The post Not all uses of equals signs are equal appeared first on Creative Maths.

]]>The sign “=” was invented in 1557 by Robert Recorde a Welsh mathematician. He was tired of having to write out the phrase “is equal to” too often. Now we cannot imagine maths without an = sign.

Using an equals sign correctly can be a challenge. Whenever an equals sign is used, it signifies that the expressions on either side are equal. A sequence of expressions separated by equals signs should all be equal to each other. For example: 4 + 5 = 3 × 3 = 21 – 12

However, the equals sign often gets used in an operational way, meaning “give the answer”. Our calculators do not help at all, as we press the “=” sign as an instruction to calculate. Many students have an operational view of the equals sign. The problem with this is that the operational use of the equals signs leads to incorrect equality statements.

How many mathematics teachers have felt their life-force draining out on reading the following:

2 + 3 = 5 × 2 = 10 – 9 = 1

The person has added two and three to get five, which they then multiply by two. Next they subtract nine to get end up with 1. The whole procedure could have been written (2 + 3) × 2 – 9

Or it could have been written 2 + 3 = 5 5 × 2 = 10 10 – 9 = 1

Either of the correct ways seems long and tricky compared with the misuse of the equals sign, sometimes called the running equals sign. My experience is that most students will do anything to avoid any more writing than absolutely necessary, which makes correctness and rigour even less attractive.

Things that are understandable as part of a narrative do not survive the permanent written form. You can see this problem in the TV programme, Countdown, where the maths person shows calculations as they are happening in narrative. Because they are explaining in real time, it seems more acceptable. And they use a horizontal line which, though it represents an equals sign, seems to be more appropriate with the narrative approach.

Spreadsheets contribute to the confusion, with the equals sign indicating that we are putting a formula in a cell. In programming an equals sign sometimes means “Check to see if these things are equal” and sometimes means assign the value of this expression to the variable.

As a mathematician I would love the equals sign to be used correctly. As a teacher of mathematics, I am finding this uphill work.

Mathematicians work with other mathematicians. Mathematicians also come back to work they have done previously. It is vital that what is on the printed page is correctly written to enable logic and computation to be checked. This is especially true in education as teachers and instructors need to be able to understand the students’ thought processes in order to identify where mistakes are occurring.

In algebra and other higher level mathematics it is vital to understand the relational nature of the equals sign. Much of rearrangement and solving of equations relies on the correct use of the equals sign. I also wonder if the difficulty students have with inequality signs is related to misunderstanding of the equals sign.

Get students to find the mistakes in a theoretical student’s work. Put something like

25 + 30 + 27 = 82/2 = 41 on the board and get students to say whether or not it is correct, how to correct it and what they would say to help the person who wrote it to understand the mistake.

Talk about the operational usage of the = sign as being like pressing the button on the calculator to find the answer.

I have been known to use writing the equal sign after an expression as an indication to students to give me an answer. Knowing better I would not do this again. I hope.

A teacher should always show correct working in examples, making explicit why they are doing so.

Say “is equal to” rather than “makes”

Here are some examples from the Equals Sign test, an instrument used to evaluate students’ understanding of the = sign. You can read more about it in the paper referenced at the end of this post.

At Grade 2 (Year 3) level

- _ + 3 = 5 + 7 =
- 4 + _ = 5
- 3 + 5 = 4 + _

At Grade 6 (Year 7) level

- 15 – 7 = _ + 5
- 6 + 3 + 7 = 5 + _
- 6 × _ = 40 – _

By providing questions that place the = sign in different positions we are helping to de-emphasise the operational use of the equals sign. I would also suggest exercises that have more than one = sign to draw attention to the ongoing equality across several expressions. If you are using a textbook check to see if there is variety in presentation.

The lack of an easily written and understood alternative to the equals sign makes it difficult to be correct without being too complicated.

I saw in a post from another country (Denmark perhaps?) that people cross out the equals sign when it ceases to be true. The equal sign was true when we wrote it, but because we are adding another step it is no longer equal. The example we have been using would look like this when it was completed.

2 + 3 ≠ 5 × 2 ≠ 10 – 9 = 1.

I suspect this would increase confusion unless it became a convention in a whole school system. Perhaps drawing a box around each completed step would be an interim solution.

Can we invent a new symbol? Preferably one that already appears on the keyboard? What about a hash? It’s like two equals signs across each other and could mean a temporary equals sign that has been superceded:

2 + 3 # 5 × 2 # 10 – 9 = 1

Or maybe 2 + 3 =≠ 5 × 2 =≠ 10 – 9 = 1

=≠ means the expressions used to be equal, but are no longer because we have carried on with other operations.

Or maybe » which points to where we are going and has two parallel bent lines like an equals sign.

(Note that these are just to stimulate thought – though Robert Recorde was the first person to use the equals sign – maybe we can invent the operate sign.)

This is a useful reference that discusses the problem of the misunderstanding of the equals sign and reports research on students and textbooks from four different countries.

This is a problem which mathematics teachers have grappled with for decades, possibly even centuries. I’d love to hear your suggestions on

- How to teach to avoid the problem of the running equals sign.
- What other symbols or conventions we have available or could invent.

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]]>The post Resources in maths and stats for a pandemic appeared first on Creative Maths.

