reasoning – Page 3 – Gareth Metcalfe Primary Maths

November 23, 2016November 23, 2016 garethmetcalfe Education General, Maths discussion

Enrichment in mathematics

In the early 20th century, psychologist Lewis Terman carried out a now-famous research project: he aimed to prove that by knowing a person’s IQ at an early age, you are able to accurately predict his or her life success. Using a series of intelligence tests, he identified an elite group of 1,470 children to study. Terman believed that it was these children (and others of extraordinary IQ) that ‘we must look for production of leaders who advance science, art, government, education and social welfare generally.’

Terman carefully monitored the progress of the ‘Termites’ over a period of 35 years to ascertain their life success. The results surprised many, including Terman himself. The group thrived in many ways, most notably being healthier, taller and more socially adept than the average American. However, their achievements were far from remarkable. In fact, a later study concluded that a random sample of people, given the same socio-economic status, would have achieved just as well. Terman himself reluctantly concluded that ‘intellect and achievement are far from perfectly correlated.’

Later studies went on to explore this idea more thoroughly, as explained in Malcolm Gladwell’s book Outliers. Generally speaking, people who achieve great professional success have an above average IQ: usually 115 or above. However, beyond this ‘intelligence threshold’, a person’s success is determined far more powerfully by other skills and attributes rather than the full extent of their IQ. Gladwell argued that it is hard to be highly successful without above average intelligence; however, beyond the intelligence threshold success is determined by other factors, for example the character and inter-personal skills of the individual.

Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content.
National Curriculum, Maths

This led me to think about the nature of the ‘rich and sophisticated problems’ that we should be providing. In order to equip children for long-term success, tasks should help children to develop personal and social skills as well as subject-specific expertise. Easier said than done for a busy and pressurised teacher!

That’s what I’m going to dedicate my working life towards: creating learning experiences in maths that engage children intellectually and emotionally; tasks create curiosity and lead to collaboration. I hope that this characterises First Class Maths, Maths Outside the Box and The Maths Apprenticeship.

I’m convinced that, if the system allows, these are the kinds of learning experiences that we as teachers want to provide for our children.

April 5, 2016April 6, 2016 garethmetcalfe Maths discussion, Maths ideas

Questions and Images for Deepening, part 4

Each half-term I’ve been blogging questions and images used to deepen learning in maths (hope they’ve been useful). Next year I’m going to write a resource for each year group made up of lots of these types of questions. I hope they’ll be the ultimate ‘go to’ tool for building deep mathematical thinking into daily lessons, enabling teachers to stretch children’s thinking in all areas of the curriculum.

This half-term I’ve been mainly based in my class, so the questions here are primarily from year 6. To begin with, finding the fraction of the shape that is blue where the shape is divided into differently sized pieces:
$fraction shape$

Then a question structure, used in two ways, that allows children to explore the size of fractions:
$fractions qs$

With negative numbers, we used spatial reasoning to estimate the size of the covered numbers:

Here is a simple negative number question structure:

And a visual representation to provide a scaffold where necessary:

Looking at rounding numbers, here’s a simple statement that children can explore and exemplify:

And another that leads to exploring patterns in rounding (adjust the number under the orange box):

To deepen, a question drawing together an understanding of rounding and finding the area of a right-angled triangle:

Finally, a question used in year 4 where spatial reasoning is used to identify a coordinate point:

This link gives information about the INSET training and school support that I can offer.

March 9, 2016April 2, 2018 garethmetcalfe Education General, Maths discussion, Maths ideas

Improving reasoning at the point of answer

Here’s a simple, cost-free, whole-school idea for improving mathematical reasoning – when children give an answer to a question, don’t tell them (or infer to them) in that moment whether the answer right or wrong.

Here are two reasons. First of all, we want to communicate that what we value is children’s thinking, their justification, their strategy; not simply whether they have the correct answer. In doing so, especially when this is a whole-staff approach, I believe that children become less anxious about making mistakes.

Also, by creating a moment of doubt at the ‘point of answer’ we give children the space to check their thinking and explain their thought process. Generally speaking, the greater the child’s certainty, the greater the seed of doubt I try to plant. This can be great fun, and it certainly gives children an incentive to justify and explain.

