What does knowledge look like in Maths

Written by Ben Rapley Monday, 24 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education can be found on Twitter @LaSalleEd @LaSalleEd #MathsConf20 on Saturday 21st September 2019 Conference with both MA and ATM, tickets still available.


The next La Salle Ed Maths Conference is #MathsConf20 in Edinburgh

'What does 'knowledge' look like in Maths?’ ' is a blog about Ben Rapley's #MathsConf19 session/workshop at #MathsConf19.


The term 'knowledge' has become a bit of a buzz word in schools. Knowledge first teaching is championed and knowledge organisers are everywhere. Ben Rapley looks at what does 'knowledge' mean in mathematics and how we might use it in our classroom.

What Does ‘Knowledge’ Look Like In Mathematics?

“It is impossible to think well about a topic in the absence of factual knowledge about the topic.” (1) Daniel T. Willingham


Knowledge is important, but what does that ‘knowledge’ look like? Is it knowing all the facts in mathematics? The angle rules, some formula and a vocabulary glossary and you are all set? Probably not, however as soon as you start trying to think of all the things you want your students to know the list can suddenly spiral out of control.

I want to look at what is ‘knowledge’ in a subject that is very practical, knowing all of the facts around expanding double brackets does not necessarily mean you can do it. Martin Robinson, in his researching of creating the perfect curriculum talks about three forgotten classical arts; Grammar, Dialectic and Rhetoric. Grammar is described as “the art of interpreting the world through foundational knowledge” (2). This can also be described as the subject disciplines, the fundamental principles of a subject. If the subject disciplines are what we call knowledge that makes a lot more sense.

Daniel T. Willingham states “Factual knowledge precedes skill” , I think skill here is defined as other subjects use it; analyse, critique, explain. In mathematical terms we might call it problem solving. Students need to be able to complete a process before they can problem solve with it.

In my previous and current schools we have had on a focus of the Core Skills, the parts of mathematics that students need to be able to access higher order mathematical thinking and problem solving. I would argue that they form a large basis of the knowledge of our subject. We practice these skills every lesson at the start (don’t tell Mark McCourt). We then have regular low stakes quizzes that help us diagnose areas we need to improve on as a class and they will become the focus of our starters. You can read more about Core Skills here. The aim is that by transferring these processes into our long-term memory then that frees up space for students to be able use their working memory to deal with interpreting a question and solving the problem.

These processes are important as they form the chunks of information that will be stored in the long-term memory; however, these will not be transferred to schema unless connections between ideas have been made and understood. Schema can then be transferred to the long-term memory through consolidation and practice (4). Creating a knowledge rich environment can balance process and understanding, making it clear what we need student to know and practicing it at the same time as providing the information to enable students to create the connections of mathematical concepts in their head.  

Here are some of the elements we are going to be using in our curriculum to try to create a knowledge rich environment:

• Core Skills
• Clear outcomes for each topic carefully sequenced
• Clear about examples we want students to see
• Multiple representations
• Vocabulary

Clear outcomes for each topic

We are separating our schemes of learning into access, core and depth. The core outcomes are areas that I expect every student to have covered and will form the bulk of our assessments. We have spent time sequencing each of these carefully, staff are then able to divide their lesson time as needed.

Clear about examples we want students to see

There can be a lot variety between question styles and they can be quite intricate and easily missed. We aim to show students all of those different styles of questions. Regular use of worked example pairs and showing boundary examples as well as non-examples. Rather than having knowledge organisers we are moving towards creating example collections.

Multiple representations

Dual coding is highlighted as one of the six strategies for effective learning by The Learning Scientists (5), one of the ways we can do this is to use manipulatives. It is clear that manipulatives can be a very powerful tool in supporting students understanding. We are getting better at using them as a department but it is definitely where we are developing. The EEF published a guide to Improving Mathematics in KS2&3 with a section dedicated to manipulatives. It says that having a strategy to remove manipulatives is just as important as the strategy to introduce them (6). We are dedicating time within department meetings to better our understanding of how we can use them and support the transition from concrete to pictorial to abstract.

