The Wisdom of the Crowd

Look, stats isn’t boring! I love doing these Wisdom of the Crowd exercises when teaching averages. Another way of doing it is to display a random scattering of dots on the screen (about 40 or so). Get everyone individually to provide an estimate then calculate the mean.
A really interesting addition to this is to do it once getting everybody to call out their estimate. Then, before calculating the mean, give everyone the chance to change their estimate and record them all a second time.
This can lead to regression to the mean, and standard deviation if you like.
See – stats isn’t boring!

The World Is Maths

Sir Francis Galton was a statistician in the 19th century. Thanks to him we have concepts such as correlation and standard deviation.  Galton, it would seem, thought through the filter of statistics, a genius who produced hundreds of papers and books on fields as diverse as meteorology, historiometry and psychometrics and who pioneered the use of questionnaires to gather better information for his statistical analyses.

Last week, at my school’s Open Evening, we conducted a mathematical experiment based on one of Galton’s observations.  

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Sum of Consecutive Numbers, a multi-layered investigtion

One of those classic investigations that gets forgotten about all too easily.  So much scope for generalising at different levels.

The fact that all odd numbers can be expressed as sum of two consecutive numbers is probably the first thing that will be established.  But why is this the case? And can students express this as a generalisation, first in the form of concise words and then algebraically?

The beauty of this is that there are then many other layers of things to discover, right down to a generalisation explaining which numbers cannot be expressed as a sum of consecutive numbers.  And maybe even a proof.

This nRich page gives away some of the answers.

Thanks to Alan Parr for reminding me about it with this excellent blog post:

The All I Can Throwers – Sessions with Den and Jenna. #1 – Consecutive Numbers

Making Groups Work

For about a year now, I’ve positioned my desks in groups of four. This trimester, my largest class is 35 students, but I was determined to make the groups fit.  I think I nailed it. For a lot…

Source: Making Groups Work

Origami for the end of term

Origami is one of those things that I think I would love to spend more time exploring but rarely do.  I have used the Origami Player with my students  (it works really well as an App within Chrome), which gives excellent visual instructions on making things. The timings have been well thought out and it gives a little timer prompt so you know how long you have got to do that fold before looking up at the screen for the next one.  It’s been an end of term, easy lesson. Nothing wrong with that.

The first session on the nRich Teacher Inspiration Day  last week where we looked at some of the activities here got me thinking about how I could make it a slightly more meaningful learning experience.  

Still not highly mathematical, but at least it gets students working together and struggling with something.  To build resilience in our students they need see the struggle as a positive and not something to be avoided at all costs.  It was a bit of a metaphor for all learning. A discussion that can be had with students when reflecting on this task might be along the lines of:

• Did you need help from someone at some point? (yes, good)
• Did you help someone else at some point? (yes, good)
• Did you struggle at some point? (yes, good)
• Did you give up? (hopefully no, good)
• Did you achieve something you didn’t think you could do before?

This type of discussion can be a powerful motivator and more useful than vague questions like “Did you enjoy it?” or “Did you have fun?”

There are lots of Origami ideas on this page of nRich’s new Wild Maths site. I really like the idea of modular Origami, i.e. each person makes a module and then they come together to create something beautiful.  I have an end-of-term cross curricular session with Year 8 to plan. My Origami paper arrived this morning.  You can use A4 paper and fold down the corner to make a square, but proper origami paper is really lovely and this pack was only £8 for 500 sheets.  It looks great so time to start practising!

I will post an update on this as planning progresses and share some pictures of the final event.  In the meantime, if anyone has an ideas for how to make Origami more mathematical (without spoiling the enjoyment of it!) then let me know.

 

Great problem solving tasks from FMSP

My Year 10 class did these 3 this week:

They have their GCSE RS and Science exams next week, so I needed to find something a bit “fun” but still wanted it to be “mathsy”.  They are a strong group with enough keen beans amongst them for me to feel confident that something like this would work.  I was impressed with their teamwork and tenacity.  At first it seems hard because you are faced with a blank sheet (their words). It took the quickest group 20 mins, 1 out of the five didn’t finish after 30 minutes.

Something I always bear in mind when doing any sort of team activity (see my earlier blog post) is that everyone should have a clear role to play and something to contribute. In this task everyone has their own set of clues, that they are not allowed to show to the others. I encouraged them to make sure they were the experts in their own clues so that when proposals were made they could say if this proposal broke any of their rules.

There is no need to cut up these sheets into “cards” as the resource suggests.  Quicker and better if you just cut them into strips:

That way you can give a strip to each student. They are less likely to drop a card on the floor which kind of ruins it! If you cut some horizontally and some vertically as shown you can cater for groups of 4 or 5 students.

We finished with 10 minutes to spare and had a pretty good discussion about the task.  Questions I asked them:

How did they start the task? (Talking or just reading)

Did it help to group the clues in any way?