I always liked Jo Boaler’s three levels of reasoning:

I can convince myself
I can convince a friend
I can convince a sceptic

And don’t be surprised if more able children can find it harder to explain their thinking in certain contexts. I remember Mike Askew saying that if children have found an answer without much of a ‘grapple’, they are likely to have almost automatised that thought process. This can make it harder (but still very important) for a child to explain their solution.

I hope this principle gives you many great classroom moments – it certainly has for me!

February 19, 2016February 15, 2016 garethmetcalfe Maths ideas

Questions and Images for Deepening, Part 3

This school year I’ve been blogging example activities, questions and images that I’ve used to extend mathematical reasoning with my class. Here goes for Spring 1:

Sorting shapes using a branching database. Children create questions to sort, position the shapes then cover their questions. Other groups have to work out what their sorting questions are. This structure could easily be adapted and used with numbers.

Similarly, classifying shapes in a Venn diagram, with children deducing the headings:

‘All sides the same length’ and ‘At least 2 acute angles’.

Here’s a reasoning question based on finding missing angles in an isosceles triangle, with two possible solutions represented visually:

The following exhibits were used to allow children to unpick and explain the most common misconceptions when measuring angles:

And finally I’ve posted various images of dot patterns that I’ve found for subitizing games. Children are asked to recognise without counting how many dots are in the given pattern (selected age-appropriately). The different ways in which the children visualised and broke down the number of dots in each image can be explored.

Since January I’ve also shut myself away and, as I’ve promised myself for ages, finally spent time writing the Y4/Y5 follow-up to First Class Maths. I am 3 tasks (and a stack of trials/edits) away from completing it, which I’m very excited about – more updates to follow on this one soon. Have a great Spring 2!

December 29, 2015February 5, 2017 garethmetcalfe Education General, Maths ideas

Questions for Deepening Part 2

I’ve been on a mission this year to post lots of examples of questions and tasks that I have used on a daily basis to deepen the thinking of the children in my year 6 class. I’m also posting activities that I’ve seen used in classes throughout the school. Below are all my posts from Autumn half-term 2. I hope that there’s something here that you may find useful!

First of all, I found this image from a Chinese textbook showing how > and = signs are introduced:

A task from Y2 that encourages children to use the = sign in different places:
A question which is very accessible but can be extended by children working systematically to find all possible solutions:

And another allowing children to explore parts and wholes:

An ‘immersion’ activity used to encourage children to see division as grouping (e.g. representing 26/3 as ‘how many 3s in 26?’):

And some questions that I’ve used with my own class.

Area and perimeter:

Percentages:
Averages:
Below is the visual representation used to unpick that final question:

My favourite resources for providing deep, meaningful mathematical challenges are those that I have published with Alan Peat ltd. These are more rich, extended and often contextualised tasks.

First Class Maths provides deep, extremely challenging and quirky tasks.

The Maths Apprenticeship gives extended challenges for deepening mathematical and personal competence skills.

Logic Squares is all about getting children playing with numbers.

October 25, 2015October 25, 2015 garethmetcalfe First Class Maths, Maths ideas, The Mathematics Apprenticeship

Questions for deepening

We’ve all read about the vision for ‘mastery’ learning in maths; I’d figured that what teachers want is to see actual examples that they might be able to use with their class. So once we got back from our residential, I decided to post pictures of questions that I use with my year 6 class (and the odd example from around the school) to deepen their learning. The reaction on social media has been very positive, so I’ve dumped them all on this blog.

At the start of the year I used one of my favourite questions:

Which two numbers have a sum of 9 and a difference of 4?

This one’s more complex than it may initially seem as the solution involves halves. An easier version is shown below:

Sara and John spent a total of 40p at the school council shop. Sara spent 10p more than John. How much did John spend?

Then came deepening within rounding:

A question exploring the difference between negatives & decimals:
How many time do you have to halve 128 until you get a negative number?

And drawing together rounding and negative numbers:

Note that for question 3, one of the numbers is negative.

