## Stem and Leaf – is there a point?

Stem and leaf diagrams, or “Those leafy stem things”, as one of my former pupils used to call them, have long been an annoyance of mine. I’d never heard of them until I was brushing up on the GCSE syllabus ahead of my PGCE and when I did come across them I couldn’t see anything that they brought to the party that couldn’t better be shown using alternative methods.

You can imagine my feelings then as the KS3,4 and now 5 curricula jettisoned them, meaning the end was in sight for the need to teach them. I let my feelings on this be known in my recent post around the new A level curriculum and this led to further discussion around them on twitter. Then Jo Morgan (@mathsjem) wrote this fantastic piece which supports their place in a classroom and gives some great activities to use in teaching them.

It got me thinking, are my feelings unfounded? Should I be writing off stem and leaf diagrams? I’ve long been an advocate of maths for maths sake, see this defence of circle theorems for one example, so why is this feeling bot the sane for stem and leaf?

Perhaps it’s that it falls under the banner of “stats”, a very applied area of maths. This suggests that there should be an application associated with it. The use mentioned in Jo’s blog for bus and train timetables is the best example I’ve seen, but I think a normal timetable will be easier to read for the majority if people, as the majority of folk aren’t familiar with stem and leaf. Hannah (@missradders) suggested that they were used a lot in baseball, but I can’t see any reason that they would be better than a bar chart or a boxplot.

Colin Wright (@ColinTheMathmo) suggested during the twitter discussion that they could be used to build understanding around data, even though they are no use for any real data sets which would be far too big. Jo also uses this idea in her defence, saying they could provide a good introduction to the ideas of skew, quartiles and outliers. I can see this argument, but I still think there are better, more visual and less convoluted ways to introduce these to pupils, such as the aforementioned bar charts and box plots along with scattergraphs and a host of other data presentation methods (but not pie charts, they’re just as bad, if not worse! But that’s a topic for another day!)

*I really enjoyed Jo’s post, if you haven’t read it I would advise you do. It made me think and look hard at my views. In the end though, I still see no need in stem and leaf diagrams and will be glad to see the back of them. If you have opinions either way I would love to hear them, especially if you have further real life uses!*

## Resilience in the classroom

Recently I’ve written a lot of posts around puzzles and problems that people have set me to solve. This is something I find fun to do and I have enjoyed solving them. I have also enjoyed sharing my thought processes and proofs here, including the dead ends I went down.

The ideas involved in doing this have led me to think a lot about which, if any, of my students would have been able to solve each of the problems.

I started thinking about my A level classed, I know that the vast majority would have given them a go. One or two might have not started because they didn’t know where to start, but the rest would have given it a try. I’m fairly certain that at least 5 of them, given enough time, would have solved most of the puzzles. I intend to test this hypothesis after the holidays.

I then thought about my year 11 class. They are a top set and I have been trying to build a resilience in them this year. When I first took the class over in September one of the students complained to my colleague, her science teacher, that I had “given a worksheet and not even told us how to do it.” It had been a starter task designed to double check the class were familiar with and able to use Pythagoras’s Theorem. My colleague then asked her “could you not do it then?” to which she had replied “I could actually, it was Pythagoras’s Theorem.” He then asked, “so why did you want him to tell you what to do?” She had no answer.

This is quite common in schools, it’s an idea a lot of pupils have. That they should be explictly told what to do for each type if question. But I think that if we are to create the mathematicians of the future we need to be building a resilience into pupils. We need to equip them with the skills and knowledge required to solve the problems, and allow them to select whichever bit of it they want/need to solve it.

I think the puzzles I’ve discussed recently are good examples of tasks that do this. Some of them have the added bonus of being solvable multiple ways, often given rise to a “low barrier, high ceiling” task that can be set to 11-18 year olds and be solvable, yet challenging, to all.

This one which started the chain is a lovely puzzle based around algebraic fractions. It is solvable to a clever yr7 student who just has a basic knowledge of fractions, or via simultaneous equations and complex numbers, which is how most people with an advanced knowledge went about it.

This triangle puzzle was particularly nice, it had a lovely solution which requires a knowledge of the sine rule for the area of a triangle and knowledge of the sine curve, or a much simpler visual one which only requires a primary school level understanding of triangles!

