Day 6- 17 August

Mathematics is an excellent vehicle for the development and improvement of a person’s intellectual ability in logical reasoning, spatial visualisation, analysis and abstract thought.

Task: Card Trick

Successful methods include arranging the cards in a row and moving cards which have been removed by moving them upwards and another method is to use drawing.

Implications for Instruction:

-          Working with teachers

-          Poker cards

-          Spelling the numbers

-          Challenging and understanding

-          Visuals (cut- out)

-          Prior knowledge

Task: Cut out figures from paper art

-          Find a pattern

-          Find similarities

-          Thinking routine (Harvard)

-          Simple to complex

What do you see?

-          Describe

-          Observe

-          Visual

What do you think?

-          Relationship

How do you wonder?

-          Reason

-          Infer

Story- telling:

Before reading- story express

-          Predict the story based on given words

-          Write predicted story in 5 to 10 lines

After reading- Circle strategy procedure

-          Retell story through pictures or text

Things I have learnt today?

-          Card trick

-          Paper art: Visualization and patterning

-          Story telling: Pattering and differentiation

Day 5- 16 August

The five learning ways in which a child learns:

1. Concrete
2. Generalization (Number Sense)
3. Visualization
4. Meta-Cognition
5. Communication

There are three kinds of triangles:

1.       Right angle

2.       Scalene

3.       Isosceles

Skemp:

1.       Instrumental

2.       Rational (you can relate to what you know before) Part of Piaget’s theory

3.       Conventional

 

 

Day 4- 15 August

Activity: Dice Problem

Subitize is to perceive at a glance the number of items presented without counting them. One of the objects that can use is a dice. Subitize is a skill that can be developed by three approaches.

How can student learn?

First approach is by:

·         Modelling and exploring by challenging students’ thinking

·         Scaffolding and provide exposure

·         Prior knowledge of activity

·         Student need to subitize by the opportunity that happened

·         Learn through observation be it intentional/ unintentional opportunities

Second approach is by Bruner’s theory:

-       Concrete objects

-       Pictorial

-       Abstract

 

Third approach is by:

·         Repetition

·         Variation of the same task

Hint: Numbers at opposite side of dice will add up to 7

 

Day 3- 14 August

Fractions:

Today’s lesson was about fractions.

What are fractions? Fractions are everywhere. Anything that is divisible into pieces or smaller units, such as a pie, a dollar or a sandwich, can be represented by fractions. They are, in addition to percentages and decimals, another way of expressing that part of a whole value that is less than one. Their ubiquitous nature makes them one of the essential skills that children will get out of school, and the more help teachers can offer, the easier the learning process will be.

Ways to help students:

Make it Relevant

-       Without real world relevance, fractions can simply seem like hard work. Help student want to understand fractions by first modeling their use in real life and make it fun and interesting

Make it Visual

-       Help student understand the concept of fractions by making them visual

Make it Tactile

-       Use concrete materials to teach

 

To teach fractions, teachers can use the Jerome Bruner’s theory of concrete, pictorial and abstract.

 

 

Day 1- 12 August

How Students Learn Mathematics?

 

Jerome Bruner’s Learning Theory. This theory talks about concrete, pictorial and abstract (CPA). Each stage is important in learning.

 

It is very important to learn through exploration and discoveries.

 

Activity: Tangrams- find different ways to form rectangle and use different number of shapes to form rectangle.

A teacher’s role in this activity:

Ø    knowing the student’s background and prior knowledge and build on it

Ø    allowing time, opportunity for discovery and exploration

Ø    build the student to the next higher order of thinking

Ø    Be a good and exciting role-model

Ø    Important to scaffold student’s learning

 

Day 2- 13 August

Whole Numbers:

The main thing in today’s lessons – the importance of exposing students to concrete hands- on experiences. Up till today, I have learnt the four ways in how students learn Maths:

1. Visualization
2. Be aware of patterns
3. Number sense
4. Metacognition 

“Children learn by manipulation of concrete materials.”

 

Today is about whole numbers: use the 10-frame to teach Maths concepts. I have learnt a skill which is to use story at the beginning of the Math lesson. I learnt how to use the 10-frame (paper, egg cartons, commercial materials, ice tray) to find the total sum of three different numbers. The usage of concrete hands- on materials when working with whole numbers. (Addition). One cannot mix nouns and adjective together. They are two different things.

