What is Mathematics? It seems like a simple enough question, but is it?

On the surface, it’s obvious that maths is all to do with numbers. But delve a little deeper, and maths is revealed to be much more complex. In fact, it’s actually something that’s all around us and involved in almost everything we do. It’s in nature, art, technology, it’s the language of science, it’s in music and dance; the list is endless!

Ernest (2000) states that it is important to appreciate “the role of mathematics in life and work, the importance of mathematics in commerce, economics (such as the stock market), telecommunications, information and communication technology, and the role it plays in representing, coding and displaying information. Also how mathematics is forever becoming more central to, but also more deeply and invisibly embedded in, all aspects of our daily life and experience.”

I stumbled across this video and feel it sums up how integral mathematics is to our lives and within wider society.

References 

Ernest, P. (2000) ‘Why teach mathematics?’ In Bramall, S. and White, J. (editors). Why Learn Maths? (pp.44-47). London: Bedford Way Papers.

The Beauty of Mathematics: available at: https://vimeo.com/77330591

For as long as humans have been around, we have been counting things and looking for ways to keep track and represent the things that we count. The Ishango Bone is a great early example of this, but over time our method for counting and tracking numbers has evolved into a number system composed of ten numerals – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 – which we know today as the base 10 system.

Understanding place value helps us determine the value of a numeral based on its position within a number, and since our number system is based on the idea of 10, to fully understand place value, children need to see how the pattern of ten repeats within numbers and how it is used to build numbers.

Children begin with rote counting, 1, 2, 3… From there they get to two digit numbers, 11, 12, 13 and to three digit numbers 100, 101, 102… And so to a child, the 1 in 1, 10 and 100 often means the same thing. However, in place value, a 1 is one, a 10 is 1 group of ten, 100, is ten tens or 1 group of 100.

This concept can be quite difficult to grasp, but base ten blocks can help children visualise place value a little better. The cubes represent one unit, strips represent ten units, flats represent 100 units and blocks represent 1,000 units. Children can easily see that 10 cubes fit into a 10 strip, 10 strips fit into the 100 flat and 10 100 flats fit into the 1000 block. Another way to understand the base 10 system is through the use of our fingers. We start with ones, being our fingers, so our counting units so to speak. Then we go to tens, being sets of fingers. Then we go to ten sets of sets of fingers, which is 10×10=100, so we go to hundreds. In other words, every time we go one place further to the left, that is, every time we go into a unit that is one times bigger than the previous place’s unit, we multiply by our base of ten.

Confused yet? Because I think I am! It’s easy to see why children struggle with this concept, but it’s a fundamental part of mathematics. Having a sound understanding of place value and our base 10 system are basic concepts that children need to be aware of, otherwise future learning will be compromised. That’s why it’s important to provide children with multiple perspectives, like the two methods mentioned above, to help highlight that there is more than one way to approach any given problem.

References

Bellos, A. and Riley, A. (2011) Alex’s Adventures in Numberland. London: Bloomsbury Publishing PLC.

‘Definitions of Base 10’ available at: http://math.about.com/od/glossaryofterms/g/Definition-Of-Base-10.htm

Mathematics stretches well into prehistoric history and knowledge of its origins and initial applications is mostly built on speculation. Because of this, it’s reasonable to assume that our prehistoric ancestors would have had a general understanding of amounts, and would have instinctively known the difference between the quantity of one thing and the quantity of two things. Basic counting like this would have likely been facilitated by the use of our hands, with each finger representing a numeral.

This would have worked well for small quantities, but how did our prehistoric ancestors count quantities bigger than ten? Evidence shows that they began using notches or marks to represent numerals. In doing so, they not only created one of the first mathematical records, but they also created the first known method of tracking larger numbers.

The Ishango Bone, discovered around 1960 on the border of Zaire and Uganda by the Belgian geologist Jean de Heinzelin, is considered by many to be the earliest mathematic artefact ever recovered. The origins of this artefact date to approximately 20,000 B.C.E. and consists of a series of tally marks carved in three rows running the length of the bone.

The bone itself has been subject to a lot of interpretation.  At first, it was thought to be just a tally stick with a series of tally marks, but scientists have demonstrated that the groupings of tallies on the bone are indicative of a mathematical understanding which goes beyond simple counting.  In fact, many believe that the tallies follow a mathematical succession, with some going as far as to say it shows the earliest known demonstration of the prime number sequence.

