Author Archives: Rebecca

My mathematical journey through Discovering Mathematics

Why choose Discovering Mathematics?

I chose the Discovering Mathematics module because haven’t had a particularly good relationship with maths in the past. I really enjoyed it from S1- S4 and I really did very well at it. I think this happened, not because I had an excellent teacher, but because I had a terrible teacher who couldn’t control the class. This meant that whilst other people were messing around there were a few of us who kept working, and, the main advantage of that was it meant we could discuss each question and work out how to do it ourselves.

I know see that this could be linked to Liping Ma’s idea of multiple perspectives where students able to work through problems together to see which solution is best. This allows the student to construct meaning from what they are doing which means they are more likely to remember it.

Maths all went down hill from S5 onwards. I had a teacher who wouldn’t let you talk and wouldn’t explain things. ‘I should “know” how to do it, so I shouldn’t need to ask. If anyone asked – they were shouted at. So that was it, my teacher stood at the class and taught ‘her way or the highway’ and me and maths fell out. This resulted in me loosing confidence in maths and ignoring it from then on.

So, I chose the Discovering Mathematics module so that me and maths could ‘make up’!  I’m going to have to teach it and it’s much easier to teach something you like than something you don’t, and, I think it shows if you don’t like something and the pupils pick up on that.

So, how did I find Discovering Mathematics?

In all honesty, I loved it! It’s opened my eyes to so many things and I truly didn’t realise how much maths is involved in our daily lives. Ok, I was aware of maths around me, but I hadn’t really thought about it!

I found the inputs where specialists came to talk to us very valuable. They gave great insight into the maths in their field. I particularly enjoyed learning about maths and medicine – we need maths to stay alive! The workshops were also very useful, they helped to bring maths to life and show how maths doesn’t have to be boring to learn – it can be fun and interesting!

Wait a minute – did I say I have been ENJOYING maths?! Yes, I believe I have! The energy and enthusiasm that Richard has shown in inputs is catching and he ensures to make many links with the real word making it relevant and interesting. I also really enjoyed manipulating maths resources in Tara’s inputs, particularly when we were practising tessellation. And, the science input about space really showed me how far apart the planets really are!

What does this mean for the future?

i like maths

 

Yes, now I like maths again!

I’m not sure I can say that I love it yet, but it’s not scary any more and I have really enjoyed being reintroduced to it.

This means that the enthusiasm that has rubbed off from Richard can join me on my teaching journey and I am going to try my best to make sure every child I teach also finds a bit of maths they enjoy. I am going to make sure I use real life examples to investigate maths, and I definitely won’t be teaching maths as stand alone topics. I am going to try to connect them as much as possible to give the children the best foundation I can for the maths they learn after they have been taught by me. (I know this all sounds very ‘I can change the world’! I know I can’t, but I can try to get people to like maths too.)

Hello maths, I missed you!

yay

Basic ideas and inverse operations in the big wide world!

What is a ‘basic idea’ in maths?

Liping Ma (2010) states that inverse operations are one of the most basic ideas of maths.  A basic idea is really just a basic attitude a person has or a basic mathematical principle. A person who holds a profound understanding of fundamental mathematics (PUFM) is able to build these basic ideas as strong foundations for more advanced mathematics they may come across. In a person with PUFM these basic ideas become so intrinsic in their mathematical thinking, that they no longer have to think about them at all! This leaves all their brain power free to work on the tricky stuff.

What are inverse operations?

inverse operation glove_puppets_operates_1123675

Luckily not one of these!

“The word ‘inverse’ means reverse in direction or position. It comes from the Latin word ‘inversus,’ which means to turn upside down or inside out. In mathematics, an inverse operation is an operation that undoes what was done by the previous operation.” (Study.com, 2015)

The four operations of adding, subtracting, multiplying and dividing are the basic foundations for all mathematics.  The inverse of addition is subtraction, and the same is true in reverse, and the inverse of multiplication is division, and the same is also true in reverse here too.  

 

math symbols_2

Let’s look at some examples…

If….5 + 12 = 17, then the inverse of this would be 17 – 12 = 5

As you can see this takes us right back to the start.

