# Can animals count?

Can animals count? This question I posed to my friend. She responded with, “Yes! No! Wait… maybe?” I had the same reaction to this question. In this blog post I will share my thoughts and findings with you and maybe you too might wonder, can animals actually count?

Horses?

In the 1900’s, there was supposedly a horse that was able to learn basic maths and could count. The owner would ask the horse maths questions that involved adding and dividing. The horse would tap its hoof with the answer. For example, if the answer was two, the horse would tap its hoof twice. However, although it seemed that the horse was counting, I think that it wasn’t, It must have been trained to react that way. As I continued to watch the video about this horse, I came to the discovery that indeed, it was found through experiments, that the horse responded the way it did, due to cues (Bourassa, 2012).

Lions?

There was an experiment carried out to see if lions could count. It seemed that the lions in this experiment were able to count if the number of lions coming (who were intruders) were greater than those in their own group. They were able to do so by listening to the lions roars. Males from the intruders would quantify the amount of females that were in the group before attacking. They could distinguish if there were one or three lionesses by the sound of their roars (BBC News, 2003). Maybe these lions are not aware that they are counting  or actively counting, as they are not thinking, one, two, three like us.  However, maybe they are counting since they could work out the exact amount of lions in prides accurately (BBC News, 2003). If lions can count, why is it that their counting is limited to five or six? (Silver, 2015).

Baby chicks?

An experiment was carried out with baby chicks to see if they could count (these chicks had not been trained). Three balls were hid behind a screen and two behind another. Chicks went to the screen that had the largest amount of balls. Then it was changed up, when some of the balls had been moved, the chicks could “count” that another screen had more balls there. It was believed that they could add and subtract (Gill, 2009).

Chimps?

It is argued that chimps can count as they could understand originality since the chimps could remember the pattern of numbers. In this experiment, I think it was purely a “memory test”, the chimps had learned symbols or, the chimps did so because they wanted their peanuts (Muldertn, 2008).

Dogs?

It seems that dogs cannot discern numbers over one. They know the difference between something and nothing. This may be because they don’t need to “count”. There is no reason to, they do not need to “count” for survival whereas wolves do and they can “count” (Silver, 2015).

Parrot?

Alex, a parrot could allegedly understand colour, shape and could count but, it had been through years of training. Therefore, parrots do not have an innate ability to count like chicks believably have (Silver, 2015). Therefore, maybe animals can count if they are trained.

However, I think that these animals are able to recognise that there are more or less of something. They can recognise if a member of their pack is missing. For the chicks, maybe they just used their memory to know that there were more behind one screen than another and again, they could recognise the difference in quantity, they know what more than one looks like. Animals who have been trained can maybe count. Could we humans count if we were not  taught or trained to? Surely I would be able to recognise the concept of more or less like the chicks? However, isn’t the topic of quantity, one of the basic concepts or principles in maths? (Ma, 2010, pp.24-25).

Although, Gill argues that animals can count, “It is already known that many non-human primates and monkeys can count, and even in domestic dogs…” (Gill, 2009).

Burns (no date, cited in Tennesen, 2009) accepts a similar idea as I do, stating that animals have an “[I]nnate ability to discern between small numbers”.  In contrast, Burns (no date, cited in Tennesen, 2009) also believes that animals can train themselves to recognise not just small numbers but numbers up to 12!

Interestingly, counting connects to longitudinal coherence, ” Counting is a combination of several skills, each building on the other” (Silver, 2015). Therefore, if animals could count, they would understand the basic elements of the key fundamental principles of mathematics! Additionally, being able to count requires an understanding of ordinality meaning, having an understanding of a series one, two, three. For example, two comes after three and three after two (Oxford Living Dictionaries, 2017).

It seems that animals can count and some are better than others (Silver, 2015). On the other hand, the idea that animals can count is contradicted by Whorf (no date, cited in Bredow, 2006). Whorf’s theory states that if you don’t have any words for numbers, you cannot understand maths and numbers (Silver, 2015).  Therefore, if animals don’t have words for numbers can they really understand numbers, can they count? The Pirahã people are a tribe that did not have words for numbers but, they could understand quantity of more or less and had words for these. When Daniel Everett tried to teach them to count to ten in Portuguese, he found that they could not count and it is not because they are any less intelligent than others. This theory is backed up by an example of The Warlpiri group in Australia who had words, “one-two-many”  for counting and when they were taught numbers past two in English, they were able to (Bredow, 2006).

