We recently received a workshop from Richard which looked at the aspect of mathematics involved in reading a map. We completed the task on google maps, because sadly we were not allowed to complete the task in the middle of the Highlands due to health and safety. We were placed in an area which we were not familiar with and were told to go from a primary school to a small street. We had to memorise the route and then complete it in street view.

I found this task very straight forward. My strategy was to memorise the general direction in which I had to travel and estimate how long to travel in that direction. Then additionally to remember distinct places to aid me. I managed to complete the task rapidly and was then asked to help another group. I feel that the reason other groups struggled was because they looked too deeply into the task and ended up confusing themselves.  By using basic ideas (Ma, 2010), such as estimation and distance, I was able to get from point A to B relatively quickly. However, I am aware that my way is not the only way to do this as there is multiple perspectives (Ma, 2010) in the way in which this activity could have been approached.

I developed map reading skills from a young age as I completed a map reading badge when I was younger in Pony Club. At horse riding, I constantly used my map reading skills to work out where I was going to ride my horse, this increased when my whole riding school moved yards and we were all in a new surroundings. While exploring our new surroundings, things often didn’t go to plan and I developed the skills to remain calm and not panic. I feel that this is one of the reasons I was able to complete the task effectively and remain calm when other people were getting themselves uptight.

I feel that this activity would be very effective in the Primary School Classroom as it allows children to do Mathematics without realising it. By teaching children mathematical concepts without labelling the class as a maths class shows children that mathematics is involved in different subjects and you may not even notice it. It might also help children who feel anxious when learning about mathematics feel more confident when doing maths.

Reference

Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.). New York: Routledge.

This summer, I went on holiday to Canada to visit family. I stayed in Markham, Ontario which is about half an hour from the centre of Toronto (which is not the capital of Canada).

On my first full day in Canada, my cousin and I ended to Downtown Toronto to do some shopping. We went to a small tourist shop as I wanted to buy a cup (I felt that I needed a new staffroom cup and what’s better than a cup with the skyline of Toronto on it). My cousin told me this confusing piece of information.

“The cup will come to about $8.93 but you will only get $1.05 dollars change”

I looked at her confusingly.

“What about the other 2 cents?”

She explained that in Canada they no longer make “pennies” which are one cent so to allow cash transactions they are now rounded to the closest 5 cents (CBC News, 2013). In 2012, the Royal Canadian Mint ceased the penny. However, the coin is still a legal tender. It wasn’t until 3^rd February 2013, that the Royal Canadian Mint officially stopped distributing the penny (Royal Canadian Mint, no date)

While in Markham, I volunteered in an elementary school. I volunteered in the class where my auntie is a support teacher. The class consists of 6 Kindergarten (Primary 1) and Grade 1 (Primary 2) children who are all very high on the autistic spectrum. When I was there, the children were learning about the names and values of all the coins which I found very interesting. They were still taught about the “Penny” though it was explained to them that they are no longer made. The children nodded to this but one of the boys clicked on very quickly.

“But Ms, what if I buy something which does not end in a 5 or a 0”

We were very impressed that he had managed to understand this concept. The teacher explained to him as simply as she could that the money was either rounded up or rounded down.

“But Ms, that’s not fair. You might get less money back than you are meant to”

This little boy was not giving up.

“But the next time you might get more money than you were meant to”

The teacher replied.

“But if you get less money both times, then you are just very unlucky”

The boy said which made us laugh.

Thinking back on this lesson, I can identify different aspects of profound understanding of Mathematics that Liping Ma identified. Having basic ideas (Ma, 2010) of mathematics is important in calculating money in Canada. Not only do you have to be able to round to the nearest 0 or 5, you also have to be able to work out and then add the tax. This then includes the aspect of interconnectedness (Ma, 2010).You have to use other knowledge which you already know to find the answer. It includes a variety of skills such as multiplication, addition and rounding. During this lesson, the teacher held longitudinal coherence (Ma, 2010) as she already knew about the system and though surprised by the child’s question she was able to answer the question effectively.

Is it fair though? I thought back to the task, when we flipped the coin. It is not a 50:50 chance. Are people better or worse off because of the abolishment of the penny? Or does it not matter because of how small a value the penny is worth?

I have thought continuously about this and can’t seem to get to a conclusion. In Canada, they just accept this but coming from a country where we have one pence and two pence. I couldn’t understand how they could just let the cent or two cent slide. Though, I do see that sometimes you would gain a cent or two but would it balance?

Reference

CBC News (2013) ‘Canada’s penny withdrawal: all you need to know’. Available at: http://www.cbc.ca/news/canada/canada-s-penny-withdrawal-all-you-need-to-know-1.1174547 (accessed on 16/11/16)

Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.). New York: Routledge.

Royal Canadian Mint (no date) ‘Phasing out the penny’ Available at: http://www.mint.ca/store/mint/about-the-mint/phasing-out-the-penny-6900002#.WCxYUYXXLVI (accessed on 16/11/16)

At first, I was completely confused during the workshop as to what we were doing but suddenly it all made sense. We do everything to the base value of 10, this is called the decimal system. Why is that? Some believe it is because we have 10 fingers but is this the only reason we could up in 10s.

Why do we use the base value of 10 and not 12?

12 is a highly composite number (Dvorsky, 2013). It divides neatly by more numbers than 10 does. It divides by 1,2,3,4, 6 and 12 leaving no remainders while 10 divides by 1,2,5 and 10 leaving no remainders (Bellos, 2012). Due to this, fractions appear easier as 1/3 of 10 is 0.333…. while 1/3 of 12 is 0.4 leaving a neater number making fractions appear easier.

