"Inspired the Uninspired"

During the 6 days of course, Dr. Yeap definitely has "Inspired the Uninspired ". He taught us skills in teaching Math building on our prior knowledge as learned during our DPT days. Furthermore, he  facilitated our learning by demonstrating it. He also gave us many "problems" to solve over the period of the course and showed us "magic" used in mathematics. These are fun activities which will truly not only engaged us but showed us what a teacher can do in teaching mathematics.

Problems leading to concept


 22 different problems were shown to given to us to solve during the 6 days. The biggest problem of all was the quiz, which we had to go out to different area around our school to answer the question given. It was a true test to our brain on other methods we can use to derive the answer without any tool used. Rachel and I used my shoe as a guide to measuring various items. We also used other tools like pen and pencil as our measuring tools. Most importantly, Dr. Yeap taught and challenge us to think out of the box, finding other ways to solve the given problems.

Some of my favourite problems which can be used with our preschoolers:

1. Poker card trick
2. Using literature book in teaching mathematics - Spaghetti & Meatballs for all
3. Salute
4. Think of two digits
5. Using toothpicks to form shapes

Concrete Learning

Learning the CPA learning theory by Jerome Bruner:

1st Phase:  Using real life objects to teach Mathematical concepts.
2nd Phase: Learning Mathematical concepts through the use of objects like counters and unifix cubes.
3rd Phase: Using pictures and photographs
4th Phase: Using drawings which are abstract

This allows us to build our current knowledge on what we know about learning Mathematics. This can also be shown clearly in Dr. Yeap's lesson as he demonstrated the concept according to Bruner. Dr. Yeap also constantly relates the activities to Bruner's theory and how children learnt. Learning Mathematical concept through manipulatives materials do involve children in their learning process thus children can remember longer.

Language and terms used in Mathematics

Language plays an important role when teaching Mathematics concepts to young children. Use of wrong word or phrases may affect the child's understanding as he/she progress to higher level of Mathematics. Some of the common mistakes made by teachers were shared as follows:

Wrong: Take away
Right:    Subtract or Minus

Wrong: Fraction - 3 out of 4
Right:    3 fourths of 4 or 3 quarters

 Wrong: Digit V.S. Number
Right:    Digit is for 1 - 9 and Number is 1 - 10

Wrong: "The shoe is the length of 5 paper clips" Measuring using objects.
Right:    "The shoe is about the length of 5 paper clips" (Reason: Because it is an estimation and the number varies if the length of paper clips are different."

 Wrong: Pen and Pencils cannot be added, even though the child refers them as items.
Right:    Things with different nouns cannot be added. This will the child to develop understanding when ones learn algebra. This is because you cannot add X & Y.

 Wrong: Cookies and stones cannot be divided as it is discreet quantity.
Right:    Only continous quantity can be divided - sugar, rice...)

Big Ideas on Numbers:

1. Knowing numbers is a "Big Idea" in Mathematics. Children learn mathematics by asking questions and teacher's facilitation. (Cardinal, Ordinal, Nominal and Measurement Numbers)
2. Children needs to know what can be and cannot be added in counting.
3. When learning to add big numbers, teach children to group and upgroup numbers into tens. This will help children to understand number bonds.
4. Number is an important concept in whole number. Children need to know how to break number up appropriately.
5. Leaning conceptual meaning of four basic opertations (+, -, x, ÷)

Lastly, teach less, learn more!

Questions:

1. What other ways can we best support children with special needs in learning mathematical concepts?

2. I head about lesson studies as shared by Lynn Heng during her talk that Dr. Yeap knows about this. Will it possible to share with us about lesson studies during our next module?

Technology and Mathematics

Integrating technology in a preschool classroom, good or bad? There are two schools of thoughts when discussing about whether or not to introduce technology to young children. However, I felt that the appropriate use of technology will engage children of different learning abilities. This also allows differentiated learning in a class.