Vocabulary

It is important to equip our students with the ability to access and understand questions, one of the ways we can do this is by explicitly teaching the vocabulary we can use. Alex Quigley says “We should avoid the assumption that the language of mathematics is simply absorbed ‘naturally’ over time by children.” (7). We can do this through exposition of a word’s etymology or morphology, or using Frayer models, amongst other methods. They can be powerful tools to help students make connections with words. We are highlighting words in our schemes where teaching the vocabulary may elevate a student’s understanding, being careful to not create cognitive overload by adding too much complexity.

Our curriculum isn’t complete and will probably continue to be an ever evolving thing, however I feel confident that we are working towards creating a knowledge rich environment that will help students thrive and understand mathematics in a way that they never thought they could.

1) Why Don’t Student’s Like School, DT Willingham, 2009, p210
2) Trivium 21 C, Robinson M, 2013, p20-25
3) Why Don’t Student’s Like School, DT Willingham, 2009, p210
4) Understanding how we learn – a visual guide, Weinstein Y, Sumeracki M & Caviglioni O, 2019, p75-76
5) http://www.learningscientists.org/blog/2016/10/6-1?rq=six%20effective
6) Improving Mathematics in Key Stages Two and Three, EEF, 2017, P10
7) Closing The Vocabulary Gap, Quigley A, 2018, 104-105


Ben Rapley spoke about "What does 'knowledge' look like in Maths?" at #MathsConf19 at the Penistone Grammar School held on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

New Functional Skills qualifications – positive changes for maths

Monday, 17 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'New Functional Skills qualifications – positive changes for maths’ ' is a blog from the exhibitor OCR (by Neil Ogden) at #MathsConf19.


New Functional Skills qualifications – positive changes for maths

In this blog on the Functional Skills reforms (following on from our blogs covering an introduction to the forthcoming changes and information about why is the choice of context so important) we’ll take a look at the reformed Functional Skills maths qualifications, due to be taught from September 2019.

The reformed assessment structure

A fairly major change in the reformed Functional Skills maths is the requirement for each level to have a section of non-calculator assessment. Ofqual requires each level to be 75% calculator and 25% non-calculator.

Another Ofqual requirement is that the overall assessment length should be between 75min – 105min at Entry Level and 105min – 150min at Level 1/Level 2.

Also informing our structure was you! Teachers were telling us they’d like a simple structure, with similar timings from level to level. Bringing all of this together, the structure we came up with can be seen below*.

*Please note OCR Functional Skills Entry Level is still going through Ofqual’s Technical Evaluation at present (May 2019).

The structure is simple, with the same timings for each Entry Level and the same at Level 1/Level 2. Overall timings are near the minimum allowed so students won’t have overly long assessment, but we have allowed a bit more time at each level to allow for students to check their work.

We know students often struggle without a calculator and so to help reduce student stress here, we have allowed a bit more time in the non-calculator papers.

The numbers of marks for calculator assessment and non-calculator assessment are in line with the required 75:25 split, however the timings have been adjusted slightly to give students slightly more time per mark when they’re without a calculator.

User friendly qualification content

Content for the reformed qualifications was published by the Department for Education (DfE) in early 2018. One option for presenting this content to teachers would be to have copied and pasted the DfE statements straight into each Specification.

We felt that the content could be presented in a more user-friendly way however, so we reworked the content into a grid structure, introducing new section headings and arranging content in rows, so that the progression through the levels and content differences can easily be seen.

Also included in each Specification is a guide to the Command Words (e.g. ‘Complete…’ or ‘Write down…’) we’ll be using in our questions too, so you can familiarise your students with them in advance.

Download the Specifications to see the structure, the content grid and definitions of our command words.


You can speak to Neil Ogden and OCR as one of the exhibitors during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Building Confidence in Lower Ability Learners

Written by Adam Mercer Monday, 17 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Building Confidence in Lower Ability Learners’ ' is a blog preview of Adam Mercer's #MathsConf19 session/workshop being run at #MathsConf19.


Do you have a bottom set that just struggle to get started? Are there pupils sat around wanting to avoid work at all costs? Do you have pupils that give-up before they even get going? This session aims to provide with ideas and the tools to help you motivate and develop those reluctant learners into more willing and able mathematicians. The session is aimed mainly for teachers with lower sets/foundation GCSE students but is suitable for all teachers of KS3 and KS4.


