Measurement

Hi.

We meet again in the 4th assessment, measurement. In this entry, we will look at the learning experience that related to teaching measurement in the early childhood. Measurement is always happened in our daily life and it is everywhere. Most of our daily activities need measurement. The examples of measurement that we encounter are measure the amount of ingredients in cooking, our height or weight. 

As an adult, if we were asked a question on "how long is this table?" we will answer, "it is about 10 cm". However, if we asked children the same question, they will answer, "it is about the length of my hand". Therefore, in the early childhood, teachers need to create learning experience that assist the development of early sense in measurement. What teachers could do in teaching measurement is famalirising the students with non standard units followed by the standard units (Origo, 2008).


However, there are some teaching sequences that could be followed when teaching measurement in the early childhood:

• Identify the attributes to be measured

Example: water (volume) can be measured using pail

2 pails of water

• Similarities and differences

Similarities- both can be used to measured volume

Eg: 1 glass of water

    1 jug of water

Differences- The volume of water in the jug is more than the volume in the glass 

• Compare the attributes

Looking at the similarities or differences that they can see in the attributes. Attributes could be spatial:length, area and volume; physical: mass (weight) and temperature; or have no obvious physical connection with objects: time (Queensland Studies Authority, 2005).

It can be direct comparison; comparing two similar objects and different objects or indirect comparison; comparing two objects that cannot be directly aligned (the length of a desk and the height of a doorway).

• Order

Ordering thing according to the attribute

long-longer-longest

• Non-standard units

Non-standard units are practical, personal and familiar, and are used in real-life situations.

Eg: Students compare themselves to see if they are shorter or taller than the gingerbread man.

• Standard units

Standard units enable more accurate and consistent measuring in different places by different people and facilitate communication that yields the same understanding about measurements (Queensland Studies Authority, 2005).

Learning Experience:

Measuring length is a crucial element that need to be focused on in the children's learning (Schwartz, 1995). Taking from this understanding, an initial activity that involved creating a measuring device which compared the length of desk to the length of a paperclip and ice cream sticks is conducted in the tutorial class. At the end of the activity, students are able to use an appropriate unit in measuring, create a measuring device and make an indirect comparison.


Activity 1: Use and appropriate units in measuring

Q: How many ice cream sticks are used to measure the length of this table?

A: It is 4 and a half.

Q: How many paper clips are used to measure the length of this table?

A: 16 paper clips

From the activity, teacher could develop students' understanding on the concept of choosing appropriate units visually. Students can see that the larger the unit, the smaller the measurement or vice versa. For example, from the activity above, students realised that the length of the same desk may need 16 paper clips long or only 4 and half ice cream sticks long. However, although the number of ice cream stick is less that the paper clips, the table's length is more accurate to be measured using the paper clips.

Activity 2: Create a measuring device

Valuing the usefulness of measuring length is a crucial element that drives children's learning (Schwartz, 1995). Therefore, in this activity students are given an opportunity to create the own device using a long strip of paper ribbon, a paper clips (act as the unit's measurement) and a pencils to measure the length of the table. By laying the ribbon out flat and marking a point (0), the student can lay the paper clip down, ensuring one end is resting on the mark, and draw another mark on the top of the paperclip.  By repeating this process, they will eventually be left with a strip of ribbon, with equal markings spaced out the length of a paper clip.Students will then need to number each mark in a sequential order.  The measuring device can then be used to measure different objects.

creating / making a measuring device

The measuring device (paper ribbon) shows that the length of the table is 20 paper clips. 

Activity 3: Making an indirect comparison

In this activity, students could use the measuring device that they made in the second activity to measure the length of the other objects. This process is called indirect comparison. In the indirect comparison activity, students could see and understand the links in all the activities conducted; using appropriate unit, making measuring device and doing indirect comparison. 

Making an indirect comparison; the length of the table to the length of the stove using the measuring device (paper ribbon).

The length of the stove is 16 paper clips (unit).

The length of this circle is about 20 paper clips (unit).

After students already understand the non standard units measurement, teachers could introduce the early childhood students to the standard unit measurement. Using everyday stand units measurement could help the students to understand the topic better. 


Standard unit measuring experiences:


1. Length

Measure of something from one point to another point. Standard unit for length are the metre (m), the centimetre(cm) and the kilometre(km).

