liping ma | My Education ePortfolio

As I write this blog, semester 1 of 2nd year is almost at its end. I get misty-eyed when I think about this model nearly being over as I have enjoyed it so much and learned things I would never have known about otherwise. Truly fascinating module. It’s nothing to do with complex equations or any horrible higher maths recaps. I have mentioned in a serious blog how i didn’t like maths in high school but ‘Discovering Mathematics’ has made me appreciate it again as i now have a deeper and broader understanding of fundamental mathematics and how this links with wider contexts.

Our first ever lecture in this module introduced us to PUFM (profound understanding of fundamental mathematics). At first I just brushed it over as I said to myself I can research into it later on in the module as I didn’t understand it and quite frankly found it confusing.

Once I came across Liping Ma’s book and read up about her theory then I was able to understand it better and see the links in all my future lectures. Ma wanted to understand why the U.S.A were in a much lower rank for test results than China was. Ma (2010) concluded that the reason the U.S.A were so behind was because teachers didn’t obtain an extensive understanding of elementary mathematics. She figured that during a teachers training they should be made aware and become habitual with basic (fundamental) mathematics as this is what the teachers in China have knowledge on from the start (Ma, 2010).

“By profound understanding I mean an understanding of the terrain of fundamental mathematics that is deep, broad and thorough. Although the term ‘profound’ is often considered to mean intellectual depth, it’s three connotations , deep, vast, and thorough, are interconnected.” (Ma, 2010, pp. 120).

To achieve the expected knowledge that Ma thought a teacher should have, she came up with 4 principles that would enable a teacher to have a profound understanding of fundamental mathematics:

Connectedness – “A teacher with PUFM has a general intention to make connections among mathematical concepts and procedures…” (Ma, 2010, pp. 122). This means being able to make links and see connections between mathematical concepts in a wide range of things in society. Also the importance of highlighting this to students when teaching so that they can discover and see these links. In the students learning this would mean that their knowledge learnt would not be fragmented but rather connected.

Multiple Perspectives – “Those who have achieved PUFM appreciate different facets of an idea and various approaches to a solution, as well as their advantages and disadvantages In addition, they are able to provide mathematical explanations of these various facets and approaches…” (Ma, 2010, pp. 122). This means the teacher should respect the multiple aspects of problems and solutions, moving away from there only being one answer. Together with allowing students to inspect these multiple aspects so that they have a flexible understanding of the topic.

Basic Ideas – “Teachers with PUFM display mathematical attitudes and are particularly aware of the “Simple but powerful basic concepts and principles of mathematics” (e.g. the idea of an equation)” (Ma, 2010, pp. 122). This simply means that the teacher should encourage children to explore the points relating to problems. Bringing thoughts back to the basics of mathematics to embolden the students understanding and make the subject less daunting. Students learning and understanding will therefore be more broad and in depth about the subject.

Longitudinal Coherence – “Teachers with PUFM are not limited to the knowledge that should be taught in a certain grade; rather they have achieved a fundamental understanding of the whole elementary mathematics curriculum.” (Ma, 2010, pp. 122). This means that the teacher needs to be able to see where the student is at in their studies and how to progress the student further or fix the problems they are having. The teacher should be of a mind to return and look at learning done in the past, but also able to plan in line with the direction of the classroom’s curriculum and accommodate the students needs within their studies.

“As a mathematics teacher one needs to know the location of each piece of knowledge in the whole mathematical system, its relation with previous knowledge.” (Ma, 2010, pp. 115).

The you look deeper into the 4 principles you can see why mathematics is so important for people to know about. In school, when thinking about the use of connectedness, we need to look across all the curricular areas to see where the links are. This is something a teacher can’t miss out on as mathematics, as I’ve discovered in this module, is in everything that we do in our lives and its important that from an early age we can see how mathematics links with the world around us. When we look into multiple perspectives a myth crops up in my mind that I have heard through out my education, that is that “there is only one process of finding an answer”. This is false. There are many ways in which a problem can be solved, it’s all about teaching the different roots and pathways. The principle of basic ideas is as fundamental to fundamental mathematics as you can get as fundamental is another word for ‘basic’. As professional teachers we need to understand that when teaching young children mathematics, we need to peel back and look deep into the roots so that children progress correctly and will enjoy maths. When we talk about longitudinal coherence, we talk about how we can progress a student further. Teachers need to recognise where a student is at with their learning and identify the correct steps in further educating the student and building on their previous knowledge.

So what have I learned fully for this module? Apart from Liping Ma’s theory about fundamental mathematics, I have learned that mathematics is precisely EVERYWHERE!! Connectedness in full power. I think it is important that when becoming a teacher you need to be aware of this so that you can educate your students on this so they can fully appreciate and enjoy mathematics to the full with the right rooted understanding. Furthermore I am a strong believer in if you’re an enthusiastic teacher while teaching your students will also be enthusiastic about the subjects you teach them. During this module my belief become more true to me as my lecturers were the most enthusiastic mathematics teachers I’d ever seen. This is contagious and made me become even more enthusiastic about the subject myself. Reflecting back on my thought of maths in high school, I’m a changed ‘learner’.