According to Bruner (Smith, 2002), children are active learners and problem-solvers who are ready to be engaged in challenging task. He also further explained the principal of learning for young children from hands on before pictorial, pictorial beofre abstract. The use of technology does bridge the gap between hands on and pictorial as most Mathematics programs are presented in a form of game which allows children to be actively engaged in play. However, children still needs concrete materials first and technology cannot be used to replace hands-on learning. Thus, teachers need to assess children's learning ability before using technology to reinforce what the children are learning.

In my school, we bought the license from this company who provide us with programs in both languages. In this program, it comprises of mathematics, language, music and other components. Activities are planned to engaged children with interesting games and activities for children to play. It also provides different concept at different level and the teachers can choose the appropriate level for each child to play.

This program comes with interactive whiteboard (the touch pen), book scanner and overhead bridge projector. Teachers can do a class learning through the use of interactive whiteboard before children work as a small group or individually.

Why did we choose to incorporate technology in our school and the advantages of it:

1. Opportunities to aid different learning styles.
2. Different form of assessment for young children. 
3. The computer also offers unique opportunities for learning through exploration, creative problem solving, and self-guided instruction. (Clements,1999)
4. Technical Competence Skills
5. More collaboration among peers as they engage in group learning.
6. Interactive learning.
7. Encourage independent learning.
8. Visual stimulation

The disadvantages of using technology:

1. Teachers need to be savvy in the use of technology.
2.  Equipment failure may disrupt the lesson. Teachers need to have back up plan.
3. Lesser attention span during other class time without the use of technology.
4. Expose to unrelated websites. Filter needed to prevent this from happening.

In conclusion, technology can be a good reinforcement for children as they learn different mathematics concepts, however, concrete learning should be done before the use of technology. Teachers' technology skills should be an issue when using technology as part of children's learning process. Teachers are encouraged to play with the programs first before using it. This will help teachers to be more comfortable and familiar with the program.

Lastly, technology should be used sparingly should not be replaced by technology. Teacher and children interaction is still critical in the early years of young children. Teachers will still need to facilitate, assess and observe children's learning when using technology.

Embrace the technology of the 21st century with open arms as we prepare our children for the future.

References:
 Clements, D. (n.d.). Dialogue on early childhood science, mathematics, and technology education first experiences in science, mathematics, and technology. Retrieved from http://www.project2061.org/publications/earlychild/online/experience/clements.htm?

Rectangle = Square, Square ≠ Rectangle

I think what interest me most is not Rectangle = Square, Square  ≠ Rectangle, but how teaching Math concepts have developed and changed since I learnt Math in school. Even though I don't hate Math, but I felt I could have done much better and scored As instead of Bs, if I was taught this way. As I recall, Math = 10years series. Please raise your hands, if you have the same experience. (I can see many hands up!)

During the past 4 lessons, Mr Yeap had demonstrated how Math should be taught according the Bruner. J's theory:
1st Phase: Real Life Objects
2nd Phase: Manipulative Objects
3rd Phase: Photographs and pictures
4th Phase: Drawings (Abstract)

This theory also affirm our methodology for teaching Math for young children. Math can be so much fun and interesting with games. We could have guess solving problem is fun when it is not paper and pencil. He also shared how we can lead children to the concept that they are learning by giving them problems to solve first.

Furthermore, Mr Yeap also shared many Big Ideas related to Math and this give us a clearer idea of the importance elements and concepts in Math. Some of the Big Ideas which gave me a better idea of Math are:

1. Number is an important concept in whole number and children need to know how to break number up appropriately. Through this children will be able to develop number sense and concept when they are not able to multiplication, division, addition and subtraction with big numbers. That is why counting, grouping and regrouping are important concept which help children to problem solve in more difficult sums.
2. To appropriate language terms when teaching mathematics to young children. Instead of saying 3 over 4, we say 3quarters. Instead of saying 2 is lesser than 3, we say, 2 is less than 3. This reminds us of the importance of using appropriate language to help children to develop skills for more advance mathematics like problem sums.
3. Math can be fun!

