Find a 10-digit number where the first digit is how many zeros in the number, the second digit is how many 1s in the number etc. until the tenth digit which is how many 9s in the number. 

 

 

Have you thought about it enough?

 

We have found this solution

6210001000

 

Did you find any more? Let us know in the comments

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With one straight cut you can slice a pie into two pieces.
A second cut that crosses the first one will produce four pieces.
A third cut can produce as many as seven pieces.

What is the largest number of pieces that you can get with six straight cuts?

Have you had enough?

 

 

The solution:

Lets start with a pie that has 0 cuts – this has 1 piece

0cut

If we cut it once with a straight line – we now have 2 pieces

1cut

If we cut it again, 2 cuts – we have 4 pieces

2cut

Cut 3 gives us 7 pieces

3cut

Lets have a look at these paying special attention to the number of pieces added by each cut.

3cuts

Cut 1 adds 1 piece to the total
Cut 2 adds 2 pieces to the total.

If we extend this on

cut 3 should add 3 pieces to the total
cut 4 should add 4 pieces to the total
cut 5 should add 5 pieces to the total
and cut 6 should add 6 pieces to the total.

6cuts

Cut 4 gives 11 pieces.

4cut

Cut 5 gives us 16 pieces.

5cut

Cut 6 can make up to 22 pieces

6cut

6 cuts can make 22 pieces of pie – I’m not too sure I’d be too happy if someone offered me some of those pieces though!

PISA tests are occasionally in the news and well known amongst teachers. They rank the 65 OECD countries according to results in standardised maths, science and reading tests. Here in the UK, we ranked significantly higher in reading than in maths (23rd in reading, 26th in maths). Incidentally, our ranking in both is dropping, described even by government as “at best, stagnating, at worst, declining.”

pisa tableThere is a perception that it is the Far East that dominates the top of PISA rankings. Whilst it is true that the top 5 are the likes of Shanghai, South Korea and Japan, the top 10 contain such nations as Poland, the Netherlands and Finland.

Similarly, the excuse that language differences inflate these differences can be quashed by the fact that English-speaking, economically-equivalent countries such as Australia, New Zealand, Ireland and Canada all perform significantly higher than the UK.

There is also little to support the myth that educational systems take generations to change: Ireland ranked 32nd and 21st in maths and reading respectively in 2008, but had improved these to 20th and 6th by 2012.

I do support the idea that cultural differences affect rankings. Talking to a few teachers who participated in the Shanghai exchange, they overwhelmingly felt that whilst teaching played a small part in the success, other factors (such as the one-child policy, the length of the school day, the emphasis placed on academic success within families, the narrow curriculum) played a far bigger part. But this is hardly a surprise, and begs the question: why did we exchange maths teachers with Shanghai when perhaps the answers were closer to home – Ireland, or Finland, perhaps?

Given the fact that the Shanghai exchange has led to almost no visible reform in teaching practice in the UK it is perhaps a fair conclusion that our teachers were doing a decent job already: what needs to change is our attitude towards learning as a nation. The issue is cultural. Whilst there are improvements that could be made in schools, I am sure that when we as a nation take the time to celebrate academic achievement more, and to avoid phrases such as “I can’t help him, I was always rubbish at maths” and “I never read as a child either”, progress will be made. Kids that don’t read regularly usually have parents who don’t either, and kids who disengage from maths usually have parents who are too scared to help properly with homework. But this can change – and needs to – although how is a whole other question.

PISA tests are occasionally in the news and well known amongst teachers. They rank the 65 OECD countries according to results in standardised maths, science and reading tests. Here in the UK, we ranked significantly higher in reading than in maths (23rd in reading, 26th in maths). Incidentally, our ranking in both is dropping*, described even by government as “at best, stagnating, at worst, declining.”

Why do we consistently perform better in reading than in maths? I think the answer lies in emphasis we put on reading (as opposed to maths) from a very early age. We read to our children at night from a very early age. Throughout primary school, children work through personalised reading schemes and bring a book home every night. And in adulthood, it is much easier to be bad at maths than bad at reading (in fact, we even have a national standard excuse for it – “I was always rubbish at maths” is almost a catchphrase, but something one would never say about reading).

The Biff, Chip and Kipper series of books is used by over 80% of UK schools

The Biff, Chip and Kipper series of books is used by over 80% of UK schools

The National Literacy Trust states that if a child does their 10 minutes of reading daily, they are 13x more likely to reach their expected reading age by the time they leave primary school. Schools invest in personalised, carefully-graded reading schemes like The Oxford Reading Tree (aka Biff, Chip and Kipper) and work hard to make sure children read them regularly. But if reading is resourced and encouraged so strongly, how does maths fair in comparison?

In fact, no such ’13x’ stat exists for numeracy. Very rare is the school that sends home 10 minutes of maths daily. In fact, maths homework generally consists of a weekly one-size-fits-all worksheet or on-line game which, to be frank, does little if anything to raise standards. Of course for a teacher to produce personalised homework for each individual would be almost impossible, which is where adaptive learning in the form of DoodleMaths comes in.

If 10 minutes a day can make such a significant difference in reading, just think what a difference it could make in maths!