So, from developing my understanding of fundamental maths and applying it to my research, do I think animals can count? Well, from the evidence above it seems that animals have the ability to recognise a difference in quantity. It is necessary for example, for survival and other animals can count if they are trained. Although, if animals can recognise and understand quantity that demonstrates a basic understanding of a basic principle or idea in mathematics, quantity (Ma, 2010, p. 104).  However, after looking at Whorf’s theory it suggests that animals cannot count because there need to be words for numbers first in order to understand numbers and to be able to count.

Please comment any thoughts you might have on whether animals can count!

References:

• BBC News (2003) ‘ Counting lions roar for help’, 19 September. Available at: http://news.bbc.co.uk/1/hi/sci/tech/3119456.stm (Accessed: 7 October 2017).
• Bourassa, P. (2012) Clever Hans. Available at: https://www.youtube.com/watch?v=C0LKN2lFWI4 (Accessed: 7 October 2017).
• Bredow, R. (2006) Living without Numbers or Time. Available at: http://www.spiegel.de/international/spiegel/brazil-s-piraha-tribe-living-without-numbers-or-time-a-414291.html (Accessed: 7 October 2017).
• Gill, V. (2009) ‘Baby chicks do basic arithmetic’, BBC News, 1 April. Available at: http://news.bbc.co.uk/1/hi/7975260.stm (Accessed: 7 October 2017).
• Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.)New York: Routledge.
• Oxford Living Dictionaries (2017) Definition of ordinal number in English. Available at: https://en.oxforddictionaries.com/definition/ordinal_number (Accessed: 7 October 2017).
• Tennesen, M. (2009) More Animals Seem to Have Some Ability to Count. Available at: https://www.scientificamerican.com/article/how-animals-have-the-ability-to-count/ (Accessed: 7 October 2017).
• Muldertn (2008) Chimps counting. Available at: https://www.youtube.com/watch?v=0cKg9D4QKCM (Accessed: 7 October 2017).
• Silver, K. The animals that have evolved the ability to count. Available at: http://www.bbc.co.uk/earth/story/20150826-the-animals-that-can-count (Accessed: 7 October 2017).

# Maths in art? Art in maths?

Previously, I’ve heard people say, “I cannot do art” or “Art is something that I just wasn’t gifted in”. This is why one particular maths workshop came as surprise to me as it demonstrated that you can do art using maths.

In class we created tessellations. Little did I know that this type of art links to basic ideas and concepts in maths such as properties of 2D shapes, angles, triangles, quadrilaterals, symmetry, proportion. It also links to longitudinal coherence since we can use these basic concepts of maths and build upon them to teach area and transformation (Ma, 2010,p. 104).

To create our tessellation, we had to look at the size of the shapes, how the regular shapes of an equilateral triangle, squares and triangles could fit together and why they did. Throughout the workshop we used a whole range of mathematical languages such as edges, sides, angles, pattern and symmetry.

We looked at the tiles in front of us and thought of what pattern we wanted to create. Once we had chosen our pattern, we placed the tiles down on the paper to see if the 2D, regular shapes would fit together. We found that there were going to be gaps however, we solved this challenge as we found that if we rotated some of the triangles, they would fit, leaving no gaps. Once we completed our tessellation, we painted it, using colours to emphasise the pattern created.

This workshop was useful as in the future if there are some pupils who for example say, “I can’t do maths but they say they are good at art, I can show them how both connect to one another, making maths or alternatively art more interesting, fun, engaging and relevant to the pupils as art can be created using mathematical concepts.

In the future, this activity would useful for teaching maths and for developing pupils understanding of fundamental mathematics. To start out, pupils could look at shape, angles and explore how shapes fit together. Pupils could be challenged by looking at why these shapes fit together. Pupils could explore this by using a protractor to see how all the angles add up to 360 degrees where the angles meet. Children could be further challenged by making their own shapes. They then could measure the angles to discover how and why they fit together. Additionally, I would like to make shapes relevant to pupils by showing them shapes in real life contexts because pupils might not think of shapes when looking at buildings for example, they could look at photos of the Pantheon in Rome, as the front of it is the shape of a pentagon.

References:

• Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.)New York: Routledge.
• Valentine, E (2017) Maths, creative? – No way! [PowerPoint Presentation], ED21006: Discovering Mathematics (year 2) (17/18). University of Dundee. 26 September.