Reflection 17 July & 18 July

Dot Cards as a Model for Teaching Number Relationships - Pg 142 - 143 (Chapter 8)

Dot cards are a set of cards (Blackline Masters) which contains:

1. dot patterns
2. patterns that require counting
3. combinations of two and three simple patterns
4. ten frames with "standard" patterns
5. unusual dot placements

(You can find Blackline Masters here: "http://lrt.ednet.ns.ca/PD/BLM/table_of_contents.htm")

Children use these cards for any activity that involves number concepts and develop thinking skills in young children. In the book, the author also suggested some activities which the teachers can use the dot cards for.

Activity 8.24 - Number Sandwiches (Using the dot cards):

1. Select a number between 5 and 12.
2. Students will need to find combinations of two cards that total that number. 
3. Place the two cards back by back and place them down with the dot side out.
4. Try to find at least 10 pairs and challenge the partner to name the number on the other side.
5. Flip the card over to confirm the answer.
6. The same pairs can then be used again to name the other hidden part.

Through this activity, children will learn the mathematical concept of:

1. Part-whole relationship.
2. Develop understanding of number concept.
3. Develop number bonds.

Chinese Proverbs

Today lesson reminds me of the following Chinese Proverbs: " I hear and I forget; I see and I remember; I do and I understand."


Explanation from: http://en.wikiquote.org/wiki/Chinese_proverbs

不聞不若聞之，聞之不若見之，見之不若知之，知之不若行之；學至於行之而止矣(不闻不若闻之，闻之不若见之，见之不若知之，知之不若行之；学至于行之而止矣)。(pinyin: Bù wén bù ruò wén zhī, wén zhī bù ruò jiàn zhī, jiàn zhī bù ruò zhīzhī, zhīzhī bù ruò xíng zhī; xué zhìyú xíng zhī ér zhǐ yǐ.)

• Literally: Not hearing is not as good as hearing, hearing is not as good as seeing, seeing is not as good as mentally knowing, mentally knowing is not as good as acting; true learning continues up to the point that action comes forth
• Common: I hear and I forget; I see and I remember; I do and I understand.
• Moral: You can only understand something by trying it yourself.
• Revised: Tell me and I [will] forget. Show me and I [will] remember. Involve me and I [will] understand.
• Also: You can't understand until you walk a mile in someone else's shoes.

Isn't this is how we learn?

Posted on 16 July 2012

Lesson 1: Interesting & Inspirational

Today is the first lesson of Elementary Mathematics. It was interesting to see how math is taught in different ways.

One of the big idea which I learn is to give children problems to solve. By giving children to problem to solve which leads to the specific mathematical concept. Through problem solving, children will be encouraged to try first. Just like what we had experienced in class. Dr. Yeap gave us problem to solve and he gave us time to work with the problem independently or with friends. As I had hands-on experienced, I was able to understand the reason behind the concept. That should be how children learn! Instead of telling them, "Class today we are going to learn about shapes or addition.." Why not place different shapes of items around and ask the children to make smart guesses on what are the similarities and differences in the items to lead to what they are going to learn and learn them to the concept. Furthermore, as the children are making the guesses, they are already observing the characteristics of the items, this will make learning easier for them.

One of the other way which I recalled doing was to place a certain number of items in a container and placed in the math corner before the children come in to class. When they come in, the children started to have conversation on why the container is there and how many items are in the container. Children started to make different guesses by deducing, counting and making random guess. Through this, children were able to make sense of what they are going to learn later.

What was inspirational was Bruner J. 's learning theory which is use by the Primary School Math Curriculum. Children learn new mathematical concept through hands-on before pictorial and through pictorial before abstract. This is what preschools are currently doing. But this leads me to wonder, whether did we prepare the children more than what they are supposed to know in Primary One as they are going towards pictorial and abstract when they are in Kindergarten Two. It will be good to share more of Primary One learning outcome so that we teachers will be better informed in teaching Mathematics to our preschoolers.