2. Area

Area is associated with coverage. Look for areas that are covered by objects that can be counted (Yelland et al, 1999).

3. Volume and capacity

    

Refers to three-dimensional space that is occupied by a substance, such as water and sand (Yelland et al, 1999). Standard units for volume are the litre(L) and mililitre(mL).

4. Mass

Mass are related to matter and heaviness of objects. The standard units for mass are the kilogram(kg) and gram(g)

5. Time

Understanding the sequence of events (eg: mathematics period is after geography period), duration of events (mathematics period is 1 hour) and the length of various units of time (eg: minutes, hours).

6. Temperature

Temperature is the state or degree of hot and cold in atmosphere, objects or body. The units for temperature is Kelvin (K), Celsius ( °C) and Fahrenheit ( °F).

Algebra

Hi everyone.

We meet again this time talking about new topic, Algebra. Before we go further, lets look at what is algebra and what does it look like in the early years.

First, what is algebra?

Algebra is the generalization of arithmetic ideas where “unknown values and variables can be found to solve problems” (Taylor-Cox, 2003; p.14). It is also the relationship between quantities and events is explained using rules (Origo, 2008). Two of the most important aspects of algebra are patterns and functions.

How is algebra in the early years?

Introducing algebra during the early years is important as it provides essential foundation for ongoing and future mathematics learning (Taylor-Cox, 2003). Hence, there is a needs for interconnected concepts in the early childhood mathematics program which are developmentally appropriate and very applicable to early childhood education(Taylor-Cox, 2003).

In algebra, pattern is defined as an arrangement of shapes or numbers that repeat or change in a predicatble manner (Origo, 2008, p. 75). Three types of patterns we looked at through the tutorial were: repeating patterns, growing patterns, and relationship patterns. According to Papic (2007), patterning involves the development of children's spatial awareness, sequencing, ordering, comparison and classification. Furthermore, pattern also provides a basis for algebraic thinking in term of teaching regularity and repetition in motion, colour, sound, position, and quantity, and be involved in recognising, describing, extending, transferring, translating and creating patterns. Thus, in assisting the children to learn pattern, it is broken down into seven stages:

• Look for patterns
• Participate in building a pattern
• Copy a pattern
• Create a pattern
• Extend a pattern
• Find the missing element
• Translate a pattern

Repeating pattern is a sequence or set that is repeated over and over. The core is the smallest part of the pattern that is repeated. It should be repeated three times. A term is each place or position in the sequence. 

Using the example above, teachers could integrate the seven stages in assisting the students to understand pattern. First, when looking at pattern, teachers should ask students to identify the core  to understand what is being repeated. In the example above, students will identify 1,2 as the core since it is repeated more that three times in this pattern. Automatically, from the core, students could identify repeats as the term. By recognising the core and the term, students could recognise the patterns. For example, looking at the pattern below, students will identify the pattern as two purple triangles one green circle, two purple triangles one green circle, two purple triangles one green circle, two purple triangles one green circle.

Furthermore, teacher could the example to extend students' understanding on pattern. Since students already able to identify the core and term, teachers should encourage the students to practise and continue the rule. Teachers could continue extending the pattern to 10th terms and remove the 7th term and 8th term by asking students to fill both of the terms with a correct pattern. After that, teachers might ask the students to guess the 15th or 20th term without having to extend the patterns. This important to evaluate students' understanding since a study found that, some children may be able to copy and extend patterns, but they may not necessarily identify a pattern as a unit of repeat (Papic & Mulligan, 2005). In other word, students must know how to translate the pattern which means changing the appearance of an element while maintaining the arrangement of the elements (Origo, 2008, p. 75).

Growing pattern occur when elements increase or decrease. Below are the example of object, number and shape growing pattern.

Using aspects of the 7 stages, tasks can be set for students to better understand gowing patterns.  These may include:

• "What is the core in this pattern?"
• "Describe the pattern?"
• "What would the 5th term look like?  6th term?  10th term?"
• "How would you translate this pattern?"

Relationship pattern occurs between two or more sets of number. It is inter-linked and affect each other. The example is shown below.

Students can be given the opportunity explore this pattern. Comparing the 2 columns, students should look at the following aspects of the 7 stages:

• "What is the relationship between figure and match sticks?"
• "Describe the pattern?"
• "What would the 4th term look like?  10th term?  20th term?
• "How would you translate this pattern?"