 

Some of the benefits of using the 10-frame:

· Teach number bonds

· To understand conservation

· To understand one-to-one correspondence (fundamental and it's important)

· Place values

· Teach counting

 

Chapter 2

Chapter 2 Entry

Mathematics is more than completing sets of exercises or mimicking processes the teacher explains. Doing mathematics means gathering strategies for solving problems, apply approaches and see if they lead to solution and whether answer makes sense. Doing mathematics in classrooms should be closely model the act of doing mathematics in the real world.

The classroom environment for doing mathematics must have the language of doing mathematics. Children engaged in these verbs and actions in mathematics classes will be actively thinking about the mathematical ideas that are involved. Students begin to take the math ideas to the next level by connecting to previous material, responding with information beyond the required response and conjecturing or predicting. When this happens, students are getting an empowering message that they are capable!

Two things that result in conceptual understanding is making mathematics relationships explicit and engaging students in productive struggle. The teacher's role in making mathematical relationships explicit is to be sure that students are making the connections that are implied in a task. he focus is on students' applying their prior knowledge, testing ideas, making connections and comparisons, and making conjectures. Students must have the tools and prior knowledge to solve a problem and not b given a problem that is out of reach or they will struggle without being productive. When students know that struggle is expected as part of the process, they embrace the struggle and feel success when they reach an answer or solution.

There are two theories which have been developed and are most commonly used by researches in mathematics education to understand how students learn mathematics- Constructivism and sociocultural theory.

Students learn through inquiry and activate their own knowledge and try to assimilate or accommodate new knowledge. The implications for teaching mathematics are:

1. Build new knowledge from prior knowledge
2. Provide opportunities to talk about mathematics
3. Build in opportunities for reflective thought
4. Encourage multiple approaches
5. Engage students in productive struggle
6. Treat errors as opportunities for learning
7. Scaffold new content
8. Honor diversity

What does it mean to understand mathematics? It means relational understanding- know what to do and why and mathematics proficiency.

The benefits of developing mathematical proficiency are:

1. Effective learning of new concepts and procedures
2. Less to remember
3. Increases retention and recall
4. Enhanced problem- solving abilities
5. Improved attitudes and beliefs

Chapter 1

Chapter 1 Entry:

The five content strands defined by Principles and Standards are:
1. Number and Operations
2. Algebra
3. Geometry
4. Measurement
5. Date Analysis and Probability

Following the five content standards, Principles and Standards lists five process standards:
1. Problem Solving
- Build new mathematical knowledge through problem solving
- Solve problems that arise in mathematics and in other contexts
- Apply and adapt a variety of appropriate strategies to solve problems
- Monitor and reflect on the process of mathematical problem solving

2. Reasoning and Proof
- Recognize reasoning and proof as fundamental aspects o mathematics
- Make and investigate mathematical conjectures
- Develop ad evaluate mathematical arguments and proofs
- Select and use various types of reasoning and methods of proof

3. Communication
- Organize and consolidate their mathematical thinking through communication
- Communicate mathematical thinking coherently and clearly to others
- Use maths language to express mathematical ideas precisely

4. Connections
- Recognise and use connections among mathematical ideas
- Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
- Recognize and apply mathematics in contexts outside of mathematics

5. Representation
- Create and use representations to organize, record and communicate mathematical ideas
- Select, apply and translate among mathematical representations to solve problems
- Use representations to model and interpret physical, social and mathematical phenomena

The process standards refer to the mathematical processes through which students should acquire and use mathematical knowledge.

The common core state standards are:
1. Mathematical practice
2. Learning progressions
3. Assessments

There are six major components of the mathematics classroom that are important to allow students to develop mathematical understanding:

- Creating an environment that offers all students an equal opportunity to learn
- Focusing on a balance of conceptual understanding and procedural fluency
- Ensuring active student engagement in the National Council of Teachers of Mathematics (NCTM)process standards (problem solving, reasoning, communication, connections and representation)
- Using technology to enhance understanding
- Incorporating multiple assessments aligned with instructional goals and mathematical practices
- Helping students recognize the power of sound reasoning and mathematical integrity

To become a teacher of mathematics, one must have the knowledge of mathematics, is persistence, have positive attitude, readiness for change and has reflective disposition.