‘Openended 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: openended 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 openended 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).
I will not write about how 101qs material should be used in the classroom, as this has already been done very clearly by Dan Meyer himself, for example in his post entitled The Three Acts Of A Mathematical Story.
Nor will I make a list of topics covered by 101qs activities: there is an excellent search engine on the 101qs website, with keywords and grades.
What I would like to do is to start from this idea of perplexing students and propose a selection that will highlight the 7 main forms of this perplexing power. (Please note: all links open a new window of the related page on Dan Meyer’s website).
This videobased resource is a fabulous introduction to ascending geometric sequences, as well as a great activity on powers or fraction multiplication.
This videobased activity plays with the same idea, but with a multiplier lower than 1 (descending geometric sequences).
Students will be surprised to prove that it doesn’t take that many reductions for a dollar bill to become microscopic.
This activity can be used when working on volume, unit conversions, speed or ratios.
In the same category, see also Super Bear.
This activity is about areas and various calculations based on ratios.
Students also get the extra pleasure of working on a real net of this incredible dollarcarpeted room.
With its triggering question (How tall would all of wikipedia be ?), this activity is purely about the joy of big numbers.
Unlike the previous activity, where the idea is to produce a plausible ballpark number, in this instance it is possible to calculate (among other things) a precise number of pages.
But for this, students have to exercise their 3D vision and be quite selective as to which information is really relevant.
This situation is also about ‘big things’, but it induces a different discussion about the nature of space. Is that a sphere filled with spheres ? what about the space between gumballs ? etc.
Many possible sequels with colours, estimated revenue, etc.
These are 2 examples from which students can work on proportionality, scale, ratios, etc.
This hilarious video is virtually limitless in its applications on speed, line equations, linear and nonlinear functions, etc.
This is a classic problem involving ratios, but the video and humour add an extra dimension. Lots of possible openended extensions as well.
On the same idea (but with a mix of continuous and discontinuous variables), see also Nana’s Lemonade.
This is an example of many possible problems involcing cuts, and therefore fractional areas of various shapes.
It is also illustrates the versatility of openended resources: for example, this video can be used either as such for Alevel students on sector areas, or with a triangle approximation for younger students.
This very simple everyday life situation raises an important mathematical question about fraction multiplication (versus addition).
There again, this activity is about various ratios, with a highly educational dimension on the absurdity of soft drinks.
On the same idea, see also Soda.
A volume estimate problem, with an interesting question about the size of a drop of water, as well as many possible crosstopic sequels.
These 2 activities are interesting for students to relate 1D, 2D and 3D concepts such as length, area and volume.
The first activity (Dandy candies) is probably more suitable for younger students, while the second (Coffee traveller) is a beauty, as it raises questions on graphs, discontinuities induced by hypotheses, etc. without going into too complex Maths.
Unlike previously selected activities, this is just one picture, but it’s absolutely packed with information and potential calculations: the situation is easy to understand and students can formulate their own questions.
As per the previous one, this picture is a concentrate of fascinating Maths question: is about length ? perimeter ? radius ? area ? crosssection ?
It also introduces students to the idea of modelling reality, as they will probably have to conceptually approximate the idea of roll to find an estimate.
On the same idea, see also Toilet paper roll.
Registration plates have long been a subject of interest for kids, and this example can be the starting point to a lesson on combinatorics.
Interesting implications on standard form as well.
Much as the Ticket Roll activity, with a different feel (more accessible to younger students, because cars are definitely 3D objects with a width)… and a more artistic touch.
Same idea, but even more beautiful… and more complex too.
This goes well beyond applying the formula for the volume of a pyramid. It questions how you measure volume, as this pyramid is actually made of cylinders ? Or is it cuboids ?
On close analysis, this is well worth a discussion, isn’t it ?
Can also be used for sequences (going from one layer to the next).
There is much to understand from this picture, which can be an opportunity for estimating the number of trolleys, but also for modelling a polygon into a circle.
Perhaps a circle is a polygon with an infinite number of sides ?
Apart from the sheer perplexing poetry of the picture, this activity introduces students to the idea of scale and how large distances can be modelled into smaller ones.
Just like the Maine solar system project, students can then choose a suitable scale and position planets in one of the school’s open spaces.
An interesting intellectual shift on negative numbers…
This activity is potentially limitless on probability, probability distributions, expected value and optimisation.