]]>In New Zealand we have yet to feel the full force of the Covid-19 pandemic, but anxiety hangs in the air. Around the world schools and colleges are closing their doors to slow the spread of the virus and students and teachers are forced to enter the world of distance learning.

Nine years ago the Christchurch earthquakes meant that the University where I worked was in similar circumstances. It is not an easy time. Here are some resources to help

At Statistics Learning Centre we have over fifty short, engaging, well-explained videos suitable for introductory statistics courses. Some videos have received over one million views and many are linked into courses all over the world. The videos are used by Australian Bureau of Statistics, World Vision, Open University and Open Polytechnic of New Zealand. You can see a summary of each of the videos here – which will help with selecting videos for your classes.

Most of these videos are available free-of-charge on YouTube. For a $5 monthly membership, the others are also available on YouTube. ALL of the videos are listed on our site.

The combination of videos and activities can really help develop learners’ understanding. Two of the videos include quiz questions – Classifying Types of Data and Choosing which Statistical test to use, practice scenarios.

We also have StatsLC online courses based on the New Zealand curriculum which instructors in other countries are welcome to use. We are happy to give a month free trial to teachers, and student subscription is just $5 a head. If there is sufficient demand we will make a tailored course for introductory college statistics.

In New Zealand we have a unique and innovative statistics curriculum, which means that materials developed in other countries to teach statistics are not particularly useful. Our StatsLC resources are specifically tailored to the New Zealand curriculum and particularly NCEA levels 1, 2 and 3. Subscriptions are currently $5 per student for levels 2 and 3 and $1 per student for level 1. Teachers can track student participation. As always, we are happy to give a two month free trial to teachers and students.

It is concerning to think of schools closing their doors. Teachers cannot just send work home and expect parents to be able to help their children to complete it. At Creative Maths we have a variety of resources, many of which are free, that can be used to keep learners engaged in mathematics at home. Games and activities are a great way for learners to keep up their maths fluency and interest.

Take a look at these:

Factor Detector – a pen-and-paper puzzle that develops fluency around multiplication facts

Guess the Multy – a free online game like “Guess Who” around multiples and factors

Spiral drawing – learn about angles – it’s mesmerising!

Ages 4 to 7 The Cat Pack – so many fun mathematical activities and games

Ages 7 to 12 Dragonistics data cards – a rich set of 240 dragons and attribute cards with many applications

Multiplication fluency – Multy Facty game

Fraction addition – Fraction Action game

There are many other sites providing a deluge of different resources. We draw attention to particularly appealing ones in our fortnightly newsletter.

If you find an appealing one, let us know in the comments below!

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]]>The post Fraction Addition and Subtraction with the Denominator-ator appeared first on Creative Maths.

]]>Many of the adults I teach are confused when it comes to fractions. It can be difficult to remember that addition and subtraction of fractions require common denominators, which stay the same when you add and subtract, while multiplication operates on both the numerator and the denominator. I have written about this: The big deal about fractions

Fraction addition confuses. A fraction operation such as 2/3 + 3/4 requires five operations to get the two fractions to a common denominator and then add the numerators. It can be difficult to explain why this is needed. Addition also seems to begin with multiplication, which further confuses. Lack of fluency with multiplication facts can provide a stumbling block, as the working memory is busy multiplying rather than focussing on the process of fraction addition.

Adult students want to go back to a rote, successful method they can almost remember from school. They revert to a method known as cross-multiplication, the butterfly method and “upside down picnic table”. Nix the Tricks speaks out against this! I have a Pinterest board of Bad Maths teaching resources, full of them.

A problem with a memorised method without understanding is that it is easily forgotten or confused. Neither does a memorised method help develop understanding of the nature of rational numbers (fractions and decimals) and rational number sense.

Early in fraction topics, time is spent finding equivalent fractions. The idea that there are multiple ways of expressing the same number (in fact infinitely many ways) is new compared with whole or natural numbers. For instance ½ can be expressed as 2/4, 3/6, 100/200, 0.5 etc. We seldom express two as anything other than 2, so whole numbers appear to have just one label.

Students are asked to find equivalent fractions with given denominators, which may seem like a mathematical exercise with little purpose. As teachers we know that they will need to change denominators for addition and subtraction, but we don’t really want to do that first. It’s a bit circular.

While pondering this, I invented The Fraction Assistant. I really want to call it the Denominator-ator but it’s a bit of a mouthful for students who are not yet sure what a denominator is. At my Maths Jam meeting, someone said they thought it was for rating denominators. To which I replied, “But why would you rate them when denominators are so common!” (What is it about maths people and puns?)

The Fraction Assistant consists of strips of equivalent fractions, which are slid up and down a frame until the denominators are the same. This provides a powerful image of what happens when converting to common denominators. The plan is not for learners to continue to use the Fraction Assistant, but rather to gain understanding. By using the Fraction Assistant for some addition and subtraction exercises, studying the equivalent fraction strips and making some of their own, students can move on to fraction addition and subtraction without the assistant.

The frame of the Fraction Assistant displays the terms, denominator and numerator, so that learners become familiar with them. In addition, the lower half is shaded for all the fractions and the frame, making it clear what needs to be made the same.

The strength of the Fraction Assistant is that it isolates the concept of finding common denominators from the mechanism of Lowest Common Multiple and conversion. Those come later once it is clear why we need to find common denominators and can build on the imagery of the equivalent fractions strips.