You can see Adam Mercer speak about "Building Confidence in Lower Ability Learners" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Ratio and Proportion

Written by David McEwan Sunday, 16 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Ratio and Proportion’ ' is a blog preview of David McEwan's #MathsConf19 session/workshop being run at #MathsConf19.


We’ll look at Ratio and Proportion, from first principles through to advanced topics in GCSE. I’ll highlight the principles which link the teaching and the assessment of Ratio and Proportion in a mixture of theory, visual representations and application to exam questions. I’ll also cover how the subject develops from Foundation to Higher and how Assessment Objectives influence the questions we write. So, if you want to know why equality of ratios is more than just a definition of proportion, how to find the multiplier for an inverse percentage from a bar model, or why ratios keep appearing as fractions in exam questions, this is the session for you.


You can see David McEwan speak about "Ratio and Proportion" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Plan learning not lessons

Written by Gary Lamb Saturday, 15 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Plan learning, not lessons!’ is a blog preview of Gary Lamb's #MathsConf19 session/workshop being run at #MathsConf19.


Pace, rigour and challenge have been lost amidst the era of curriculum change and in this workshop we look into how maximum 'up-time' can be achieved by phasing learning using an 80 : 20 principle for planning. At St Andrew's we strive for mastery with every pupil and through professional reading and CPD, we delve into effective explicit instruction, how to foster mathematical thinking, how to use visual modelling for understanding, and how to do all this in a time efficient way.


You can see Gary Lamb speak about "Plan learning, not lessons!" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Variation theory - more than minimally different questions

Written by Chris McGrane Friday, 14 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Variation theory - more than minimally different questions’ is a blog preview of Chris McGranes's #MathsConf19 session/workshop being run at #MathsConf19.


Variation Theory is a an aspect of mathematics pedagogy which has had increased focus over the past couple of years. This presentation will, through examples of tasks, questions and prompts explore this topic more fully. There is much more to variation theory, beyond lists of minimally different questions - come along to find out what!


You can see Chris McGrane speak about "Variation theory - more than minimally different questions" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Helping maths to get back to the future Texthelp

Friday, 14 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Texthelp - Helping maths to get back to the future’ is a blog post by Alan Sharpe, Education Sales Executive and exhibitor Texthelp who will be at #MathsConf19.


The maths department has often been overlooked by edtech developers, but change is coming...

Today’s ‘always-on’ digital culture means students are rarely found without a smartphone in hand, and the virtual world at their fingertips is just as important as the physical. Technology in the learning environment is important for creating experiences which engage today’s modern learners, and that’s why educators are keen to harness the value of technology in the classroom. In fact, UK schools spend more than £900m a year on education technology - in some cases, iPads are replacing notebooks, mobile apps are being used for collaborative group work, and teachers are providing valuable 1:1 instruction through technology. While education technology is rapidly transforming teaching and learning, it sometimes seems that maths has been left behind in the rush.

Getting the maths class back to the digital forefront

Despite maths being the first subject to incorporate tech with the pocket calculator in the 80’s, it is often left behind in today’s tech-inspired classrooms. Complex coding language makes it challenging for both teachers and students to create digital maths, and therefore, the valuable supports that EdTech tools provide across the rest of the curriculum cannot be easily integrated in the same way. Supports that help students to access and process content, such as text to speech; prediction, inputs in multiple formats; and, the flexibility to work collaboratively and instantaneously in shared workspaces. At Texthelp, we wanted to make sure that access to EdTech is available across the entire learning journey. That’s where EquatIO fits in.

With benefits for both students and teachers, EquatIO digitises maths. Teachers can quickly and easily create digital maths content that is engaging and accessible for all students, while students can explore maths concepts in a way that best suits their unique needs and learning preferences.

It also offers a huge leap forward for accessibility and inclusion, allowing learners with diverse needs to engage and interact with content in ways that are easier for them.

EquatIO helps all students express themselves

When it comes to expressing their knowledge with EquatIO, students can choose to handwrite, dictate or type their solutions.

