about oblongs. I asked what she meant and she said that she’d accidentally called one a rectangle and had to correct herself and had informed the class that at her last school she’d had to call them rectangles but at this school had to call them oblongs and sometimes got higgledy piggledy about this. I asked my daughter why they couldn’t call them rectangles and she said that it was because squares can be rectangles too.

This set off a lengthy chain of thoughts in my head. Firstly, I was quite impressed by the fact a 5 year old could articulate all this about knowledge about shapes so well. Then I thought, does it really matter whether they call them oblongs or rectangles? Then I thought, wait a minute, why are we prohibiting the use of rectangle because it can also mean a square, but we are not prohibiting the use of oblong when it can also mean an ellipse? My chain of thought then jumped down a rabbit hole questioning whether we should actually be referring to regular or equilateral rectangular parallelograms and non – regular/equilateral parallelograms. Why are we allowing children to call a shape a triangle, when it is one possible type of triangle in a family of triangles, but not allowing them to call a shape a rectangle when it is only one possible rectangle in a family of rectangles. These thoughts stewed around in my head for a while and I thought I’d ask the twittersphere for their opinions on the matter.

These opinions fell into a couple of camps. The first cam thought that oblong was a nice enough word and they didn’t mind others using it but preferred not to themselves. The second camp felt that it was important to distinguish between an oblong and a square so important to use oblong not rectangle and the third camp thought that actually it was better to use rectangles due to the elliptical oblongs. I questioned some of the respondents from the second two groups a little further to see why they fell into these groups. Those in the second seemed unaware that the word oblong also meant ellipse and those in the third thought it was more important to excluded ellipses than squares. Stating that it was easy enough to explain away the special case that is the square.

I’ve spend rather a lot of time considering this, and am now not really sure what I think on the issue. I can’t see a problem with using a rectangle and explaining away the square as a special case. We call all triangles triangles and expand as and when required. No one bothers about calling a non-rectangular parallelogram a parallelogram, despite the fact that that could mean a rectangle. But again I’m not sure I’m massively strongly against the term oblong either. It could open up a good discussion about the term and how it could apply to ellipses, although this probably is a little too much for a year 1 classroom. I think I’m leaning towards rectangle as a preference though, as explaining away a special case is, for me, much more preferable than ignoring a whole class of oblongs.

*If you have views on this, whichever way you lean, I’d love to hear them, either in the comments or via social media.*

## 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.

## A surprising find

The other day I my timehop showed me this lovely little post from last year. It includes “Heron’s Formula” for calculating the area of a triangle, as I read it I remembered thinking it was a little strange that not many people had heard of it before.

Today I was looking through a number of textbooks trying to find a decent set of questions on area, perimeter and volume for my year nines as I wanted to consolidate their learning at the start then move onto surface area. *I’m not a fan of textbook misuse- ie “copy the example and try the questions” but I do sometimes use them for exercises as we have a very limited printing budget and some of them have superb exercises. For a fuller picture on.my view of textbooks, read this.*

I was looking in one of my favourite textbooks:

And I happened across this:

There it is! Plain as day! Heron’s Formula! In a KS3 textbook!

I was disappointed that its function was described and its name wasn’t and there was no mention of why this worked. It basically reduces the question down from a geometry one to a purely algebraic substitution task and I would question the appropriateness of including it in an exercise on area, but still, I was incredibly exciting to find it there!

*Are you a fan of Heron’s Formula? Had you even heard of it? Do you have a favourite textbook? I’d love to hear your views.*

## Math(s) Teachers at Play – 83rd Edition

Hello, and welcome to the 83rd Edition of the monthly blog carnival “Math(s) Teachers at Play”.* For those of you unaware, a blog carnival is a periodic post that travels from blog to blog. They take the form of a compilation post and contain links to current and recent posts on a similar topic. This is one of two English language blog carnivals around mathematics. The other is “The carnival of mathematics“, the current edition of which can be found here.*

It is traditional to start with some number facts around the edition number, 83 is pretty cool, as it happens. Its prime, which sets it apart from all those lesser compound numbers. Not only that, its a safe prime, a Chen prime and even a Sophie Germain prime, you can’t get much cool than that can you? Well yes, yes you can, because 83 is also an Eisenstein prime!!!! Those of you who work in base 36 will know it for its famous appearance in Shakespeare’s Hamlet: “83, or not 83, that is the question…..”

Firstly, to whet you appetite, here is a little puzzle, courtesy of Chris Smith (@aap03102):

The solution can be found here. *And there are tons more puzzles here.*

So, what delights do we have for you within the carnival?