We have made a prototype “Print-and-Learn” that you can download from our website, along with suggestions for teaching.

We would love feedback from teachers and learners, and suggestions for further development.

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]]>The post Creating and critiquing good mathematical tasks with variation theory appeared first on Creative Maths.

]]>- Careful selection of exercises can turn purposeful practice in maths into a task that also develops conceptual understanding.
- Poor, off-the-cuff or random selection of exercises can create barriers, feed misconceptions and at best miss out on opportunities for better learning.
- Using a framework of variation theory can help teachers examine and improve their practices and tasks, preferably collaboratively

If students can learn a spurious rule for answering questions rather than the desired concept, they will grab it with both hands. In my class a student worked out that if the scenario involves faults in tyres, it must be a binomial distribution. Another rule was that if the question gives a mean and standard deviation, then it must relate to a normal distribution. These rules often work in that they help students get correct answers in assessments, but they can hinder learning. When I became aware of this in my online resources, I made sure to vary the contexts and the ways that scenarios are introduced in order to go against any spurious rules.

Variation is inevitable, so we need to let it work FOR us. In creating experiments, as statisticians, we learn about variation, and we use control groups to limit the variation as much as possible to what we are focussing on. Similarly when we are making up exercises for students, we need to consider carefully what we wish to vary and what we wish to control.

Variation theory in the teaching of maths is a framework to examine our tasks and teaching sequences. I have found it fascinating to examine the examples I give when teaching and see the variation that I miss if I do not take time to prepare well. Kullberg et al state: ” The variation theory of learning emphasizes variation as a necessary condition for learners to be able to discern new aspects of an object of learning.”

The following quote from Watson and Mason speaks to me:

“We claim that if the teacher offers data that systematically expose mathematical structure, the empiricism of modeling can give way to the dance of exemplification, generalization and conceptualization that characterizes formal mathematics.”

For a fuller summary, leading to some more great articles, see the Cambridge Espresso by Lucy Rycroft-Smith

Recently I have been working with the Theorem of Pythagoras. I wonder if the adults I teach have been shown non-examples of right-triangles and seen that the theorem does not hold there? Is it really clear that the rule only applies to right-triangles? Do we have examples where we cannot use it – there is no right-angle, so will need other tools?

When we are teaching about binomial distribution, we need to give instances where the scenario is not suitable to model as a binomial distribution.

We examine a pair of box plots for evidence of difference between two groups. One example of a pair of boxplots gives little indication as to what matters and what does not. We need to see multiple examples of pairs of box plots, with varying degrees of overlap, differences in location and spread, different contexts and even different orientations, in order to see the commonalities. And coming back to the contrast – students need also to see pairs of boxplots that do NOT show evidence of difference between the groups.

In order to be able to provide a good range of examples and exercises, we need to think carefully about the potential aspects for variation.

My adult students often find it difficult to multiply by powers of ten. I have explored the different variations that are possible, and came up with the following list, which may or may not be exhaustive:

- Different powers of 10 (10, 100, 1000)
- Other number has trailing place value zeroes. (eg 340, 7300)
- Digits after the decimal point. (eg 5.7, 29.33)
- Number needing to have zeroes added vs just change in position (eg 5.5 × 10, 6.5 × 100)
- Zeroes inside the number (eg 805 × 10)
- Staying less than 1 (eg 0.0072 × 10)

This is invaluable when I am creating a learning task made up of a number of exercises. I can make sure that I include all the variations in a way that helps conceptual development.

I call this the binomial faulty tyre effect as one of my students told me that if the question is about tyres (tires for US audience) then the distribution must be binomial. We can unknowingly set up patterns that create false learning. Using certain contexts for different distributions can provide a hook on which to hang the distribution, but students can entrench unimportant or coincidental effects. Sometimes use the same context for different distributions. Here is an example of using the same context for two different probability distributions.

- About 50% of all calls to an emergency centre are false alarms. Out of twenty calls to an emergency call centre, how likely is it that there will be fewer than five false alarms?
- Emergency calls occur at the rate of thirty in one hour. What is the probability that there will be a call in the next five minutes?

This is talking about combinations of factors. For the boxplot example, we make sure that we look at different overlaps, different spreads and different combinations of both.

This does seem like a lot of work, but once a set of tasks is developed, it can be used and refined by many teachers over many lessons. This is a benefit of the Japanese “lesson study” approach, where several teachers focus on one specific lesson and work together to create a teaching sequence, observe the teaching and learning and develop the plan further.

This is my initial take on variation theory. If you found this interesting I would suggest following up the references on the Cambridge Espresso. There is also an interesting podcast with Craig Barton talking with Watson and Mason about variation and task development. Craig Barton has a whole website around variation theory. He is clearly a fan!

I find the ideas discussed here around variation and structured exercises compelling. I have long been concerned that problems made up in response to children’s input may well lack the required variation to develop full understanding of a desired concept. Conversely I am also concerned that questions made up “on the fly” by a teacher who is not confident in mathematics may lead to confusion over concepts for which the teacher has not yet got a strong understanding. I would be interested to see how variation theory would fit in with teaching mathematics through student inquiry.

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]]>The post Talking Maths in Public appeared first on Creative Maths.

]]>There are three types of people in the world: those who can count and those who cannot.

Just kidding.

But the ways people respond to mathematics can be put roughly into three groups – the maths-likers, the perplexed and the traumatised. See “Writing about Maths for the Perplexed and Traumatized” by Steven Strogatz.

