## Pi Day Activities

Pi day yesterday was a welcome break from the current global chaos – a lovely last day before a full month of online learning begins. Here’s what we organised if you need some inspiration for next year. I decided to ensure sustainability by only involving Year 9 (Grade 8) students so that we can run it in a very similar way next year. All Year 9 students had to sign up for one activity 2 weeks before the day.

1. A Pi cake competition

2. A Pi Memorisation competition

3. A Pi Skyline competition

4. A Free-hand circle drawing competition (https://vole.wtf/perfect-circle/)

5. A Live Buffon’s Needle Experiment

6. A “10,000 Colour’s of Pi” Mural

Lesson Task: I created this task for all Year 9 students on the day with inspiration from Dan Meyer – How round is your circle?  (Free-hand circle drawing task)

Opinions on this task? Any amendments or additions would be gratefully received before trialling it with my Year 9 group. Happy to receive comments below or via twitter (@DanielPearcy). We have completed a considerable amount of work on straight line graphs and I would like to move their thinking forward. The aim of the task is to connect their understanding of straight line graphs with sequences initially, and then move that new found connection onto more interesting sequences.

In regards to John’s comment below, I will now start by visually representing the first two sequences on the board before setting them off on the task.

Posted in Uncategorized | 2 Comments

## Cognitive Load Theory: Questioning the Intricacies

To give some context for this post, I’ll start by saying that I absolutely love Craig Barton’s podcasts. Whenever I listen to one I almost always have to re-visit given the ridiculous number of ideas and claims made. One thing’s for sure – Craig Barton deserves a bloomin’ medal for the work he’s done on these podcasts and his contribution to maths education in general. It’s crazy just how good these episodes are for PD – I always learn so much – but then I always leave with more questions than I know what to do with.

With that in mind, last night I re-listened to the podcast with Greg Ashman; like every other Craig Barton podcast it’s absolutely brilliant! Greg is highly knowledgeable about Cognitive Load Theory and I took a lot away from his responses. Since the start of my career, I’ve been a proponent of a balanced view of educational theory in that I happily use explicit instruction, but also think that there’s a good chunk of mathematical knowledge that can be internalised through structured investigation or inquiry (I rarely use open investigations or inquiries). More recently, proponents of Cognitive Load Theory (CLT) have begun to swing my thoughts towards the balance between explicit instruction and investigation, and if I’m honest, I’m starting to come round to changing elements of my practice to take more account of CLT.

I’ll also be additionally honest and say that I’m a bit worried by all of this. I’m worried that teachers aren’t questioning every element of CLT and ensuring, as far as possible given that we’re in the social science game, that they dig deep into CLT to fully understand how it might affect their practice. With this in mind I have a few questions about CLT, specifically based around the Greg Ashman’s responses. I could look it all up in the Literature – assuming it’s present – but that wouldn’t be taking advantage of the benefits of social media.  So here are my major questions from this podcast – hopefully some awesome person will answer these without much effort and then I’ll be happy to move forward.

Working Memory and Discovery

Greg explains a piece of interesting research conducted by Sweller that students were given some problems in which they either had to multiply by 3, or add 29 (something like that anyway). Sweller cleverly put these problems together so as to ensure that the students were simply alternating from adding 29, to multiplying by 3, to adding 29, etc. He did this to determine whether they would discover this underlying structure to the problems, or whether their working memory would be filled up by the act of determining what to do. It turned out that the wide majority of students didn’t see the underlying structure, and their full attention was on the surface operations of the problems.

Questions:

• Did Sweller notify the students that there was an underlying structure to the problems? This would clearly have an effect on their mindset during the task and might shift the focus to a higher level of discovery for more students.
• What if Sweller had simply told the students to track their calculations by just writing them down – and said nothing else about discovering anything. Students wouldn’t have to hold extraneous information in their head, and this would not affect their processing of working memory. In this case, I assume that their attention during the task would be on solving the problems, and then after some time it would be simple to recognise the deeper structure. The whole premise behind CLT is to reduce the processing of working memory, and tracking your process in a written form does not conflict with that.

Explicit Instruction vs Pure Discovery

Greg explained a separate piece of research where science students were explicitly taught concepts, and in contrast, other students were given the opportunity to discover the same concepts. When it came to the assessment, those students who had discovered those principles performed no better than those students who had been explicitly instructed during the assessment task.

