Category Archives: Recommended reading

Analogue Clocks: Pointless and Confusing?

Living in a digital world, I ask the question: do we need to teach pupils how to read an analogue clock in schools?

Although most of us have converted to digital clocks in our online, digital world with our smartphones, computers and televisions; we cannot deny that our encounters with analogue clocks are not completely non-existent. There is an argument that in most places we visit – schools, work offices, supermarkets, restaurants and hotels – we most likely still encounter analogue clocks (Merz, 2014). Therefore, is teaching how to read analogue clocks not a necessary skill to teach pupils in school?

According to Merz (2014), many teachers are frustrated with the idea of this skill being disregarded, with the argument that analogue clocks can provide a vivid representation of time that digital clocks cannot – which can aide visual learners. Analogue clocks can also teach concepts including time management, the passage of time and how much time we have left to complete something (Merz, 2014).

However, with our fast-developing technological advances, it is difficult not to wonder if eventually analogue clocks will disappear in our society. Nowadays, we see plasma screen televisions or digital billboards nearly everywhere we go – displaying digital time. Although analogue clocks are often visually appealing and provide nice décor, they don’t really provide any use other than telling the time. It is therefore arguable that digital screens are much more valuable in society as they are multi-purposeful and allow for more creativity (The benefits of digital billboard advertising, 2015). For example, recently in a shopping centre in Edinburgh, I passed a large television screen which displayed the current top news stories, multiple adverts for new products which could be found in the centre, whilst also displaying the time.

Moreover, one of the key issues with teaching pupils about the analogue clock in schools, is how complex it is for pupils to grasp and understand. This light-hearted, comical video highlights the difficulties for young learners learning how to read time:

https://www.youtube.com/watch?v=0QVPUIRGthI

(Dave Allen – “Teaching Your Kid Time” – ’93 – stereo HQ., 2009)

I partly decided to write this blog post as I was one of the learners in primary school who had difficulties learning about time. I could not wrap my head around the idea of ‘quarter past’, ‘half past’ and ‘quarter to’ (considering we represented every other number on the clock as a number). I also struggled with the concept that there were different ways of reading the clock (e.g. saying 35 past 7 or 25 to 8) which would both be correct. This raises key issues of problem solving and looking at a mathematical concept from multiple perspectives (key skills which are transferrable across all mathematical topics.)

It is important to note that these are aspects of telling the time which apply to both reading the analogue AND digital clock. It is therefore my opinion that the real issue with teaching time to pupils is the concept itself, rather than teaching pupils how to read a particular type of clock. The video above does highlight the difficulties of learning to read an analogue clock – however with the fundamental understanding of the concept of telling the time, I believe that most pupils would welcome the challenge of applying their knowledge to reading an analogue clock. For example, it is vital that children have a strong understanding that 6 is half of 12 to be able to understand why we use the term half past. Another skill which would benefit children before reading an analogue clock is knowing the 5 times table. According to Drabble (2013), without knowing the 5 times table, “anything beyond the o’clocks becomes almost unotainable.” This relates to the idea of longitudinal coherence, introduced by Ma (1999) who states that teachers should use children’s prior knowledge to enhance learning in the topic at hand. It also links with what she writes about basic ideas, meaning that children should revisit the basic concepts they have learned (i.e. fractions and times tables) to understand that they are required for other areas of mathematics (Ma, 1999).

In conclusion, after doing research online and through my own experiences, I believe there is a necessity for teaching pupils about digital and analogue clocks. I believe that we currently live in a world where analogue and digital clocks are both relevant and should therefore both be exposed to pupils. I have realised since studying this issue, that it is important to ensure that pupils understand the principles behind telling the time before introducing them to an analogue OR a digital clock. Furthermore, learning how to read two types of clocks reinforces pupils’ understanding about the concept of time and allows them to practice telling the time from different contexts. This reflects the work of Ma (1999), who highlights the importance of connectedness – meaning that children can link what they have learned to different contexts.