# Teaching mathematics progressively!

In the “School maths or intriguing maths?” workshop, I experienced mathematics being taught in an maths in an intriguing, creative and progressive way. I saw how teaching maths in a progressive way gives more depth to children’s learning which is one of the principles in Curriculum for Excellence (The Scottish Government, 2008, p.30) which furthermore, suggests using acting learning to do so in order to deepen understanding.

In the workshop we were learning how to teach the fundamental basics or principles of volume and capacity using active learning. To do so, we made a cube using squared paper. Once we drew out out cube, (which we choose to be 3 x 3 x 3)  then folded the cube up, sellotaping the sides together. In order to find out the volume of this cube, we filled our cube with 1cm^3 cubes. This was a much more creative way of learning about volume than rote learning a formula by a procedure. By doing this experiment, we were surrounded in mathematical language and had more questions whilst exploring volume.

I found that learning through experience is much more beneficial than just sitting in class writing on paper. It is much more memorable and exciting! An idea that I hope to take forward in teaching! For example, in class an idea was given that if there is a question about money for example, use real money. If it’s about buying and selling, I could get the pupils to set up a pretend shop or they could go to the secretary office for an exchange of money and before the practical, pupils could make a hypothesis, then observe what happened and can reflect on their learning.

There are various ways or perspectives from which maths can be taught, linking to advice given by Ma (2010, p. 104) which is to solve a problems by looking at things from multiple perspectives. However, Eastaway (2010, p. 33) states that even though using for example, the fun finger method for multiplying which is learning multiplications using a different way, it still might be better to memorise the times tables and he claims that memorising them may be the “…best method” for some. So, from this I am aware that even though there may be more fun ways to learn maths and they are fantastic, for some people, learning maths by just using paper may be the best way for them.

Alternatively, Mason, J. Burton,L. and Stacey, K (2010, p. 134) suggest making maths more creative and relevant which it was through making the cube and by finding the volume of the elephants. This was as Ma (2010, p. 104) states, building upon basic ideas and foundations which creates questions and developes understanding  throughout the experiences since we had to think of how to find the volume of the elephants which were not square. This also demonstrates how to view the problem from another perspective to aid understanding (Ma, 2010, p. 104). I never would have thought of teaching how to find the volume of an elephant in such a creative and interesting way which encourages a pupil to think (Mason, Burton and Stacey (2010, p.138). Maths can be fun when you use creative, active learning to teach it.

In addition to the workshop, I read how to teach children to think mathematically and to approach a maths question or problem without panicking about it. Mason, Burton and Stacey (2010, p.135) suggests firstly looking at what they know, what they want to know and how they can check. This approach, increases pupil confidence, as they can check whether their answer is correct without having to ask the teacher as they can prove themselves that their answer is correct.

Pupils can develop a positive mindset when approaching a question and they can think of what they do know when looking at it instead of looking at what they don’t know. If stress or panic is the first result then that can block your thinking. (Mason, Burton and Stacey (2010, pp.135-136). This is advice for pupils that I would like to take forward when teaching children maths to help both them and myself develop a fundamental understanding of mathematics.

However, what I found interesting was that Mason, Burton and Stacey (2010, p.139) state that reflecting on successes increases confidence. Yes, I do believe it can but at the same time I think there can be a potential for future stress in order to keep up with previous success and self-confidence in the subject could decrease if the next time you are not successful. Therefore, as a teacher successes are to be celebrated but failures should be supported too and this is why having a positive mindset is important!

In conclusion, teaching maths can be fun! Pupils can learn maths with a positive mindset, through a progressive way, that is creative and interesting, that stirs up questions from children and deepens their understanding through active learning. Maths doesn’t have to be just on paper, it is all around us!

References:

• Eastaway, R. (2010) How many sock make a pair?. London: JR Books.
• Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.)New York: Routledge.
• Mason, J., Burton, L. and Stacey, K. (2010). Thinking Mathematically. 2nd edn. Harlow: Pearson Education Ltd.
• The Scottish Government (2008) curriculum for excellence building the curriculum 3. Available at: http://www.gov.scot/resource/doc/226155/0061245.pdf (Accessed: 25 September 2017).

# More than memorising!

Can I really link everything back to basic mathematics?  This is what I am keen to explore and learn about through the Discovering Mathematics module.  In this blog post I am going to discuss how connecting mathematical errors made in other areas of maths can be linked back to basic, fundamental mathematical principles.