And, if….7×3 = 21, then the inverse of this would be 21 ÷ 3 = 7

And lastly… If 10² = 100, then the inverse of this would be √100 = 10

You can find a short clip to explain this in more detail here.

Lets see if we can find inverse operations in the big wide world…

maths-bar

A man, let’s call him Frank,  goes out to a bar for the night. On his way there he stops off at the cash machine and takes out  3 X £10 = £30. When he gets to the bar he orders a drink while he waits for his friends. It costs £3. His friends are late. Frank orders another drink and some peanuts, it costs £3.95. He’s still waiting… He orders another drink now, a double whisky, that costs £7.50. Then he decides he’s had enough and he’s going to go home where drinks are cheaper!

Frank gets outside and realises he can’t drive home because he’s been drinking (sensible Frank!). So he has a choice of walking or getting a taxi but he’s not sure if he has enough. Frank uses the inverse operation of addition to work out if he has enough:

So, when he took money out addition was used:

£10 + £10 + £10 = £30

Now he wants to work out how much is left…

£30 – £3 – £3.75 – £7.50 = £15.75

Which means, using the inverse operation of addition (which is what he did when he started) Frank has worked out that, not only does he have enough money for a taxi, he also has enough to buy a kebab too!

But, but do inverse operations only occur in the pub?! 

No, inverse operations are everywhere!

We can take this basic principle of inverse operations to an even more basic level.

Black – is the inverse of – White

inverse zebra

 

Heavy – is the inverse of – Light

heavy light

Hot – is the inverse of – Cold

hot cold

The list of inverse operations that can be found in the big wide world could go on and on.

I wonder how many you can identify next time you have some spare time.

Multiple perspectives and Santa’s logistical nightmare

What is ‘multiple perspectives’?

Liping Ma describes one of the properties of someone who has a profound understanding of mathematics as a person who has ‘multiple perspectives’. What this means, in terms of teaching maths, is that as a teacher you would have many different ways of coming to a solution instead of just one way that is fixed. This means that if you are given a problem, that is slightly different from ones you have done before, you will not be hampered by only knowing one way to find a solution, and it may not be the best or easiest way, you will have many ways to find a solution and even if you don’t you will still try and work one out.

multiple perspectives

Sometimes looking at things another way will give you the answer.

How do multiple perspectives fit into the ‘real world’?

Lets look at logistics…

Logistics is the effective planning and implementing of the storage and distribution of goods and services. This can be done on a small scale, such as a local delivery service for meals on wheels to an enormous scale such as…

santa logistics

…THIS!

(Now, obviously this isn’t a true representation of Santa’s delivery routes as some countries don’t seem to get anything)

However, let’s just think how this delivery driver would use a multiple perspective and fundamental mathematics to help him on his way.

He would need to plan his delivery route, taking into account the basic mathematical principles of:

Time and Distance

To work out the quickest route between all his stops whilst taking into account the time zones he will cross.

 

time distance

Ordering

To pack his sleigh correctly, keeping the presents he needs first in easy to get places.

To make sure the heaviest parcels go on the bottom and the lightest on the top.

full sleigh

Tessilation

To work out if the shapes will fit together to save space.

Shape and size

To work out how many presents he can fit in his sleigh.

pile of presents

Probability

To take in to account likely factors that will impact the deliveries such as heavy traffic, bad weather, diversions and hungry reindeer!

In addition a multiple perspective will allow this delivery person to use his multi-faceted problem solving abilities to successfully plan around any difficulties or delays that may occur.

santa flying

Longitudinal Coherence and the ‘Concept Knot’

What is ‘longitudinal coherence’?

Liping Ma (2010) describes longitudinal coherence as a teacher’s extensive knowledge of the whole mathematical curriculum and they are able to draw on this to inform their practice. Teaching with longitudinal coherence builds a “conceptual understanding” that connects “new ideas to existing knowledge” that will “build ideas across time” (Krajcik et al., 2009). This allows the basic principles the children have been taught previously by other teachers to be reinforced as the teacher has vertical knowledge of the curriculum. It also allows the teacher to have an excellent knowledge of future stages of the curriculum allowing foundations to be built to support more advanced stages mathematics the children will learn as they progress.