**Maths Fun**

Firstly, we had a few submissions that were based around having fun learning maths.

Firstly, Mike Lawler (@mikeandallie) submitted this on a 242 sided Zonohedron: *This project plus the follow up project, were projects out of Zome Geometry that we did in the open space in our new house (i.e. we don’t have much furniture yet!) Really fun project for kids. Lots to learn about geometry, symmetry, and especially perseverance! Really shows how amazing the Zometool sets are as learning aides, too.*

Our Carnival Founder, Denise Gaskins (@letsplaymath), has submitted this on “Fun with the impossible Penrose triangle.”

**Pedagogy and Reflections**

There’s a few post around the pedagogy of teaching mathematics, including reflections on what’s been tried in the classroom.

Cody Meirick submitted this on “Maths Investigations”: *The developers of this series argue that math should not be viewed as a history lesson, teaching formulas and concepts that mathematicians “invented” centuries ago. Instead, math time should be an active and even creative process, allowing students to learn through experimentation and exploration*.

Benjamin Leis asks “can we get there from here?”: *I’ve been blogging about my experiences running a math club for the first time. This one was a planning exercise to figure out how to make a particular problem accessible to the kids.*

Rodi Steinig has submitted this nice little post around tessellation.* In the 5th of our 6th Math Circle session about Escher and Symmetry, middle-school students make some discoveries about assumptions, and also discuss the pros and cons of inventing your own math.*

The superb Ed Southall (@solvemymaths) has produced more posts in his excellent complements series, aimed at helping to further subject knowledge within the profession. The latest instalment is on Highest Common Factors (or Greatest Common Divisors, to those of you across the pond) and Lowest Common Multiples.

Manan Shah (@shahlock) wrote this excellent piece entitled “100 days of school“. He looks at the maths behind animation.

Emma Bell (@el_timbre) looks at maths hubs and some of the issues involved with them.

Kris Boulton (@krisboulton) explores the question: “If we cannot see the learning win a lesson hat are exit tickets for?”

Colin Beveridge (@icecolbeveridge) has written this post on the three “tricks” he wishes his GCSE students would use. *(SPOILER: They aren’t really tricks!)*

Bodil Isaksen (@bodiluk) has written this fine piece on embedding maths into all aspects of school life.

**Resources **

Also this month many people have shared great resources, here are some brilliant posts on that.

Jo Morgan (@mathsjem) has produced another of her maths gems series. The series looks at great ideas and resources Jo has discovered recently, and it is another of my favourite series. This is issue 25.

Mel Muldowney (@just_maths) has put together this lovely post entitled “Two is the magic number” sharing some resources aimed at wiping out misconceptions and checking answers.

Sam Shah (@samjshah) has shared his thoughts, experiences and resources from a recent lesson on angle bisectors in triangles. It is well worth a read.

Paul Collins (@mrprcollins) has shared these fantastic mathematical plenary sticks.

Bruno Reddy (@MrReddyMaths) shared this great post which includes hints, tips and resources to aid peer tutoring.

Tom Bennison (@Drbennison) shared this superb post on using geogebra to teach conic sections.

Martin Noon (@letsgetmathing) shared this excellent trigonometry calculator, check it out!

**Teachers at play**

February saw maths teachers from across the UK gather at “A celebration of maths“, which was a superb event put on by the White Rose Maths Hub (@WRMathsHub.

Dave Gale (@reflectivemaths) and Colin Beveridge (@icecolbeveridge) have produced another sterling episode of “Wrong, but useful” the nations favourite maths podcast (well, mine at least).

*Well that rounds up edition 83, I hope you have enjoyed it. If you want to submit a post to the next carnival you can do so here. If you’d like to host contact Denise (@letsplaymath). And make sure you catch next moths carnival which will be hosted by John Golden (@mathhombre) over at mathhombre.*

## Brilliant – An app review

Over the weekend I read this post by Jo Morgan (@mathsjem). In it she refers to a website “brilliant.org” which offers some brilliant stuff to be used in maths teaching. I liked the look of it so I set off to investigate and when I accessed the site I was presented with a pop up asking if I wanted to download the app, which I promptly did.

When I accessed the app I was presented with a number of mini quizzes which covered a range of early KS3 material, but on further exploration I discovered a treasure trove of fantastic questions and explanations of topics ranging from very basic up to degree level.