Two fractions questions with multiple solutions:

A year 4 question looking at deepening vertical addition:

And finally reasoning in multiplication:

I’m hoping to keep these posts coming (Twitter @gareth_metcalfe or FB Gareth Metcalfe Primary Maths) and, in the fullness of time, to make freely available a set of deep reasoning questions for all year-groups. If you’ve found it useful, or you know someone who might be interested, please share this blog post!

My maths resources, published by Alan Peat ltd, have also been designed to deepen children’s experience of maths.

First Class Maths provides deep, extremely challenging and quirky tasks: https://garethmetcalfe.wordpress.com/2015/02/01/first-class-maths-curious-and-challenging/

The Maths Apprenticeship gives extended challenges for deepening mathematical and personal competence skills: https://garethmetcalfe.wordpress.com/2014/05/22/tma-the-how-to-guide/

And Logic Squares is all about getting children playing with numbers: https://garethmetcalfe.wordpress.com/2015/10/03/the-learning-in-logic-squares/

October 3, 2015June 7, 2016 garethmetcalfe Maths ideas

The Learning in Logic Squares

There are lots of maths apps on the market: most are visually appealing and allow children to rehearse key mental maths skills. What more does Logic Squares have to offer? This blog gives six reasons why Logic Squares takes children’s learning to a deeper level.

Accessible challenge

At the start of each level, the player can select ‘kick-start’. This will put one of the numbers into position, giving a significant clue to help the player to unlock the level. In doing so, the levels are made more accessible for children of different abilities. There are a range of solutions for each level.

Useful feedback
When the player clicks ‘check’ any lines that are incorrect are highlighted in red. Then the player can re-position the numbers having seen their mistake(s), allowing them to address any misunderstandings in a non-threatening way.

High threshold
By the end the levels will stretch even the most able primary child. Take level 18 (using 1-20). Can the children reason as to why the number positioned has to be a 2? Children have to identify the lines with fewer possible solutions and learn to complete these parts first.

Supports understanding of = and > signs

Children’s understanding of = as ‘same as’ rather than ‘makes’ will be secured. The example on the left shows how the common structure ___ – ___ = ___ is reversed to ___ = ___ – ___. The grids from level 16 onwards are 7×7, allowing for two numbers either side of the equals sign. < and > sings are used throughout.

Subtraction as difference

Let’s assume the children start level 11 by filling in the bottom line. They may complete a vertical line using the most straightforward calculation that comes to mind (4-3=1). Soon they’ll realise that they need to be more selective in their use of the smaller numbers; larger numbers can be used in subtractions, considering differences.

Understand how to manipulate numbers in calculations
On level 12 the player has got to this point and has realised they are unable to complete the last line. Which is the first line to try to alter?The vertical line on the right hand side? Unlikely to work. The middle horizontal line? The 6 and 5 aren’t interchangeable; also, neither number could be used in the position of the 1. But switch around the 4 and the 2 and we have a solution!

So that’s how Logic Squares gets children playing with numbers and exploring key mathematical concepts in a fun, non-threatening way.

To see level 10 in detail, read the blog:
https://garethmetcalfe.wordpress.com/2015/08/31/logic-squares-the-strategy-maths-app/

Logic Squares for the iPad is on sale in English:

https://appsto.re/gb/nF2S9.i

And in Welsh:

https://appsto.re/gb/apng-.i

August 31, 2015August 31, 2015 garethmetcalfe Maths ideas

Logic Squares: the strategy maths app

Logic squares is a maths app soon to be released by Alan Peat Ltd. Players will learn to manipulate numbers within calculations, becoming increasingly strategic in their thinking as the levels progress in difficulty.

The rules are simple: fill the gaps using the numbers provided to make each line correct. To make the levels more accessible, a ‘kick-start’ button can be pressed at the start of every round which positions a number automatically. Mistakes are highlighted in red when the player clicks ‘check’, directing the player to any errors so amendments can be made.

Levels progress from more straightforward:

To the very challenging:

Let’s look at level 10 in detail and the kind of thought processes that the children will need to go through to answer it successfully.

