## Fractions! Did you just call for pizza – and a party?!

We have had a few inputs now on mathematics: and they have all been most useful in helping me to realise that imagination is maths and maths is imagination. Long way to say it; basically think to the limits of space. Endless? Yes. Claire needs to enter the classroom and not just see the earthly game of “ooo, where are the ticks and crosses on my worksheet?” Children need to be exclamation marks and rocket to the next planet of discovery. As Juliet Robertson (2011, 00.33) says: “Maths isn’t about a pencil and paper… it is about using numbers as a tool to understand the world around us. Our lecturers and Molina (2004) emphasise that children need to realise what the sum is about – and not just recite the steps required to reach the answer (conceptual – former vs. procedural – latter). The depth of understanding must be enough that the child can build their own marshmallow spaghetti tower to hold an egg… and not just to fit the ‘it’s a tower that stands.’ I appreciate that I need to understand how best to deliver subject knowledge in a way that caters for every student too, if my children want to leave as confident individuals. And, doesn’t such success start with consistence in the little habits? Here’s a bad stereotypical student habit: ordering the red and yellow triangles! That’s the pizza habit – that maths can actually solve my guilt with!

There are times to throw a party, celebrate and dive into a box of Dominos. Maybe student life? Perhaps (okay, true). But, a classroom also benefits from having pizza. Children like to socialise and discover things. So… why not ask them to budget for their own party and introduce fractions at the same time? I am back to developing confident, creative and conceptual-focused maths learners: remember the Scottish Government (2019) design principles are for children to have depth, relevance and choice in their learning. By holding a pizza party, a child can be introduced to the fairness of fractions as well as budgeting for their needs. Let me explain (whilst I distract my mind from its desire to seek out the nearest pizza shop). Perhaps I’ll use the technique that we were taught through a sweet story shared in our latest tutorial: count the objects around you! Whilst typing this up, you find myself counting all the flowers in my window view! Sadly, I’ve only found one. Dundee’s library has more leaves (better for symmetry, though!)

Fractions Call for Fairness

Our lecturer was talking about making everything exciting for young children: the same goes for teenagers – which I realised when taking a group of 16-18-year-old Italian teenagers around a fairy trail this Summer! They. Loved it – seriously. Walking can be fun (if you don’t particularly enjoy it) when you have small goals to work towards: so, fractions can work the same way too, I presume! Here it goes (my plan for a pizza party at the end!).
Pre-lesson Activity (Starter):
We introduce what fairness is about – f for fractions and f for fairness. In the end, fair and equal are almost synonyms (kind of!). This could be done using a cake. Oops, food again! Give one student a bigger piece than the other… and watch the chaos unfold or cake end up in your face! Not really.
Lesson One:
The students could then explore the meaning of a whole (perhaps look at percentages.) What exactly is the meaning of the denominator compared to the numerator. How do you show how much of something you have using fractions? Visual resources are more important than worksheet questions for a start. Why? Penner-Wilger et al. (2007) found that finger representation whilst counting will increase a child’s ability to estimate well and understand the number-system. The same idea could be applied to fractions: mental representation goes along way. Maybe that’s why it’s the old-story that a face-to-face conversation is better than an email?! Anyway…
Lesson Two:
Recap fractions and introduce the idea of budgeting: that’s the next stage. Depending on the level of knowledge and depth of understanding of the students, the pupils could explore the cost of pizzas using greater than/less than signs or even percentages! It would be important to also look at some questions on fractions at the end as a formative assessment – okay, well just so you know what to plan next to teach them.
Sequential Lessons and the End Goal
Obviously, you cannot plan weeks and weeks in advance as you are unable to predict the students’ learning progress. UNLESS: a crystal ball has been inserted into your brain. I remember, when on my first-year placement, the teacher saying that they never plan longer than a few weeks because: you must change your plans; AND therefore, your time is most likely into space (and you use up fuel!) However, the end goal should try and remain the same or be appropriately adapted to the pupils. Hello, differentiation!
For instance, some complete the budgeting and other students just focus on fractions. Still, by the end, the class should come together to hold a pizza celebration. That would show that (I think) that we use maths all the time – and that you can have a choice in your learning. And, could a community of enquiry be created all whilst doing this? Maybe!