Point:

• I agree with that fact that, probabilistically speaking, more students in the explicit instruction group would have attained the knowledge required for this specific assessment. That clearly has implications for teaching knowledge that we all have to be aware of.

Question:

• When did the assessment happen? Straight after the learning experience, a few weeks later? I have always thought that when learning is attached to our emotions – in this case the pleasure of discovery – that we are more likely to retain that knowledge. Of course I agree that deliberate and spaced practice, retrieval cues, etc. will also improve long term memory,  but memory seems more intricate than a basic knowledge of the Forgetting Curve. Indeed, a quick google search provides a large body of research on this (Example Link). Therefore, is it plausible to think that those who discovered these principles, whatever they were, are more likely to retrieve them at a later date?

One last point I’d like to make is that I’m pleased that Craig asked Greg about the continuum of direct instruction because most things in education aren’t binary. If I say that I prefer an explicit instruction approach, then that could mean a whole world of things. I could be on the “lecturing” end of that continuum, or the more liberal end of the continuum. Some people prefer to call the more liberal end dialogic instruction (Danny Brown introduced me to dialogic teaching in this blog), where there is constant dialogue between yourself as the authority in the classroom, and your students. In this case, if you’re introducing anything, such as the formula for the area of a trapezium, you lead and steer the construction via a mixture of explanations and student responses.

So if anyone can help with the questions above I’d massively appreciate it. I can’t help thinking it’s our professional duty as educators to maximise outcomes for our students (and I think to excite their curiosity and interest in mathematics). I therefore really want to fully understand the true implications of Cognitive Load Theory on my practice.

Posted in Uncategorized | 3 Comments

## Does an equation exist to describe a square? What about any polygon?

Yesterday I saw some awesome Mr Men Math posters created by @solvemymaths (www.solvemymaths.com) and have already stuck some of them up in the corridor at school. Mr Astroid particularly caught my eye as one of the interesting Mr Men shapes.

You can see, and understand intuitively, that the equation is a variation on a circle. In fact, the shape of the astroid looks quite close to a square. This got me thinking…does a function exist to describe any given polygon? So here goes…

Posted in Uncategorized | | 2 Comments

## Moving towards more Meaningful Mathematics Assessment in the 21st Century

I’ve always believed that there’s nothing wrong with a sit down test if the test items assess the core philosophy and values of the subject. Hence, I’ve tried to make an assessment for my Year 7 students (grade 6) which I believe is more meaningful in the 21st century. It’s so easy now to do a quick search on Google to find the formula for anything. Indeed, during a planets investigation today with Grade 9, we simplified the problem of planetary orbits to consider them as circles instead of ellipses – pretty standard. However, one of the students put his hand up a minute later to ask why we weren’t just using one of the many formulae he had found through a Google search for the perimeter of an ellipse. This is great but it means that we need to teach students how to derive formulae and analyse formulae instead of simply using them without thinking. I guess this is similar to when students just pick information from a website without critiquing it – so frustrating and meaningless.

Anyhow, here’s the assessment. Any feedback to improve would be appreciated.

## Allowing Students to Discover the “Mystery” of Square Roots (Surds)

Here’s my attempt at investigating operations with square roots (surds) when you’re not prescribed by a GCSE/IGCSE syllabus. This is only a general outline of the lessons we’ve had so far but it’d be great to hear how other people go about teaching surds.

To start off, we hadn’t spent much time discussing irrational numbers but I thought the question below would push us into a discussion/debate about them and then things would start flowing:

(An irrational number) x (An irrational number) = (A rational number). Discuss

Posted in Investigations, Number | Tagged , | 1 Comment

## Percentage Increase and Decrease Applet: Any improvements?

Here’s a Geogebra app I made to help students who were struggling with percentage increase and decrease. Can anyone think of any improvements? Click the picture to go to the link.

Posted in Number | Tagged , , | 3 Comments

## Estimation Calculator (Multiplication) on Geogebra

Click the picture to go to the Applet.

Posted in Uncategorized | 2 Comments

## Why I like Mathematics and why Students should Discover and Create their Own

An old video – and in some points cringe-worthy now I look back – but I do sometimes like to look back at it. [By the way, I don’t have a clue where I plucked out the 90% statistic towards the end – extra planning was needed here I think!]

Posted in Teaching Ideas | 3 Comments