Category Archives: 2.3 Pedagogical Theories & Practice

Horology – Importance of Time 🕑

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Horology is defined by the art or science of measuring time, such as watchmaking. Horology is a complex process which is full of mathematical processes in order to succeed in creating an accurate watch. Ilan Vardi describes mathematics as language with the use of equations and formulas however what distinguishes it  from other subjects is the idea of philosophy. For example, the understanding and completion of any mathematics problems tends to be by discovering the order in things. Mathematicians and watchmakers share very similar interest;his job is to establish a sequence that tracks the measurement of time. the series of events that allows the clock to show the correct time and continue the constant measuring of time.

But what is time? I believe its important to emphasis to children the actual value of time. We all know that counting 60 seconds = 1 minute and 60 minutes = 1 hour.  Similar to place value, this can be a very  difficult process for children to first understand.

Dave Allen demonstrates how complication teaching time can be.

Giving children a background to why time is important (keep track of days, weeks, seasons etc.) would be a more efficient form of guiding their understanding. If they understand the importance of time keeping and why humans use the units we do, they would see the relevance of the subject rather than being told how to read a clock. Before discussing how we record time nowadays, we could consider teaching about sundials or Egyptian water clocks. By learning about other measurements of time can highlight how we can came to using the clock/watch today.

Egyptians waterclock were one of the first timekeeping devices that did’nt use the sun, in the 16th century BCE. It worked by recording the flow of water dripping at a constant rate from a bowl through the a small hole in the bottom. As the water levels went down and several markings in the container allowed them to know how much time has passed. i found it particularly interesting to discover that the Egyptians also used an instrument called merkhet, that followed the alignment of the stars in order for them to know what time of night it was. They identified 10 hours of night because of the 10 stars and then gave 1 hour for the sunset and sunrise, giving a total of 24 hours day and night. This is an interesting fact as to why we still use 24 hours today!

This finding again, highlights that maths is everywhere and has been for a very long time. This encounter of the origin of time has gave me a deeper understanding of why we record time today and how essential maths has been to creating the world and life that we live today.

Liping Ma’s Discovery

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Liping Ma has undergone research to compare the differences on how mathematics is taught in China and the USA. Her findings were fascinating; even though US students go through more education to become a teacher than those in China, the Chinese teachers have a deeper understanding of mathematics and therefore are able to teach more efficiently. She found that American teachers taught more in a procedural way rather than using the logic of mathematics.

“About 10% of those Chinese teachers, despite their lack of formal education, display a depth of understanding which is extraordinarily rare in the United States.” (Liping Ma, 2010)

I was edger to find out more about Liping Ma’s discover; How do the chinese teachers teach more efficiently? and what can I learn from this for the future?

In Liping Ma’s book ‘Knowing and Teaching Elementary Mathematics’  she explains her theory(Cuarezma, 2013) and suggests how the Chinese teachers understanding how maths contributes to the students success. This is an important theory that i would consider while teaching children mathematics.

so why is it that Asian children consistently outperform American students? there are many factors that researchers have found that contribute to this “learning gap”;

  • Difference in cultural (parential expectations)
  • School organisation (time spent on maths)
  • The content within the curricula.
  • Teachers Knowledge.

Researcher Deborah Ball identified teachers knowledge and understanding of mathematics should be connecting ideas of and about the subject. The concept by the knowledge of mathematics meant; “comprehension of particular topics, procedures, and concepts, and the relationships among these topics, procedures, and concepts.” (Liping Ma, 2010). therefore, understanding all these things and making it clear to the students will assist them reaching success.  The meaning behind the knowledge about mathematics is aiming at the comprehension of the nature and discourse of mathematics. Additionally teachers consistently thinking about the ‘substantive knowledge’; correctness, meaning and connectedness. Mathematics should always be open to more than one way solutions. Students who can solve a problem with a variety of methods will be able to achieve higher, as the skills can be applied to similar situations. I believe this is important for myself and teachers to consider while teaching because if we notice a child struggling to grasp a process, it may be beneficial to teach another method that they could use. Hopefully seeing another method will be the situation clearer for the child rather than becoming frustrated. 

As Liping ma was researching she drew on how teachers taught; subtraction, multiplication, division by fractions and perimeter. I mainly looked into subtraction and found her way of teaching a lot more enlightening. She studied the American the method of borrowing or exchanging in subtraction, so for example, discussing that when subtracting 21-9 that they had to borrow unites from the tens column. she realised that teacher were expecting students to know, based on their knowledge, from a very procedural method of teaching.

Chinese teachers mainly use a regrouping method of subtraction and in contrast with US teachers,  35% of them demonstrate multiple ways to carry out regrouping. Liping Ma states that teachers address the standard algorithm as well as discussing other ways to solve a problem. The main method of regrouping is by decomposing a unit of higher value, so breaking down the hundreds, tens and units number. for example using 21 again, Chinese teachers would decompose the number rather than suggesting to borrow. This highlights that the language a teacher uses is crucial to help students understanding. Using ‘borrowing’ can confuse children as it acts as a metaphor whereas decomposing highlights that the higher digit can be broken down. Linking what happens when the children done addition is also important to help them understand how 10’s are formed and taken away.

