How to excel in Mathematics?

Mathematics is considered to be a rather challenging subject to many students. For the majority in this group,studying mathematics is time consuming and frustrating.


However, there are those who love mathematics, they know that once you are familiar with the patterns in the questioning techniques, solving mathematics problems gives them a sense of satisfaction. Some said that you are either born to do mathematics, or not. I tend to disagree.


To be good in mathematics, you need to have the conceptual understanding, the strategy and the discipline to practice. Think of mathematics to be like a competitive sports like football. Firstly, you need to understand the rules of game. Then you need to have a strategy on how to win in this game. Lastly, you need to practice regularly in order to execute your strategy with precision.


Mathematics is not a passive subject. You can’t just read about it. You need to tackle and solve those problem sums regularly, in order to get real expertise and excel in this subject. Once you understand these principles, you will begin to see the beauty of this subject. As Bertrand Russell, the great philosopher and mathematician,once said, "Mathematics posses not only truth, but also supreme beauty".


Lastly, if you aspire to be at the top of the game, you need to have that passion for learning and desire to excel. Using football as an analogy again, the top footballers are able to achieve success because they have a passion for the sports. They truly believe that they can make a difference in the game. If you have such passion, you are on your way to success. If you are still struggling, fear not. You can always look for a mentor, someone who is able to guide you in your journey to excellence.

Challenging PSLE Math Question 2/2012

24 more girls than boys signed up for a holiday camp. 
3/4 of the girls and 2/3 of the boys turned up for the camp. 
20 did not turn up. 
How many children signed up for the holiday camp?

Challenging PSLE Math Problem with Video Solution

Challenging Math Problem (PSLE) with solution

•88 children took part in a swimming competition.

•1/3 of the boys and 3/7 of the girls wore swimming goggles.

•Altogether 34 children wore swimming goggles.

•How many girls wore swimming goggles on that day?


Solution below:

Role of the Teacher in Mathematical Discourse

Teachers play an important part in orchestrating mathematical discourse in the classroom. In designing math lessons that promote discourse, the following should be considered:

1.    Problematising the content – By turning content into problems, it creates opportunities for students to discuss and present possible solutions. Instead of offering standard solutions, the teacher can facilitate learning by posing questions to elicit and challenge students’ thinking. Through appropriate questioning technique by the teacher, students would be actively engage in clarifying and justifying their ideas, either orally or in writing.

2.    Scaffolding – To support the students, teachers have to decide when to provide information, when to clarify an issue, when to model, when to lead, and when to let a student struggle with a difficulty. When misconceptions are surfaced during their discussion, teachers should seize the opportunity to turn it into a teachable moment to explain a certain concept with more details.

3.    Feedback – Providing timely feedback to students on their performance will enhance their learning. Teachers could also create opportunities for students to provide constructive feedback to their peers. Giving peer critique involves higher order thinking skills such as analysing and evaluating, which are desirable outcomes of education.

4.    Monitoring- During learning activities in the classroom, teachers should monitor students’ participation and decide when and how to encourage each student to participate. It is also a good practice to reflect after the lesson to identify what students have achieved, where they make errors, when they have misconceptions, when they are not engaged - in this way, teaching and learning can be synchronised and powerful.

With students taking more responsibility of learning, teachers would have to consider adjusting their role in the classroom to complement the change in teaching approach. From delivering instruction in the past, teachers have to assume the position of a facilitator, who will provide appropriate scaffolding and feedback to guide the students in the right direction.

What is Mathematical Discourse?

Mathematical discourse is a teaching approach that engages students in discussion about math in a manner that articulate their understanding of concepts. Through the lesson activities, students also sharpen their skills in mathematical reasoning and debate. This approach to teaching mathematics is a more constructivist and student-centered approach that involves teachers asking probing questions; think “Socratic Questioning”, to make student thinking visible by eliciting from them the steps needed to arrive at the solution and why a certain strategy was used. Through strategically designed learning activities, students are given opportunities to critique their own and others’ idea and uncover efficient solutions to mathematical problems. 

The Theory 
Classroom discourse is based on the theory of constructivism, which states that learning happens when the students interact with the environment. Through these interactions, knowledge is created. This is also aligned with the social-cultural views on learning which advocates that learning is a social process whereby students working collaboratively are able to reach new understanding that otherwise could not be achieved by working alone.

Make Thinking Visible Through Flip Classroom Model

What is the flipped classroom?

An innovative teaching model that delivers instructional content outside class through interactive, teacher-created resoruces (eg. Podcast, online digital videos, power-point presentation, etc) and interaction with the content – skill development, practice and related learning activities – is done in class. Moving lectures outside of the classroom allows teachers to spend more 1:1 time with each student. Students have the opportunity to ask questions and work through problems with the guidance of their teachers and the support of their peers - creating a collaborative learning environment.

In a flipped class, instructional activities like explanation and lecture are done at home, while homework activities like math problem sets or writing practice activities are done in class. Thus homework becomes classwork and vice versa.

What are some benefits of this method?

