Category Archives: 2.3 Pedagogical Theories & Practice

What is a Number?

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We come into contact with numbers every day. Time, working out how many portions of dinner we’ll need to make or simply “hey what’s your number?”. But have we ever taken the time to think, what is a number and why do we even use them?

My first question is, why does the number three collectively represent other items in threes? Surely three dinosaurs would be more than three peas, even though they represent the same amount collectively. But it’s not due to weight, height or mass, its due to the value of the number three. Numbers are a language. For value, comparison and equivalence. They help us understand how many there is of something (value), help us compare that five dinosaurs are two more than three peas and that three peas and three dinosaurs are equal.

Numerals play a significant part in this. A numeral is the symbol or collection of symbols used to represent a number. A child’s first experience of a number is most likely going to be an adjective that would collectively describe a set of something e.g. five cats or two arms, this is more commonly known as the ‘cardinal aspect of a number’ (Haylock, 2014). This highlights that children can recognise an equivalence between two sets of objects and is why we understand that three dinosaurs are equal to three peas. At this stage numerals are the link to numbers. Therefore, children can look at numerals such as pictures of shapes or objects and count how many are there of one thing. This will therefore help them connect to real life situations and when shown beside numbers, they can begin to connect numerals and numbers together. By beginning to connect the two, children will have a deepened understanding of numbers and therefore their longitudinal coherence will develop. Starting their journey with numbers and maths simply and positively to prevent future maths anxiety (MA, 2010).

Numerals have been shown throughout our history and teach us a lot about the development of mathematics. One of the oldest is hieroglyphics, used by the Egyptians, dated as far back as 3000 BC. Their numerals were shown through drawings and symbols such as a bird and an Egyptian man (O’Connor and Robertson, 2000). Maths was commonly used by Egyptians if they were dividing food, solving problems for trade and market and most importantly for the pyramids (Mastin, 2010). This highlights the first use of maths for economy and trade. Wealth played a significant part in Egyptian life and social class was divided by money but more than anything, maths. Through looking at our history, we can see the similarities in the importance of economy then and now. We live in a world that’s economy changes daily and has the power to change and impact upon people’s lives. If we looked more in depth at historical economy such as Egyptian trade and social classes, we could learn and reflect on our economy nowadays and therefore maths is a historical aspect of economy that involves counting and numbers.

All factors of numbers have a place in fundamental maths, but most importantly teach us a lesson for now. We can apply what we have seen previously in history and the importance of numerals and numbers to aid our development with maths and see that it is essential for our wider society and not just basic classroom use. Relevance is a fundamental principle of Curriculum for Excellence (Scottish Government, 2010) and therefore it is important that teachers bring this forward from historical findings into our everyday maths lessons and make children think about their future with maths in society.

Number Patterns and Sequences

Another way of teaching maths can be through number patterns and sequences. This can be another way of making maths interesting for children whilst developing their relationship with numbers. Vale and Barbose (2009, p9), stated that the use of patterns in maths can challenge students to use “higher order thinking skills and emphasise exploration, investigation, conjecture and generalisation.” Therefore, basic skills in maths are used within problem solving to develop not only longitudinal coherence but also multiple perspectives as they can use different methods to find the solution to a problem.

An interesting number pattern is Pascal’s Triangle, named after Blaise Pascal, a French mathematician. This is a triangle that starts with the number ‘1’ and then below are the sum of the addition from the numbers above. For example, if 1 is beside 2 in the triangle the sum (number written below) would be 3. An example of this can be seen below.

Pascal’s Triangle makes maths fun for children whilst learning about addition and number patterns. Addition is a fundamental aspect of maths that it taught as early as ages 4-5 and can be continued to be taught throughout primary school through ways such as Pascal’s Triangle. The introduction of colours can also produce other findings within the triangle. For example, pupils can colour odd and even numbers or work out the horizontal sums of the triangle (Maths is Fun, 2017). This therefore keeps maths relevant and allows for differentiation in a classroom, as some can complete the main body of the triangle (addition) and some can move forward looking at other aspects of the triangle. Therefore old maths techniques and problem solving continue to find a relevance in our everyday maths and classroom.

References:

Haylock, D. (2014) Mathematics Explained for Primary Teachers. 5th edition: SAGE.

L, Mastin. (2010) Egyptian Mathematics https://www.storyofmathematics.com/egyptian.html (Accessed: 5th October)

J, O’Connor and E, Robertson. (2000) Egyptian Numerals. http://www-history.mcs.st-andrews.ac.uk/HistTopics/Egyptian_numerals.html (Accessed: 5th October)

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. (Anniversary Ed.) New York: Routledge.