These are 2 complementary activities on equivalent fractions.
This simulationbased activity is how about a volcano and how long will it take the lava to reach a village called Tarata.
It is about speed and circles, but with potentially many implications on graphs, imagining and modelling the impact of slope on speed, etc.
Beyond the teen story, this video raises the interesting question of how to measure the depth of a hole in the dark and with no tool.
Much to discuss about the relevant information on speed, weight, gravity, acceleration, etc.
In Maths, and particularly geometry, students are accustomed to working with theoretically perfect objects, like circles for instance. But what about imperfect objects ?
This activity introduces students to error percentages, and more importantly to the idea of distance criteria.
There are also similar activities for best midpoint, best square and best triangle.
There are dozens more resources on the 101qs website, which also features a search engine with keywords and grades. Here is also a link to Dan Meyer’s own bank of ThreeAct Maths tasks.
It should be noted that, among other advantages, many 101qs activities (and particularly the most visual ones) provide excellent practice for estimating (cf. feel for numbers in Maths ability pyramid). Openended Maths also reduces the fear of numbers (cf. post on Three things your Maths students should not be afraid of).
Beyond that, 101qs resources and investigations often defy both topical classification and level classification. They are so rich that most of them can be used at different levels with different objectives.
However, there are several areas of the curriculum where 101qs is definitely a goldmine, for example:
My daughter’s top 5:
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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 problemsolving skills.
In order to do this, it is necessary to identify and name these skills. This post covers 10 problemsolving skills, which you can see in action in UKMT JMC 2015 (cf. my JMC 2015 teacher’s notes).
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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 78 students, can be used for educational purposes up to GCSE, sometimes even with A level students with the addition of relevant extensions and investigations.
In order to help teachers, I have indexed each UKMT question by Maths skills and problemsolving skills.
With the Maths skills index (which will ultimately be laid out in an Excel file when more of these teacher’s notes have been published), a teacher can access all questions related to a particular topic (fractions, for example, or simultaneous equations, etc.).
With the problemsolving skills index, it is possible to select questions according to which specific key drivers you want students to learn in order to solve problems more efficiently, such as:
(NB: these problemsolving skills will be fully explained in my next post: Introducing some problemsolving skills)
The JMC 2015 paper can be downloaded here. We all have favourite questions. For me, the prize for JMC 2015 is Question 24, a highly educational problem on palindromic numbers.
Here is a recap table of JMC 2015 questions indexed by Maths skills and problemsolving skills, please click here to download the full PDF UKMT JMC 2015 (teacher’s notes).
JMC 2015 Teacher’s notes – Recap table
Competition  Year  Question  Skills  Problemsolving 
JMC  2015  1  · Number / Adding and subtracting integers
· Number / Negative numbers 
· Elimination
· Deduction · Alternative strategies 
JMC  2015  2  · Measuring / Time units  
JMC  2015  3  · Number / Estimating
· Number / Properties of number 
· Elimination
· Alternative strategies 
JMC  2015  4  · Number / Adding and subtracting integers
· Algebra / Forming and solving linear equations 
· Deduction
· Economy 
JMC  2015  5  · Fractions / Adding and subtracting fractions
· Algebra / Forming and solving linear equations 
· Name / Label
· Bar modelling

JMC  2015  6  · Geometry / Angles in a triangle  · Deduction 
JMC  2015  7  · Numbers / Factors and multiples  · Economy
· Deduction · Alternative strategies 
JMC  2015  8  · Numbers / Factors and multiples  · Elimination
· Deduction · Alternative strategies · Easy way out 
JMC  2015  9  · Number / Estimating  · Bar modelling 
JMC  2015  10  · Number / 4 operations on integers  
JMC  2015  11  · Number / Properties of numbers  · Systematic list 
JMC  2015  12  · Fractions / Adding and subtracting fractions
· Algebra / Forming and solving linear equations 
· Name / Label
· Bar modelling 
JMC  2015  13  · Number / Adding and subtracting integers
· Number / Factors and multiples 
· Systematic list 
JMC  2015  14  · Number / Properties of numbers  · Elimination 
JMC  2015  15  · Number / Factors and multiples  · Elimination
· Alternative strategies 
Competition  Year  Question  Skills  Problemsolving 
JMC  2015  16  · Geometry / Polygons / Angles in a triangle
· Algebra / simultaneous equations · Algebra / Rearranging formulae 
· Name / Label
· Alternative strategies 
JMC  2015  17  · Binary logic  · Tree
· Economy 
JMC  2015  18  · Fractions / Equivalent fractions  · Easy way out 
JMC  2015  19  · Number / Properties of numbers  · Systematic list
· Elimination 
JMC  2015  20  · Measuring / Area and Volume
· Algebra / Sequences 

JMC  2015  21  · Geometry / 2D shapes
· Algebra / Forming and solving linear equations 
· Name / Label

JMC  2015  22  · Geometry / 2D shapes
· Geometry / Symmetry · Fractions / Introducing fractions 
· Complete the grid
· Economy

JMC  2015  23  · Measuring / Area  · Deduction

JMC  2015  24  · Number / Place value
· Number / Factors and multiples 
· Deduction

JMC  2015  25  · Geometry / Angles in a triangle
· Algebra / Forming equations · Algebra / Rearranging formulae 
· Deduction

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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.