On top of this there are some fantastic puzzles to solve and the app tracks what you have done and how well you have on it. I found it really fun, I enjoyed testing my knowledge on some of tge further mechanics and am looking forward to attempting some of the puzzles.

I think this app is brilliant for anyone who loves maths, whether they be studying it at school, at university, teach it or just enjoy keeping themselves smart.

There is also the chance to subscribe to additional content, I haven’t yet but I am tempted. The additional content seems to be more worked solutions to the quizzes, but I think I’ll see how much I use the app before I upgrade.

I do love the prices too, it’s £tau pounds a month, but if you sign up for a year they reduce it to £pi – a lovely touch.

A great app, if you love maths, get it and if you’re studying maths it will help immensely.

I’d also advise following the app page on Facebook, you get a constant stream of little beauties like this:

## Tests of divisibility

Today I was marking my year 9 classes books and came across some work on prime factor decomposition and tests of divisibility. Yesterday I had been arguing with Colin Beveridge (@icecolbeveridge) about the use of formula triangles, (see the comment section on this post) and any other method of anything that was presented in the fashion “do it this way and don’t worry about the why.” I felt a wave of hypocrisy flow over me. I had never explained the tests of divisibility to the class, and furthermore I didn’t even know myself why they worked! I thought I’d explore them and see what I came up with.

If the last digit is divisible by 2, the number is divisible by 2.This test of divisibility is obvious, the definition of an even number is that it has 2 as a prime factor, and all even numbers have a last number divisible by 2. No further exploration needed. I know why this one works and I’m certain the class do too.

If the sum if the digits is a multiple of 3, then the number is divisible by 3.This is a fact I’ve known since I was at primary school back in the 1980s, but I can honestly say that I’ve never thought about why it works. I wrote a couple if equations out, realised I should be working in modulo 3 and came up with this:

Which basically boils down to the fact any number can be split into the digits multiplied by a powerful of ten.

A= (x0)10^0 + (x1)10^1 +… + (xn)10^nAs (10^n) mod 3 = 1 for all natural numbers (and 0) then it follows that:

(A)mod 3 = (x0 +x1+…+xn) mod 3Which implies our test of divisibility. This also implies the

test if divisibility for 9(ie is the sum of digits is divisible by 9 then so is the number) as (10^n)mod9 = 1 for all natural numbers and 0. To prove you just follow the exact working but in mod 9.A number is divisible by 4 if it’s last two digits are divisible by 4.This one uses the fact that if two numbers are both divisible by a number then so is their sum.

We know any number bigger than 100 can be expressed 100a + b where a and b are natural numberso and b<100. We also know that as 4 is a factor of 100, 100a is divisible by 4 if a is a natural number. Hence the whole number is divisible by 4 iff the best is divisible by 4, and b is the last two digits.

From this we can deduce the

test of divisibility for 8(ie a number larger than 1000 is divisible by 8 iff the last three digits are divisible by 8) as 1000 is divisible by 8. The proof is the same, but you split the number into 1000a + b where a and b are natural numbers and b is less than 1000.Testing for 5 and 6Testing for 5 (the last number is 5 or 0) is another obvious one that needs no further exploration. And the test for 6 is simple, if 2 and 3 are prime factors 6 must be a factor, as all numbers can be expressed as a product of their prime factors and 6 is the product of 2 and 3.

Testing divisibility for 10Again, this is obvious as we are talking about a base ten system! That leaves just one more test.

To test for divisibility by 7, take the last digit, double it. Take this away from the number you are left with if you subtract the last number and divide by ten. If the answer is 0 or divisible by 7 (ie if the answer mod 7 = 0) then the original number us divisible by 7.What a ridiculously complicated divisibility test. It looks much nicer algebraically:

(10a + b) mod 7 = 0 (a and b are natural numbers, b < 10)iff

(a -2b)mod7 = 0Once I expressed it algebraically I had two simultaneous equations and solved like this:

I enjoyed working through them and now I’m confident I know why these tests work, but I’m fairly sure that the ones that aren’t obvious would be too complicated to explain to year 9. This made me think about their use, and I certainly think it’s fine to use them even without understanding. Which brought me back to the formula triangles debate. This is one that will rage, but as I explained in my post on it, I don’t have a great issue with people who understand the algebra using them. I also don’t have a great issue with people who struggle at maths and aren’t going to pursue it past GCSE using them. My issue is with higher attainers who could and should understand the algebra being taught then with no deeper conceptual understanding. I guess it’s a topic I will need to think more on.## Share this via:

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