Mathematical Miscellany #15

From Colin Foster on Nrich we have Mathematical Etudes where he discusses lovely rich tasks and tedious exercises!

Colin Foster

Colin Foster – Mathematical Etudes

An important read as we begin another school year I feel, the more our students have to think about a task, the more they will learn. I agree with Colin liking the factors task, I have used this approach successfully in class too. In fact we can extend the task and look at finding numbers with varying numbers of factors.

Factors nzmaths

How many Factors – nzmaths

How Many Factors on nzmaths requires students to  find ways to group numbers, which numbers have only two factors and which have only three factors?

For more from Colin Foster his KS3 Instant Maths Ideas (3 books) are freely available online; these contain a wealth of ideas to try in the classroom. Colin Foster is an Assistant Professor at the Centre for Research in Mathematics Education in the School of Education at the University of Nottingham.

Colin Foster - Instant Maths Ideas


From PhET Simulations one of my favourites is now available in HTML5 collection: Projectile Motion.

Projectiles
PhET Projectile Motion

I have included a page on PhET simulations in the Tools and Calculators series of pages, part of the Lesson Planning collection.


Educas has many very useful secondary and vocational resources for Maths. Note the series of resources for topics new to the GCSE specification.
Educas - Venn Diagrams


Dr Frost slideI have mentioned Dr Jamie Frost and his plans for the new A Level specifications. Dr Frost has been very busy producing a complete set of resources for Pure Mathematics 1, I do like the fact that Dr Frost has included extension questions for each chapter. These resources reference an Edexcel text but of course the content is common to all qualifications so all the explanations and worked examples for A Level. I have included a page for Dr Frost’s resources as part of the A Level series. Note the collection of extension questions mapped to the new specification.

Dr Frost has some brilliant ‘Just for your interest’ posters; see this for example on e. He has put the complete collection of these together. Follow Dr Frost on Twitter.


For more A Level (UK 16+) resources, Danielle Moosajee has a growing collection of A Level resources on her site PixiMaths, Check her KS5 Teaching Resources Index. The resources include very clear PowerPoints, associated questions and student self-assessments.


A consistently popular post on this blog is one on online whiteboards. All resources / links have been checked and I have removed resources which have not been updated for some considerable time as I think this is a cause for concern regarding the longevity of the resource.

doodle.ly

Doodle.ly



With UK examination results recently announced I have put links to the results statistics and grade boundaries on the Results 2017 page. Also included are some interesting links from Schools Week, see A-level results 2017: UK subject tables which compare results from 2014-2017. Also from Schools Week, on GCSEs – GCSE Subject Tables and “The 7 most interesting things we have learned”

An important read I feel on such Statistics from Cambridge Mathematics, Examining our own statistical literacy on results day. which makes such important points about the size of the dataset required for it to be explored in context.


And finally, here’s a great read from Shaun Allison on Getting off to the best start with a new class.


 

 

Lesson Planning

With the new school year fast approaching, I have reorganised some pages and resources in the never ending quest to make things easy to find! This is an ongoing project.

There are several additions to the Lesson Planning Pages.

Lesson Planning PagesA new series on Tools and Calculators includes for example GeoBoard Activities with GeoBoard resources.
Trangles on a 3x3 piboard

There are many excellent PhET simulations from The University of Colorado Boulder. Look at Trig Tour for example. Note that this is HTML5 so available across platforms.
Trig Tour

I do like to use resources in class that my students can use at home, so for example in the calculators and tools collections (see Calculators – Collections) under Equations – Linear and Quadratic, we have from Mathisfun this very clear and easy to use interactive illustrating the solution of linear equations.
balance-equations-mathisfun

You will also see on the same page this calculator from Math Warehouse which shows the solution to any quadratic equation with full working using the formula.

Quadratic Formula

Math Warehouse

The Problems and Activities page has had updates including the addition of Transum, a site I like a great deal as do many students. From Transum, we have the very wel known Maths Starter of the Day. Note that there is a complete index of starters including the topic of the starter. The Transum site is easy to navigate, there is a clear Topic Index for Teachers also, for students a Maths Map with numerous activities to support their learning. The example shown here is one of the Algebra Activities found on this page.

Transum

Transum – Algebra Activities

An essential part of every lesson is of course that we check that students are indeed learning, hence my reminder on Study Strategies on the Lesson Activities page.

retrieval-practice-learning-scientists

The Learning Scientists

Calculators

Calculator Pages

Calculating FinancesTime for a series of new calculator pages to make it easier to find information and resources on calculators. The pages in the series:

Some highlights from these pages…

The collection includes manuals from Casio and Texas and also some very clear guides from Dr Frost.

