Tuesday, September 28, 2010

My Overall Reflection

Can Mathematics be meaningfully introduced to young children? For any mathematical learning to take place, it has to be linked to the child’s experience. For example, he or she cannot find “2” meaningful until he has seen a pair of shoes or other things in pairs. In addition, for him or her to clarify the concept of “2”, he or she has to know that singles are not twos, nor threes, nor more. In other words, his or her knowledge of “WHAT IS” must also be linked to “WHAT IS NOT” experientially. 

Children can learn mathematical concepts meaningfully (This is not referring to the chanting of mathematics concepts by rote). However, as educators we have to be mindful of the way very young children learn such abstract concepts.
Throughout the course I have learnt many significant concepts to teach young children. Like what our friend Jerome Bruner has said, “Concrete” and “abstract” have to tie hand-in-hand to carry out learning objectives. 

My E-L-P-S of “Abstract Thoughts” in young children would be:
Experience: Children see, feels, and explore objects
Language: Math is not just numbers, models, children recognise spoken words representing their experience and abstraction of reality
Pictures:  Children recognise pictures of these experience- another abstraction
Symbols: Children associate written symbols related to them.
Children learn mathematics better if they grow up in an experience-rich and language-rich environment. Their learning is also enhanced through the use of technology (graphic intensive and /or multi-sensory media.

On the whole, I felt that mathematics is not something which children can learn and acquire on the spot. It is through practice and practice again in order for concepts and manipulative skills be internalised. As educators, we must remember that problem –solving skills cannot be developed without a good grasp of the mathematical concepts manipulative skills. It is important to pace children, do not rush them nor allow them to rush.
The emphasis is on the process and not merely the answer. The development of the process takes time, effort and adults’ involvement.
In essence, children can learn meaningfully if the learning is meaningful to them!


For children to do well in Geometry, they need to develop spatial visualization skills. This means that they have to mentally see plane and solid shapes in different orientation, and do not depend on what they actually see on paper or on the computer monitor. The goal of geometry is to help  students understand the properties and dynamics of space, shape, size, proportions etc so that they can be applied to practical real-life experiences such as identifying the related properties in a structural design class and for future construction knowledge. My school does not have tangram sets to teach children Geometry. However I felt that it is essential to begin these attribute blocks to children when they are in pre-school level. It lays the basic foundation of introducing all kinds of shapes to children.

video video

Geoboards are  also an excellent hands-on manipulative tool for discovering the concepts of shapes, angles, measurement, area, and perimeter. 

1. Making shapes

Give each child a geoboard and a large elastic band and a picture of a square and have the students try and make one on with their elastic and geoboard.
Don't assume that this is easy for all students, some need you to guide their hands to begin with. Do this activity with various shapes and teach vocabulary, such as triangle, square, rectangle, corners and sides as the children form the shapes.

2. Copying pictures

Give children dotted paper (same number of dots as geoboard) with shapes drawn on them. With this kindergarten activity the students try and reproduce the pictures.
 IT lesson on Right Angle
Different angles

Introduction for Regular Pentagon Angle measures:
              Any plane figure which is enclosed by its sides are said to be a polygon. There are different kinds of polygon. Basically, the polygons are named on the basis of it number of sides. Any polygon which possess 5 sides are said to be a pentagon. If all the sides and the angle measures of a pentagon are equal, then it is said to be a regular pentagon. The angle measures of any polygon is basically classified into 2 types. They are the exterior and interior angles. In this article, we shall discuss about the regular pentagon angle measures.
Interior Angle Measures for a Regular Pentagon:
Normally, the interior angle measures of a regular polygon can be obtained by using the formula,
180(n) - 360
  where n is the number of sides of the regular polygon.
  The number of sides for a pentagon is 5.
  We can substitute n = 5 in the formula to determine the interior angle measures of a pentagon.
             = 180(5) - 360
        = 900 - 360
= 540

Therefore the interior angle measures of a regular pentagon is 108°.
Sum of interior angle measures of a regular pentagon:
   The sum of interior angle of any polygon can be obtained by using the formula.
                                                            180(n - 2)
   where n is the number of sides of a regular polygon.
   We can substitute n = 5 to determine the sum of interior angle measures of a regular  
                 = 180(5 - 2)
                 = 180(3)
                 = 540°

Whole Numbers & Number Sense

It is important for children to achieve competency with numbers because of the relevance of numeracy and application of mathematical operations in everyday life.
In my school, teachers would play the CD-Rom to Nursery children, learning the concept on “Whole Numbers” teaching number words as they recite the poem “One, Two, Boo!” and singing “The Number Rumba”. Through this activity and some follow-up, they are able to differentiate the ordinal and cardinal sense of a number.