Now, back to circle time. Well, reflection! Our recent maths lectures have really cemented the idea of discarding textbooks more and more – and trying to make as many things applicable to real-life where necessary. Obviously not everything can be done in that way… but a lot could. Juliet Robertson (2011) discussed having an outdoors master-chef cooking lesson, involving mud and measurement! That does sound like a rocket-fuelled idea – that even astronauts might want to explore. I’m getting excited for teaching maths. Even more exited than I was before. If we make the students realise the everyday value of actually knowing why we carry out a mathematical operation, then our we are on the right route. Ollerton (2004, p.81) does a rather good summary for me: “The ultimate prize is for students to recognise they are learning for their own benefit.” That’s a tad better than pizza, I’d say.

References for this Post:

Molina, C. (2014) ‘Teaching Mathematics Conceptually ‘, SEDL Insights, 1, pp. 1-8. Available at: http://www.sedl.org/insights/1-4/teaching_mathematics_conceptually.pdf (Accessed: 26 September 2019)

Ollerton, M. (2004) Creating Positive Classrooms London: Continuum.

Penner-Wilger, M. et al. (2007). “The foundations of numeracy: subitizing, finger gnosia, and finemotor ability,” in Proceedings of the 29th Annual Cognitive Science Society, pp. 520-525. Available at: https://pdfs.semanticscholar.org/cc16/9648993eccb49088b511d37590f93e3fc1a2.pdf (Accessed: 27 September 2019)

Robertson, J. (2011) Messy Outdoor Maths. 31 March. Available at: https://www.youtube.com/watch?v=Nh_4SEUpSrA (Accessed: 27 September 2019).

Scottish Government (2019) Scotland’s Curriculum: How We Do It. Available at: https://scotlandscurriculum.scot/5/ (Accessed: 27 September 2019).

## Let’s think about (ma)thinking, okay?

Mathematics? Oh so fearful of it…are you? Or… look… there’s a sum! And… the answer? Let me spend time sourcing it. Right now. Some people leave education with this desire to crawl under the table when 5,6,7 and all the other numbers approach them. Others, well others, carry maths in their head – and would prefer to speak in digits if possible. The myth (according to the results of a 2013 research study) of ‘I’m right brained so call me a problem solver’ vs. ‘I’m the more rational leftie’ means that children should no longer be brought up to consider themselves as naturally mathematical or a born linguistic. Indeed: we have talents, we have strengths, we have preferences. But – of course, there’s a ‘but’ – mathinking is vital if we are to efficiently carry out everyday tasks. From recipe reading to saving our red squirrels ‘in the animal superhero cape,’ the digits that surround us must be of comfort. And, most certainly not, be treated as those spheres in bubble wrap. Our toddlers, our children, our adolescents ought to grow up ready to fail. Then succeed. Then… realise… that mathinking is about the effort and determination and less about the green-tick solution. That’s confidence-fuelled mathinking… in my head.

Recently (in fact only a matter of less than 72 hours ago) MA2 – that’s moi included – received their final STEM input. Nope… not how to arrange flowers for Valentine’s Day  or for the Chelsea Flower Show. However, we did receive a Happy Valentine’s on the top of our sign-in sheet from a lecturer with the last name of Valentine! And, that was topped off by creating our own game involving mathematical concepts. (Smiley face!) Lots of discussions were held on how to take maths out of the textbook and away from standardised assessments. A connectionist-belief approach involving open conversations was found to be the most effective way: and well, a recent starter-activity on consecutive numbers confirmed that. Sometimes, our inputs involve being the children – or attempting what the primary students have to do but with a university student’s mind on. Once (a very long time ago… not really, but I like to pretend I’m a storyteller sometimes) we had to figure out a word-problem involving consecutive numbers. At first, we all panicked. By the end, however, it was ‘Happy Ever After’ and everyone was wearing fairy-tale gowns. How lovely. Let me tell the story without imaging that a bunch of five-year olds were sitting beside my seat – here it goes:

Many of us stared. We realised there was no prince-charming. We put our hand up.#

The teacher (very friendly) told us to keep persevering through the rubber hail and rain.

We succeeded in writing down an answer.

And that’s the ending for you.