This triangle puzzle is the best example of one with multiple solutions. I used Heron’s Formula (which no one else seems to have heard of!) But it is equally solvable using accurate drawing, similar and congruent triangles rules and/or Trigonometry (including Pythagoras’s Theorem).

These problems are great, and will build resilience, but the two most recent ones are the ones which illustrate this best.

While discussing constructing a proof with my brother he was asking how algebraic proof worked. He has two A levels in maths (at A no less) but he stopped studying maths a decade ago so is a little rusty. His questions, though, made me think about my classes. I know my year 13 pupils can construct proofs, but I’m not sure about all of those pupils who are younger. I am going to ensure I build more opportunities for this into my lessons.

In solving this problem I noticed a pattern in the numbers, I expressed this pattern algebraically and manipulated the algebra to prove the pattern was true for all numbers. This is what mathematicians have done for centuries and how theorems are born. And this is a skill I need to instil in my students.

In the root of the problem I discussed solving a problem which involved searching for integer solutions of an equation in two variables. It was fun, and again I was asked, “how on earth did you work that out? I wouldn’t know where to start.” Well I didn’t know where to start either, I just tried things until I got something that was right. This is what the best of my students do. This post gives some great examples of this. It needs to be more though. As maths teachers we need to make sure our students are willing to do this, if they don’t know what to do to apply things they know until they get an answer. I have been using this approach with my pupils. I won’t help them unless they have tried something first.

The way the maths exams have been previously has made this spoonfeeding possible and far too common. In the last couple if years the exam boards have thrown a few curveballs, which has meant that students have had to apply their knowledge in different ways to the past papers. I think this is the way forward, and hope the new GCSE and A Level exams address this.

## The great calculator debate

Back in December I posted this blogpost about calculators. It caused quite a stir and prompted many responses. Dave Gale, aka reflective maths, tweeted back with this video, Colin Beveridge, of flying colours maths, responded with this blogpost and there was a much wider debate on twitter with tons of people getting involved on each side. It was brilliant to see. I thought though, that I needed to write a further post to clarify and review what had been said.

In the first instance, I selected a sensationalist title which was intended to catch the eye. I do think, though, that the title may have led people to think my stance was a little more hardline than it actually was. And having seen the views set forward by the alternate position, I think my view has softened further still.

When I wrote the original post I was certainly advocating the banning of calculators in primary classrooms, and I would stick by this now. The opposing case to this was that an inability to subtract two and three digit numbers from 360 was causing a barrier to teaching angles. I would counter this with the statement that subtracting two and three digit numbers from a three digit number is such a basic skill that it needs to be mastered either before moving on to angles, or with angles providing a great opportunity to hone this skill. The other argument was that the government were banning calculators from KS2 tests, but using the same test. On the face of it, this is silly, but I don’t think it is a valid argument for keeping calculators. Rather it is valid argument for altering the tests.

Colin wrote in his post that calculators are not the enemy, but rather it is their misuse. I can see his point here, but I wasn’t advocating we destroy them all, I was advocating that we eliminate their use in primaries and cut it down radically in secondaries. He questioned the necessity of adding 4 or 5 numbers with 5 or more digits together, and this is a point I will concede. My hard line of only using it for trigonometry was perhaps too hard. But I still feel a vast reduction in their use would produce better mathematicians in the long run.

The video Dave sent was of Conrad Wolfram talking about why the future of maths should basically be entirely computational. Conrad feels that we need to stop teaching hand calculation and start teaching only computational mathematics. I feel that this would be an entirely wrong move. Computers can only do as they are told. If we are looking to prove a theorem generally, then we need to be able to hand calculate. Computers can check case after case, but this is not enough for a “proof”, as it is impossible to check an infinite amount of cases. A computer would not have been able to come up with mathematical induction or infinite decent.

A number of people responded along the lines of “What’s the point in learning how to do this when you can use a calculator?” This seems to me to be a ridiculous argument, like saying “Why learn to write when you can use a word processor?” or “Why learn to walk when you can use a mobility scooter?” If we head down that path is won’t be long before we are like the fat oafs in “Wall.E” (see this video) or even completely plugged in, a la “The Matrix”.