Some other big ideas branching from "How to teach mathematical in a learning environment":

1. Dienes. Z theory on variability.
2. Math is about patterns and sequence.
3. 4 types of numbers: ordinal, nominal, cardinal and measurements numbers
4. Using different tools in teaching mathematical concept: poker cards trick, egg trays, ten frames and dice to teach tens and ones.
5. How to do differentiated lesson for diverse learners in class?
6. How math and language is interrelated?

Posted on 16 July 2012

Politically right?

Expectation

Chapter 2 Reflection

Mathematics and language is inter-related. Nowadays, mathematics is a test of language, understanding of words. Besides knowing adding, a child needs to know that, the sums, altogether, in total and etc means the same. Because of this, math is no longer straight forward. You have to understand the language, the logic, the patterns and the sequence in Mathematical concept. 


Theories
In the textbook, it discuss the two theories which can be used to explain how we learn mathematics.

First, it's the constructive approach of Piaget, we learn math through constantly building on our prior knowledge and creating new knowledge out of it using the tools and materials which we know. It's like building blocks, we build our foundation before building the top but at times, the blocks may fall and we build it up again based on what works and does not works on our previous experiences.

Second, it's the sociocultural theory of Vygotsky. In sociocultural theory, mental processes exist between and among people in a social learning setting. Secondly, the learning depends whether it is within the learner's zone of proximal development, a range where the learner is able on ones own and learning from the support of others. Next, it's semiotic mediation which refers to the use of language and culture which is interrelated which includes the use of mathematical symbols and it is through learners' interaction in class through activities which develops the meaning of these symbols.


Implications for Teaching Mathematics
However, it is good to be informed of the theories but it is not a teaching strategy. Theories give us an idea how children learn and process their knowledge and how we teacher can support their learning process:


Providing opportunities for children to talk about mathematics in their normal routine and classroom activities. One of the good example is to do counting while waiting for children to clean up or to line up for the next activities. Different numbers can be use to count down or even shapes. Another good opportunity is to get the children to keep learning materials by sorting them according to their characteristics. Children will have opportunity to reinforce what they have learn in a different context. This is a funny activity which children of different age group can do. Teachers can also find opportunities during their outdoor play too.


Building in opportunities for reflective thought through planned activities which engages children in interesting problems in which they use and build on their prior knowledge. Learning corner activities and construction play are good opportunities for children to engage in independent reflective thinking. After all, play is the most natural setting for all children.


Encourage multiple approaches through different variations of activities from hands-on to visual cards to paper-pencil activities. These will encourage children to transfer and relate their knowledge and skills across different activities. Children will be able to learn how to relate what they have learnt and apply in different context. In our school, besides learning mathematical concepts through hands-on activities and activity sheets, we also engages the 21st century technology in using computer software in engaging children in math. Math concepts are presented in interactive games which children can play as a group or as an individual. This is use to reinforce the concepts whiche the children had learnt during class time.


Engaging students in productive struggle which allows children to work with different level of activities and to challenge them to the higher level when possible and giving them appropriate time frame to work on it. This will give time for children to try to on it on their own, rather than depending on their teachers for solution.


Treating errors as opportunities of learning and exploring different ways of solving the errors. Confidence and positive learning needs to be cultivated to young children to persevere in their learning and treating mistakes as learning point. Teacher needs to reinforce by affirming and encouraging the child to try other ways instead of jumping in to give the right answer to the children. Facilitation is important, which is the role of the teacher. 

Scaffold new content and remove when the children are able to be independent in working on the mathematical concept. Teachers are encouraged to do observation of children in order to assess the readiness of a child in the mathematical concept. This will allows the teacher to know the zone of promoxity for the children as to move them to new learning concepts when they are ready.

Honor diversity learning in young children has they all have different prior knowledge and background in their learning. Teacher should be sensitive and provide opportunities for young children to learn in different stages. For myself, I uses exercise books for differentiated learning. Example, different math sums and problems will be given to the weaker and stronger children and the average children will work on the common sums given. The sums will be written into their exercise book to challenge the children at the right learning level. The use of computer software programs also provides opportunity for children to learn math at their own level. The program which we use has different learning levels for different concepts.