Strogatz uses the term “naturals” for this group. The maths-likers are people who liked maths at school and find it interesting. Some maths-likers go on to become maths teachers or accountants or statisticians or work in some other area that uses mathematics. A few become research mathematicians. Whether or not they continue to use maths in their careers, they like it. They may even play with maths in their spare time or go to MathsJams.

The traumatised I have written about before. The traumatised had experiences in their schooling that left them feeling that they were not good at maths and would never understand maths. These are the people who step back from you when you explain that you teach maths. (As an aside, do people think we like it when they cheerfully tell us how much they hate the thing we do?)

And there are the perplexed – the ones who were able to answer the questions in the maths book and in assessments, probably by working hard, following directions and suspending disbelief. Maths was not something that they really got, but it was not a problem to them. They are perplexed because they can do maths, but they do not understand why anyone would find it more than useful.

I recently attended a conference in Cambridge England for mathematics communicators. It is the second such biennial conference, and brought together over a hundred people, most of whom would fit happily in the maths-likers category. It included bloggers, writers, YouTubers, teachers and students. Organisations such as the Royal Institution, the Museum of Mathematics in New York and Open University were represented. Matt Parker entertained us as Master of Ceremonies and we heard from Simon Singh, Brady Haran from Numberphile and a whole range of other communicators. You can see the exciting programme here: http://talkingmathsinpublic.uk/programme/

You can also hear an interview with Jesse Mulligan on Radio New Zealand National here.

I was able to attend, all the way from the other side of the globe, with help from the Canterbury Mathematical Association. In order to make the most of the opportunity I gave a workshop entitled “The Magic of Statistics – with Dragons”, which was well received despite my lack of voice. It was nearly “Miming Maths in Public” from me. The delegates loved my dragons!

It was an amazing experience which helped me gain confidence and ideas in my own efforts to talk maths and stats in public through this blog, the YouTube channel and games. I particularly enjoyed the presentation on being inclusive to all people. The magician Neil Kelso reinforced the need for kindness. Magicians tend to make me nervous, but his genial and warm approach was a wonderful model for me to build on.

On a totally unrelated note, I loved being the subject of so much Prime Minister envy. (NZ’s Prime Minister, Jacinda Ardern, has a big following in the UK.)

One thing the conference made me think about is who is our audience, and who should be our audience. I will focus this on YouTube, as that is currently my main area of influence.

The internet and Wikipedia are facilitating the democratisation of knowledge. Almost anyone can find out almost anything by looking it up using a search engine (most often Google). YouTube takes it a step further in democratising understanding. Any time I want to find out how to do something, I go to YouTube.

Some mathematics communication is aimed at maths-likers. It shows cool and beautiful and exciting abstract and applicable mathematics. Through these videos we can encourage more learners to feel happy about mathematics. Numberphile and the Royal Institution are of this type.

Some mathematics communication is aimed at the perplexed. These are sites like Primrose Kitten (though why a kitten would be yellow escapes me) which help with exam preparation. It was fun to see the massive spike in views the day before GCSE exams in the UK. My own statistics videos help millions of people who are struggling to understand their statistics course. I hope, as I’m sure PK does too, that our help will possibly move the learner out of the perplexed category so that they see a bigger picture. Khan Academy was an early arrival on the YouTube educational scene. It has a procedural focus and helps people who want answers. I have written elsewhere about Khan Academy.

My question is, what is there for the traumatised? Their lives have been blighted by bad mathematics experiences and we need to make sure there is help for them. In order to develop a more mathematically capable citizenry, we need to be working at all levels. Parents and teachers who feel better about maths will help young learners avoid catching maths trauma.

Success helps people to feel better about maths, so the helping videos embraced by the perplexed may help the traumatized also. However, I would like to do more work in this area, to find out what would help. What topics are relevant and exciting? Will it help to provide links between mathematics and science, art, music, nature, computing, probability, economics and business?

Education does not begin and end at school and higher education. People continue to learn throughout life, and the opportunities to do so through the internet and video are mind-boggling. As a maths communicator, I am keen to provide opportunities for all, including the traumatised, to grow in their understanding of mathematics throughout their life.

Along with Strogatz and my favourite magician, I believe the key lies in the heart. We must show empathy and kindness when reaching out with mathematics.

The enduring memory from my attendance at Talking Maths in Public was the kindness of strangers. The people were an interesting lot and neurodiverse. Everyone was kind, and everyone wanted to make the world a more mathematical and better place.

If you have ideas that can help me in my quest, please comment below. You can also support my work by becoming a YouTube channel member.

And big thanks to Katie, Sam, Ben and Kevin who did such a great job of organising TMIP.

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]]>The post How to help your child with her multiplication facts appeared first on Creative Maths.

]]>Fluency with multiplication facts makes learning later skills easier. When simplifying fractions, it is helpful to know multiples of numbers. When learning division algorithms, fluency in the basic facts means that the brain is free to learn the new procedure. In algebra, it is extremely helpful to be able to recognise common factors of two numbers, such as 36 and 24. Being fluent with multiplication facts is invaluable for estimating in many areas of life. A recent survey of high school teachers reported that they value knowledge of tables highly.