Were they tempted to share the clues? (I told them I’d give them 30s time penalty if they were caught sharing.) Or were they happy having their own? (most were)

How did you work well as a team? What did you think went well?

Was it hard? What made it hard?

 

Creating data by learning your prime numbers

Here’s a little idea for a team activity that could get quite competitive and hopefully “fun”. I haven’t tried it yet, and it might be a while before I use this. When I do, I’ll try to update this post with any tweaks depending on how it runs.

It’s based a on this really simple website Is This Prime created by@christianp which I saw on Jo Morgan’s MathsGems.

It presents you with numbers and you click YES (i.e. it is Prime) or NO.  It’s not an app so it can be used on a laptop / desktop although it works really well on browser on a tablet.  I’ll be doing this in a computer room as a group activity.

I reckon this could work with classes from Year 5 to Year 8, but most pupils in the class will need to have a reasonably good grasp on their times tables or it could be frustrating. It provides consolidation of times tables and primes but I think the real objective here is actually to use this as a lead in to various data and averages topics.  I always try to teach KS3 Statistics using data that the students have created themselves as they are far more engaged and care about what the data is telling them. This not only provides that meaningful data, but does so in a way which consolidates some fundamental number facts at the same time

I plan to use Google Sheets to collect the data which we will then analyse in a later lesson.  Google Docs in general is great for this type of collaboration.  I have created a template for a group.

Each group has a separate sheet that they fill in as they go.  Just duplicate the sheets for as many groups as you have, making sure that each group is working on their own sheet before you start.

Talking of groups, here are my general rules for planning any group activity:

1. I chose the groups.  I have nothing against pupils working in friendship groups but I know who to avoid putting together and the process of self-selecting can be painful for some.
2. Everyone has a role. Some pupils will see group work as a chance to sit back and some will naturally dominate.  Assigning specific roles reduces this.
3. Everyone contributes equally. By rotating the roles I will try as far as possible to make sure that everyone ends up doing the same activities by the end of the session.

 

To get some excitement going, I’ll keep a running commentary on the highest score.  I also plan to write up the “Errors” that the Error Recorders give me as we go. I want to make sure we have some time at the end for reflection on how it went, i.e.

• Did you work together as a team? How did you support each other?
• What was a good strategy for a high score? (When I play it, I rarely use all 60 seconds as I am trying to go too quickly and so I am often tempted to guess ones I don’t know)
• As a team what did you do to make sure your scores were improving (Write down the errors on a big piece of paper? – I didn’t say you couldn’t!)

I would definitely leave the data analysis part until the following lesson.  There is lots you can do with this and it could form the basis of a series of lessons on Averages, Data representation including Box plots. We can start with the question, “Who was the winning team?”,  which in itself is open to interpretation.

Some great problem-solving tasks

FMSP isn’t all about A level Further Maths.  They have some great GCSE resources too.

This page contains some superb resources for problem solving and group work with some very helpful discussion prompts and worked solutions for each question.

These look like nice resources for introducing group work, as each member of the group has “their” cards but they work collaboratively to complete the task.

Also, this post from Number Loving has some good suggestions.

Ways of Working

Something that has never come naturally to me in the classroom is establishing and following routines. I’m not drawn to following routines myself. Before I was a teacher my work had very few routines in it.  But I’ve seen how effective routines can be in developing good classroom management at all ages. So it’s something I know I need to work at.

Whilst there are lots of brilliant and inspiring ideas out there, a few of which I have tried, I think coming up with your own ideas in this area means they are more likely to stick and become, well, routine.  So here’s something I’m going to introduce from next term.

WoW 1

The one person talking could be the teacher, or another student.  You should be actively listening and thinking about what they are saying and what it means.  It is simple – if someone else is talking and you then start talking over them, it’s highly disrespectful, even if it is about the work.

WoW 2

There will be something specific that I want you to discuss and I’ll tell you how long you have to discuss it. Make sure everyone gets a say and that you don’t chat about irrelevant stuff that doesn’t help your learning.

WoW3

Sometimes all we need is a little nudge in the right direction.  See if you can help each other before asking the teacher. It needs to be quiet enough so that everyone else can concentrate on their own work, so use a whisper if you need to discuss something.

WoW4

There will be a specific reason why I want to be sure that this is your own work, e.g. during a rest or a piece of work I want to mark to assess your understanding.

 

Great Fractions Resources from Solve My Maths

This blog post from Solve My Maths is a treasure trove of deep thinking fractions questions.

This image inspired me to plan a group activity with a class.

Give each group a set of coloured stickers (8 different colours – getting ones that exactly match the picture could be a challenge! I might need to make my own and then take a photo.  Maybe a grid in the background would be helpful if I do this…)

Show the left-most and the right-most column of numbers on this image and hide the stickers in the middle.  Their challenge is to fill in the ones in the middle.  They can use calculators if they like – seeing a fraction as the equivalent of the operation of dividing numerator by denominator is useful.  Work together, only place the sticker if everyone else in your group agrees.  I think this will really get them talking.