N/A (2017) Pascal’s Triangle.  https://www.mathsisfun.com/pascals-triangle.html (Accessed: 6th October 2018).

Scottish Government. (2010) Curriculum for Excellence Building the Curriculum 3 A Framework for Learning and Teaching: Key ideas and Priorities. Available at: http://dera.ioe.ac.uk/1240/7/0099598_Redacted.pdf  (Accessed: 5th October 2018).

Vale, I. and Barbosa, A. (2009) Multiple Perspectives and Contexts in Mathematics Education. Available at: https://www.academia.edu/1485703/Patterns_multiple_perspectives_and_contexts_in_mathematics_education  (Accessed: 5th October 2018).

The Art of Tessellation

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When Jonathan first said the word “tessellation”, I immediately thought what on earth is he on about?! Yet, following this lecture I now understand that tessellation is something that surrounds us.

Tessellation is the arrangement of identical shapes that fit together perfectly to create a pattern. These shapes have to fit precisely beside one another, meaning they’ll leave no gaps. If we look closer at items that we come into contact with on a regular basis, such as chocolate, footballs and kitchen tiles, we can see that there are shapes such as hexagons, squares and triangles that are joined together to form tessellation.

But how does tessellation link into a classroom setting?

Tessellation can occur through two different types of shapes. The first are regular. These include squares, hexagons and equilateral triangles and therefore form a more simplistic tessellation, for example, in the form of chocolate squares. Regular shapes, unlike irregular, have the ability to interconnect as all the vertices meet one another and therefore create the sum of 360 degrees. The second are irregular shapes, these are shapes such as pentagons, octagons and isosceles and scalene triangles (Maths is Fun, 2018). These work similarly to regular shapes, however, the shapes must be cut and pasted to a different part of the shape to be able to interlock with the other identical shapes. An example of this is shown below in the creation of a horse.

A form of tessellation can particularly be seen in Islamic religion through mosaics and geometric patterns (Hames, 2017). Islamic art focus on the creation of stars through tessellation. For example, they particularly use equilateral triangles to create 6 to 12 points of stars. These represent and symbolise harmony and hum consciousness. These features can be introduced into a classroom. Using maths (tessellation) and interconnecting it with art is a great way of introducing a calm and settled environment to the classroom. Boaler (2009), states that completing tasks in different ways therefore allows children to see that there are different methods to learning maths and therefore maths can be enjoyable for everyone.

Liping MA’s idea of inter-connectedness is highlighted through the use of maths and art. By using a mixture of the two, children who feel anxious about maths will therefore find a task such as creating Islamic art, as a more relaxed approach to maths. For example, if they enjoy art they believe it is more about art than the maths. Furthermore, this will lead to their longitudinal coherence. This is because they will have the basic understandings of shapes and therefore children have a sound enough understanding to bring this information forward to more complex areas such as tesselation.

An example of a lesson that could be used for tessellation is multiplication to create stars. By finding the different digital roots e.g. 4 times 6 = 24 which therefore this simplifies to 2 + 4 = 6, you can start at a point in the circle and continue to connect to the following dots (answer). When completed the pupils can colour these in and therefore maths and art have been interconnected in a lesson, helping those who have a passion in art have a profound understanding of maths.

 

Example of tesselaltion star from the digital root of the 4 times tables.

 

 

Overall, tessellation is a great lesson to introduce differentiation within a classroom. It allows for both art and maths to be taught at the same time, making maths fun and achieveable for those suffering from maths anxiety. Tesslation links into our classroom setting through a number of different lessons and has a major link to pupils’ understanding of shapes. A basic concept of maths that is learnt thoroughly to bring forward. This lecture in particular is one that I will continue to revisit when teaching, as I have not only learnt how maths can be fun, but have learnt about a different culture in the process. Therefore, I think this topic could be integrated into the classroom in a number of ways such as a class topic or investigation task.

References:

Boaler, J. (2010). The Elephant in the Classroom: Helping Children Learn and Love Maths. London: Souvenir Press.

Giganti, P. (2010) Anatomy of an Escher Flying Horse. Available at:  https://www.youtube.com/watch?v=NYGIhZ_HWfg (Accessed on: 25th September 2018).

Hames, S. (2017). Tessellations in Islamic Art. Available at: https://classroom.synonym.com/tessellations-in-islamic-art-12085299.html. (Accessed on: 12th November 2018).

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. (Anniversary Ed.) New York: Routledge.

Maths is Fun. (2018). Tesselation. Available at: https://www.mathsisfun.com/geometry/tessellation.html (Accessed on: 24th September).

Warner, M. (no date) Digital Root Patterns Available at: https://www.teachingideas.co.uk/number-patterns/digital-root-patterns (Accessed on: 26th September 2018).