Teaching Volume and Capacity

Learning Objectives:

1. Choosing appropriate units of capacity and making sensible estimates with them in everyday situations.
2. Extending understanding of the relationship between units converting one metric unit to another.
3. Choosing and using appropriate measuring instruments.
4. Interpreting numbers and reading scales to an increasing level of accuracy.
5. Developing mathematical strategies and looking for ways to overcome difficulties.

Experience and Outcome:

• I can use the common units of measure, convert between related units of the metric system and carry out calculations when solving problemsMNU 2-11b
• I can explain how different methods can be used to find the perimeter and area of a simple 2D shape or volume of a simple 3D objectMNU 2-11c

Resources: basin, different sizes of containers and measuring jugs

The idea of this activity is to let pupils experiment with different sizes of containers to develop their understanding of volume and capacity. By providing a basin of water and containers, you can let the pupils create their own investigation, so let them decide what they want to find out and how they are going to do this. However, it is recommended that the teacher provides guidelines so the pupils complete a productive activity.

Activity adapted from Blinko, J. & Slater, A. (1996). Teaching Measures: Activities, Organisation and Management. London: Hodder & Stoughton.

Teaching Area

A good way to initially introduce area is through the use of concrete materials, as this helps the pupils grasp a more tangible understanding of the concept. While reading about teaching measurement I came across this activity which I feel would be a beneficial way to introduce the concept of area or to reinforce it.

Learning Intentions:

1. Choosing appropriate standard units of area and making sensible estimates with them in everyday situations.
2. Extending understanding of the relationship between units.
3. Finding areas by counting methods.
4. Developing mathematical strategies and looking for ways to overcome difficulties.

Experience and Outcome:

I can estimate the area of a shape by counting squares or other methodsMNU 1-11b

Resources: three sets of all the different shapes possible from one, two, three and four squares, a spinner with the numbers one to four and a five-by-five grid for each player.

Instructions

Teach the children how to play a grid game. Each player needs a board and there should be plenty of shapes they have made available.

Players should:

• take it in turn to spin the spinner
• take a shape with the appropriate number of squares
• place their shape on their board

The first player to fill their grid with shapes is the winner. The players must decide whether or not they are going to allow each other to move the piece around on the board during play. It doesn’t matter what they decide, as long as they all stick to the rules.

Activity from; Blinko, J. & Slater, A. (1996). Teaching Measures: Activities, Organisation and Management. London: Hodder & Stoughton.

Teaching Mass and Weight

‘Plasticine Models’

Learning intentions:

1. Comparing objects using appropriate language by direct comparisons
2. Understanding the language of comparatives

Experience and Outcomes:

I can estimate how long or heavy an object is, or what amount it holds, using everyday things as a guide, then measure or weigh it using appropriate instruments and unitsMNU 1-11a

Resources: plasticine, pan balance

Ask the children to make two plasticine balls which weigh the same. Let the children decide how to check that both balls weigh the same. The idea that they should balance is often surprisingly difficult from them. Once it has been established that both balls really do weigh the same, ask them to make a pancake with one and a snake with another. Discuss what they have done:

• will they still balance?
• why or why not?
• how do you know?
• check to find out

Activity from Blinko and Slater (1996) Teaching Measures: Activities, organisation and management. London: Hodder & Stoughton.

The Final Post…

The lectures and workshops have finished, the assignment has been handed in and this module has come to an end. It has been an absolute whirlwind of a semester and time went in a lot faster than I thought it would.

When I first picked this module I actually thought it would be similar to higher maths – completing equations and such. When I found out it was an assignment we had to write, to say I was shocked and panicked is an understatement, all that was going through my head was; ‘how do you write 1500 words on maths?!’. I am not a fan of writing essays and was actually looking forward to sitting an exam which involved doing calculations (I’ve actually missed doing maths).

So, I think we’ve clarified that this module was not what I expected it to be. Nonetheless, I have thoroughly enjoyed exploring the principles and concepts of mathematics and through this exploring I’d definitely agree with Liping Ma that teachers should have a profound understanding of the fundamentals of mathematics. I believe this, because I think you need a deep understanding to explain things and properly break them down and to be able to successfully and confidently answer questions. I concluded my assignment by saying that I still don’t have a profound understanding of the fundamentals of mathematics, although I do have a greater understanding than when I first started. Thus, I need to continue to explore maths and I will do this through reading and attending CPD such as those held at the science centre.

Something else I have gained from this module is an enthusiasm to explore mathematics. I never really possessed this before so I guess Richard’s enthusiasm must have rubbed off on me! I am confident that this new found enthusiasm will propel me and my curiosity into further reading and I will continue exploring mathematics in the future. I am hoping that when I next teach a maths lesson, this enthusiasm will come across and will make the lesson more engaging.