Firstly (and crucially) children need to think carefully about their starting point. If they start with the addition and subtraction lines and ‘fill in’ the multiplication line last, they are unlikely to find a solution as there are far fewer ways to complete the multiplication. And which multiplication to choose: 3×2? 5×2? When you look at the right hand column (__+__), probably the one with the higher product. Let’s assume that the player’s started with 5×2=10:

So a logical next step would be to find two numbers that add to make 10. Which combination of numbers is best? And which way around should the numbers be positioned? Consider this:

The 4 can be made with the 1 and the 3, but with the larger number (the 6) being on the top line and all the smaller numbers used up, it’s not possible to make the top line correct. An adjustment’s needed. The player may realise that it’s easier if the larger number’s used as the answer to the addition rather than the subtraction. Say 7 and 3 are tried next:

Now the player’s in a position to find a solution (there are multiple solutions for all the levels):

I’m going to produce videos demonstrating the process of completing 3 of the levels that teachers can show to children, to make the learning from the tasks explicit. However, the thing that I love about Logic Squares is that it’s in the (highly addictive) process of trial and error, seeing links and spotting mistakes, being frustrated then finding a way to overcome, that the majority of the learning takes place. By the time the children get to level 25, they’ll have honed a wide range of crucial mathematical skills and ideas. Enjoy!

March 10, 2015March 10, 2015 garethmetcalfe Maths discussion, Maths ideas

Developing reasoning in daily maths lessons

The new maths curriculum requires children to become fluent with number whilst developing the ability to reason mathematically and problem-solve. To achieve this, children will need a broad range of mathematical experiences. Here, I will share a small piece of this jigsaw: how a ‘traditional’ maths lesson – a lesson aimed at developing fluency – can be tweaked to incorporate reasoning and problem-solving skills.

The procedural skill introduced in the lesson centred around the use of brackets. However, there are two fundamental mathematical principles that are also being developed here: the use of inverse, and the understanding of the = sign meaning ‘same as’ rather than ‘makes’. This is reflected by the questioning (mid level of difficulty) as shown below. There is a gradual progression in the structure and depth of the questions, challenging the children’s understanding of the concepts in a non-routine way.

Question 5 is then used to extend the reasoning element by using the ‘how many ways?’ structure. This challenges the children to work systematically to find all possible solutions.

These principles can be used for children of all ages. For example, presenting subtraction calculations in the following order will encourage children to reason about the underlying structure of subtraction:

13 – 8

12 – 7

11 – 6

Equally, consider how the following equation helps a child to develop their conceptual understanding of multiplication and of the = sign:

4 x 5 = 4 + 4 + 4 + 8

Also, the ‘how many ways…’ question structure is enormously adaptable, allowing you to build reasoning into maths lessons on a daily basis. I hope it’s a little technique that some people may find useful!

October 13, 2014October 13, 2014 garethmetcalfe Maths ideas

My favourite properties of number question

‘Do all odd square numbers greater than 1 have 3 factors?’

I love this question. Let me explain why, and the train of thought that it can generate.

Children usually start by generating the odd square numbers, identifying that square numbers have an odd number of factors (1, the number and the square root). This helps to underline the uniqueness of square numbers. Nothing new here, really.

So children tend to identify the first two or three odd squares greater than one (9, 25 and 49) and realise that these numbers only have 3 factors. This, I tend to find, is enough evidence to convince most children that the answer to the question must be ‘yes’.

However, this is of course a false presumption, and by making it children realise a crucial mathematical principle: that finding examples to support a theory is not the same as finding a proof. There must be reasoning as well as examples to generate a proof!

So I then ask the children to consider 81. And they soon realise that it is also divisible by 3. What’s happened to the pattern? And why?

Children then investigate further examples, noting that 5 of the first 6 odd square numbers greater than 1 (9, 25, 49, 121, 169) have only 3 factors. But then 225, the square of 15, has 9 factors! There must be some logic here, and of course there is. I ask the children to consider the square roots of each number, as underlined below:

After much discussion and deliberation, and a healthy dollop of struggle, someone makes the breakthrough: the numbers with prime square roots have 3 factors; other numbers can be further divided by the factors of the square.

But does this pattern continue for all odd squares? And why was 1 excluded from the list?

It’s an amazing question for exploring the very nature of the properties of number, and particularly primes and squares. And, in my opinion, for experiencing the joy and beauty of mathematics!