Open-ended Maths: a selection of 101qs favourites

The 101qs website was created in 2012 by Dan Meyer, a great Maths teacher and dedicated apostle of open-ended Maths.

‘Open-ended Maths’ is an expression that sounds pretty scary to many Maths teachers. A lot worse than ‘out of the textbook’, which is rather trendy and flattering. Dan Meyer’s first paragraph on his About page does not seek to cajole them back to their comfort zone either:

We don’t care how well you lecture. We don’t care how well you engage us. We aren’t impressed by your fancy slide transitions or your interactive whiteboard. We care how well you perplex us.

This sets the tone perfectly: open-ended Maths is about stimulating students by presenting them with a perplexing picture or video — perplexing in the sense that it raises questions. Students have to formulate the questions themselves and subsequently solve them, which often means developing appropriate resources on the way.

If you are a regular reader of my posts, you know by now that this blog is not about being ‘for’ or ‘against’, but understanding precisely WHY and HOW an idea is potentially useful for teaching Maths. In this instance, it so happens that open-ended Maths is not only cool, it is also an essential teaching tool — for reasons I shall explain in a later post — on the condition that it is not used systematically (for reasons I shall also explain).

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Introducing (some) problem-solving skills

Problem-solving is not only a prominent Maths activity, as shown in the Maths ability pyramid. It is also a discipline of its own, with its specific know-how. In other words, the specific skills of problem-solving can be learnt too. By doing so, students will not only learn to solve problems more efficiently, they will also make the best of problem-solving’s high educational value.

For Maths teachers, it means that it is possible to choose problems for students not only according to a particular Maths topic (fractions, algebra, trigonometry, etc.) but also with a view to practise one or several problem-solving skills.

In order to do this, it is necessary to identify and name these skills. This post covers 10 problem-solving skills, which you can see in action in UKMT JMC 2015 (cf. my JMC 2015 teacher’s notes).

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UKMT (teacher’s notes) – JMC 2015

This is the first post presenting my teacher’s notes of past UKMT papers, starting with JMC 2015.

The objective of these teacher’s notes is not to provide solutions — UKMT already provides an excellent Pdf of solutions and further investigations, which you can download here — but to provide insights to teachers as to how UKMT questions can be used in the classroom.

In other words, these teacher’s notes are about making the best of the educational value of UKMT questions. Which, by the way, extends the scope of UKMT questions beyond their target age group. For example, some JMC questions, although intended for Year 7-8 students, can be used for educational purposes up to GCSE, sometimes even with A level students with the addition of relevant extensions and investigations.

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What is a logical progression ?

Maths makes a lot of sense when it is taught in a way that makes sense. If you teach something that makes sense in a way that doesn’t, at the end of the day it doesn’t make sense.

In other words, if we teachers don’t get our conceptual thinking right, then we can’t really expect students to, can we ?

The most important dimension of this conceptual thinking depends on the age group you are teaching to. However, there is definitely one dimension of conceptual thinking you cannot overlook from secondary school on: the logical progression.Read More »

Are Maths exams robbing students of the fun ?

After weeks of writing and talking to teachers about behaviour management, I am back (with relief) to posting on my favourite topic: Maths teaching. That is no doubt the effect of meeting two highly inspirational people (one Maths teacher and one Headmaster) within a few days.

I would like to start this post with a seemingly remote comment which was kindly sent to me by a reader concerning the Maths ability pyramid, pointing out that the pyramid does explain ‘what we do in Maths and why we do it’, but not HOW we do it.

Which is absolutely true: the HOW question is extremely important and it will require several posts to develop this (inexhaustible) point. One could even say that, for Maths students, it’s the ‘HOW we do Maths’ that makes a huge difference.Read More »

Go for hybrids ! (How to spark off a Maths topic ? – Part 3)

This is the last part of our review of possible ways to spark off a Maths topic (preliminary note: it is recommended to read the 2 previous posts: How to spark off a Maths topic ? – Part 1: concept processors and How to spark off a Maths topic ? – Part 2: memory reinforcers).