There is already some examples of 12 base systems in the world;

• There is 12 months in a year
• Grocers deals are often in dozens
• Our day is divided in two sets of 12
• 360 degree in a circle which is 12 sets of 30

A blog by George Dvorsky (2013) states that historically, someone thought about this. Twelve is the last number before we go back to derivations of 3, 4, 5, etc. This suggests that there is some recognition of the 12-base system.

It is still possible to count in 12 using your fingers, this is shown below.

To use 12 as the base system two additional symbols would need to be added instead of 10 and 11 as they have two units. This was the where I confused myself the most. Once I changed the numbers to letters, I understood what we were doing and when I moved back to using numbers it became clearer. As much as I now understand the 12 base system, I do not think it is suitable to change completely from using the base value of 10 as it is what we are used to and it has been used for so long that it is what people understand

Reference

Dvorsky, G. (2013) ‘Why we should switch to a base-12 counting system’ Available at: http://io9.gizmodo.com/5977095/why-we-should-switch-to-a-base-12-counting-system (accessed on 17/10/16)

Bellos, A (2012) ‘Dozenalists of the world unite! Rise up against the tyranny of ten!’ Guardian. Available at: https://www.theguardian.com/science/alexs-adventures-in-numberland/2012/dec/12/dozenalists-world-unite-tyranny-ten (accessed on 17/10/16)

When we were given the opportunity to choose our elective for second year, I instantly knew that “Discovering Mathematics” would be my first choice. I have always enjoyed maths, through both primary and secondary school and I wanted to learn more about it. As comfortable as I am in my ability to do mathematics, I was less comfortable with the thought of teaching it. During my placement, I noticed that I was teaching a lot of mathematic lessons. I introduced new topics and although I knew what to do, I struggle to engage some of the children, particularly the ones who complained that they “couldn’t do maths”.

During our workshop, we learnt that you must have these three things to develop mathematical thinking (Burton, Mason and Stacey, 2008). These areas are competence, understanding concepts and confidence.

This got me thinking about my placement class. During one of my whole class lessons, I was revising co-ordinates with them and then introducing 4^th quadrants. It had been a year since the class had learnt about co-ordinates and we knew that not all the children would understand plotting points in all four quadrants due to not having a strong understanding of negative numbers. I looked up different resources and finally decided upon a “Homer Simpson” co-ordinate sheet where the children had to plot the points and then join them up. As I wanted all the children to be able to participate in this activity, I knew that I needed to come up with a way to make it inclusive to all. The children are seated in mixed ability groups so I decided to work with this. The children would complete the worksheet in pairs.

I started the lesson and tried to make it as interactive as possible having children up plotting points and telling me what quadrant it was but I still noticed some of my reluctant learners zoning out. I handed out the worksheet and they quickly got to work. At first, it took a while to get the children engaged but once they realised what was going on they changed. It was rewarding to see the more reluctant so engaged in the activity and the pride that they held when they were able to do it correctly.

Reflecting on the activity, I realised that although some of the children lacked the understanding, they gained it in confidence and with this confidence, it aided their competence and their understanding. I am glad that all the children were able to complete the same worksheet. I never really thought about the affect different worksheets have on children. They realise if they are doing different work to others and it does affect their confidence. By allowing children who struggle more the chance to work with the children who are more confident in maths, it allowed them to feel a sense of achievement.

As a teacher, I will work on ensure that all children in my class are confident in their ability to do mathematics. By allowing children to work together and share their skills and knowledge, it helped both children’s confidence and understanding which helped them as learners.

Reference

Mason, J., Burton, L. and Stacey, K. (2010) Thinking Mathematically (2^nd ed.).  Harlow: Pearson Education Ltd.

Having been around horses most of my life, it never occurred to me that they may or may not be able to count. During a recent workshop, we were asked if animals could count and it got me thinking about my horses.

When I was younger, I had a pony who would knock his leg three times on the door of his stable if he wanted some attention when we were in the yard. Each time he did this he would get a reward as I would go over to him and give him attention. At first, I thought he was counting but now I think back to it, was it a counting or just brought on by positive reinforcement? Skinner (1953) believes that positive reinforcement encourages a behaviour to be repeated. Could he count or was three just such a well-known beat? In horse riding, I constantly count down from 3. I put my leg on three strides before a jump. I hop three times before I get on my horse. A canter stride has 3 beats to it. I feel that it was just out of habit that he did it three times and not that he could count.

However, I do believe that horses have an understanding of mathematics. Perhaps not counting, but they are able to estimate and have some type of understanding of patterns. They are able to correct their feet to ensure that they could take the correct amount of strides to make the jump. They, also, have an understanding of shape.

There has been many articles written about horses being able to count. I found an article by Nic Fleming explaining his views that horses can count based on a study. He noted that researchers have found that horses were able to detect the bucket with the most amount of apples in it which is the same skill that a baby develops at 10 months. Psychologist Claudia Uller (2008, quoted in Anonymous, 2008) who conducted the experiment believed that horses had a “rudimental ability to count, process the information and make a decision”.

References

Fleming, N. (2008) ‘Horses can count new study says’. Telegraph. Available at: http://www.telegraph.co.uk/news/science/science-news/3338537/Horses-can-count-new-study-says.html (accessed on 08/10/2016)

Anonymous.  (2008) ‘Horses are as good as babies at counting’. Daily Mail. Available at: http://www.dailymail.co.uk/news/article-556329/Why-horses-good-babies-counting.html (accessed on 08/10/2016)

Skinner, B. (1953). Science and human behavior. New York: Macmillan. Available at: www.bfskinner.org/newtestsite/wp-content/uploads/2014/02/ScienceHumanBehavior.pdf (accessed on 08/10/2016)