The latter may be more like reality, if we are setting the picture straight and not at an 89 degrees angle! At first, many of us in the lecture room were not at all sure about how to attempt the problem-solving question. Deliberately, purposefully and cleverly was the question slide jammed packed with information – and the only picture remained to be a table of consecutive numbers in blue and white. For me, lots of writing was the ‘tying the shoes tight enough’ hurdle. But, for others… the fear of making a mistake prevented them from taking the leap of faith. What reminisced with me was the impact sides, pictures, diagrams (and all the visual jazz) can have on a student’s learning. During my first year placement, my teacher advised me to keep it simple! Yes.. nothing like the Mona Lisa or a Magic Eye picture… from Miss Smith. And almost a year later, the same point was made again. Kids prefer simplicity when learning (although I personally believe that the occasional rainbow cannot go a miss). That pot of gold still does exist… Somewhere Over the Rainbow! Jokes aside, it’s easy enough to alter the layout of questions for our students however… calming a raging maths anxiety monster in them may take more than a few kind words (or a ROY G. BIV smiley – acronym lovers… you’ll know what I mean!)

The past few lessons, lecturers (or whatever name they should be termed) have taught me something more than ‘cut away’ all that text. As a future educator, I really do hope that my pupils will be like mathematical bees. Buzz, buzz, buzz… isn’t that algorithm so full of pollen? Understandably, some children will have a dislike for certain areas of maths (like me and symmetry and learning R and L) but overall, the wonder of maths must be ignited in them. A fire (with infinite logs) for problem solving does not always come from the pupil. Nope. Instead, as Ofsted writes, it is about the maths spirit adopted in the classroom:

“[Teachers] made conscious efforts to FOSTER A SPIRIT OF ENQUIRY [deliberate capitalization from your blog-post author], developing pupils’ reasoning skills through approaches that saw problem-solving and investigation as integral to learning mathematics. They checked that everyone was challenged to think hard and they adapted how they were teaching to achieve this. As a result, their classrooms were vibrant places of learning.” (Ofsted, 2008, p.12).

Spirit of enquiry. That is it. How do we develop that? Well, there would be many theories including Askew et al 1997’s research which deems that teachers tend to lean towards a particular set of believes (either connectionist, discovery or transmission). Most effective practice flourishes from the first mentioned belief-set, in which the most effective teaching happens when students can appreciate that all areas in maths are, in some way, related to each other. For instance: fraction word-problems are sometimes better solved by involving decimal conversion. It’s that simple thing like… why buy fresh bread when you already have some in the freezer? Enabling students to use the skills they already have is also of priority in the teaching of maths. Connectionist-orientated teachers set aside time (maybe in a circular fashion if Early Years students melt their hearts) to talk. Just openly talk.

#Why do we solve it this way? Is there another mathematical route that we can take to reach our destination? What about a storyboard?

That leads me on another path to bringing up storyboards but before… I’ll let you know that I want to be a connectionist teacher. That’s a seed already sown – thanks Dundee University. (That phrase was most certainly not sarcastic bee the way).

Storyboards (in the past – okay!) reminded me of ‘how to stretch’ and satisfy the students whose brains make them wish to reach for the nearest paint pot and brush. They were a time-waster. Just show some grit and keep moving forward with the sum. Write the working out, Claire. (That what previously motivated me in maths). But, well, you live’n’learn and realise that 1 + 1 doesn’t always make a two. Sometimes, a window is the result!!! After hearing about the various strategies and understanding the purpose of drawing out the number sentences, storyboards are on my Pinterest. (Sorry, I’m one of THOSE teachers.) Uh-huh, daydreaming about my classroom (maths) displays can occasionally happen with moi. Back to the point now, it is important not to overlook resources that you didn’t enjoy as a child yourself. There’s no excuse for not using something in your classroom because it was futile to you as a learner. Open-mind please. Keep considering all the options, Miss Smith.

Maths, I’m admitting, has a soft-spot in my heart. Whenever I was stressed out about an essay, me would run to do maths. Yes, on occasions, scuttering down the stairs to the kitchen for my textbook… but well I loved when you just got the answer after trying hard. There was a tick or cross – and you knew the result. Maybe it was a sense of control in the subject? After all, sciences were my thing until long reports became involved. I have always been torn between the arts and maths, yet teachers need to be enthusiastic for them all. There is something good, something interesting, something positive in every subject: my own maths students who enter with a dislike for maths need to feel that way. Mathinking… thinking abstractly… can be done by everyone – not just males too. Hemree (1990) found that maths anxiety stems from previous failure in maths examinations and is more common among female students than male students. The gender gap in STEM subject is already evident and it is common knowledge that our generation is trying to remove the female electric fence that surrounds females in careers such as engineering. A survey (conducted for the United Kingdom WISE Campaign) highlighted that 89% of engineers were male in the workplace – and so carrying out a subtraction leaves us realising that only 11% were female. Only 11. As I see it, that is a gap that an elephant would struggle to sort out. It really is.