Actually… the best sparks are often hybrids. It’s quite nice to have logic (conceptual processor) + emotional engagement in order to activate memory (memory reinforcer).

Let’s look at a few examples…Read More »

Memory reinforcers (How to spark off a Maths topic ? – Part 2)

We continue our review of possible ways to spark off a Maths topic (it is recommended to read the previous post: How to spark off a Maths topic ? – Part 1: concept processors).

Today, we are talking about the second category of sparks: memory reinforcers.

The idea of memory reinforcers is to spark off a topic with something that engages beyond the intellect in order to reinforce memory. What you choose may not seem logical, but that is beside the point if you reach your goal of memory reinforcing.

As memory champion Josua Foer explains: ‘We remember when we are able to take a piece of information and experience it. We remember when we pay attention. We remember when we are engaged’.

A memory reinforcer can be…Read More »

How to spark off a Maths topic ? – Part 1: concept processors

This post is about sparking off the core topic of a Maths lesson. (NB: This is different from starting a lesson with a Do Now or Starter, which are often used (and quite rightly so) as retrieval activities or stimulating/settling activities, but are not necessarily related to the core topic of the day’s lesson). Therefore, the spark will often be the second item in the lesson, but it is the first point of contact with the main topic.

Choosing the right spark to introduce a topic is important for 3 reasons:

  • First of all, the spark should totally serve the core topic, either through raising a question that will inevitably lead to the new concept or skill being introduced, or by catching attention and reinforcing how the lesson will be remembered.
  • Secondly, the spark should be in direct correlation with the way the main point of the lesson will be wrapped up (more about that in a future post).
  • Thirdly, the spark should introduce the first increment on prior knowledge, i.e. building up on what students already know.

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How to develop fundamental abilities

This post provides a few practical tips on how to develop fundamental abilities (i.e. the first level of the Maths ability pyramid), thus helping students to become more confident by increasing their awareness and fluency with the mental manipulation of objects and processes such as order, numbers, causes and consequences.

There is a double benefit in working on this development: not only does it help teenagers to focus and develop mental resources, but it does so by involving them in a series of lively exercises that look very much like collective games with relatively little Maths involved. In other words, developing fundamental abilities is both low-cost and high-benefit.Read More »

The magic of Q&A beyond expectations

It is always very stimulating when questions and answers in a Maths lesson suddenly go beyond all expectations.

This might sound like this is something that only happens exceptionally. Not so. In a class where a climate of questions and answers has been set in mutual trust, exceeding expectations happens almost every day.

In this post, I would like to show two such examples, one where my expectations as a teacher were exceeded, and another where a student improved his own conceptual understanding starting from a misconception – which was of great benefit to the whole class.Read More »

Three things your Maths students should not be afraid of

For many students, Maths is not fun, it’s fright !

That is not necessarily what they will show in the classroom. When students experience difficulty in Maths, they might appear bored, annoyed, rebellious, ironic, puzzled, defiant, etc. but not afraid, because they are proud.

It’s adults that tell you how frightening the experience was when they were kids. Adults are not in the classroom anymore, they can let the fright out and even laugh about it.

But where does this ‘Maths fright’ really come from ?Read More »

Why is it so important for students to have a sound number foundation ?

Yes, you’ve heard it before… ‘Students can’t do sums anymore’.

You had heard it before, hadn’t you ? Well, just in case you hadn’t, that’s exactly what someone was telling me last night at the pub: ’Oh, you’re a Maths teacher, eh ? Well, I’ll tell you one thing, mate: kids can’t do sums anymore !’

That was just in case you hadn’t heard it before.

True or not true ? Important or not ?Read More »

The Maths ability pyramid

Pyramid

The Maths ability pyramid is a communication tool I have created in order to explain more easily to students and parents what we do in Maths and why we do it.

The initial idea behind the Maths ability pyramid is not only to access a better understanding of what learning Maths is, but also to get rid of this highly misleading fiction: the one monolithic so-called ‘Maths ability’ (you know, when parents or students tell you: ‘Sir/Miss, I’m not good at Maths anyway’…).Read More »