As student’s type, EquatIO predicts the expression the student is typing, so they don’t need to memorise every tricky formula in the maths language. As students speak, EquatIO ignores the ‘umms’ and ‘errs’ so only the maths appears on the screen, just like magic. And for those students who prefer to scribble down their thoughts, the handwriting feature transforms student writing into digital (legible), accessible text.

Not only that, but students can use EquatIO mathspace digitises maths. Teachers can quickly and easily create digital maths content that is engaging and accessible for all ​EquatIO mathspace,​ a web-based application similar to a digital whiteboard, to add annotations and manipulatives to their work. This allows them to engage with maths concepts in a variety of ways, encouraging them to demonstrate their thinking on a deeper level. In turn, the teacher can provide feedback directly into the mathspace, creating an avenue for meaningful one-on-one insight, without using up valuable class time.

Integration that’s not only simple, but engaging

Whether you are a Google, Windows or Apple school, EquatIO can be integrated simply and easily. It can be used within the G Suite for Education environment and in Microsoft Word on Windows and Mac devices. Plus, with EquatIO mathspace, we also offer a smart workspace where maths sheets can be stored and shared between teachers and students time and time again. Through this robust integration, students can now use the same devices and digital applications they’ve become familiar with in their other classes across the entire curriculum. Rather than resorting back to pen and paper when they enter the maths classroom, students can use EquatIO in all its forms to engage with maths digitally and benefit from all that EdTech has to offer.

If you would like to know more about using digital technology in the maths realm, stop by stand G3 at MathsConf19 on 22nd June for a chat with the Texthelpers. In the meantime, visit https://text.help/BiQftv for more information about EquatIO.


You can see Texthelp as an exhibitor at #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Yes But Constructions

Written by Ed Southall Thursday, 13 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Yes But Constructions...’ is a blog preview of Ed Southall's #MathsConf19 session/workshop being run at #MathsConf19.


An interactive workshop looking at the topic of constructions - tips on how to teach it more effectively, ideas for interesting and purposeful task design, alternative approaches to methods and a bit of "yes but why" does this work? If you have a preferred pair of compasses, bring them!


You can see Ed Southall speak about "Yes But Constructions..." during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

How to Change the Subject of a Formula

Wednesday, 12 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'How to Change the Subject of a Formula, 100% guaranteed’ ' is a blog preview of Joe Berwick's #MathsConf19 session/workshop being run at #MathsConf19.


Changing the subject of a formula is a key part of secondary school mathematics, and yet it’s a topic so many students struggle to master. Some techniques like ‘flipping’ can cause misconceptions. Function machines break down when the rearrangement has to deal with the subject on both sides. There must surely be a way that is consistent and works every time? This technique builds on Kris Boulton’s techniques of Direct Instruction from the conference in Birmingham to create a faultless approach to changing the subject of a formula. Students can struggle with changing the subject of a formula because they get confused with the method that has to be applied. This ‘new’ technique aims to eradicate that struggle. More research will have been done compared to the workshop given in March and more situations will be added.


You can see Joe Berwick speak about "How to Change the Subject of a Formula, 100% guaranteed" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

3P Learning MathsConf19

Written by 3P Learning Wednesday, 12 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'3P Learning - Mathletics is an online maths resource’ ' is a blog post by exhibitor 3P Learning who will be at #MathsConf19.


Mathletics is an award-winning, online maths resource used by thousands of schools, specifically designed for personalised teaching and learning. It is proven to significantly increase levels of student engagement, confidence and motivation and to improve attainment and progress in maths from KS1 to KS4. Mathletics is aligned to the White Rose Maths scheme of learning with dedicated courses for Years 1-6.

Students have access to a vast range of curriculum-aligned learning resources; activities, eBooks and videos, the ‘Multiverse’ times tables challenge and new Multiplication Tables Tests to support the Multiplication Tables Check. With excellent visibility of progress and results, Mathletics provides students with all the tools they need to become successful learners.

For teachers, Mathletics offers great flexibility and time saving features such as Dynamic Results provide an immediate snapshot of student results. Automated reports, customisation capabilities and the ‘Assign’ functionality allows teachers to schedule and plan ahead. Teachers can assign work for a single student, a group or a whole class based on results, enabling effective data driven teaching and learning.