The advantage a parent has over the class teacher is the opportunity to work one-to-one with a child. However here are multiple traps in parents helping their children. Maths anxiety is catching. If you have a bad relationship with maths, it can be difficult for you to mask this so that it does not pass on to your child. This is particularly important for mothers with daughters, as this has been found to be the relationship that best transmits and maintains intergenerational maths anxiety. As a parent you need a growth mindset around mathematics.** “I used to find mathematics difficult, but I know if I work at it, I will get better and so will you”**, is a much better message than “I was no good at maths and you get that from me.”

Conversely a parent who has a particular inclination towards mathematics can also be intimidating to the child. It can be difficult to be patient when you have no idea why they are finding something so difficult. This impatience is counter-productive.

Another problem is that the child may be taught in school in a way that makes no sense to the parent and teaching the “old-fashioned” way may confuse the child or annoy the teacher who is trying to embed a different method. When it comes to long addition, multiplication or division, parents are best to find out what method the teacher is using and try not to teach tricks. (see Nix the Tricks)

Many of us have emotionally charged memories of being tested for multiplication facts and worrying that we would be too slow. It is very helpful for learners to gain automaticity with multiplication, but speed tests do not develop speed, nor do they test for fluency.

Flash cards are ok – it’s what you do with them that matters!

The nice thing about flash cards is that you can choose just a few to focus on and you get instant correct answers. Speed is NEVER important. Using a timer creates anxiety and blocks the kind of thinking that is needed. I am fluent and confident with my mathematical facts but put a timer on and that is all I can think about. How much worse is it for learners who are already struggling?

A commonly encouraged order to learn your multiplication tables are: 2, 10, 5, 0, 1, squares, 3, 4, 6, 8, 9, 7.

The foundational facts are the 0, 1, 2, 5, 10 and squares. Do not move on to the other tables until these are mastered.

From these foundational facts, the other facts can be built up. Three times is two times plus one time. Four times is double then double. When learners build on a foundation, their understanding grows at the same time as mastery. Learning unconnected facts by rote does not help long term conceptual understanding, and the facts are more easily forgotten.

Stories help us to understand what is happening with multiplication, and having a sensible context makes very wrong answers more obvious. Here are some examples of multiplication contexts that can occur around the home.

- “There are three children in this family. I need to make enough cookies for one each day for each of them for a week. How many cookies am I going to need?” “What if I need two each for each day?”
- We have two pets and each pet has four legs.
- How many legs altogether on the table and chairs?
- Every day for the next five days I am going to walk for twenty minutes. Is that going to be two hours altogether?
- We are having KFC and want two pieces each. How many pieces?

The first way children are introduced to multiplication is the idea of equal groups. Two equal groups of ten blocks gives a total of twenty blocks.

Another meaning is repeated addition. Adding four fives together is the same as multiplying four times five.

Arrays occur when equal groups of items are arranged in rows. We would think of 2 times 5 as being two rows of five objects. We use this representation in our Multy Facty resources, poster and game.

We all have access to objects for rearranging. I like milk bottle tops, but Lego bricks, counters, stones and beads are all good for counting and arranging.

Take a number of objects – for example 21 and count it in different ways. Can you count it in twos? There is one left over. Count in threes, count in fours. Rearrange in arrays and in groups. This helps build number sense and the idea of “tidy” numbers and primes.

Mastery requires substantial and enjoyable practice. Well chosen games help apply multiplication facts and procedures. We recommend our own game Multy Facty, which embeds mathematical principles into the game and develops fluency. Well-designed games encourage flexible use of number facts and remove the pressure from “doing tables”.

I was not a good maths parent for my older son, W. His father and I both had mathematically-oriented careers so it was a mystery to us that he did not seem to grasp mathematics as intuitively as we did. Sadly I did not keep that to myself. I can not remember doing much except helping with algebra when needed. He was grateful then to have a maths teacher for his mother. W edited many of my statistics videos and managed to stay wilfully ignorant of most of it.

Our other son, J, is blind and autistic. He could subitise (tell how many something is at a glance) with his fingers at a very early age, and is a calendar counter who can tell you the day of the week of any date. He was also remarkable at converting between ways of expressing time – we would say 6:45 and he would say quarter to seven. My greatest mathematical teaching achievement with J was teaching him to solve linear equations.

How have you helped your children with maths?

Let me know in the comments below.

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]]>The post Fluency in maths appeared first on Creative Maths.

]]>I can recite Latin verbs: the present tense of love is amo, amas, amat, amamus, amatis, amant. I recited them as I swam up and down the pool forty years ago: Amabo, amabis, amabit (breathe) amabimus, amabitis, amabunt (breathe). But if I were suddenly faced with an ancient Roman and had to express my affection, it would take a bit of thinking. I lack fluency in speaking Latin.

When we are fluent in a language, we can respond and converse without having to think too hard. The language comes naturally, and we do not use up space in our brain thinking about what word to use. Fluency comes from using the language in multiple settings, from trying things out, and failing and trying again. In order to learn a language we need to overcome the fear of looking stupid, and just give it a go. We also hope that the native speakers around us are kind.

The idea of fluency applies also to mathematics. The National Council of Teachers of Mathematics provides this definition of procedural fluency:

“Procedural fluency is the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another.”

You can read more about their position here: Position Statement

Fluency and automaticity with basic facts can ease later maths learning. When learners are grappling the division algorithm, it can derail the process if they have to stop and look up the seven times table, for instance. Conversely, when learning to add fractions, a student who is automatic in their multiplication facts is at an advantage when thinking of the Lowest Common Multiple of two denominators.