There are some excellent step by step instruction videos on using the Casio FX991EX calculator.

MEI have many tasks for older students using Casio Graphical Calculators (they also have tasks using GeoGebra and Desmos).

Normal
On the Mathematics for Students pages you can find everything from a game to help students understand order of operations to a Normal Distribution Calculator.

 

Multiple Choice Mathematics

Following an earlier post on multiple choice questions, I thought it would be useful to put a selection together for older students. I have reproduced this below and it has been added to the A Level 16+ series of pages.

From OCR (MEI) their Foundations of Advanced Mathematics level 2 qualification covers arithmetic, algebra, graphs, trigonometry and statistics. Assessment is by a two hour examination which consists of 40 multiple choice questions. As OCR suggest these questions could be used for diagnostic tests.

OCR MEI Foundations of Adv Maths
Choose Past Papers, mark schemes and reports. Papers and mark schemes from June 2007 are freely available (access to the most recent papers is available for schools only via OCR Interchange).

OCR Foundations of Advanced Mathematics

We could of course illustrate the solution well with a little use of technology!
OCR MEI Multiple Choice example

Diagnostic Questions
A source of multiple choice A Level questions, particularly for Pure Mathematics is Diagnostic Questions; see for example this quiz I created on  Logarithms and Exponentials.


DrFrost UKMT Algebra
Dr Frost Maths

The UKMT Maths Challenge questions are excellent for students of all ages. This post includes a selection of links and resources to access the Maths Challenges.


R9452
For superb resources for the Oxford Admissions test multiple choice questions see these Underground Mathematics Review Questions where you will find not only the questions but suggestions and complete solutions.


June 86
A trip down memory lane! (Edexcel’s Emporium has some papers from this era.)
University of London June 1986 Mathematics 1

At the start of my teaching career I really liked the first paper of the Mathematics A level from the University of London School Examinations Board – thirty multiple choice questions to complete in one hour, 15 minutes.

Multiple Choice Example

For questions 1 to 20, candidates had to select one answer from 5 and for questions 21-30 the instructions were as follows.
June 86 Mathematics 1

University of London June 1986 Mathematics 1

The pdf file here has the paper, followed by the exam board answers followed by notes from the 1986 version of me! These days I would illustrate with Desmos and/or WolframAlpha for example as well where appropriate.
Colleen Young answers

Note the comment from Graham Cummings below, there are further papers available in the Edexcel’s Emporium:

The Emporium has some 17 multiple-choice question papers from the period 1988-1992 – by no means a complete set, but they range across the Mathematics, Further Mathematics, Pure Mathematics and Applied Mathematics syllabuses. You can find them in the “Pre-C2000” cabinet within GCE AS/A Level.

Signing up to Mathematics Emporium is highly recommended, note that it is a free website intended for the use of teachers of mathematics in secondary schools, regardless of what board you useRegister for an account and ensure you supply a correct centre e-mail address in your name for verification, your centre name and centre number.

edexcel-emporium

Know Thy Impact – John Hattie

Hattie - Visible Learning

So – back to school again and I thought I would make a final and rather important update to Resolutions for (Mathematics) Teachers. Reading John Hattie’s Visible Learning for Teachers is such an important reminder that we should really be looking at the impact of all we do on our students. We might think a particular method or resource is amazing, but do we think so because we have considered very carefully how it will help our students learn?  
For a summary of the book, read this from The Main Idea.

The five dimensions of Expert Teachers Hattie identified were based on a review of the literature.
In summary:

  1. Expert teachers identify the most important ways to represent the subjects they teach
  2. Expert teachers create an optimal classroom climate for learning
  3. Expert teachers monitor learning and provide feedback
  4. Expert teachers believe all students can reach the success criteria
  5. Expert teachers influence a wide range of student outcomes not solely limited to test scores

Dimensions 4 and 5 remind me of Carol Dweck, these points she made struck a chord with me:  for teachers to develop a growth mindset in their students they need to develop their own growth mindset; do we ever judge our students too quickly? Also, such a useful reminder that we may sometimes worry too much about ‘teaching to the test’ when we just need to remember that ‘The outcomes are natural byproducts of engaging in good practice’.

I have sometimes listened to audio books as I do like to hear authors read their own work, I believe it helps understanding. You can hear John Hattie himself on the principles discussed in Visible Learning in these two videos: Visible Learning Part 1: Disasters and below average methods  and Visible Learning Part 2: effective methods. If you are in a hurry you might want to skip straight to the last part of the second video! For anyone who can’t get enough of Hattie, he was interviewed as part of Radio 4’s series The Educators.