Mathletics has also been shown to effectively support a mastery approach, help foster a growth mindset towards maths and build parental engagement. For more information, and to try Mathletics for free, please visit uk.mathletics.com


You can see 3P Learning - Mathletics and see the online maths resource #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Emile Education MathsConf19

Wednesday, 12 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Emile Education - Free Multiplication Tables Check (MTC) Resource’ ' is a blog post by exhibitor Emile Education who will be at #MathsConf19.


As you may know, the MTC will be an online check in a three week period June each year for pupils in year 4 that checks their knowledge of the 2-12 times tables. Schools can decide when their pupils will sit the check within this 3 week period. The school can also decide on what device type – tablet or computer (there’s no paper option). The MTC will be different for each and every child as the test features a set of 25 generated times tables questions.

Brilliantly, a company has developed a resource which mimics the tests and so allows pupils to practice and teachers to track progress – MTC with Emile – which is free for all schools.

Pupils have 6 seconds to read the question, understand it, and enter a response. They will then receive a 3 second pause before the next question is shown.

So with 9 seconds for each question and 25 questions, the test will take 225 seconds or 3 minutes 45 seconds. (A bit of maths for me to do there.)

Questions will be made up from the 2-12 multiplication tables. The table shown here gives the maximum and minimum number of questions that will be presented to the pupils.

Please remember that these checks will only become compulsory from June 2020. So that means your current year 3 students will be expected to undertake the check.

The results will not form part of the league tables but Ofsted will have access and they will more than likely come up during an Ofsted inspection in a similar way to phonics checks.


You can see Emile Education - Maths Table Check (MTC) Resource as an exhibitor at #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Making Mathematicians

Written by Kate Milnes Tuesday, 11 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Making Mathematicians’ is a blog preview of Kate Milnes's #MathsConf19 session/workshop being run at #MathsConf19.


Students see mathematics as answering questions, yet the jobs mathematicians do require them to start with a scenario, ask themselves questions and then pursue a line of enquiry. This session will look at how we can build these skills in our students taking into account some of the challenges we face in our classrooms. We will look at ways of introducing elements such as conjecture and mathematical thinking in students within our current structures and schemes of learning.

Some further thoughts from Kate Milnes

Over the past two years we have been working hard to improve the culture and outcomes of a Maths Department in the centre of Blackburn. The difficulties students present again and again are with problem solving questions and when faced with exams they struggle to identify what knowledge to draw on or skills to use. I sat in the Complete Maths ‘Mastery’ CPD day in March and Mark McCourt helped me think about why they struggle so much with these things.

Our students see maths as answering questions. When they sit in an exam and don’t immediately see a starting point, or a path through the question they are considering, they feel disheartened, frustrated and think they ‘can’t do maths’. Mark’s statement that Mathematicians are presented with a scenario, ask themselves questions and pursue a line of enquiry made me realise that this approach is where our students fall down. This problem not only affects our students, it affects students up and down the country.

There is a plethora of excellent Maths CPD available nowadays and this session brings together the ideas I have seen and used that help students to think like mathematicians. I will describe some of the tasks I have seen and tried out that allow students to ask themselves questions, pursue a line of enquiry and generally think like a mathematician. These are ideas that can be built into pre-existing schemes of learning and tried in lessons immediately, they don’t require any extra planning. I will discuss my experience of trying some of the ideas and resources I have come across and where more like this can be found.


You can see Kate Milnes speak about "Making Mathematicians" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

The Evolution of Vocabulary in Maths Education

Written by Joanne Morgan Monday, 10 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'The Evolution of Vocabulary in Maths Education’ is a blog preview of Jo Morgan's #MathsConf19 session/workshop being run at #MathsConf19.


Language changes all the time. Words become obsolete, words are superseded, words fall out of fashion. Through my research into maths textbooks from the last 500 years I have discovered all sorts of delightful and surprising vocabulary. Come along to develop your subject knowledge and (hopefully) to be entertained and amused.

Here are some further thoughts from Jo:

Whatever happened to vulgar fractions?
My Longman maths textbook from the 1960s tells me about the ‘Fraction family tree’. Its members are the brothers Vulgar Fraction, Decimal Fraction and Percentage, plus of course the illegitimate son Ratio who we don’t like to talk about.