This picture reflects what I saw on my first year placement and shows how to make reading the time on an analogue clock more visually appealing for pupils, whilst also acting as a visual aide (however it is important that pupils realise that they cannot rely on this, as every other analogue clock they see will not be represented in this way!):

Image credit: Teacher’s Pet (2014) www.tpet.co.uk (http://displays.tpet.co.uk/?resource=1507#/ViewResource/id1507) 

References

Dave Allen – “Teaching Your Kid Time” – ’93 – stereo HQ. (2009). (Video) YouTube: davidwrightatloppers.

Drabble, E. (2013). How to teach … telling the time. The Guardian. [online] Available at: https://www.theguardian.com/education/teacher-blog/2013/aug/05/telling-the-time-teaching-resources [Accessed 4 Nov. 2017].

Ma, L. (1999). Knowing and teaching elementary mathematics : teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates.

Merz, S. (2014). Should We Still Teach Analog Clocks?. [Blog] Stories From Schoolaz. Available at: http://www.storiesfromschoolaz.org/still-teach-analog-clocks/ [Accessed 4 Nov. 2017].

The benefits of digital billboard advertising. (2015). [Blog] Signkick. Available at: http://www.signkick.co.uk/blog/the-benefits-of-digital-billboard-advertising/ [Accessed 4 Nov. 2017].

The System

Following a workshop on the idea of counter-intuitive mathematics, I was intrigued to find out more about how we can look past our intuition or gut feelings to understand mathematical concepts such as probability.

I am someone who struggles to grasp this mathematical concept, as I cannot see past the black and white picture: that everything that happens is completely random and cannot be controlled. To gain a more open mind about this topic I watched the documentary by Derren Brown – The System. This documentary focuses on trying to defy our understanding of probability and to see “the bigger picture”. Derren demonstrates this through inviting people to participate in horse race betting – telling them that he can guarantee they get a win every time.

We follow the journey of Kadesha, who has won every bet since being given anonymous tips from Derren Brown. When Kadesha eventually meets Derren, he explains to her that he has come up with the first “system” which will guarantee her a win for every single horse race she places a bet on. A mathematician worked out the probability of a system being possible to predict the horses accurately and came back with the result 1.48 billion:1 (which is not completely impossible but highly improbable.) So then how can Derren guarantee a win every time?

The truth is, he can’t. What we don’t know as we’re watching Kadesha’s winnings is that there have been 7,775 losses in the process. Altogether, Derren contacted 7,776 random people to take part in this experiment, knowing that someone was bound to win every race. This is because of the extremely high number of people taking part. For each race Derren randomly divided the participants into groups for each horse in the race (so inevitably, one group had to win each round). Kadesha was lucky enough to be part of the winning group every time until the 6th and final race when she was on her own and Derren could only choose one horse who he thought would win (unlike every other round when he could choose all the horses and at least one person would win their bet.)

This is something which didn’t even occur to me as I was trying to work out how this system was possible. I was honestly convinced Derren was a magician! But now, it makes perfect sense. The system works because Kadesha is not alone. This is exactly the way probability works. The more chances or opportunities you give yourself the more likely you are to be successful.

Derren used a similar example with flipping a coin. He explained that he would flip a coin 10 times and every time it would land on heads. What we don’t realise watching the show is that he had been flipping a coin all day until it came to the point when it landed on heads 10 times in a row – so inevitably it was going to happen eventually.

As stated by Derren: “to work out the system you need to understand that we can only know what comes from our own limited experience and our experience can be very far from the truth.” In my opinion this a great way of explaining counter-intuitive mathematics. If I was someone who had a greater understanding of probability, maybe I could have figured out the hidden secret behind Derren Brown’s “system”. This documentary has also highlighted the importance of looking at concepts from a very open mind (which is something I’ve always struggled with in mathematics and have mentioned in previous blog posts). These are both very important aspects of learning mathematics that I will take forward with me in the future: to look at mathematical concepts from a very open mind and to see how the concepts link with my own experiences or experiences of others.