When teaching my further maths class numerical methods, I often have to field questions as to why we are doing this when if it were needed in the research world a computer would just do it. My answer is always simple, and always the same. “If no-one learns the theory, it will be forgotten, and no-one will be able to programme the computers to do it.”

No, calculators are not the enemy. But if the world becomes too reliant on them then we lose the skills we have built up over the centuries, we lose the ability to construct proofs for general cases, and we lose the beauty and the satisfaction one can get from solving a problem with nothing more than a pencil and paper.

Since the original post, I have realised that this is a wider issue than just calculators. Discussions with colleagues have highlighted that this problem occurs in other subjects when scaffolds are used. Thesauruses can lead to nonsensical sentences in English, for example. Scaffolds can also just mask a problem, pupils can get round something they cant do in lesson (ie subtract 197 from 360), but if it comes up on a non-calculator exam then they will not be able to obtain the correct answer.

Further reading:

From Mark Miller: Removing the cues

and Revision before redrafting (which includes the “greatest” sentence known to man: “a quantity of the most evil inscription is fashioned subsequently to a lexicon”.)

## “Manglish”, and a mastery curriculum

Today I attended #pedagoowonderland, it was a wonderful event with some superb sessions. One of the workshops I attended was by @lisajaneashes about “Manglish”, this is her philosophy on maths and English across the curriculum (you can read her blog or pre order her book if you are interested in learning more).

During the session Lisa said something that got me thinking about a whole host of things and I wanted to share these thoughts. She said during the session that it would be really effective to cover certain topics in other subjects at the same time as you are doing them in Maths.

There are a number of things happening at the moment, and this idea, to me, links them together.

Firstly, with the curriculum overhaul coming out of Whitehall, (see what I feel is missing here) we have the opportunity to develop a new and exciting curriculum for our school. Secondly, we are trying to look at whole school numeracy, and thirdly, we are hoping to increase the number of our pupils who go onto further study maths.

**Mastery Learning**

I’ve been reading a lot about New curricula recently, and something that strikes me as interesting is the idea of a mastery curriculum. (You can read Joe Kirby’s (@joe_Kirby) blog here. Michael Tidd’s (@michaelt1979) here, and check out this website). The basis of mastery learning seems to be to spend longer on each topic, covering fewer areas each year and ensuring that classes have mastered a minimum level of learning before moving on. This strikes me as exciting. A SoW with short units means you cover a topic for a fortnight, complete a unit assessment, and move on. This can work really well, especially for the high achievers, but it has its draw backs. If students have failed in year 7 to fully master how to solve one or two step equations then when equations next come up you have to revisit that. As they haven’t managed to learn it in two weeks the first time, they may not have retained much and they may fail to fully grasp the topic again. This can be come a cycle and can lead to pupils in year 11 becoming stuck on problems they should have solved at a younger ages. A mastery curriculum would enable deeper learning, and give pupils more time to learn these skills, offering those who master them quicker to mover further on in the curriculum. The theory being that the longer, deeper, covering of the topic would ensure retention rates were higher and when the class returned to the topic they could move on.

**KS3/4**

I’ve been involved in a few discussions recently on the need for separate keystages, do we need a specific KS3 and KS4 scheme of work or could we have a five year scheme of work? In the absence of levels, I’d imagine many secondary schools are looking at moving to GCSE grades as a way of reporting from yr7. If we are using these grades from the start why not a singles scheme of work?

**Feedback**

The shorter scheme of work system gives rise to a lot if summative feedback. (You can read more about our feedback here). This means that formative feedback happens in lessons, but written feedback tends to be summative, with pupils receiving written feedback on the topic they have completed, an extension question (or consolidation question) for them to try and then move on. A move to the mastery curriculum would mean that marking with the same frequency would give more chances for formative written feedback which could create a much better dialogue in the pupils books.

**Maths Across the Curriculum**

To start with, I think we should call it maths, rather than numeracy. I don’t think it should be just about numeracy. There are many other areas that can link in, rather than just simple number tasks. Similarly, I think we should talk about English across the curriculum, as it shouldn’t just be kept to “key words”.