My recent survey of high school teachers listed fluency with multiplication as one of the key skills they would like students to have when they arrive in Year 9.

Fluency in mathematics is a hot topic in mathematics teaching. There is **universal agreement** that fluency in applying basic mathematical facts enables later mathematical learning. There is **considerable difference** in how fluency is achieved.

For generations children have learned their tables by rote and been required to answer questions involving basic facts and multiplication quickly, often with some jeopardy (the ruler across the knuckles or public humiliation). Speed was taken as evidence of fluency. Most adults will be able to recall instances of having to recall facts under pressure in a classroom. For many this evokes memories of considerable emotional trauma. As a competent and keen mathematician, I still remember the grip of fear that I would be too slow. I was never quick with my tables at school, though I was fluent enough. I was fortunate as I felt good enough at mathematics for it not to put me off the subject, but this is not usually the case. I have written about maths trauma, which is related to maths anxiety. One of the traumatising traditions listed was “using speed to measure understanding”.

Jo Boaler is waging a war against time-pressure tests of basic facts. Her paper Fluency without Fear summarises research and outlines her concerns around some ways of developing fluency. She encourages the development of number sense through games, activities and number talks.

It is important to give learners opportunity to develop fluency in procedures and automaticity in number facts.

But

It is damaging to drill learners in maths at speed, as the effect of time pressure causes blockages in the very part of the brain that is needed for processing.

So how should we develop fluency? We need to use activities that apply the number facts in meaningful ways. We need games that use the mechanism of mathematics. Learners need distributed practice at the right stage of conceptual understanding. Conceptual understanding and procedural fluency need to proceed in tandem.

At younger levels there are Number talks and Choral counting that can be used to explore and develop number sense.

One way is number explorations. For example twenty-four is a multiple of 1, 2, 3, 4, 6, 8, 12 and 24. Time spent in the company of twenty-four will embed in learners the idea that it has multiple factors. Physical, pictorial and abstract representations will aid memory and provide hooks. Then later when they are factorising a quadratic involving 24, it should trigger memories of its factors.

My favourite resource in learning about procedural fluency is a webinar by Jennifer Bay-Williams, which you can see here: Research-based strategies that build procedural fluency. It changed the way I think about teaching maths facts.

Jennifer Bay-Williams has since co-written the book: Math Fact Fluency.

One of the five fundamental principles espoused in the book is that “students need substantial and enjoyable practice.”

The best form of substantial and enjoyable practice is games, games and more games. In this context I am not talking about computer or on-line games, but games between real live people. Games encourage flexible thinking and discussion. They are social and develop other skills. Mathematics underpins all games. Maths games can be played using minimal equipment, such as cards, dice and counters. These games can be competitive or co-operative, they can be individual, small group or between teams. They can pit the class against the teacher! Games can also be a home-school link. Teach the game at school and get the learner to play it at home or with their friends or at the after-school care. If it is fun, it will not seem like homework.

Some games are better than others and we have developed a checklist for evaluating games. You can read more about it on this post: Evaluating Mathematics Games. And there is a copy of the checklist available here: Checklist for Evaluating games

Here at Creative Maths we are committed to building a world of mathematicians. To help with this we have games and free resources that will help to develop fluency in multiplication and fractions. Hundreds of people have downloaded our free resource for developing fluency in multiplying by three. There is a low cost “Print and Play” file for all the multiplication facts available also.

We love to invent games as well as make videos. In the comments below or our Facebook page, let us know what particular skill or procedure you are wanting to develop and we will recommend a game, or invent one if we cannot find one.

If you have a game you find really works for your learners, let us know in the comments also or on our Facebook page.

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]]>The post Achievable challenge in teaching maths appeared first on Creative Maths.

]]>I always choose the most difficult Sudoku puzzles. I like it best if I get really stumped and have to leave the puzzle and come back later. If I do manage to crack it, I feel a sense of achievement, and completion. From time to time I have tried “The most difficult sudoku” but have never managed to place more than one number. There isn’t a lot of fun in that. Fun exists in what is sometimes called “The Goldilocks zone” – not too easy, not too difficult, but just right. I have also seen similar ideas called “achievable challenge” and “the zone of proximal development”. I like the idea of achievable challenge.

Achievable challenge is personal. My adult learners enjoy maths when I can set the work at the right level. For them it needs to be only just challenging – more towards the achievable end of achievable challenge. This is because they lack self-efficacy with regards to mathematics from years of not being taught mathematics in a way that worked for them. Possibly the greatest motivator is success – past success gives self-efficacy, which encourages the prospect of future success. My learners have blossomed as they have experienced success in a subject they had previously not succeeded in. There is a virtuous cycle in which success develops self-efficacy which in turn leads to more resilience in the face of challenge, leading to greater success. We need to help our students stay in this virtuous cycle.

The challenge for classroom teachers is finding tasks that provide achievable challenge for all their students. Sometimes I wonder if this is an unachievable challenge, which causes moral exhaustion for teachers feeling unable to meet the diverse needs of all their learners.