For further reading of current ideas, see Tom Sherrington’s excellent collection: Contemporary educational ideas all my staff should know about.

If you are about to return to school (or have already done so) then I wish you and your students a great year.

Writing Mathematics Online

I have written various posts on the available tools online for writing Mathematics and this is a topic which remains consistently popular. Time for yet another revisit and update as new possibilities are now available – all resources / links here have been checked. I have removed resources where blogs / twitter have not been updated for some considerable time as I think this is a cause for concern regarding the longevity of the resource.

Something I use a lot. I should explain my requirements – I want tools to communicate Mathematics online, for example I may wish to provide some model solutions or answer students’ questions. Writing mathematics can be a pain (and yes I know about LaTeX). Note that there are various possibilities – sometimes just a static picture is required, sometimes you may want to display how to solve a problem in stages, or perhaps you require a collaborative space. You will also need to consider if you want the examples to be permanent or whether you just want a collaborative space for discussion. A graphics tablet is essential.

My favourite method for illustrating Mathematics online (and in fact the one I use most often) when I just need a series of static displays is to turn an interactive whiteboard flipchart (or a PowerPoint) into a pdf file; the pdf file can then be sent to students or uploaded to whatever virtual learning environment or online storage your school uses. If you do not have access to interactive whiteboard software there are alternatives, one could use Windows Paint for example; there are also various free online tools available; see some of the resources below.

flipchart to pdf example

flipchart to pdf example

For sharing resources, it is possible to upload a PowerPoint or pdf file to Slideshare. There are many examples on this blog of my SlideShare slideshows – see this for example

I should mention that I find Slideshare excellent – I use the free version which offers me everything I need – it works every time – I use it a lot!
I created the PowerPoint for the slideshow above by writing on the interactive whiteboard software using my graphics tablet and taking a picture of each page using the Windows snipping tool (it’s in Accessories) – this takes seconds – the snipping tool is something I use every day! (Alternatively I could have saved the interactive whiteboard flipchart as a pdf).

There are as always several options:

Screencastomatic

Screencastomatic

If you wish to record a screencast of the moving pen / step by step solution variety and save your work, Screencast-o-matic is an excellent option. It is very easy to use to capture the screen and your recording can then be uploaded to YouTube if you wish.

Illustrating how to simplify an algebraic fraction : simplify-algebraic-fraction

Screencast-o-matic offers everything I want in this category. It is very easy to use indeed – I can write very smoothly whilst recording.

Further resources offering various solutions for writing Mathematics online:

Twiddla

Twiddla

For a collaborative board, try twiddla which seems excellent for collaboration – voice as well if you want. Twiddla offers some very sophisticated features including the ability to use mathematical formulae and upload files and images. Use of the board with all it’s features is free but you cannot save any of your work (possible with a subscription). This would be good for working online with a group of students.

doodle.ly

Doodle.ly

Doodle.ly offers a very easy way to have a quick scribble and then share your thoughts! See quadratic example (and note that I created that just to test Doode.ly years ago – a good sign that it’s still there!) Select Doodle Now, doodle, then if you want to publish you will need to sign in. You can sign in with Twitter or Facebook. I like the different pen tools available, the full screen option and the ease of publishing. Apps are available for Android and iOS.

Writing RepeaterFinally – your students may find this amusing – the Writing Repeater from ICT Games – write something and play it back – now this is a lovely tool for little ones learning to write but I’m sure we can think of some uses!

Handwriting recognition, LaTex and more!

A consistently popular post on this blog is that on online whiteboards. If I want to communicate mathematics online to answer a student query for example I find it quicker to use a graphics tablet and an online whiteboard.

I do keep an eye on various LaTex generators, one that has come to my attention is MyScript. In this demonstration, handwriting is turned into LaTex (one line at a time). The handwriting recognition is impressive and I found it easy using my graphics tablet to enter expressions accurately; see the quadratic formula below for example.

MyScript

So you scribble an expression and it get turned into LaTex for you – it works:

MyScript

MyScript

But I must confess I was just as excited to note that immediately see a graph where appropriate, powered by my favourite Desmos graphing calculator.

..and finally if you wish to be distracted by some more fun applications there are some other great demonstrations from MyScript. Try Web Shape for example and turn your sketches into vectorized shapes. This should work well on the interactive whiteboard.

improve your scribbles!

Improve your scribbles!

MyScript now has apps for iPad and Android tablets.