I have linked below the documentary if you would like to find out if Kadesha won the final race or lost her £4,000 worth of savings… I have also linked an article which explains/criticizes the show.

http://www.independent.co.uk/news/media/sport-on-tv-hes-never-wrong-but-brown-fails-to-frighten-the-bookies-777421.html

References 

Derren Brown – The System (Full). (2011). (video) YouTube: ScepticaTV. (https://www.youtube.com/watch?v=9R5OWh7luL4&t=2506s)

Tong, A. (2008). Sport on TV: He’s never wrong but Brown fails to frighten the bookies. The Independent. [online] Available at: http://www.independent.co.uk/news/media/sport-on-tv-hes-never-wrong-but-brown-fails-to-frighten-the-bookies-777421.html [Accessed 30 Oct. 2017].

Place Value, Number Systems and Their Complexities

Following a workshop this week on the concept of place value, I was inspired to write a blog post on the difficulties and complexities found in these seemingly simple, basic concepts of maths.

As someone who learned mathematics in a procedural fashion, it is very difficult for me to wrap my head around the concepts and understanding behind the maths itself. I therefore found this workshop very challenging when I had to look past the mathematics I have learned and replace it with concepts either made up or used in minority areas across the globe. A good example of this was when we had a look at our number base system… 

Understanding place value 

Universally the most common number base system is 10 i.e. 1, 2, 3, 4, 5, 6, 7, 8, 9; then using the concept of exchanging or “borrowing” (which is the common term used in schools) to make the number 10 – where the number 1 represents 10 units (Russell, 2017).

I feel like in my experience at school, exchanging (or borrowing as it was called) was never really explained to me. I understood the concept of hundreds, tens and units – but I could never picture that the number 654 could be the same as saying 65 tens and 4 units or that the number 6717 could be the same as saying 67 hundreds and 17 units. To be able to see this and understand it, a key understanding of place value is required.

This was something which I saw as a maths stater activity when I was out on placement. The teacher would ask the children to break down a number and to think of as many ways as possible to express it. This then helped the children as they moved onto their multiplication and division work (where the concept of exchanging is extremely important). I therefore think it is vital that when I go out to teach, I also touch on the theory and concepts behind the topic the children are working on.

It is also vital to use the correct terminology when teaching mathematical concepts like place value. Liping Ma (2010, p.37) explains that 86 percent of Chinese teachers made the change of saying “composing” or “decomposing” a number, rather than using the term “borrowing”. This mathematically makes more sense because when we do a sum like 32 – 15 we need to break the number 32 down to see how we can possibly take the number 5 away from 2. We are not simply “borrowing” but taking 10 units from the tens column. This is something I will take into consideration in my own teaching practice in the future.

Particularly for place value, I have found some great videos online which show how you can teach the concept to children. This is a great example of teaching place value and also using a child’s mistake to create a new learning opportunity:

(Singapore Maths Place Value Lesson, 2015)

The binary system – a great way to visualise place value! 

As I feel I was not given the opportunity when I was younger to fully explore the idea of place value, it is difficult for me to see past a 10 base number system when using the concept of place value. In our workshop we were introduced to the most basic system – the binary system. The binary system is a system mainly used for computer systems and technology. A binary number is made up of only 0s and 1s (Mathsisfun.com, 2016). So, the way I explained it to myself, if the highest unit you can have is 1 then the number 2 must be 10, the number three 11, and the number 4 100. This took me a while to grasp as I just couldn’t imagine the hundreds, tens and units columns when working it out in my head. Luckily, we got given a spreadsheet that made the concept much easier to visualise, which was then when I managed to wrap my head around it. I have inserted a similar picture which helps to demonstrate this concept:

Image credit: J.D. Casnig, Knowgramming.com (http://knowgramming.com/nanosemaphore/a_bit_about_binary.htm)

As you can see, the two base system (binary system) is at the bottom. This picture also gives you examples of what it would look like for other number base systems.

10 base system – the best system?