Number Sense Development
The common concepts taught to preschoolers in Singapore are:

1)      Simple “Addition & Subtraction”
Children learn to add and subtract just by counting together at first and then, with practice, fairly quickly learn to recognize by memory. For example, children can learn to play with dominoes or with two dice and add up the quantities, at first by having to count all the dots, but after a while just from remembering the combinations. Children can play something like blackjack with cards and develop facility with adding the numbers on face cards.
Is it effective to draw sticks for counting?

It is important to practice counting things with children. Counting can be incorporated into daily activities: counting money, counting toys or game parts, counting Cuisinaire rods or Unifix cubes, counting on a monthly calendar, counting how many utensils are needed for dinner, etc. Children can count candy, poker chips, the hearts (spades, clubs, diamonds) on face cards in a deck of cards, dots on dice, beans, or pennies. Games like Chutes and Ladders or Monopoly will give lots of practice counting.

Children often have trouble learning "transition" number names, such as the number that comes after the 9s in the two digit numbers; e.g., after 29, 39, 49, even though they can count by tens: 10, 20, 30, 40, etc.  It takes extra practice to learn this. You have to help them understand that what comes after, say, 69, when counting by one's (70) is the same thing that comes after 60 (70) when counting by 10's. They need lots of practice to understand that when you finish the forties you go into the fifties, when you finish the twenties you go into the thirties.  So give extra practice in counting, starting at 7 in each "decade"; i.e., 27, 28, 29, 30, 31, 32 or 57, 58, 59, 60, 61, 62, 63.

Model Drawing is an effective teaching aid to help children understand the problem.

2) The Relationships of More, Less, and Same
 Learning how to put numbers in order and compare them in less than and greater than guidelines is an important math skill. The important ideas are to make sure we use discrete materials that are movable when teaching this concept to children. Then, emphasize moving left to right when comparing groups is also essential in developing this concept. I would usually model estimating prior to actual comparing. For instance, I would initially cue students to differences in groups by using groups that are unlike in many attributes (size, shape, color), gradually fade these differences and have students compare groups with like attributes. Eventually, the learning objective would be to invite children to identify if a given group of objects has more than, less than or the same number of objects when compared to another group of objects.

2)      Numeral Writing and Recognition
Most nursery children are able to count to twenty or even higher and kindergarteners up to 100; and most recognize numerals to twenty in isolation and can match one to one. I agree with the book that in most cases this concept is taught using matching exercise such as matching sets (count the number of cubes first) with numerals or words. 
Number Lines
3) Counting on and Counting back
I felt that this skill can only be practiced if children have strong recognition of number sense.  Number lines is a great tool to enhance this development. 

 The uncommon concept is:
Anchoring Numbers to 5 and 10

Tens frames are a great tool for teaching guided math lessons.  The ten-frame provides a spatial representation that supports children’s visual understanding of  “five-referenced, ten-referenced, and doubles-referenced conceptions of numbers up to ten and the development of mental imagery for such numbers. It also supports development of partitions of ten.
Teachers should start off by using 5 frames to make sure that students learn the complements of 5. Then they can move on to using 10 frames.  These frames can be used to teach students at the concrete, pictorial and abstract level.  Use two color markers (if you don’t have these then just spray paint some lima beans so they are two-colored). See the ideas and links below for resources.  
More Ideas and Games:

Use of Technology

Computer Software Programs

Math CD-ROMs are very useful because they can be used for learning a concept or for practice. On the Internet, many Math games are explained; these can be used with a class to review and practice a Math concept. Games are very motivational for students so they will be eager to participate and gain a better understanding of the concepts. Just having a computer involved can make a huge difference in students’ attitude and feelings towards Mathematics. For my school we have a Math software program which is very beneficial in learning Mathematics. Our teachers usually show/explain Math concepts using concrete materials and then use this software for students to practice the skills. This program also keeps records of students’ progress, reducing the number of worksheets needed.

As an educator, I felt that using computer to teach and practice Mathematics reduces disruptions in the classroom as it engages students with behavioral/emotional issues in a different way. Using computers is also more fun for children so they will put in more effort and thus learn concepts more quickly and retain these concepts better than in a typical classroom setting.

There are tons of computer games that make learning basic math facts and concepts exciting and enjoyable. For this instance, the teacher is using the CD-Rom to teach “Fractions” and “Whole Numbers”. The computer provides fabulous visuals of fractions and manipulates them in ways that create understanding of  
this difficult topic. On the other hand, the computer endlessly patient if the child need re-teaching of any subject. 

Monday, September 27, 2010

Place Value

Place Value

1)      Number in Numerals
    In a preschool setting, a child cannot understand the arithmetic of whole
    numbers without a basic understanding of the numeral system. Children
    should be exposed to concrete materials such as pop-sticks when learning
    how to count. So, the first activity which I will use with children is to
    make their own bundles of pop-sticks. This draws attention to the
    importance of ten.