The logical progression is the order in which you choose to expose and develop the successive aspects of a new Maths topic or chapter. Rather than being theoretical about it, let’s look at a practical example from HIgher GCSE and ALevel Maths: quadratic equations.
Here is a progression (taken from a widely circulated textbook):
What are the issues with this progression ?
Then what would be a good logical progression for quadratics ? The first quality of a logical progression is that it should be powered by necessity.
Considering the five issues below, a sound logical progression for quadratics could go like this:
Are we saying that, for any given Maths topic, there is only one acceptable logical progression. No, of course not. But there is certainly not a great number of those…
Why is a rigorous logical progression important ? Because it brings:
So, ever wondered why it’s actually better to have specialists teach a specialist subject ? Well, this is an example. Among many others…
]]>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.
However, we have to be more specific: which students we are talking about ? Indeed, the ‘HOW we do Maths’ question is quite different whether we are talking about primary schools, secondary schools or postGCSE education ?
In this post, I would like to focus on secondary school students, because that is where the erosion of Maths ability and performance is highest. It is no big secret that many kids entering secondary schools with a fairly good Maths foundation lose interest in the subject, accumulate gaps in their Maths knowledge and finally end up with a lower GCSE grade than they would have got in Year 7…
So, of course, the big question is: how can we avoid that ?
Leaving aside behaviour management issues (which we really shouldn’t, because they actually play a major role in this erosion), the answer is very much in the ‘HOW we do Maths’.
There again, the Maths ability pyramid helps us to understand the situation. If you think of exams, i.e. Maths GCSE, then you are focusing on the second layer of the pyramid: skills. If, as a teacher or a student, you are focusing on the skills layer only, then it’s skills for skills’ sake; or skills for GCSE’s sake, which is pretty much the same in the end. And that, of course, can be pretty boring.
A quick look at Maths GCSE papers will only confirm this. There is nothing particularly creative about GCSE questions. That is not their purpose, anyway. GCSE questions are standardized and repetitive chunks designed to test a number of basic Maths competences. Period.
In other words, it’s not Maths exams that are wrong, it’s the way Maths exams are considered and used as a base for Maths teaching.
The same could be said of Maths levels in secondary schools. I am not saying levels are absurd and shouldn’t be used. There is ‘some truth’ in levels: on the whole, a level 5 student ‘performs better’ than a level 4 student. But to regard it as a measure of a student’s ability in Maths is wrong. It would be like looking at the Maths ability pyramid and saying that only the middle layer (skills) is real. What levels measure is not students’ ability in Maths, but their response to the way they have been taught Maths.
Does that mean that skills shouldn’t be practised ? Of course, they should. Does that mean that massed practice should be banned ? No, it shouldn’t, but there is now enough available evidence to manage practice in a much more efficient and subtle way (see Daniel Willingham’s works — particularly his book Why don’t students like school ? ). And more importantly, to subordinate skills practice to the more exciting dimensions of Maths.
There are many parameters that make a difference between one Maths lesson and another, between one Maths teacher and another. It’s a complex alchemy and nobody has the absolute right answers. But there is one parameter that is fundamentally important: time.
Ultimately, at the secondary school stage, it’s the way you allocate time to the various Maths activities that reflects your vision of Maths teaching.