To finish off, it is clear that the style of teaching is all-important. As mentioned above, discussions are vital to student’s success in maths. However, we ought to consider our own underlying nervous system when we approach numbers? Do we shake? Our hands: do they turn red when seen by a thermal camera? Are we still (like I was) centred on achieving those ticks? Or… the process… do we strive when something makes us want to scribble or shred the paper into as many miniscule pieces as possible. According to Finlayson (2014) our own experiences can make us teach with a straight ‘yes’ or ‘no’ style or be flexible. Bee like a ballerina who is trying to bee a flower. Don’t get me wrong… I’ve failed many times in maths (and still use the trick to remember my left and rights)… but well it’s amusing to buzz around afterwards. That’s what I want my students to do so I ought to love a struggle myself. That’s what teaching is really: setting a true example. But, someone please, is there a faster way to stop mixing up left and rights? Maybe my students could teach me that. I’m up for a role reversal every so often!

References:

Finlayson, M. (2014) ‘Addressing math anxiety in the classroom’, Improving Schools, 17(1), pp. 99-115. Available at: https://journals.sagepub.com/doi/abs/10.1177/1365480214521457(Accessed: 18 February 2019).

Hemree, R. (1990) ‘The Nature, Effects, and Relief of Mathematics Anxiety’, Journal for Research in Mathematics Education, 22(1), pp.33-46.

Nielson, J. (2013) ‘An Evaluation of the Left-Brain vs. Right-Brain Hypothesis with Resting State Functional Connectivity Magnetic Resonance Imaging’, PLoS ONE, 8(8). Available at: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0071275

## Evolutionary Experiences

My learning experience (so far) has been examined under a microscope, with the results below. You’ll also learn a little on behaviour (but only if you are eager to)!

When a word in a dictionary ‘spots’ a new reader: Hello, my name is Floccinaucinihilipilification.Nice to meet you – what’s yours?

The reaction when the reader struggles to comprehend the word: Bonjour, je m’appelle Claire. Encantada de conecerte.

That feeling. Complete bewilderment. The first time I encountered the above 29-letter word, that is indeed English, my neurons fooled me by befriending my breakfast cereal. Snap. Crackle. Pop. The native fuse had blown within grapheme number two: mother tongue was forced to re-circuit to the foreign languages, before realising that wire was also home to many flaws. No conduction shut access to the main street, so information alleyways were ‘helpfully’ blocked off too! The logical words inside me were spat out as jargon. To (temporarily) swap my brain matter- like we swiftly switch between our WIFI and 3G – would have removed the obstacles; evolution is a placid tortoise, however. I suppose that’s better than this biological process being like an over-zealous, super jealous (MacDonald loving) ‘rabbit.’ Slow and steady wins the…
…but, the race was cancelled. For stormy weather, that is. The next sun ray never beamed down until Semester Two. Poor tortoise and her many miles left to doddle. Left foot, right foot, left foot – and so it went on. Just if trudging one foot in front of the other was so weightless! Looking on the bright side, the weather-proof shell provided more than sufficient shelter whilst my brain muscles went through the trek of adapting to my first term at university.

As we all do, you are now most likely donning your black Sherlock Homes’ coat with ‘evidence’ of this slog as merely the typical freshers’ homesickness. Stop now. Claire’s brain has a mobile home, the shell – remember? Leaving family never threw me off kilt, yet surprisingly penning my first academic essay did. Informative essays – the dry mixture – were never flavour of the month for me but throughout school, I had learnt to cope with them. English teachers only insisted on one being written every academic year (bearable) but I was soon to realise that university has its own agenda. Higher educational establishments, in general, treat these fact-driven essays like classroom Starters of the Day. Draft one for this project; scribble another for that. I knew brushing these aside would only surmount to another pile of problems, especially since they constitute as our summative assessments. A failure to submit sets off vexatious alarms: no-one craves a crab-pinching headache or the prospect of a degree bursting into snake-tongue flames.

