A couple of weeks ago I started adding a selection of random items of news, suggested resources and so on to the end of a blog post, basically anything that had caught my eye that week (thanks to Doug Belshaw for the idea; I enjoy his weekly Thought Shrapnel). This week there seem to be several so I’ll make this week’s post a collection of such items.
Graphing the Sine Function from Math Open Reference
My Year 11 students (age 15-16) are studying for a second GCSE in Mathematics. We are using AQA’s Level 2 Certificate in Further Mathematics, a course which we are all enjoying; it extends these able students and is excellent preparation for further study of the subject. We are currently looking at trigonometry and I want a good demonstration for graphing the sine function. Looking at various applets I came back to a site I use a great deal and have written about before, John Page’s Math Open Reference. Here you will find many very clear applets including exactly what I want, have a look at Graph of the Sine function. Note that is possible to display full screen which is great for the Interactive whiteboard. Drag the point A around and watch the graph and try Progressive mode. I have put the links for my students in this post.
I must start this post with something Dan Meyer said at the MEI Conference 2021 that really struck a chord with me, he said “There are no mistakes or misconceptions, just takes and conceptions.” Dan Meyer mentioned WW Sayer who said:
Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong’. Rather he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole art of teaching.
For a starter addressing common misconceptions try the excellent Classic Mistakes resources by Nigel Hopley.
A superb resource to use in class (or for students to use at home) to address misconceptions is of course Craig Barton’s and Simon Woodhead’s Diagnostic Questions site. The site has many thousands of questions with carefully designed multiple-choice responses to address common misconceptions.
The Insights feature is so helpful for learning about misconceptions, suppose we look at a White Rose Quiz on Algebraic Notation, for example, looking at the Insights we can see for any question the number of responses for each option from the many students who answered this question.
From NCETM, these videos and resources for teaching Key Stage 3 maths topics include common misconceptions and pitfalls; looking at Directed Numbers for example we find slides and a pdf support document including as illustrated here, “What things typically go wrong?”
Some years ago a website, counton.org which is now no longer available published a very useful document on misconceptions. In 22 sections, in each section misconceptions are given along with the correct version. Further explanations are also provided and also follow up exercises with answers.
The above pdf document includes all 22 sections. The first 8 of these documents, by Ilan Samson & David Burghes, are on the CIMT website.
Malcolm Swan’s excellent ‘Improving Learning in Mathematics‘, includes a section (5.3) on exposing errors and misconceptions. An activity suggested there is to let your students become examiners and mark the work of others, this works very well, I have highlighted some excellent resources for this on the ‘Spot the mistake!‘ page.
On the SERP website before see MathByExample and AlgebraByExample which is a set of Algebra 1 assignments that incorporate worked examples and prompt students to analyze and explain. These resources can provide prompts for discussing common misconceptions.
From Michael Pershan, see his Math Mistakes site, to quote the Author:
The purpose of this site is to collect, organize and make sense of the mistakes that students make while doing math. I’m also increasingly interested in using mistakes to help us create worked examples that students can learn from.