I also think that Maths across the curriculum needs to be a culture embedded in a school. Lisa spoke today about how she wasn’t good at maths at school and how she didn’t care about it. She told us how this was compounded by her English and Art teachers telling her they were rubbish at it and that as long as she got the c it didn’t matter and she could just forget about it. This is a problem which is still rife today. Last year one of my year tens informed me that one of his teachers had told him she could never do algebra and it hadn’t had a negative effect on her. This infuriates me. A lot of pupils tell me they hear things like that at home, which is bad enough! The whole grade C culture is detrimental too, as my sixthformers are finding out when unis want Bs. (You can read more on this here).

Once the culture is embedded, maths links can be made with other subjects. This sort of link could be strengthened, as Lisa suggested, by covering these at the same time. Logistically, this would be a nightmare to embed with the 2 week unit scheme of work, but I think it would be more doable given a mastery curriculum which covered topics in more detail for a longer time. The whole school would know that in this half term year seven were looking at representation of data, and they could build that into their lessons accordingly. If in geography pupils were collecting some data, they could analyse that in their maths lesson. If the scheme of work was written in such a way that pupils in each year group were covering the same strand of maths, this could provide exciting whole school opportunities. Assemblies could tie into the topics. Cross curricular projects could be in abundance. Pupils would be seeing the links, seeing the importance, seeing the context and having the learning consolidated and embedded.

**Drawbacks**

There are drawbacks to this idea. There is the worry that pupils may get restless and lose interest if the same topic was covered over and over again, although I think this is avoidable with planning. Set changes would be much harder to implement as different classes would have reached different points in each area. It may be harder for pupils to catch up if they moved from another school partway through the course.

**So is it the answer?**

In short, I don’t know. I think there are many plus sides to moving to a mastery based curriculum and I am currently swaying towards thinking it would be a great way to go. But to be sute I need to read more on it and discus it more.

**What do you think?**

Have you implemented this sort of curriculum? Did it work? Are you thinking of it? Does it sound good to you? Or do you think it’s daft? Whatever your opinion, I’d love to know.

## Resources

Today I decided to try look into uploading resources to the TES website. I figured that I get so many off there from such fantastic contributors (my favourite ones can be found here: https://cavmaths.wordpress.com/resources/ ) that it was about time I started to put some of my own up.

My page can be found at http://www.tes.co.uk/mypublicprofile.aspx?uc=1383503&tab=resources&profileTab=resources

So far I haven’t put much up there, but my first upload is a collection of my favourite starter puzzles (http://www.tes.co.uk/teaching-resource/Starter-Puzzles-6334156)

In this collection is my 4 pics one word starter, which created a whole raft of other similar resources by other superb teachers. I have blogged in more detail about that here: https://cavmaths.wordpress.com/2013/02/25/4-pics-1-word/

Also included are the “wedding maths” starter (https://cavmaths.wordpress.com/2012/12/23/wedding-maths/) and an old favourite (https://cavmaths.wordpress.com/2013/01/29/an-old-favourite/) .

I have included two show-me activities based around reflection, rotation and enlargement which i find fun and engaging. They included pictures of celebrities they will know, and some they may not know. I have found this can be a good way to get some wider discussion into the lesson, perhaps around the prime minister. I do find it slightly worrying that more pupils recognise the President of the USA than our PM!

The final one I have included is a whiteboard activity called “Whats wrong with this chart”, which was inspired by a series of tweets from ~~@~~nyoungmath (a maths teacher from Canada) which included the pictures I have used and the answers from one of her classes about what was wrong with them. I have used this today as a revision activity with Yr 11 and they loved working out the mistakes!

## What a week!

This week has been a good one. Monday started with the late notice news that my trainee was ill and as such would not be able to teach my year seven class. He had, however, emailled me a lesson he had planned. I taught from his lesson, and he had some good activities, but I didn’t feel as confident teaching the lesson as I do when teaching my own, and i don’t think it went as well as it would have done if I’d planned it. There were things I would have done differently, but a good experience all the same.

The rest of Monday was pretty good, and I got the first taste of one of my new y11 groups. (the foundation one). The lesson started a little messy, as some of the pupils felt like they were seen as the “dumber” ones, but we had a Frank discussion about the fact that a smaller group of them targeting the topics they can’t do on the foundation paper gives them the best chance of doing well. They accepted this and hot on quite well. The lesson I has with them on Friday was amazing, they worked really hard and I think they now all want to do their best.