One of the big ideas of statistics is that of variation. There is variation in all areas of life. Scientific experiments are designed for some aspects of variation in order to explore other areas of variation. You can see a video about variation here:

When we teach we are confronted with a wide range of students with variation over multiple dimensions. These dimensions include but are not limited to motivation, self-efficacy, prior knowledge of the relevant content, cultural understanding, health, mental health, stereotype threat, resilience in the face of challenge, mental processing speed and time available.

Board games, card games, computer games, sport all include challenge, success and loss. I have recently been playing Pandemic, which is a co-operative board game where the players play together against the game to save the world from disease. It has the perfect level of achievable challenge for me and we win about one in every four games. As I have high self-efficacy in this sphere, I almost prefer it when we fail. If it were too easy, it would not be fun.

In contrast, when things are too difficult they cease to be fun. One year in a fit of delusion I joined a social basketball team. I practised and practised and practised, but I did not get much better. The team was more competitive than I had anticipated, and I spent most of the season on the bench or in brief spurts on the court. One time I did manage to get a goal, everyone was so surprised even the opposing team clapped. I suspect I was in the wrong level of team for me to experience achievable challenge. I failed too often for it to be fun, and I often felt I was letting the side down. My self-efficacy in the realm of basketball was low and went lower. I decided running was a better sport for me.

When people are having fun, they are more likely to continue doing what they are doing. This is why games can be powerful way to help people learn things such as basic maths facts. The best games can cater to a wide range of people. For example our Fraction Action game teaches about adding fractions and equivalent fractions. However, the graphics are designed such that a child who can add by counting can play. (I tried it out on my 5 year-old grandson). It also has a level of challenge and choice that engages people who are adept at adding fractions. Using a game like this in a family or classroom enables practice and conceptual development at multiple levels. You can find out more about Fraction Action here: Fraction Action game

You can download a free accompanying resource here: Free fraction resource

There is interest currently in developing a growth mindset around learning, and specifically learning mathematics, with Jo Boaler and Carol Dweck as two of the main researchers and proponents. As with all good ideas and research, the theory can sometimes be misapplied with counter-productive effects.

I was very keen to get my adult students to experience productive struggle, telling them that this will help them to learn. However, I failed to consider their history and feelings around mathematics. They were not ready for any struggle at all, until they had been successful. I have gradually increased the challenge and given students choice, which has further developed confidence. Several of them also had a perfectionist streak and once they started getting 100% in quizzes, nothing less would satisfy. We spent a bit of time looking at why this was the case and getting them to accept that getting things wrong is a step towards learning. They still see it as failure though.

Earlier I wondered if teachers have a task that is an unachievable challenge. It depends on how much we are concerned about the individual learners. With a university class of 800 students I could not be concerned with each individual student but endeavoured to provide every opportunity for them to learn and get the help they needed. When marking exams I would remind myself that I was not failing them, but rather assigning a grade to the level of understanding they displayed. I felt satisfaction that most of the students learnt most of the material.

Now I have a class of ten adult mathematics learners. Now it is personal. I think about each one and their history, and design tasks that will suit them. I rejoice in their success and worry over their concerns. It is both demanding and rewarding and within my range of achievable challenge. I am learning to cope with productive struggle in my teaching rather than perceived instant success which may not lead to long-term success.

This is what makes teaching what it is.

How do you provide for achievable challenges in your learners?

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]]>The post Like, Share, Comment, Subscribe, Join – YouTube! appeared first on Creative Maths.

]]>YouTube has an enormous impact on the lives of millions. Creators, young and old, are supplementing their income, or making a fortune through posting an unthinkable range of content. Some is uplifting, educational, funny, entertaining, diverting and nostalgic. Some is less positive. My son who is blind gets a great pleasure from ‘watching’ YouTube clips of old game shows, children’s programmes, and even Sesame Street. When I want to know anything – how to crochet, how to fix a tap, how to use Adobe Illustrator and so much more, I look for a video on YouTube.

My channel is Dr Nic’s maths and stats Youtube channel. Its main content is about statistics, particularly introductory College statistics, though I have dabbled in other areas such as skip counting and spreadsheets. There are over fifty thousand subscribers, and nearly two million views a year. People really love and benefit from my videos and ask me how they can help me to keep going.

There are a multitude of ways to respond to a YouTube video, and it isn’t always clear what they are.

Click on the thumbs up symbol. This endorses the video for other users, the creator, and the YouTube search algorithm. It matters if you like a video – you can use the power of the click to encourage good content.

You can tell people about useful and uplifting videos and the most direct way is by using the Share facility in YouTube itself. This lets the algorithm know what you value, and what it will recommend to other viewers.

YouTube is a surprisingly intimate platform. Your comment, particularly if it is helpful and constructive, helps the creator. I love comments. I read all my comments and often reply. You can see from the heart symbol that I have read your comment and thank you for it. The fact that a viewer makes the effort to comment endorses the video.

If you have your own “channel” on YouTube you can make your own playlists. You might have a collection of the statistics videos you want to watch in preparation for a test, or for easy access when you are doing statistical analysis. If you are a statistics teacher, a playlist can help your students know which videos to watch. You can send them a link to your playlist. Playlists endorse videos and help the YouTube engine know what to suggest.

This is often misunderstood to imply that you are paying something. It means that you are interested in what this channel does, and makes it easier for you to find and watch. Subscription does not mean you will get told about every single video that channel posts. You need to click on the notification bell for that. Subscription means that you can take part in the community chat. For the Creator, subscriptions are gold, as they add validity to their channel. When a channel gets 100,000 subscribers, they get a nice trophy from YouTube. I encourage anyone who watches any of my videos to subscribe to the channel.