Vulgar fractions are simply two integers (the numerator and denominator) placed above and below a fraction bar. They are also referred to as ‘common fractions’. I know the terms vulgar fraction and common fraction from my childhood, but I never say them in my own classroom.

A Bostock and Chandler textbook from the 1980s tells us that “Usually we refer to common fractions simply as fractions and to decimal fractions simply as decimals”.

This alludes to the fact that during the latter part of the 20th Century the terms ‘vulgar fraction’ and ‘decimal fraction’ fell out of common usage in schools, leaving us with ‘fraction’ and ‘decimal’. With this change in vocabulary I don’t think we have lost any precision – the terms fraction and decimal are fit for purpose. So, although some lost mathematical vocabulary fills me with sadness, I don’t mourn the loss of the word vulgar.

The term vulgar fraction had been in use for centuries before it died out – it can be found in mathematics texts from all the way back in the 1500s. Here we can see it appear in a snappily titled textbook from 1741:

Textbooks from the 1700s and 1800s not only show that the term vulgar fraction was in common usage, but also provide some interesting definitions and explanations.

For example, in “A Treatise on Arithmetic, theoretical and practical” (Loomis E, 1856), we have this explanation:

It’s the last part that I particularly like: “This mode of expressing decimal fractions is used to avoid the inconvenience of writing the denominators’. I never thought of it like that. Just like the percentage sign is used as a shortcut to avoid writing a denominator of 100, decimal notation is just a shortcut.

All this talk of vulgar fractions made me wonder about the word vulgar. It’s a rather odd word to use in mathematics. A dictionary gives us the following definitions:

1. lacking sophistication or good taste.
2. making explicit and offensive reference to sex or bodily functions; coarse and rude.
3. [DATED] characteristic of or belonging to ordinary people.

Now I’m pretty sure that fractions don’t go round making offensive reference to sex and bodily functions. Certainly not in my classroom. Clearly the third definition is the relevant one here – the idea of being ordinary or common.

It turns out that the word vulgar comes from the Latin word vulgus meaning ‘the common people’, and it was first used in English in the fourteenth century. It then referred to something that was in common or general use or something customary or done as a matter of everyday practice.

The meaning of the word vulgar changed over time. It started as “relating to the ordinary people” but came to mean “commonplace”, and by the seventeenth century it had become “lacking sophistication or good taste”. One of the only places that the word continued to mean ‘ordinary’ was in mathematics. I think it’s a bit sad that a word meaning “of the common people” came to be associated with crude, uncouth and obnoxious. But I’m not sad that I don’t have to say the word ‘vulgar’ to a room full of teenagers anymore! That was asking for trouble.

In other news, did you know that it’s totally uncool to call a fraction improper these days? Apparently, it’s a bit of an insult to fractions greater than one to suggest they’re not proper… Who knew?

Come to my workshop at #mathsconf19 to hear more about vocabulary that’s changed over the years - lost maths vocabulary can be insightful, sometimes amusing, and always fascinating.


You can see Jo Morgan speak about "The Evolution of Vocabulary in Maths Education" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Harnessing the "power" of maths

Monday, 10 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Harnessing the "power" of maths’ ' is a blog from the exhibitor Pearson at #MathsConf19.


Harnessing the "power" of maths

"Maths is not just a subject studied in the classroom. It is relevant and important in our lives beyond the school gates, whether we are looking to understand our economy and natural world, engage with society or manage our personal lives. Maths can also unlock doors by giving us the tools we need to access new learning... or pursue a career that increases our earning power."

This is one of the key themes showcased in our new Power of Maths Report, which details the steps needed to enhance Britain’s approach to maths education, and help inspire confidence and success among children.

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The report draws upon the insights gathered at our inaugural Power of Maths Roundtable, which took place in late 2018. The roundtable saw leading practitioners and academics join school, business and third sector leaders to explore how we can collectively transform perceptions of maths, both inside and outside the classroom.

With the support of Roundtable host and TES editor, Ann Mroz, and speakers including maths teacher and comedian Kyle Evans; leading academic, Dr Natthapoj Vincent Trakulphadetkrai and maths practitioner, Danielle Ashley; delegates discussed everything from mastery, maths anxiety, promoting STEM careers - and lots more.