Mathematics Proficiency

Mathematical Proficiency requires teachers to put in alot of effort as mathematical concepts and connections are developed over a period. Activities are selected to help children to progress developmentally.

Learning math in an instrumental manner will help children to be effective learner of learning new concepts and methods in solving mathematical problems. Children are constantly engaging in building on their existing knowledge. Through this children can make more connections based on what they are learning and what they had learnt as we constantly challenge them to think further.

Children will also have less to remember as they learn in an instrumental manner. Children will be able to connect and "big ideas" (Brooks & Brooks, 1993) which are interrelated concepts.

In order for children to connect to more concepts, children needs to retrieve inoformation and connect to their ideas. As children connect, it increased retention and recalling of information for young children.

As children work and connecting with more concepts, it enhances problem-solving abilities. When children is familiar with the concepts adn the relationships between a situation and a context, and they will be able to identify and solve the problem.

When children are familiar with the concepts as they work with over a period of time, children will be more confidence in their learning. Children can also improve their attitudes and beliefs as they make sense of what they are leanring. It also develops a positive self-concept and confidence in the child's ability in learning mathematical concepts.


Math is like connecting different dots together to make sense of what they are learning and they are going to learn. It is how teachers facilitate children's learning in connecting the dots.

Chapter 1 Reflection

In chapter 1, the author high-lighted different types of principles and standards for school mathematics.

It mentioned the learning principle for learning mathematics. The first learning principle is the importance of understanding the concept and to have the ability to think and to reason the mathematical logic to solve new problems. The second principle is that student can learn mathematics with understanding as as learning is enhanced in classrooms where students are required to reflect on their learning through evaluation to develop their mathematical understanding and reasoning skills. This is what we have always been taught in school, to understand the concept and not to memorize the sum. On the other hand, we are taught to learn and practice our mathematical concepts and skills through endless practice on 10-years series. So, how do we really understand the concepts if we are just mainly doing the same types of sums over and over again? From numbers to algebras to geometry to measurements and etc... It puzzles me how I have passed my math just by memorizing formulates and sums.

Thus, as an early childhood educator, I believe the importance of learning the concepts through hands on activities and not just through worksheets. Through manipulative materials, children are free to make mistakes and correct it when they get it wrong. This also allows children to self-assess and self-correct when the completed the activity given. Unifix cubes, counting bears and 2-coloured beans are good manipulative materials. These are materials which I constantly used it in teaching mathematical concept to young children.

As mentioned by Walle, Karp and Williams (2010) the five process standards through which students acquire and use mathematical knowledge:

Problem solving learn mathematical knowledge through solving sums.

Reasoning and Proof emphasizes the logical thinking on what makes sense and what not.


Communication is important in mathematical as one needs to know how to talk about, write about, describe and explain mathematical ideas.

Connections made within and among mathematical ideas and mathematical should be connected to others disciplines and to real situations.


Representations uses graphics to present the mathematical ideas and relationships as a form of communicating mathematical ideas to others.

Through the principles and standards, it allow us to understand how to become a teacher of mathematics which was mentioned in the textbook. It mentioned the importance of teacher to have knowledge of mathematics in order to teach mathematics to young children. I agree with this as teacher we must first have fear the subject that we are teaching because the children will feel it. Just like me, I love math because of my favourite math teacher in Primary School, if without her, I will not have survive math. Persistence and positive attitude are other characteristics mentioned in guiding children in solving mathematical sums which they are not able to understand over and over again. Thurs, persistence is needed to demonstrate to students that it is alright to make mistake and re-trying it again. Teachers should always be ready for change in their teaching approach to cater for different learning styles and needs. One approach may not work for all. Lastly, teachers should always be reflective in their teaching approach. Research and reading are important for teachers to constantly be more informed of other teaching methodology, to be a better Teacher of Mathematics.

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