The discussion of a binary system also led to the discussion of a dozenal system. It is argued by many that this makes more mathematical sense than our 10 base number system. Some good examples can be found on the dozenal society website, including: packing/packaging (e.g. we get a dozen eggs), a clock dial is numbered 1-12, and the factors of 12 being more useful than the factors of 10 (Dozenalsociety.org.uk, 2017). This is something which really should get us thinking. I assumed that everything we learn in maths is what makes the most sense but this has highlighted to me the importance of questioning the concepts we learn as there could always be better ways…

So what would a 12 base system look like?

Numbers 1-9 stays the same. The number 10 looks like a rotated 2 and is called dek. The number 11 looks like a rotated 3 and is called el. The number 12 looks like the number 10 and is called doh  (Dvorsky, 2013).

An interesting argument for the 10 base system is that it is easier for learners to use their hands to help them count. However, as argued by Dvorsky (2013),  we have three separations on each finger, which if used separately would be extremely useful for counting using the dozenal system. If we use our thumb as a pointer, and start with the index finger we can work our way from the bottom of our index finger until we reach the number 12 (or 10) at the top of our pinky. This is difficult to explain without a visualisation so I have attached an illustration below. Using this system, gives us a total of 24 numbers to work with using just our hands!

As noted, there are many complications and arguments for and against a dozenal system that I could write a whole blog post about it. However, I wanted to touch upon it here as something which has inspired me to open my eyes in terms of not seeing mathematics as a set and fixed structure.

Place value and number base systems are both concepts which I have never considered to be so complex until now. Even after doing my research for this blog post and revisiting my notes from the workshop, I am still struggling to pick these concepts apart and look at them from a very open mind. As I develop my thinking throughout this module, I am hoping it will become slightly easier to be able to analyse and question the mathematical concepts I have learned.

References 

Casnig, J.D. (2013). A Bit About Binary. [online] Knowgramming.com. Available at: http://knowgramming.com/nanosemaphore/a_bit_about_binary.htm [Accessed 4 Oct. 2017].

Dozenalsociety.org.uk. (2017). DSGB. [online] Available at: http://www.dozenalsociety.org.uk/ [Accessed 6 Oct. 2017].

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

Mathsisfun.com. (2016). Binary Number System. [online] Available at: https://www.mathsisfun.com/binary-number-system.html [Accessed 4 Oct. 2017].

Russell, D. (2017). Basic Math Concepts: What Is Place Value?. [online] ThoughtCo. Available at: https://www.thoughtco.com/understanding-place-value-2312089 [Accessed 4 Oct. 2017].

Singapore Maths Place Value Lesson. (2015) (video) YouTube: Singapore Maths Academy (UK).

Stand and Deliver

As part of our introduction to this module it was recommended that we watch the film Stand and Deliver and to analyse the character of Jaime Escalante – a maths teacher at Garfield High School, East Los Angeles, California. We had one question to answer:

Why is Jaime Escalante Different? 

Jaime is different because he takes into account each and every pupil in his class; not only their ability but their individual personalities and lives beyond the classroom. His relationship with the pupils is what encourages the pupils to work with him, rather than see him like all their other teachers (who see them as nothing more than pupils in a classroom.) I believe the class gel so well together by the end of the film because Jaime uses each of their strengths to solve problems together as class, rather than only helping those in the class who seem the most interested and able. Jaime also teaches the pupils fundamental problem solving skills before teaching them anything difficult which provides the pupils with the foundation they need in order to go onto advanced calculus. On the other hand, a lot of the time teachers focus on the topic at hand and not the skills which are needed in order to make the topic manageable for pupils. Jaime also regularly relates what he is teaching to real-life concepts which makes the classes more relatable for the pupils. He doesn’t shy away from taboo subjects, such as sex, but instead uses these to facilitate learning.

This is a great film which really highlights the importance of the difference a teacher can make when it comes to teaching topics like maths, which a lot of students shy away from. It also shows that bringing your own enthusiasm and confidence to a classroom can encourage other pupils, who have previously had a bad relationship with maths or school, that they can do well and succeed. These are all key attributes that I will ensure I bring forward with me when teaching mathematics in the future…

Image credit: http://www.ranker.com/list/stand-and-deliver-characters/reference