You can speak to Pearson as one of the exhibitors during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

Teaching and learning with algebra tiles

Written by Bernie Westacott Sunday, 09 June 2019

Edited and compiled by Robert J Smith @RJS2212

La Salle Education will be hosting @LaSalleEd MathsConf19 #MathsConf19 on Saturday 22nd June 2019, tickets still available.


#MathsConf19

'Teaching and learning with algebra tiles: no tricks, no gimmicks, just correct mathematics’ ' is a blog preview of Bernie Westacott's #MathsConf19 session/workshop being run at #MathsConf19.


The correct use of algebra tiles is based on an understanding of the field axioms, particularly the existence of an additive inverse (zero pairs). We will work through the learning trajectories for aspects of algebra from UKS2 to KS4, using algebra tiles in online apps. Please bring your laptop/iPad/tablet so that you can play along – wi-fi access will be available.

Some further thoughts from Bernie:

Teaching and learning with algebra tiles: no tricks, no gimmicks, just correct mathematics

Many of the problems my pupils experienced in algebra were actually due to them not having a sound grasp of dealing with positive and negative integers. Over the 39-year period where I was a full-time Maths teacher, I think I tried every which way to get my pupils to understand how we deal with integers - I was never truly happy with any of these ways.

I then discovered how this was being taught in Singapore when I came across a paper that described how the Singapore Ministry of Education and the National Institute of Education developed the ©Algetools programme for use in secondary schools. I came to realise that this approach was based on the Field Axioms and was for me an accurate representation of those underpinning principles. The transition from integers to algebra was seamless – I wish I had discovered this approach decades before!

After reading further, and spending months getting to understand the app (no manual available), I slightly adapted the introduction to be accessible to our Year 6 pupils. This ran successfully for a few years in our school until the app disappeared from the internet. In the UK, teachers and pupils are using algebra tiles/discs but I think that some of the strengths of the Algetools app have been lost. I also worry that there may be some misunderstandings surrounding the methods explained in Singapore textbooks (or UK textbooks based on these) currently being used in the UK - some of these issues revolve around the use of the ‘-‘ sign. The diagrams in these textbooks are based on the Algetools app (the use of which is a requirement of the Singapore curriculum and its use is stated clearly in the Singapore textbooks), but UK teachers do not have access to this app. We will be using an algebra tiles app, but I will briefly refer to the original Algetools app.

I intend to cover as much ground as I can in the 50-minute slot, hopefully from integers up to some aspects of KS4 algebra. However, as is true for other areas of maths, it is the beginnings that need to be carefully and thoroughly presented if the later elements are to make sense.

The extracts below, taken from a paper I was recently reading, reflect my own experiences and are a good introduction to the challenges of using manipulatives (be they real or virtual) to teach integers and certain aspects of algebra. My experience has been that once the integers are fully understood, then the related aspects of algebra are grasped more quickly and with fewer errors occurring.

Making Sense of Integer Arithmetic: The Effect of Using Virtual Manipulatives on Students’ Representational Fluency

Johnna Bolyard This email address is being protected from spambots. You need JavaScript enabled to view it. Patricia S. Moyer-Packenham This email address is being protected from spambots. You need JavaScript enabled to view it.

Making Sense of Integer Arithmetic

Many mathematics educators argue that students over-learn “take away” as an interpretation of subtraction in whole number arithmetic (Moses, Kamii, Swap, & Howard, 1989)…When students expand into arithmetic situations with negative integers, the complexity increases and “take away” does not adequately model subtraction with positive and negative integers. Thus, a more flexible interpretation of subtraction (i.e., one that includes comparison, difference, and other contexts) that allows for both quantity and direction features of integers to be made explicit is needed (Moses, Kamii, Swap, & Howard, 1989).


You can see Bernie Westacott speak about "Teaching and learning with algebra tiles: no tricks, no gimmicks, just correct mathematics" during #MathsConf19 at the Penistone Grammar School on Saturday 22nd June

Don't forget in July we also have our 'FREE' Maths Teacher Network events in association with Oxford University Press and AQA.

We look forward to seeing you at our next La Salle Education Event if you don't already, follow us on Twitter @LaSalleEd

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