In the first workshop of discovering mathematics, we discussed how many things around us can be linked back to basic principles and concepts in mathematics. Some basic principles and concepts of maths are weight, quantity, addition, subtraction and division. It is interesting how there are some things that we have just accepted when we are taught mathematics. A word to describe this is, axiomatic. (English Oxford Living Dictionaries, no date).  For example, one add one is two however, what is the number one? It is universally agreed that one is one.  One is a number that represents or symbolises for example, one pen, a quantity of something. Quantity being a basic concept in maths.

However, the number one is something that I just seem to accept, one is one.  It’s universally accepted. This can’t be the same for all things to do with maths. Pupils need to understand why things just are, why they are accepted

More complex maths or maths errors that children or even some adults might make can be linked back to fundamental basics. Ma (2010, pp.24-25) looks at errors in pupils’ long multiplication answers. Children when doing long multiplication where not lining up numbers into the correct positions. Teachers had different opinions as to why this was a problem. Some teachers said that it was because pupils just did not line the numbers up correctly whilst others argued that the the errors are because the pupils do not understand the reason or relevance of what they are doing in their algorithms. Therefore, an argument between algorithms being  just a process that children need to memorise and the need to understand why they were doing something.

I agree with the latter argument, that the mistakes were due to a basic principle and concept not having been  understood. This being, place value. For me personally, I know that when I was taught how to do long multiplication I was taught a process, a procedure that I had to memorise. I was not told why I was doing the procedure and some pupils may not have understood the basic principle that connects to the solution that is necessary in order to solve the maths question.  The children in this example given by Ma, is that the pupils did not understand the value of the the numbers.

Therefore, since this mistake can be connected to a basic mathematical principle not being understood then maybe other mistakes can be too! This is why when teaching in the future, I should analyse the mistakes made by pupils since once basics are understood then they can be applied to other mathematical concepts and teach children to further their own understanding of fundamental mathematics by being able to work back to the basic concepts to see where they may have made an error.

In chapter 5, Ma (2010, p. 104) gives advice that I want to take on board in my teaching practice such as, making connections between the current topic or concept of maths to other concepts in maths, to show pupils more than one way to solve a problem so that they can take various approaches to solve a problem thereby, looking at things from multiple perspectives, to focus on the basic ideas and to build upon these foundations.

In conclusion, to aid my understanding of mathematics and the reason behind pupils errors, I will demonstrate the rationale behind the maths that pupils are doing and the connection that it has to other concepts so that it’s not just a memorised procedure that is accepted. Therefore, pupils can understand why they are doing what they are doing, the logic behind it and how it connects to fundamental mathematical concepts that they need to understand first. Pupils can therefore see how the maths that they are doing makes sense.

References:

• English Oxford Living Dictionaries, (no date) Definition of axiom in English. Available at: https://en.oxforddictionaries.com/definition/axiom (Accessed on: 20th September 2017).
• Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.)New York: Routledge.

# Three paper clips, a sticky note, an elastic band and a pencil.

Do you remember back when you were at school and there was a certain pen, pencil, rubber or ruler with the multiplications on the back of it, that almost everyone wanted to have? For me, it was Crayola’s Twistable Crayons. They were the “cool” stationary that nearly everybody wanted to have! I remember sitting in class watching my friends colouring in their work with these magical crayons. I compared my work to theirs, thinking “their work is so much better than mine”. This is a reality of school; some people will have more resources than others. Ultimately, my colouring pencils were just as good as the crayons but sometimes scenarios similar to this one, can affect a child emotionally.

On Tuesday, I experienced a very valuable lesson, one that I will always remember. As the group I was in sat down at a table, we talked among ourselves, ready to start the workshop. We were informed that within our groups we would be coming up with a University welcome pack for freshers, which would be useful for them starting out university. However, we had to create the starter pack by using only the materials in the envelope that was in middle of our table. My table, had the envelope with the group number 4 written on it.

We were so intrigued to see what was inside our envelop. We eagerly opened the envelop to find we only had 3 paperclips, a sticky note, a pencil and a rubber band. We looked at what we had before us in shock. We watched the other groups around us excitingly open their brown envelopes which were full of items that we didn’t have. All the other groups were talking enthusiastically, coming up with so many creative ideas for their freshers pack, as we sat looking at our minimal resources.