Here’s the key message: the shape of the time pyramid should be very different from the Maths ability pyramid. Ideally, it should be pretty much upside down: spending more time on problemsolving, then skills, then explicitly fundamental abilities (I do say ‘explicitly’, because skills and problemsolving constantly draws on fundamental abilities anyway).
This inversion of the Maths pyramid puts exams in the right perspective. To suggest that Maths GCSE should be treated in an offhand manner might seem a little exaggerated or iconoclastic, but at least it helps understand that Maths competences required for GCSE papers are only a ‘chemical residue’ of a much wider (and happier) Maths learning experience. In other words, the point of learning Maths is not to pass Maths exams, and to consider it this way boosts student performance at Maths exams considerably.
A few years ago, Sir Ken Robinson made a number of videos and written contributions trying to get his key message across: schools tend to kill creativity. At the time, this was rather badly received by the education establishment. I think his message was widely misunderstood.
Here, we have an example of what I think Sir Ken really meant. Creativity is not a lofty objective for idealists. Creativity doesn’t mean there shouldn’t be repetitive practice and exams. Opportunities for creativity exist in tiny little moments of classroom life when pupils are given an appropriate challenge. And THAT is precisely one of the many benefits of problemsolving in Maths. This is a particular kind of creativity, among many others. More about that later…
]]>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…
Card tricks are among the most efficient ways to introduce algorithms. They are also essential to understand the difference between the world of probabilities (an event which is more or less likely to happen) and the world of algorithms and computer science (a process that NEVER fails). Card tricks are at the same time totally rational and spectacular, making them ideal hybrid sparks.
This picture (from 101qs) is a good example of a hybrid spark for any lesson on angles or trigonometry. To start with, there is a striking and irrational element to it, which reinforces memory. Then reason takes over and concepts/skills provide the explanation.
Stories provide an immense reservoir of hybrid sparks, particularly when the story seems totally mysterious to start with, but the lesson loops back on it and things appear absolutely logical and obvious in the end. More on that later…
An officiallooking letter from the Minister of a fake African country usually works very well.
‘Conceptual’ does not necessarily mean ‘exclusively mental’. On the contrary, sparks were the body is involved can be very effective:
Example: if you keep halving the distance between your fingers, will they eventually touch?
Example 1: There are 10 kinds of people in the world: those who understand binary numbers, and those who don’t.
Example 2 (to introduce vertical asymptotes): The right to swing my fist ends where the other man’s nose begins. (Oliver Wendell Holmes)
Building 3D shapes from videos, nets, etc.
Games are not ideal sparks as such (they are actually ideal driving forces — more about that later…). However, they often prove efficient when embedded as a spark because there is a natural engagement when playing them and they provide opportunities for easy incrementing.
Let’s look at an example: countdown. This is a common game in the classroom, but there are different ways to build on it:
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One last point about sparks: from a great number of conversations with teachers, as well as observing lessons, I have often found that Maths teachers are not necessarily aware of the considerable variety of sparks that is available to them. And even then, they do not use this variety that much. Should they ? The answer is: yes if possible, they probably should. But why ?
Why is it important to vary the types of sparks from one lesson to the next ?
One of the bestkept secrets of using sparks to their full variety is that it enables Maths teachers to stabilize the structure of most of their lessons to a nice routine that students will feel comfortable with. In other words, the more varied the sparks, the more routine you can have otherwise, which will benefit the whole class.
]]>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…
Example n°1: …an image that students will remember
Example n°2: … an unusual way to emphasize the main point:
For example, when explaining how to add fractions, I use a sequence of capital letters, each on a separate Powerpoint slide: E – N – O – R – M – O – U – S… ‘That makes ENORMOUS, OK ? Today’s main point is: ENORMOUS. It is so enormous that I want you to write it in your exercise books in capital letters: NEVER ADD OR SUBTRACT FRACTIONS BEFORE THEY HAVE THE SAME DENOMINATOR !’.
Of course, that spark does not dispense from explaining the rationale of having to work with same denominators. This is explained during the lesson, but at least the rule is remembered even by students who are not completely clear about the mathematical rationale.
Proceeding this way is often disregarded by ‘fanatics of understanding’, who claim that students should only memorize what they understand. In an ideal and rational world, they are right. But the world we live in, and particularly the world of teenagers, is neither ideal nor rational. So it is sometimes best for students to memorize the main point and understand it later through further practice.