*There is a perception that it is the Far East that dominate the top of PISA rankings – or that it is language differences that cause these changes. But English-speaking economically equivalent countries such as Australia, New Zealand and Canada all perform significantly higher than the UK. There is also little to support the myth that educational systems take generations to change: Ireland ranked 32nd and 21st in maths and reading respectively in 2008, but had improved these to 20th and 6th by 2012. Why do we exchange maths teachers with Shanghai (where cultural differences such as the one-child policy do skew results) when perhaps the answers are closer to home? For more on our PISA ranking, click here

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I first came across inductive learning on a training course early in my teaching career. We were given individual diagrams of 2-D and 3-D shapes and asked to group them. What was fascinating was how everyone came up with a different outcome. Through this, and by communicating this to each other, we developed our understanding of the properties of 2-D and 3-D shapes. I have used this with great effect many times since in my classroom.

Inductive learning is defined as allowing students to “generate their own information, organize that information, make sense of what they have collected, and communicate their understanding to others” (Dell’Olio & Donk, 2007).  It is the most powerful way for children, and especially young children, to learn things for themselves.

The traditional ‘chalk and talk’ teaching style (teachers explains > students practice > teacher reads out the answers) is probably still the best way to manage a maths classroom with 30 students, and if had to estimate, I’d say that across the country right now, 75% of maths lessons would be taught in this or a similar style. However, most of us learn best through doing things for ourselves. Now it is difficult to let a class of 30 students loose on a set of problems without being sure they have a decent understanding of what’s coming (we probably all recall that lesson at school where we thought we understood, but when it came to reading out the answers, we’d actually got them all wrong). But with technology, not only are we able to intervene as soon as a child gets something wrong, but we can also actively encourage children to try things out, experiment, and derive their own conclusions from a question.

With DoodleMaths, when we want children to learn something new, we have developed what we call “closed-outcome inductive learning” questions. In such questions, children are asked to sort, link or order information that is carefully selected to provide a single desired learning outcome. There are approximately 1500 such questions within the app. Below is one such example:

KS2 (5)

In the example shown, two of the fractions are in quarters, two are in fifths. This should be sufficient to induce a child into understanding (or recalling) that the denominator represents the number of divisions a fraction is split into.

Most digital maths products have not put technology to its best use. Typically, tutoring apps and websites tend to replicate the chalk and talk style but on a one-to one basis. There are two flaws here. First, and as previously stated, this is not the best way to learn on an individual basis. Second, a video or animated explanation is inferior to a tutor, and typing in an answer on a keyboard is generally inferior to writing it. If we are to use technology to genuinely raise standards in maths, we need to embrace what it is good at and use it to its advantage: with DoodleMaths, children learn through doing, through instant feedback, through regular practice at a work program that we are able to adapt to their evolving level, strengths and weaknesses.

Children still occasionally need support, which is of course why we have the help button – and the help button is where most of the development of DoodleMaths will come in the next year. But we believe that children should try to learn things for themselves first, and only listen to an explanation when they need to. This produces more engaged, confident learners who progress demonstrably faster.

In fact, the question most people ask is, “what on Earth is an adaptive algorithm???”

An algorithm is a formula, process or set of rules that you (or a computer) follow in order to complete a task. An algorithm for brushing your teeth might be:

open bathroom cabinet > take out toothbrush and toothpaste > open toothpaste > measure needed amount onto toothbrush > brush teeth etc.

Miss a step, or do it in the wrong order, and the task is not completed satisfactorily.

We use algorithms in DoodleMaths to analyse responses and set work that is adapted to suit each user’s individual needs.

Why do we need these adaptive algorithms? The answer is simple, and something we have touched upon in other blogs: making the right choice is extremely difficult.

Children don't always choose what's best for them

Children don’t always choose what’s best for them…

Give children the choice of what to learn and most will choose the simplest task that will give them the greatest reward most rapidly. Ask parents to choose a topic for their child and they are usually baffled by the choice of topics they are confronted with (the website shown below invites parents to choose from the content for Year 4. Not only is this overwhelming, it also neglects the fact that many children in year 4 still haven’t mastered Year 3 topics, whilst others need to be stretched by Year 5 work). Whilst teachers are not baffled by such a choice, they are still likely to select work on the basis of what is best for the class as a whole and not each individual. The truth is, the exodus towards digital resources over the last 10 years has not raised standards in maths. Common sense will tell you that simply replacing paper-based content with digital will not accelerate a child’s learning.

Parents are often baffled by the choice that some websites offer

…and parents are often baffled by the choice that some websites offer

The advent of ‘big data’ and the power to process it has opened up a better way to do things. We collect information on every question answered by every child  – every minute of every day. This is analysed on an individual basis to determine the level a child is working at, their strengths, weaknesses, and the pace at which they need to learn. The data is also analysed on the basis of the population as a whole in order to hone our questions and algorithms and make them more effective.

Of course, there is nothing new about we are doing. Any good tutor will assess a child to determine their level, strengths and weaknesses prior to teaching them.

And it’s fairly easy to determine a child’s weaker areas. What is more difficult is to respond to these in a way that maintains a child’s motivation and doesn’t knock their confidence – particularly when you can’t see their face or listen to their voice. This is where our content is vital. Questions are very carefully written to work alongside our algorithms such that they are always slightly, incrementally harder than what has been previously mastered. All our content has been written by leading maths teachers.

To be clear: with DoodleMaths, children get no choice at all in what question comes next – it is determined purely on the basis of what they most need to learn. This, though, is what gets – and guarantees – results. Our competitors, in the form of digital resource banks, are only as effective as the individual selecting the work. With DoodleMaths, on the other hand, we can say, with confidence, that any child earning 100+ doodlestars per week for four weeks will increase their maths age by three months.