…and the concept knot

Longitudinal coherence therefore builds on the lateral knowledge held by a teacher using connectedness by adding vertical progression to the knowledge package a teacher with PUFM holds. Liping Ma describes this interwoven use of connectedness and longitudinal coherence together as the ‘concept knot’.

knot

 

Connectedness and how to always get a bargain

What is ‘Connectedness’?

Connectedness within mathematics is where each concept is linked to a number of other concepts. Liping Ma (2010) describes how a teacher who does use connectedness to teach a mathematical topic may teach it as a ‘stand-alone’ topic, not linking it to anything else in the curriculum or what the children have learned before. Teachers with using connectedness do not just focus on one particular part of mathematics, they draw from a “knowledge package” they have developed through their own learning. This means that when a teacher uses connectedness in their teaching, they will not only have an idea of the concept to be taught, but also all of the interlinking concepts relating to it.

This is connectedness…

connectedness 1

If you’d like to see a bigger picture – have a look here.

But what is it connecting? Galaxies? Pathways in the brain? Let’s have a look…

connectedness2

It’s connecting all the different concepts of the GCSE maths curriculum together to demonstrate exactly how interlinked they all are. The larger the word the more that maths it is used in the classroom.

Each dot represents a maths topic. There are 164 spots on the diagram representing all topics in the curriculum. The spots are connected by 935 which represents a connection between two topics where one topic must be learned before the other can be taught. For example, you must learn how to add fractions before you can learn how to do equivalent fractions. 

The size of the spot represents how many links they have and therefore how many topics rely on them for prior learning. The larger the spot the more topics rely on it. Therefore largest spots represent the fundamental maths skills required to be able to fully understand all parts of GCSE maths.

The colour of each spot represents the part of the curriculum they relate to:

Number and calculating – red, Shape space and measure – purple, Algebra- turquoise,           Data handling and probability- light green

Click here to download a high-resolution pdf of the network diagram.

The largest spots are red showing that if the number topics have not been mastered you will struggle to teach them much else. If you removed all the red nodes, and those they are connected to, you would have little left that you could teach.

Let’s compare this to a linear way of teaching maths…

mathstable3

(Source: http://www.beaumontleys.leicester.sch.uk/maths/)

Using the timetable above as an example (I am in no way implying that the timetable above has been taught with no connections!) we can see if the children were taught in a linear fashion and told “today we are learning about co-ordinates”, then the next lesson “today we are learning about the properties of 3D shapes” then the children may never understand the links between all topics, so therefore not have any depth to the understanding they do have.

I feel the diagram of connectedness shows how important it is to ensure firm mathematical foundations are built during maths lessons, and that these are revisited and reinforced regularly to allow for a secure base for further knowledge to be built on and connected to. You could teach 164 stand alone topics with no links between them, or, you could teach with connectedness giving the children a firm foundation of maths whilst allowing them to create links themselves and understand how it all links together.

(Source: Great Maths Teaching Ideas. Available at: http://www.greatmathsteachingideas.com/2014/01/05/youve-never-seen-the-gcse-maths-curriculum-like-this-before/)

How can we use connectedness to get a bargain?

Imagine, you are in the supermarket and wanting to buy orange juice. There are three different brands of orange juice for sale and they are all on special offer. The bottles are all 1 litre and all three brands usually cost £2.50 for a bottle. However, today bottle ‘A’ has 25% off, bottle ‘B’ has 1/3 off and bottle ‘C’ has an extra 0.25ml bottle free (let’s call this 0.25 free). The question is – which one is better value, per ml of orange juice, to buy? If you did not know that fractions, percentages and decimals were connected then you couldn’t use proportional reasoning to solve the problem. This could have a negative impact on your budgeting and bank balance. So, it just goes to show that connectedness can help you get a bargain!

third off  fractionsdecemals percentages25off