Our team worked together so well despite what we had, each of us coming up with ideas. We knew our welcome pack wouldn’t be as impressive as other groups. Despite our situation, our group came up with using the envelope to draw on a map of the campus. The sticky note could be used to identify what building a student would have to go to the next day and the pencil could be hung from the map by using the paper clips and the elastic band. We became more enthusiastic with our idea and quite pleased with what we had come up with.

Meanwhile, our group had observed that the tutor was paying more attention to groups 1, 2 and 3. We weren’t even being acknowledged or talked to. We got a few unimpressed looks. We discussed among ourselves that we were trying our best with what we had, what else could we do but our best?

Then the time came to present our ideas to the class. We felt subordinate to all the other groups as they presented their amazing ideas. We progressively got more and more nervous, knowing that our idea wasn’t as “cool” as the rest of the class.

The tutor then informed as that we were instructed not to use the envelope. We were to only use the contents inside. We felt frustrated, knowing that we had tried hard. So we then had to change our idea. We put the campus map and the important notes section on the post-it and attached it by the paper clips and elastic band to the top of the pencil. However, we received the anticipated reaction. Our idea wasn’t good enough. Group 1, 2 and 3 all scored higher than us.

At first, we didn’t even get a score after all our effort! Then we were given 2 out of 10. Our group felt disappointed and disheartened. Group one having the most resources, got the highest score. In second place was group two, in third was group three and in last place was our group, group four. We felt disapproved of as most attention was given to group one, they received the most praise for their work. Yet, it was unfair because they were able to make a better welcome pack due to all the items they had received in their envelope.

Next, we were informed that this workshop was all a set up! It was to demonstrate to us that some people will have more resources than others. Therefore, illustrating that the greater access schools have to resources can help improve children’s learning and experience at school and these resources should be equal! For example, some children may have access to extra tutoring whilst others don’t. This can result in some pupils getting higher in tests, potentially leaving those who don’t have the same opportunity and don’t score as high, to feel inadequate and less capable than others in their future! This is not an equal opportunity for all children.

Furthermore, I experienced first-hand what it’s like to feel as the least favourite in the classroom, no child should feel this way in school. This can cause some children to be less enthusiastic about learning and maybe even dislike school. Therefore, teachers should be fair in their support and interest into each pupils learning.

As a result of this workshop, my awareness increased of what it felt like as a pupil to be in a different position than others in the classroom by having less resources. As well as receiving less attention than others in the classroom. This is not what any child should experience or feel like in school. They should feel valued, respected, included, they should have justified support and resources to help them learn and to enjoy learning.

By Rachel Noble.

# Why teaching?

Ever since I was a child I always dreamed of being a primary school teacher. I would set up all my teddy bears in a circle around me, hold up a sheet of paper and explain an imaginary worksheet. I wanted to be like my teachers in school. My teachers inspired me, they always motivated me to do the best I could. They celebrated my success and supported my weaknesses. My teachers cared about me. They wanted to see me succeed. I always wanted to be the caring, helpful and inspiring teacher. The teacher that pupils would not be afraid to turn to for help.

I have thoroughly enjoyed my experience in both primary and secondary school, which has motivated me to want to give others the same delightful experience in school. I want to have a classroom that is a fun and creative learning environment. I have always had different learning styles, primarily aural and visual learning styles, through which I would want to embed in my future classes. I am excited to use the many different teaching approaches to cater to the different learning styles of children.

Furthermore, teaching is a very rewarding profession. I have experienced this first hand in my work experience. Whenever I seen that a child was struggling I gave them extra attention to ensure that they would be able to understand the work. Many times I had seen that when children understood and were able to do the work, they had a smile across their face as they raced through the next questions with enthusiasm. From this, I learned that teaching is a very satisfying and fulfilling profession.

Throughout secondary school I volunteered for several years and still continue to volunteer every summer in a local summer youth club. Every year I look forward to the days were I’ll be engaging with the children, living out my gift, and my dream. Furthermore, I have an excellent rapport with children and I have always received great feedback from professional youth workers and teachers. I wanted to continue working with children, so I volunteered for more and more summer clubs each year which grew my passion to work with young children and my desire to be a teacher.

My desire to pursue a career in teaching was born out of my personal learning experience, from the beginning of primary school, together with my natural ability of being able to communicate effectively with young children. Teaching provides rewards such as inspiring children and seeing them grow and develop. I am so thrilled to now be embarking on my dream journey to be a primary school teacher!