For me, there were many highlights, although if I had to choose my top two it would be Simon Reynold’s lecture on the solar system and Anna Robb’s lecture on maths and music, for both completely different reasons. During the lecture on the solar system, Simon highlighted the importance of using scale diagrams and not pulling pictures off the internet without explaining that they are inaccurate. Without pointing these things out and using different activities to aid understanding of the vastness of space, children will never have that understanding. The use of acting out the solar system with different sizes of balls was suggested and then demonstrated, and seemed possible to do within the classroom. Simon also touched on using big numbers, or numbers with powers, and told us not to be frightened of using them, but should try to give an understanding to the children of them so they become familiar and comfortable with them.

I play a few instruments and I think this is why the music input appealed to me. The music side, I was already comfortable with so I felt safe within this environment and it made it easier to embrace the maths side – highlighting the importance of a comfortable environment! I had never really thought about the maths element in music but it really captivated and excited me. From now on I will looking for patterns and symmetry and Fibonacci’s numbers within each new piece of music I play!

There is no doubt this was a thought provoking and enjoyable module and I am sad that it has come to an end. I would definitely recommend it to future second years and I will never forget Richards enthusiasm. I think a lot more people would enjoy maths if they had such enthusiastic teachers!

Place Value

Throughout the centuries there has been many variations of writing numerals and counting. An example of something completely different, and outdated, Yan, Tan, Tethera method which was most commonly used to count sheep, although has also featured in a Scottish dance. Today, in Britain, we use the decimal system for counting which is based on ten; the number of fingers and thumbs we have. This system has derived from an ancient Hindu system which was developed by Arabs during the ninth and tenth centuries before quickly spreading throughout Europe.

When teaching, to explain the concept of place value, we write out the names of columns to demonstrate the value of each number:

 Thousands Hundreds Tens Units 3 4 5 2

This number is then 3452. We can describe this number as consisting of three thousands, four hundreds, five tens and two units. When written out in words it would read; three thousand, four hundred and fifty-two. Zero is an important feature in that it is described as a place holder; if there were no tens in three thousand, four hundred and fifty-two, it would become three thousand four hundred and two which would be written as 3402.

Thus, in our decimal system the concept of place value can be explained as; the place of a digit in a number tells you the value of that digit.

A very significant feature of the place value system is the principle of exchange; it is described in Haylock as being “the principle at the heart of our place value system”. It is the idea that a ten in one place can be exchanged for one in the next place to the left, for example 10 hundreds could be exchanged for 1 thousand.

It is this principle of exchange which makes the place value system fundamental to the way we complete calculations, which is how I can introduce the element of Connectedness (Ma, 2010, page122). My understanding of Connectedness is that there are a lot of areas of mathematics which overlap, such as a concept being shared within different areas. So when I was reading about place value it reminded me of the element of Connectedness, and this is due to the principle of exchange. Within addition calculations you ‘carry’ numbers, this is the process of replacing ten in one column by one in the column to the left. The importance of place value here is imperative as without an understanding of it, you would be unable to complete the calculation.

An understanding of place value is also vital to complete subtraction sums consisting of ore than 1 digit. It is needed to complete column method of subtractions using the principle of exchange to overcome the difficulty caused when the digit to be subtracted in a particular column is less than the one it is being subtracted from. This demonstrates the element of Connectedness.

I have taught a few lessons on place value before moving onto multi-digit addition and subtraction calculations. From giving those lessons, I understand how difficult it is to understand sums using ‘carrying’ and ‘borrowing’ if you do not have a knowledge of place value. I found that the best way to teach place value was by using base ten blocks, which most schools have access too. They consist of units, tens and hundreds, and makes it a lot easier to demonstrate how you can exchange 10 units for 1 ten and 10 tens for 1 hundred. It is also possible to use these blocks to practice addition and subtraction with exchanging. The physical aspect to the lesson definitely helps the child understand what is happening.

Maths and Music

I love music and I always have. I was never able to sing so I turned to instruments instead, just like my dad and my granddad. From a young age I played the trumpet in jazz bands, brass bands and windbands and have never doubted it wasn’t for me. Since moving to university, I have stopped playing as much but this input reminded me just how much I miss it.

Within music, there are a lot of scales and the whole tone scale is  particular favourite of mine. After this input I started researching further  the relationship between maths and music and discovered there was a link between the whole tone scale and Pythagoras. The whole tone scale is demonstrated at 1:01.

The whole tone scale was developed my Pythagoras by using the ratio of the frequency of the sound waves of two tones. Octaves have a ratio of 1:2, and so Pythagoras used smaller intervals to develop the whole tone scale.

Also mentioned is the symmetrical nature of the scale which can be seen here:

Symmetry in music is common. A well-known composer who used it was Bach, and it can be seen in his Goldberg Variations at 3:11.

Another Mathematician, who’s work featured in music is Fibonacci. His work features in the design of the piano keyboard as well as in pieces of music. You can read about this in my blog about Fibonacci.

I never thought that one day my love for music would increase my enjoyment for maths, but it definitely has! I now know longer view maths as a scientific and analytical subject, instead, I can see the art and emotion and beauty of it.