YouTube has recently (early 2019) enabled channel membership for some channels. This is the only action on this list that will cost you money. Joining the channel is a very real way to support the creator. Only some channels have this facility, though Youtube will be allowing more channels. At present you would pay about $4.99 US a month, depending on where you live, 70% of which goes to the creator. There are various “perks” associated with channel membership. It is similar to Patreon, but less versatile.

For Dr Nic’s Maths and Stats channel you will be able to have input into the direction the channel takes, such as voting on the topic for the next video. You get a membership badge, and you can ask statistics questions. You can cancel at any time.

Please consider becoming a member. For less than one cup of coffee a month, you can help me keep making high quality content that helps people all over the world to understand maths and statistics.

Here is a link to join the channel: https://www.youtube.com/channel/UCG32MfGLit1pcqCRXyy9cAg/join

Any one of the actions listed above will help Dr Nic’s Maths and Stats channel to grow and make more and better videos. You can help! Please do.

Here is a video explaining this, from Dr Nic’s Maths and Stats channel:

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]]>The post The big deal about fractions appeared first on Creative Maths.

]]>When you ask people what topics in maths they like and dislike, fractions tend to appear in the dislike column – often vehemently. Recent polling found that maths teachers think fractions are important for students entering High School. I teach in a career course in maths for people who have missed out on maths on the way through school, and a large component of that is fractions.

I love fractions. I prefer them to decimals because they are more exact. One seventh is so much neater than its decimal equivalent. I like adding fractions and multiplying fractions and drawing pictures of them. But that is nerd-speak. I probably should have deleted this paragraph.

I previously wrote about what makes decimals difficult and many of the same things are true of fractions. (It was exactly a year ago!)

- It is difficult to work out relative size for fractions – the bigger the denominator, the smaller the value
- There are infinitely many expressions for the same fraction
- When you multiply by a fraction between zero and one, the result is smaller than what you started with – which is different to whole number multiplication.
- Adding and subtracting fractions requires transforming fractions to a common denominator, but multiplying and dividing doesn’t.
- Multiplying fractions is probably the easiest operation – except for mixed fractions.
- Dividing by fractions is just silly – apart from in algebra, who ever divides by fractions?

Jo Boaler and many other maths educators encourage visual representations of fractions. Often textbooks use circles to show fractions. Circles are pretty and remind us of cookies and pizza (a staple of fraction examples.) Circle representation suffers from the same problem as pie charts, in that it can be difficult to tell the relative size of a fractional value. For example, which is bigger, 3/5 or 5/7? For this reason bar models, also called tape diagrams, are useful for expressing fractions. So long as the bars are the same length, you can compare different fractions. I love bar models, and have a supply of graphics for all sensible fraction amounts.

A customary way to teach fractions is to start with equivalent fractions, getting people to convert between, say thirds and sixths. I have recently seen students complete these exercises accurately, and yet when faced with adding thirds and sixths, they do not transfer their skills from the “equivalent fraction” page. It is as if the two things were unrelated.

It is particularly interesting teaching adults, as many already have entrenched ideas on how to do fractions, some of which work, but have no meaning to them, and many of which are mixed up together. Though my knowledge of “Nix the Tricks” makes me want to avoid the whole “cross multiplication” method, it works for many of the students. When I try to get them to understand what they are doing, they express impatience, and often end up more confused. So in the interests of getting a correct answer, they do use “cross multiplication”. To be fair, that is what I use. It works.

I do find myself wondering why we bother with adding and subtracting fractions. These days calculators can perform fraction operations in an almost miraculous way. Prior to calculators, it was important for some people to know how to calculate with fractions, but is it now? Maybe it should wait until we get to algebra, where it really is needed.

On the other hand, it has been found that “Early proficiency with fractions uniquely predicts success in more advanced mathematics.” Hugues Lortie-Forgues, Jing Tian and Robert S. Siegler This statement needs some exploration of the role of causation, prediction, lurking factors etc. Is it that proficiency in fractions actually encourages success in later maths, or is it that people who understand fractions in the way they were taught at school also understand advanced maths better, or is it that people who find fractions difficult are put off mathematics?

I asked my friends of all mathematical persuasions if and when they use fractions. Many, but not all said they tended to think more in terms of percentages. Common uses of fractions were in cooking, craft, distance travelled when running or flying and time to complete a task. Americans tend to use fractions more as they have yet to embrace the metric system and use cups rather than scales for measuring when baking. None of the examples my friends gave involved operations on two fractions such as adding or multiplying but rather they were used as ways to state an amount, or to find the fraction of a whole number.

One area where fractions are particularly relevant for me is in probability. Probability is an area where fractions are particularly suitable for expressing likelihood and also for calculating complex probabilities. Decimals just don’t work as well for me.

As usual I have no answers, but one useful thought from my reading:

“Interventions that focus on rational number magnitudes appear to be especially effective in helping children learn fraction and decimal arithmetic.”

So maybe the takeaway lesson is to spend plenty of time ordering and comparing fractions before attempting any other operations.

What do you use fractions for?

Why do you think fractions are so feared and disliked?

How important are fractions, really?

We have created a game to help learn and develop conceptual understanding around equivalent fractions and fraction addition. You can find it here:

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