·   Allow teachers to spend more time to faciliate students’ learning

·   Foster stronger student/teacher relationships

·   Offers a way for teachers to share teaching resources with other teachers, students and parents easily

·   Enable students to “rewind” lessons and master topics at their own pace

·   Creates a conducive environment in the classroom that promotes collaborative learning

Engaging Students in Mathematical Discourse

Engaging Students in Mathematical Discourse
by Chris Siew


An effective mathematics programme is more than just concepts and skills. Problem solving is central to mathematics learning. The development of this ability is dependent on five inter-related components, namely Concepts, Skills, Processes, Attitudes and Metacognition(Mathematics Framework, MOE).
This article highlights the notion that mathematics should be taught in a way that mirrors the nature of the discipline (Lampert 1990), instead of practising concepts and skills in isolation. How can teachers design lessons that create opportunities for students to make conjectures and engage in constructive argument about problems in order to discover important mathematical concepts? What are the attitudes and processes that we need to develop in our students to help them understand the essence of what it means to do mathematics?

Designing the Learning Environment
There is no doubt that accuracy is important in mathematics. However, teachers have to communicate to students that understanding concepts and communicating mathematical ideas are more important than just getting the right answers. Promoting mathematical discourse is a strategy to develop students’ understanding and communication skills in mathematics.
Let us start by examining the kind of structures that need to be in place to facilitate discourse in a math class.

1. Setting expectation for learning – The first step in transforming a traditional classroom to one that promotes discourse is to set the right expectation for learning. Students need to understand that their role in the classroom is not just listening to the teacher and waiting for answers. In the new learning environment, they will be exposed to open-ended tasks and they would need to play a more active role to complete the task. The teacher will assume the role of a facilitator in these activities, who will scaffold and point them to the right direction but not necessary show them the correct answers all the time.
2. Establish classroom relationship – There will also be a shift in relationship between students. Learning will become more social and ideas will be exchanged freely. Their peers will no longer be just someone who sits beside them, they will be their learning partner and collaborator to build each other’s knowledge. The relationship between the student and teacher will also be changing. The teachers will not be the only source of information. Instead, they will direct students to various resources to uncover the key learning points of each topic.
3. Establish rules and management system – Teachers have feedback that conducting collaborative learning activities can be quite chaotic; students are noisy and teachers are not in control. The key to good classroom management is to establish clear rules on the dos and don’ts when students are engaged in learning activities. It is also important to put in place a system to manage conflicts that may arise during interaction among students.

Teacher as facilitator
It is a misconception that in this new learning environment, teachers just present the problem and wait for things to happen. Do you think students will be able to discuss, solve the problem and present the solution automatically? Learning does not happen by chance, it happens by design.
In order to achieve participation and conceptual understanding, teachers have to design lessons that provide opportunities for students to:

1. articulate their understanding,
2. engage in peer-critique and
3. build knowledge collaboratively

To make students thinking visible during discussion and promote engagement, teachers could consider two aspects of teacher discourse: cognitive discourse and motivational discourse. (Stein, 2007)
Cognitive discourse refers to things that teacher says to promote conceptual understanding of mathematics by making their thinking visible. The objectives of teacher discourse are to

1. help students make connection,
2. help them learn from mistakes and
3. stress individual accountability

Motivational discourse refers to things that teacher says to promote students’ engagement and addresses the affective domain. Students need an affective support structure in the classroom that makes it a safe environment that allows them to learn through mistakes and take risks.
The objectives of supportive motivational discourse are to:

1. Promote learning

1. Focus on the process of learning
2. Challenging students
3. Viewing errors as constructive
4. Supporting persistence

• Promote positive emotions

1. Using enthusiasm
2. Using humour
3. Reducing anxiety
4. Addressing emotional needs

• Foster peer collaboration 

1. Building collaboration
2. Emphasizing joint goals/shared responsibilities

Discourse Assessment
Teachers need time to develop competency in using mathematics discourse to advance learning in class. To help track the progress as a class goes through this transformation, Hufferd-Ackles, Fuson, and Sherin (2004) created a framework to describe and evaluate the level of mathematics discourse in the classroom. (Refer to table below)

Components	To compare   lessons from Term 1 to Term 4 based on analysis of lesson observation videos
Questioning	Was there a shift from the teacher as the main questioner to students & teacher as questioners?
Explaining mathematical/ science thinking	Did students increasingly explain & articulate their ideas?
Source of mathematical/ science ideas	Was there a shift from the teacher as the source of all math/science ideas to students’ ideas also influencing the direction of the lessons?
Responsibility for learning	Did students increasingly take responsibility for learning and evaluation of others and self?

Adapted from Hufferd-Ackles, Fuson & Sherin (2004)

Four components are examined – questioning, explanation of mathematical thinking, source of mathematical ideas, and responsibility for learning. Teachers can use this framework to assess the discourse level of the whole class. It can also help them identify individual students who are not actively participating in the discussion so that additional scaffold and support can be provided to encourage this group of learners to be more active in class.
Conclusion
Promoting mathematical discourse in the classroom can help to strengthen students’ processes in mathematical reasoning and their skills in communicating mathematical ideas. As students learn to make and test hypothesis, question, and agree and disagree about problems, they are learning the essence of what it means to do mathematics. As teachers, we need to re-examine our practice and re-think how we can use mathematical discourse to better engage our learners.

References

1	Hufferd-Ackles, K., K. Fuson, and M. Sherin. "Describing Levels and Components of a Math-Talk Learning Community". Journal for Research in Mathmatics Education 35 (March 2004) : 81-116
2	Lampert, M. "When the Problem is not the question and the solution is not the answer: Mathematical Knowing and Teaching." American Education Journal 27 (Spring 1990):29-63
3	Stien, C. "Let's Talk, Promoting mathematical discourse in the classroom."Journal: Mathematics Teacher. Issue: November 2007, Volume 101 , No.4