Example n°3: … a farfetched and excruciating repetition
One of the virtues of memory reinforcers is that they can work on repetition and unexpexted logic. In fact, the more excruciatingly repetitive and illogical they seem at first, the better.
To get students to memorize the main point about equivalent fractions, I have used the following 7 pictures, which I show to students one by one on a slideshow with questions and answers in between.
On showing Picture 1, the dialogue goes this way: what do you see ? do you notice anything ? etc. Then you show Picture 2 and ask the same questions, which usually prompt more or else the same answers (how else…?). Then Picture 3 etc. until it gets really irritating, because it’s always the same … Then, after the last picture comes an excruciating illogical conclusion: with equivalent fractions, it’s exactly the same: many fractions are equivalent because they have the same value, but only one fraction is in its simplest terms.
That conclusion is so farfetched and perplexing that students will usually remember it forever (particularly boys, for some reason).
The beauty of it is that, because it is so illogical, it creates a logical void in the class, which has the effect of arousing curiosity: some students will want to know what ‘simplest terms’ means, or why ‘only one’, etc. whereas they might not have been so interested if you had presented a mathematical proof right away.
]]>Choosing the right spark to introduce a topic is important for 3 reasons:
But beyond these three criteria, the choice of the ideal spark is very personal. There isn’t one right answer. It depends on your teaching style, it depends on the students, it depends on how long you have taught the class and how far you feel you can go with surprising ways to reinforce material.
The potential diversity of sparks is vast, but it is useful to have 2 categories in mind: concept processors and memory reinforcers. I will start with concept processors because, due to their virtuous inbuilt logic, they are often most favoured by Maths teachers.
Concept processors are sparks that make the new concept or skill appear totally inescapable, necessary and logical.
The most common concept processor is to present students with a problem that can’t be solved without introducing the new concept/skill of the day. Because the new concept/skill appears so inescapably necessary, this way of sparking off the topic is often regarded as virtuous, and it actually is, although that does not always make it the most efficient solution.
Let us look at 6 examples of concept processors.
Example n°1: a simple word problem
A great classic is the bow and arrow question, which is often used to introduce simultaneous equations.
A bow and an arrow cost £11 in total. The bow costs £10 more than the arrow. What does the arrow cost ?
More challenging and much more rewarding in terms of teaching and learning: UKMT questions. UK Maths teachers are very lucky to have UKMT, because many questions from Junior and Intermediate challenges are great stimulators of Maths thinking processes.
Ideally, if it wasn’t for monotony, every lesson could be sparked off with a UKMT question, as they cover a broad range of Maths notions.
Example n°2: exploration and deduction
To introduce straight line equations, students are asked to cluster the following straight lines.
In a second step, students pick up clues that lead to writing the equation in each of the identified situations.
Example n°3: error games
Using another kind of deduction, error games are also an interesting possibility:
This can be used for a whole range of techniques, particularly where numbers are involved (multiplication/division by powers of 10, etc.)
Example n°4: a picture
‘Conceptual’ does not necessarily mean ‘verbal’. Sometimes, a couple of pictures are enough to make the concept selfevident, like in the following example on compound units (speed).
This example, by the way, comes from the 101qs website, which I strongly recommend. It is a key resource in the world of openended Maths.
Example n°5: collective processing
Another highly efficient concept processor is to organise a collective process in order to learn skills that require sequential thinking, like Bidmas or solving equations.
Let’s take Bidmas as an example. When this is learned for the first time, it is possible to work with groups of 6 students, each student representing one of the Bidmas steps (Brackets, Indices, Division, Multiplication, Addition, Subtraction). Each student is only allowed to perform his own part of the process. Then roles are switched. Then each student combines 2 roles, then 3, etc., until each student is able to perform all operations in the right order. This is quite thrilling because students gradually discover they can do very complex operations, with the support of others to begin with, then on their own.
This works for any skill that requires sequential thinking, like solving linear equations. There again, one student can be a Collector (collecting like terms), another can be an Expander (expanding brackets), another will be the Adder (adding the same number to both sides of the equation), and another will be the Divider (dividing both sides of the equation by the same number). By proceeding in this way, students will be surprised to find they can easily solve equations which would normally be regarded as complex, like algebraic fractions.