Overall, i think if we suppose a child to have a deep understanding of a specific subject, then so must we.

References:

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

In Richard’s lecture today we looked at the fundamental mathematics within sport. Before the lecture I knew there was a lot of maths within sports but I had never looked deeply into each part of a sport and how it’s applied to mathematics.

Firstly in the lecture, Richard had us look at an old football score table for 1888-1889 and compare it to the ones we see today. We looked at the rankings, the wins, loses and draws and how many the won for and lost against. To try and make sense of the old table we redesigned it by firstly putting the football team names in order from rank 1-12 and followed that with their point beside them. By ordering the team we used the fundamental mathematics category, from Liping Ma, of basic mathematical principles (Ma, 2010). We used counting to help us order and matching to match up the points form the old table to then new order of teams.

Once we had created the new table and a few minutes of problem solving time we came to understand by using algebra we could figure out the total number of point by multiplying the amount of games won by 2 and adding the number of games they drew. We wrote this out as Wx2 + D = P.

After speaking a bit about maths in sport, we worked in small groups to come up with ways in redeveloping an existing sport based on its fundamental mathematics. My group chose to redevelop the sport of netball. In the redevelopment of our version of netball we considered the length of the pitch, where the players pass to/move to during the game and the point systems, considering the goals/basket heights.

By extending the court length, this increases the chance for changeovers and more passes to up the chances of conflict with the other team. Thus makes the game more exciting and unpredictable.

In normal netball the ball is passed to attack, who are closest to the goal. But we decided to make it less simple and create a new rule stating that the defence player now stand on goal side and the centre pass has to pass to either WA or GA (who are now of the other teams goal side). This means there will be a longer distance to pass the ball to get a goal.

Instead of having just 1 hoop to score, we added 2 more hoops creating 3 hoops for possible scoring (a little like the Harry Potter sport of quidditch) which all symbolise different scores. Highest hoop equals a higher score, the lowest hoop equaling a lower score. We also considered from where the player throws the ball to score. We thought by splitting the key in two would make a more fair goal as it is easier to score when you are closer to the hoop. Therefore the closer half on the key gives you a lower score than the further away half of the key.

In our new version of netball there are a lot more fundamental mathematics taking place that the player needs to be aware of. The player will have to use basic mathematical principles (Ma, 2010) when deciding what hoop to try to score in. The use of addition and subtraction would be used to put the hoop score along with the place they have shot from together. They would have to think about what angles would give them the best chance of scoring a goal. They would have to consider the distance they would have to throw the ball as the pitch is a longer length now, perhaps the consideration of an extra pass in place in the midfield?

In my own time I wanted to research about mathematics used in lacrosse as i play it at university. There are many basic mathematical principles relating to lacrosse:

Perpendicular lines are used in lacrosse. To avoid your stick being check by an opponent you must keep it perpendicular to the ground as you can protect the whole stick with your body.
Horizontal line or 180 degrees line are used when starting a match in the middle of the field, when two players from opposite teams throw the ball up in the air.
The speed of the ball when it is passed. The speed can differ whether you are doing an air pass or a ground ball pass.
The speed of the ball can also be effected by the force you put into pulling the bottom of your stick back and pushing top of you stick forward to pass.
The angle of the stick when you pass and receive the ball. Your arms should represent an acute angle when pulling back to pass the ball.
The weight of the ball.
Body weight distributed between your feet when passing, receiving a shooting.
The motion and curve of the swing when throwing/passing the ball.
The motion of cradling the stick.
The diameter of the ball (approx. 25 inches).
Segment bisectors, the line that cuts the pitch into two equal parts goes through the middle of the midfield third.
Symmetry – the pitch is symmetrical when split directly down the middle.

In the future, during P.E lesson at school I would like to link the curricular areas and demonstrate to pupils how basic mathematical principles can be seen in the different types of sports they will be playing. I would encourage pupils to do sports outside of school as well and bring in their discoveries of maths within it. I remember in school we did a topic of DST (distance, speed and time). I could look at this with the pupils and apply sports to it. This would make maths a bit more enjoyable if they are passionate about a certain sport.

Overall, while playing a sport if you think about it mathematical and apply this is strategy it could improve your chances of winning for your team! The fundamental basic concepts that I spoke about (weight, motion, symmetry, angles, position, counting, adding, subtraction, multiplication, simple algebra, force, distance, speed and time (Ma, 2010, p.104). This therefore proves mathematics can be applied everywhere including playing a sport.

References:

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

Edwards, A. (2012). [Website]. https://prezi.com/59smjt77b-sm/math-project-lacrosse/. (Accessed 09/11/17).

Liping Ma’s Profound Understanding of Fundamental Mathematics

The Winning Equation in Sport 🥇