The first row (a) can be divided into 4 groups, where each group has 9, 19, 21, and 11 tallies.  The sum of these 4 numbers is 60. The second row (b) can be divided in 4 groups, with each group possessing 19, 17, 13, and 11 tallies. These are the 4 successive prime numbers between 10 and 20.  This constitutes a quad of prime numbers. The sum of these is 60. The third row (c) can be divided into groups of 8 and features two sets of numbers (3 & 6 and 4 & 8) that may represent multiplication by 2, and one set of numbers (10 & 5) that relates to division by 2. Lastly, the first row (a) and second row (b) both add to 60, while the third row adds to 48. Both 48 and 60 are divisible by 12.

These mathematical trends suggest some basic understanding in terms of grouping numbers, but some people think they are not significant enough. The argument is strengthened by the absence of any other archaeological discoveries that indicate an understanding of the prime number sequence dating from that time period. However, despite the considerable debate surrounding the actual implications of the Ishango Bone and the fact we’ll never really know what these tallies represent, I’d like to believe that the tally marks are evidence of the first primitive counting tool.

The notion of humans understanding mathematics can also be illustrated in the Lascaux cave paintings in France.

Some of the markings indicate a basic human understanding of the 29-day lunar cycle. In one case, there are 14 dots followed by an empty square, which experts believe represent the 14 days that the moon is visible in the night sky and the day that it disappears. This suggests that part of our early mathematical understanding of counting derived in part from observational astronomy.

And so, whether it be counting or making sense of objects in the sky, it’s clear that early mathematics developed as a result of our ancestors’ need to understand and make sense of the world around them. And although the mathematics used today is vastly different, the same necessity still stands.

Drawing on history in this way can show children the need for mathematics and links in well with the four principles of PUFM. Being able to see the connections between the tally marks and their potential purpose helps us to understand and make sense of how our basic foundation for counting and recording has developed. Also realising that there’s no one right answer in terms of knowing what these tally marks or cave paintings mean highlights the need for multiple perspectives in terms of fostering a more flexible way of thinking.

References 

‘A History of Mathematics’ available at: http://www.storyofmathematics.com/

During our first input, we were introduced to Liping Ma’s concept of profound understanding of fundamental mathematics (PUFM) and its importance in terms of enhancing teachers’ knowledge of, and ability to better teach, elementary mathematics.

Ma states that to “fully promote mathematics learning, teachers must first have a profound understanding of fundamental mathematics”. But having a PUFM is more than being able to understand elementary mathematics, it’s about being aware of the theoretical structure and basic attitudes of mathematics that are within elementary mathematics, and being able to use this as a foundation for which to instil the same awareness in the children that we teach. It’s important to have both a conceptual and procedural understanding – to know how and why we do something – that is deep, broad and thorough.

Ma identified four principles that are essential to gaining PUFM:

(Inter) connectedness: refers to being able to see connections between concepts and procedures. These connections allow for the use of prior learning and knowledge to be applied to new mathematical situations. This helps to ensure learning is not fragmented, but viewed instead as a unified body of knowledge.

Multiple perspectives: refers to seeing and appreciating different approaches to solving a problem. This encourages a much more flexible way of thinking as it is not restricted to any specific learning style.

Basic concepts: refers to being aware of the basic concepts within mathematics. It is important that these central ideas are revisited and reinforced as they provide the foundation upon which future concepts are learned.

Longitudinal coherence: refers to being aware of the entire mathematics curriculum and how this can be used to link previously learned knowledge with newly acquired knowledge (one basic idea/concept builds on another). This allows for there to be much more understanding and flexibility in terms of where learning is headed as lessons can be tailored with this in mind.

While I’m still getting to grips with what PUFM really means for me as a teacher, it’s amazing how being made aware of these four inter-related principles has really changed the way I look at mathematics, and will definitely influence how I teach mathematics in the future.

References 

Ma, L. (2010) Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. 2^nd edition. New York: Routledge.

An interview with Liping Ma: https://www.eduplace.com/intervention/knowingmath/liping_ma/conversation.html

Maths is a subject that I never really had much confidence in, and thinking back to when I was a child, I remember feeling such pressure to keep up with those who managed to grasp concepts that I found myself so often struggling to get my head around. I feared the thought of failure more than anything else and convinced myself that I was just one of those people who couldn’t do maths.

Now in my early twenties, I still feel the same way. During my placement in first year the thought of teaching maths filled me with such dread. I often found myself doubting my ability when explaining even the simplest of concepts. This made me realise that to teach maths successfully, I needed to have a deep, broad and thorough knowledge and understanding of maths, and by extension, the ability to explain concepts in a meaningful way. This meant going back to basics and trying to pinpoint the reasons for my lack of confidence in maths.

I chose this module as my elective for this exact reason; Discovering Mathematics is designed to help explore, reflect and critically examine various topics in fundamental mathematics and enhance subject knowledge.

I hope this module helps me to become a bit more confident in my ability to teach maths so I can be a better teacher, and one who has children that are more confident in their ability to learn maths.