2)      Number in words
       After counting children would normally write the numbers followed by words    
       because they feel that numbers are easier to write and learn then words.

3)      Place Value Chart
   The term place value means that the value of each digit depends on its place in the numeral symbol.  

4)      Hundreds tens and ones notation
      I felt that Number Expanders are useful tools because they offer a hands-on
      way of manipulating the symbolic representation (numeral) of a number. They
      make a bridge from physical to symbolic models for number. Opening and
      closing the number expander act as a reminder of the actions with materials
      (but not as a replacement for this). Because they do not physically model
      the size of the number, they can be made with any number of place value
      columns and so can represent very large or small numbers at more advanced
Example of Hundreds tens and ones notation

5)      Expanded Notation
   I am giving an example to explain why expanded notation should be taught the
   last. The symbol 2 in the number 2134 stands not for 2 but 2000. In fact,
   2134 means
                            2000 + 100 + 30 + 4

   i.e, two thousand one houndre and thirty-four. As is well-known, the ten
   symbols 0, 1, 2,…, 9 are called digits. the digit “2” in 2134, being in
   the fourth place (position) from the right, stands not for 2 but 2000, i.e.,
   two thousand, as mentioned above.
An example of expanded notation done by a 6 year old boy


Environmental Task on Problem Solving

The Environmental task we did was motivating and meaningful. I could remember my money concept was learnt from my aunt who used to bring me to supermarket to explore grocery items and I have acquired the knowledge of using money to purchase items which I need. On the whole, I felt that money is an important concept which students should be taught because money is essential in everything we do.

The two main skills that children need to know when it comes to money—and the earlier, the better—is being able to identify money and being able to tender money and make change. Using children’s books and poems can help students learn how money is used in everyday life and provide a starting point for a lesson on money. Teachers can draw on this knowledge to introduce the values of coins and notes and relate their value to items that children recognize, such as candy, toys, or movie tickets.

According to Jerome Bruner, environment is one issue which concerns educators as it creates the willingness to learn in children. In our task, we have applied the three modes of instructions which Jerome Bruner talks about in his theory of mathematics.

First: The Enactive Mode. We are using real-life experience to carry out learning objectives such as developing the responsiveness of money in children; enhance their counting skills; the values of money; cognitive skills as they engage in thinking about, and acting within, a financial environment.

Second: The Iconic Mode. The different sections Daiso supermarket serves provide a set of images for children to visualise what they would like to buy. Instruction list by the teacher is also very important as it helps children to easily grasp the information they need to know.

Third: The Symbolic Mode. The children can do beyond the instructions level through abstract thinking. Children think about what kinds of materials are good for this D-I-Y gift. Problems become more complex as students learn to combine coins and notes to reach certain values. For this instance we have given them $10 which includes coins and notes. The reason why we chose Daiso was because everything costs $2 and the children can practice multiples of two. We felt that preschoolers should not be handling decimals at this point, perhaps when they turn 8 years old. 

Sunday, September 12, 2010

Reading Reflections on first two chapters of the textbook

     Principles and Standards for School Mathematics is so important that all teachers and prospective teachers should be familiar with what it says and have read at least relevant important sections.
National Council of Teachers of Mathematics (NCTM) provided detailed discussion of each of Principles and Standards for school mathematics.These principles and standards presented a vision of school mathematics—a set of goals toward which to strive. Throughout the principles and standards, the vision for mathematics education is expressed using words like "should, will, can, and must" to convey to readers the kind of mathematics teaching and learning that NCTM proposes.
     The NCTM Principles and Standards for School Mathematics (2000) clearly defines five process standards (Problem Solving, Reasoning and Proof, Communication, Connections, and Representations) for the learning of mathematics. These processes function as simultaneous goals for student learning, activities, habits, and processes through which mathematical content is learned. Thus, for instance, while it may be a goal for teachers to get students communicating mathematics to one another, the process of communicating also leads to the learning of mathematics.

     Developing a solid mathematical foundation from pre-kindergarten through Primary One is essential for every child. In these years, students are building beliefs about what mathematics is, about what it means to know and do mathematics, and about themselves as mathematics learners. These beliefs influence their thinking about, performance in, and attitudes toward, mathematics and decisions related to studying mathematics in later years. Children develop many mathematical concepts, at least in their intuitive beginnings, even before they reach school age. 
     From my teaching experiences, toddlers spontaneously recognize and discriminate among small numbers of objects, and many preschool children possess a substantial body of informal mathematical knowledge. Hence, after reading the 2nd chapter, I felt that it was crucial for educators to foster children's mathematical development from the youngest ages by providing environments rich in language and where thinking is encouraged, uniqueness is valued, and exploration is supported.