Fibonacci and the Golden Ratio

Named after Leonardo of Pisa, the Fibonacci sequence is a sequence of numbers defined by the fact that each term is the sum of the previous two terms. Thus, the first fifteen terms are as follow:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377

Leonardo of Pisa, nick named Fibonacci, was an Italian Mathematician who was a major figure in spreading the Hindu-Arabic numerals to the rest of the world. These numerals are the ones we use today and replaced the previously used Roman numerals.

The Fibonacci sequence was first presented as a solution to a rabbit problem:

Fibonacci’s sequence also frequently appears in nature in the form of a spiral. It can be seen in flowers, pine cones and pineapple fruitlets, giving the seeds the most efficient and even distribution in the space available. The most common example is in a sunflower; the seeds appear in a spiral with the seeds coming out of the centre which usually grow in formations of 55 clockwise and 89 anticlockwise; both Fibonacci numbers.

Within a lecture focusing on Fibonacci and art we drew our own spirals using Fibonacci’s numbers:

We also investigated the mystery of Phi and the Golden Ratio, it relates to the spiral also, as with each 90 degree turn it gets 1.6180 times wider. From Fibonacci numbers, you can get the ratio by taking any two successive numbers and putting them into this formula:

The Golden ratio has been commonly used in art and these are only a few examples:

My main interest, however, is in music rather than art and after the past few years of playing piano, I was amazed to find out Fibonacci played such a large role in music. Fibonacci’s sequence is a framework which appeals to many composers, possibly because the golden ratio. The Golden Ratio often features to generate rhythmic change or develop a melody line.

Fibonacci’s numbers feature in the piano keyboard:

Examining the scale of C to C, you can identify thirteen keys; eight white and five black. This can be further related to Fibonacci using the layout of the keys, with the black keys in groupings of twos and threes.

Fibonacci’s numbers also feature in an octave. Within a notes octave there is a span of thirteen notes and a scale is made up of eight notes. If you investigate a chord, the finding is that the chord’s foundation is the third and fifth notes which are based on a tone made up of the two steps, and one step from the root tone, as demonstrated below:

The Mathematical Horse

A horse, named Clever Hans, was put on display in 1891 by his owner William von Osten and became known around the world for his inexplicable abilities. Apparently Hans could count, tell the time, read and spell (in German). He communicated through stamping one foot on the ground, for example, if he was asked what five add two was, he would stomp seven times.

Led by Carl Stumpf, a philosophy professor, a group of researches examined Clever Hans to try and unveil his secrets. However, in 1904 a statement was issued saying they could find no evidence of trickery. Although he didn’t find any evidence of trickery, Carl Stumpf and one of his students, Oskar Pfungst, noticed a link; Hans could rarely answer a question von Osten didn’t know the answer to. Eventually, after lots of careful testing and observation they came to a conclusion; Hans could not count, he was actually responding to unconscious cues from his trainer. Von OSten had been making subtle movements, sometimes merely a change in facial expression or a shift of stance, when Hans reached the correct answer that would cue Hans to stop.

There is no doubt that Clever Hans was clever, just much less so than both von Osten and the public believed.

There are many opinions regarding the question ‘can animals count?’. Personally, I don’t believe animals can count using numbers in the same way as humans. However, I do believe that animals have an ability to evaluate the amount or quantity of something in the sense of whether it’s a large amount or small amount.

Another study, completed by Kevin C. Burns and his colleagues of Victoria University of Wellington, New Zealand examined robins. To do this the burrowed holes in fallen logs and stored varying numbers of mealworms in these holes. They then witnessed the robins fly straight to the holes with the most meal worms first. Burns conclusion was that; ‘they probably have some innate ability to discern between small numbers’. However he also thinks that they use their number sense on a daily basis, therefore, through trial and error they can train themselves to identify numbers up to 12.

Watching the video of Ayumu the chimpanzee it is easy to believe that Ayumu understands numbers. There is no doubting Ayumu is extraordinary, being able to recall the order of eight digits when displayed for 0.21 seconds and the order of five when displayed for 0.09 seconds. You cannot deny he has fantastic reaction speed and memory, however is he doing maths or is he just recalling from memory.

I, Science Website seem to believe he is doing maths as they brand him; ‘Counting Chimp’ and it states that his job is to order the numbers 1-9 when they are displayed on the computer as is shown in the video. However, the numbers are then covered by white squares so he has to memorise the numbers position and then order them.

For me, the key word here is ‘memorise’. I believe that Ayumu is just memorising the order of the numbers and doesn’t actually understand them and there is no evidence he has an understanding of numbers. Therefore, without this understanding, it is impossible that Ayumu would be able to count.

It would be an amazing thing to see animals be able to count. However, after this research I still believe that they do not posses that ability and cannot see them possessing it in the near future. I do believe however, that animals all have an awareness of the quantity of something and that is how a mother duck is able to tell if all her chicks are with her and the same for the robins with the mealworms. It is nothing more than that however, and the truth is animals cannot count, no matter how much humans wish they could.

Reference List

http://www.isciencemag.co.uk/features/counting-chimp/