The robotic, methodological approach to academic writing boxes up any expressionist. Jack (my brain’s creative animal) is not easily dispelled, however – oh yes, his nostrils catch those oxygen bubbles every time. Air forcefully weaves through the mouse-nibbled holes in the wafer-like layers of carboard for ventilation. His spring’s metal remains sturdy and shining, but four years of these conditions could be idealistic for rot and rust. Dead. Jack would be… Isn’t it a (table)spoon full of sugar that my degree programme has, in a way, ‘adopted’ him? In clearer (and other) words, personal reflection has become embedded into my coursework through GLOW Blogs. The online space starts out as bare ‘walls,’ but slowly and steadily we can hang up ‘pictures’ to create a gallery of our progress as teachers-in-training. Seeing others’ exhibited work twists any frown around as honest answers are given to hot-topic issues. This platform puts Brookfield’s Lenses into this cheetah-paced, techno-centred century; the truth magnified in everyone’s discourse considerably helps to settle any teaching niggles. Pinning up my first post… with the hammer of a mouse… made me realise that points can be argued in other ways than emotionless (but logical) essays. Jack hardly needed any ice to recover from this mental ordeal – literally, his rest and recovery constituted army-style star jump drills. Up and out, simply stretch about. Was he pretending to morph into a starfish in my head? At least he can gain credit for knowing seventy-percent of our brains are water-tanks. It’s only sad he loses my brownie point for idolising a brainless species.

The fear of harnessing in my creativity eased off by the end of my first term at university. Tortoise (or to the biologists, evolution) had gained courage – and for sure, some strong ‘biceps’. Today, tackling academic essays isn’t an arduous adventure into the unknown because expanding my blog and writing skills is more of a hobby. Assuredly people will judge my opinions, my style, my whole empire: irrelevant. As much as feedback is any author’s energy drink, it is the mental stimulation, clarification and justification that continually sharpen our pencils. Recently, three learning theories – behaviourism, constructivism and social constructivism – peaked my interest. (Fun fact: the suffix -ism is also the noun for a distinctive theory, doctrine or practice.) These theories must be underlined more often; our preferred learning styles as teachers affect our success in classroom management ( Wray, 2010). No identification as to how I best assimilate knowledge could quickly escalate into a convergent earthquake: the entrance of placement would powerfully rise, and time could do little but subduct. Since the earthquake’s focus would be myself, my students would dreadfully be at the epicentre of this disaster. What a magnitude of a problem. Aren’t we all just glad it wasn’t under the watchful eye of nature? Preventative research and reflection: taken.

As by literacy’s (more than ten) commandments, the next paragraph would succinctly follow on with a written debate as to which teacher-ism approach I will adopt on placement. However, the floccinaucinihilipilification of words sitting row upon row is evident when I then admit that my learning style weighs up to be that of a social constructivist. People who are like-minded hold this worldview because we are satisfied by actively seeking out information collaboratively; transmission of knowledge constructs didactic robots. A chance to extend beyond the margins of the paper is when our brains’ glue guns heat up. So, for that reason, this blog post will have a line drawn under it soon. Fret not, lovely readers: my Sway presentation is the firefighter ready to rescue those confused and curious neurons from sparking to extreme explosions. Cliff-hangers are everyone’s bug-bearers, so respectfully sharing my reflections is simply of common courtesy. Don’t let it slip your mind to hold down that ‘off’ button on your mobile phone (copyright rules do apply!) and enjoy the silent ‘movie.’ It’s never too late to dash for that bag of popcorn – or bowl of Rice Krispies!

Dry: This word is notoriously synonymous with derogatory terms – boring, uninspiring, fruitless – however my usage does not aim to convey that academic writing is tedious. In fact, factual essays are the golden sponge in a Victoria sandwich. Regarding other literature styles, personal compositions fill us up like the oozing jam and cream whereas creative pieces dust the icing sugar on top (with a pick of strawberries if we’re lucky.) As a constructivist, my preference lies in creating subjective-based work that is less associated with a specific end-goal. Nonetheless, there are still hundreds and thousands of sprinkles in the reading of informative work by those who kindly lead knowledge discovery: my mind’s schema is like Rainbow land. Point is: saying you prefer blog writing is not scientific proof for your peers’ believing you loathe studying the ‘meaty’ works, the protein.

Due acknowledgements for this blog post:

Arthur and Cremin’s book (2nd edition)-  Learning to Teach in the Primary School

Wray, D. (2010) ‘Looking at Learning’ in Arthur, J. and Cremin, T. (eds.) Learning to Teach in the Primary School. 2nd edn. Oxon: Routledge, 2010, pp.129-145.