Tuesday was a brilliant day. Every lesson went as well as it could possibly go. I’m not sure why, but I was really on form. I had to pick up two lessons the trainee usually takes my classes, but I had notice this time, so was able to plan my own lessons. I did inequalities with my year 7’s, which stretched and challenged them and they loved it. My year 13 lesson was on polar curves, a superb topic, and as with many recent further pure topics one I didn’t have to teach much in. The class have become accomplished mathematicians, and have a superb ability to see the links from the skills and knowledge they already have to the new maths. Year 11 did ratio, which results plus tells me is one of the areas they did badly on in march, the lesson went well and they will all get full marks on it next time! Year 8 did revision, year 9 fractions and year 10 changing the subject. All lessons that were planned well for the class and that came off brilliantly. (I will try upload some if these resources to TES soon!)

My lesson for the rest of the week have gone quite well too, but not as well as Tuesday’s, which might well be my best day since I started teaching!

The other highlight of the week was getting to see the school production of Sondheim’s “Into the woods”, but that’s a topic for another blogpost, so watch this space!

## Area the Hero’s Way

This morning I posted this about a lovley simple triangle puzzle I had found and explored. One of the responses came from Mrs Watson (@MrsJMWatson) who tweeted this reply:

The link is to this page from Nrich and contains this puzzle:

Triangle T has sides 6,5,5 and Triangle Q has sides 8,5,5 What is the ratio Area T:Area Q?Now this is a great example if a “low barrier, high ceiling” problem. I think your first instinct are probably to go with what you are most familiar with. In today’s earlier problem the respondents who are used to working with the sine rule for area instinctively went for that, but others who are more used to working with other things went other ways.

For this problem my instinct was to use the cosine rule to find an angle, and then the sine rule for area ((1/2) absin(c)) to find the areas and simplify the ratios. My friend Steve, also a math teacher up to A level, actually worked through this, but didn’t need the sine rule for area as the ratio is easy to spot when you see the angles add up to 180 degrees (we’ll let him off for using degrees just this once).

I didn’t go down this route, I was thinking about generalising for all triangles and thought Heron’s Formula would be better for this.

I used it and found this lovely solution:

T) a=5,b=5,c=6 s=8Area: (8.3.3.2)^1/2=12 U) a=5,b=5,c=8 s=9

Area:(9.4.4.1)^1/2 = 12

So Area T: Area U = 12:12 = 1:1Discussing my method and answer with various people today I have been shocked at how many people (most it seems) haven’t heard of “Heron’s Formula”!

Heron’s FormulaHeron’s Formula is a fantastic piece of mathematics. I don’t know how I know about it. I have known about it so long it didn’t occur to me others wouldn’t. I guess I had assumed I learnt it at school, but if that was the case others would know about it. I’ve read and watched a lot about maths over the years, so I guess I must have picked it up from a book or show.

For those of you who don’t know, Heron’s Formula states:

For a triangle with side lengths a,b and cArea=(s(s-a)(s-b)(s-c))^1/2

where s is the semiperimeter (ie s=(a+b+c)/2)A truly marvellous formula. It’s named after Hero of Alexandria, who along with this formula is credited with being the first person to envisage imaginary and complex numbers.

Hero’s own proof is pretty cool, and involves cyclic quadrilaterals and properties of right angles triangles. There are lots of other proofs too, my favourite is the trigonometric proof, which I think would have been what I ended up with if I had decided to generalise this problem using trig!

Later in the discussion Mrs Watson said the phrase “Pythagorean Triples“, and I instantly saw both triangles could be cut into 2 3,4,5 triangles. I think this is the nicest solution. When I checked back to another discussion I noticed that Andy (@andycav_25) had also had this realisation. His instinct was to draw a perpendicular height. I wondered if I would have gone down this route if I’d had paper to work on instead if working mentally?!

Finally, Mrs Watson mentioned that her year 7 class do it via construction. She didn’t elaborate much on this, but I imagine she means draw both triangles accurately, measure the height, work out the areas. This is, in itself, quite nice, and shows that pupils can tackle the problem using whatever tools they have at their disposal.

This is a lovely problem, I’d love to hear how your instincts would tackle it, how your students for if you try it, and if you previously knew about Heron’s Formula!

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