An example of the regrouping by a Chinese teacher from Liping Ma’s (2010) book-                             “How come there are not enough ones in 53 to subtract 6? Fifty-three is obviously bigger than 6. Where are the ones in 53? Students will say that the other ones in 53 have been composed into tens. Then I will ask them what can we do to get enough ones to subtract 7. I expect that they will come up with the idea of decomposing a 10. Otherwise, I will propose it. (Tr. P.)” 

An example of borrowing from An American teacher-                                                                 “Where there is a number like 21−9, they would need to know that you cannot subtract 9 from 1, then in turn, you have to borrow a 10 from the tens space, and when you borrow that 1, it equals 10, you cross out the 2 that you had, you turn it into a 10, you now have 11−9, you do that subtraction problem then you have the 1 left and you bring it down.”

Students that understand why higher value units need decomposed is more efficient than borrowing as they can also apply this knowledge when working with three digit equations. Once they’ve learned the facts and the procedure they an apply this to any situation they are in.

What I take away from Liping Ma and Deborah Balls’ theory is that teachers must be able to anticipate children’s response in order to encourage their way of working and not to simply restrict them to one method. Displaying multiple methods of solving a problem is essential as it promotes more independent learning and also a better variety as each child learns differently. I will also be more cautious of the language i use in the classroom to ensure I provide the best learning situation and avoid confusing the children.

References 

Ma, L 2010, Knowing and Teaching Elementary Mathematics : Teachers’ Understanding of Fundamental Mathematics in China and the United States, Taylor and Francis, Abingdon, Oxon. Available from: ProQuest Ebook Central. [4 October 2017].

Cuarezma, A. (2013). Q & A with Liping Ma. The New York Times. [online] Available at: http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma.html [Accessed 4 Oct. 2017].

Abacus- Important or Not?

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An Abacus is one of the first counting devices invented and I was intrigued why it was used?

It is a simple wooden/ plastic framed tools with beads used to show a visual representative of place value and assist addition, subtraction, multiplication and division. It can be used as a calculator by sliding the beads along the rods (which is a lot funnier than typing on a calculator!). The abacus has been around for around 2500 years and as primarily used in countries such as China, Korea, Japan etc but it’s now recognised by most of the world.

The abacus tool is not used as often in schools anymore as devices change and evolve however, an assistant psychology professor at the University of California in Riverside, Aaron Seitz, (South China Morning Post, 2011) states that children should not just be writing to learn maths but should be using other senses to consolidate their learning.

“Maths is not strictly verbal, tactile or a language; it is a kind of an abstract representation and so if that abstract representation is built upon information from multiple sensory modes, it is going to be more developed and more effective than coming from a single sense.” (South China Morning Post, 2011)

The experience that children gain with using the abacus is more successful as they can use physical touch, sound of the beads and visually looking at them as more efficient tool to grasp rather than simply writing down a sum on paper.

The following youtube video demonstrate how a child uses the basic abacus to assist with his maths.

(Addition and Subtraction by Abacus, 2016)

The abacus is especially useful in early years so the children can understand the concept of the number holding a quantity by the visual representation of the beads. I believe the abacus is still important to use today as it is one of the easiest and more fun ways for children to understand addition and subtraction.

Although abacus is not used as often anymore, i believe the reason behind the abacus is important to ensure the best learning opportunities for young children. Many other ways teachers can use maths in a physical form such as getting the children to count (add or subtract) money, sweeties or each other. After, researching about the abacus, I will take into consideration while teaching the basics of addition and subtraction that if need be, i know it is an effective tool that i can implement.

References 

South China Morning Post (2011). Ancient abacus still has a place. [online] Available at: http://www.scmp.com/article/971772/ancient-abacus-still-has-place [Accessed 10 Oct. 2017].

Addition and Subtraction by Abacus. (2016).Mostofa Saadi.

Teaching Science

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Prior to our input with Richard on science, we were all to prepare a 2-minute science demonstration to show to our group. There were great ideas which can stem great lesson plans for the children in the classroom. This picture shows the “lava-lamp”
experiment, img_7992the oil and water demonstrates to the children that materials don’t always mix due to the oil being less dense than the water, this is called “intermolecular polarity”.  Add food colouring for some colour and a fizzing tablet for the bubbles. You can then take about the creation of gas.

 

Other experiments included the “Floating egg”. so we got to see that an egg does not float in normal water, however, the egg

img_7995does float in salt water. This can get the children asking why it has happened! children tend to think that its only heavy items that sink but with this experiment you can teach them that it all about the DENSITY of the object compared to the density of the water.

Another experiment shown in my group was the colour chromatography. A simple line drawn by a marker pen on kitchen roll that’s dipped into the water can show the separation of components in a mixture. The mixture separates because its components travel across the paper at different rates, based on their attraction to the paper or solubility in the solvent. You can do this experiment with multicoloured pens and it should show the three priimg_7997mary colours that are built to make that colour.

In another science lesson, we were thinking about highlighting the importance of fair testing to children. that a test cannot be accurate if more that one variable is changed in an experiment. for this lesson, we were linking unfair testing into other curricular subjects. A good one is in PE children can be given different equipment or resources, some children can be given advantages in order to make it easier which promotes unfairness. Timg_8006he children would be able to see it a lot clearer if they can identify how it is unfair to do a sport with more than one variable changed. For example, we created an “unfair lesson” for throwing an object into the target. Each group had a different object (varying weights and sizes), they all had different angles and distances from the target and they had different methods of throwing. The children can quickly identify which team were getting more in and why then you can relate the importance to science.