Example n°6: conceptual discussion
Although this may sound frighteningly intellectual, it isn’t: this is actually a concept processor where students are fully taken on board, and in a way that they immediately buy into. This processor can only be used from time to time, but its results are likely to trigger other processors.
The idea is simply to show students 3 or 4 videos explaining the same concept/skill. Here are 4 videos explaining how to expand brackets:
This is just a selection from many possible videos available on YouTube on the topic of expanding brackets. For the longer videos, you would only use the relevant section.
The point is to have students discuss these videos. Which they will, because they love to compare, argue, vote, etc. It is widely hoped that students will criticize these videos. You, as their teacher, will learn a lot from what they like and dislike. There are even secret hopes: for example, that students may come to realize how excessive scaffolding can make things more complicated for them. In the end, teacher and students will agree on which rules and examples should be copied in their exercise books.
]]>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 lowcost and highbenefit.
To emphasize this point, one should remember that the stumbling block for many Maths students is the howto, or the strategy. When faced with a typical Maths activity, they must first understand what to do, and then know how to do it. Students that are intimidated by these stumbling blocks will thrive in fundamental activities because these activities are repetitive and immediate.
The exercises I describe below are among the easiest to introduce to a group of students, but needless to say that teachers can imagine new exercises depending on the needs they identify in the class: there is virtually no limit.
Some of these exercises are oral, some written. If oral, you usually use cold calling, which adds pace and thrill. If written, students usually use miniwhiteboards, or exceptionally paper if you want to collect and assess results. They are best done at the beginning of a class, or at the end, or to gather attention before a wholeclass moment. They are usually done with students seated, but some teachers prefer a standing position, this is up to you.
These exercises are divided into 4 categories:
Attention / instant memory :
For many students, working on the development of fundamental abilities means first and foremost settling down. This is really the first stepping stone to reach for, the one stage that conditions much further development. That’s why it is so important to start with this first category of games.
Mental maths, quick calculations:
For the importance of regular Mental Maths practice in the development of a number foundation, see my earlier post: why is it so important for students to have a sound number foundation ?
Mental images:
Getting students to work with mental images is important, but should be introduced very gradually. Using the same activities (instant memory, mental maths), three levels are possible.
Logic :
This category of exercises aims at getting students to practise basic deduction based on clues and hypotheses.
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Referring again to the Maths ability pyramid, please note the following additional benefit of these activities: implicit learning.
Implicit learning is an intermediate learning level located in between the first layer (fundamental abilities) and the second layer (skills) of the pyramid. As indicated by the word ‘implicit’, it consists in learning something totally new without being aware that you’re actually learning. It’s a way to remove stumbling blocks, for example as a preliminary to algebra.
There are two examples of this implicit learning in the activities described above:
Back in May 2015, thousands of people gathered at Sydney Opera House for a talk by Stephen Hawking. Appearing in 3D hologram form, beamed in from Cambridge University, the physicist was asked the following question by an audience member: ‘What do you think is the cosmological effect of Zayn leaving One Direction, and consequently breaking the heart of millions of teenage girls across the world ?’
‘Finally, a question about something important’, he replied. ‘My advice to any heartbroken young girl is to pay close attention to the study of theoretical physics, because one day there may well be proof of multiple universes. It would not be beyond the realms of possibility that somewhere outside our own universe lies another different universe – and in that universe, Zayn is still in One Direction’.
Stephen Hawking is the ultimate teacher. Not only does he manage to think up a witty answer to an unusual question, but he also transforms it into an opportunity to make a wider public think differently about how they can relate to his own field of investigation, i.e. cosmology.
The fact that this answer made it to YouTube, as well as widely circulated newspapers (Daily Mail) and magazines (The Week), proves that this opportunity has been used successfully.
I think this is a lesson for all of us who teach, and particularly for Maths teachers whose role it is to answer well beyond the apparent scope of a question. Maths teachers often have to answer ‘the question behind the question’, as it were. This is due to the fact that most often students who don’t get it cannot explain what they don’t get; they just don’t get it, and it’s the teacher’s task to wind back to the source of the problem.
Thank you, Mr Hawking, one of the greatest scientific geniuses of all time, a truly human person and a living example of the 3 dimensions of kindness (more about that later…).
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