In order to understand the relationship between the two tables, students could expand the table into a numeric line pattern as below:

Functions are rules that describe the relationship between two sets of numbers (Origo, 2008, p. 43). One set of numbers is the values of the independent variable (input),where the other set is the values of the dependent variable (output). Functions recognise how "things" change in relation to each other. In the tutorial, we learnt that teachers could start exposing students to understand function by describing words or pictures. Through the "function machine", students could  clearly illustrates how we start with something, and end with something different.

The example above uses a concrete object to explain to students how does function machine works. It starts with a red circle and it turns into red triangle when it comes out from the function machine. The second example starts with dark blue circle and ends with dark blue triangle when it comes out from the function machine. From here, students could guess what will comes out (output) from the function machine if it starts (input) with green circle. Similarly, what do we start with (input) if it will comes out (output) as a light blue triangle? From this activity, students will realise that the function machine only change the shapes without changing their colours.

As the student moves through the years, functions will begin to move from the concrete to the abstract, and finally, standard prepresentation (Willoughby, 1997, p. 314). Next, teachers use numbers to illustrate a standard representation.

In the example above, it starts with number 2. It goes through the function machine and comes out as number 5. As for the number 4, when it goes to the function machine, it comes out as number 9. When it comes to number 7, it comes out as number 15 from the function machine. Here, students realise that every number that goes into the function machine will be doubled and add one. Therefore, teachers could ask students to explain the process and the output for number 8. 

Another aspect of function is equal as balance-equalities and inequalities. Equality (shown by the symbol =) occurs when two or more objects or expressions are identical in a particular attribute that can be counted or measured (Origo, 2008, p. 35). Inequality (shown by the symbol ≠) occurs when 2 or more expressions are not equal (Origo, 2008, p. 35).

In the tutorial class, we learnt to understand the topic through scale. This is because through scale, students are able to visualise and experiment the concept of equality. 

In the example above, on the left side, we have a set of 4 yellow balls and eight green balls. Otherwise, on the right side, we have three yellow balls. By looking at the equation 4+8=3+?, we know that both of the side are equal.  Therefore, with a clear understanding in equality,students will solve the problem by solving the left side first. By adding 4 and 8 students will comes out with an answer 12. As both of the sides are equal, on the left side 12-3 and the students will get 9 as an answer. Thus, students will come out with an equation (4+8=3+9).

Other examples of equation carried out in the tutorial class can be seen below.

Q: What is the equation derived from this scale?

A: (10+6=7+9)

Q: Is that equal or not and give reason to your answer.

A: It is equal because both of the sides have the same answer which is 16.

Q: What is the equation derived from this scale?

A: (7+5=6+6)

Q: Is that equal or not and give reason to your answer.

A: It is equal because both of the sides have the same answer which is 12.

Number and operation (counting, number word sequences and substising)

Hi everybody..

We meet again in the second entry..

In the previous entry we learned about how the young children learn mathematics looking at the beginning process illustrated by Iron (1999). In this entry, I will focus on how the children learn numbers looking at the substising, counting and number word sequences.

Before, we go further, do you know that in the early number sense, children actually developing the concept of number through the beginning process? For example, teacher can introduce the early children to the number concept through sorting, comparing, ordering and patterning. 

Example: Sorting

      1) Ask the students to group the buttons according to the same colour (attribute)

2) Ask the students to give the number of the button for each of the colours.

In the early number sense, the teacher must create activities that develop numeral skills in recognizing, identifying and writing. Recognizing-teacher selects a named numeral from a randomly arranged group of displayed numbers; identifying - students are to state the name of the displayed number; writing - students practice writing the numeral, focusing on numeral shape. This process helps the children in matching the symbols to the name and to the quantity.

Thus, in the teaching model, learning must be consisted of the concrete object, symbol and language.

According to Doverborg, Elisabet and Samuelson (2000) as stated in Gelman and Gellistel (1978), there are five principles that can help the early number sense children to develop counting skills.

One-to-one principle

When counting, only one number word is assigned to each object. On the other word, children must be able to match and pair objects according to the same quantity.

Stable-order-principle

When counting, number words are always assigned in the same order. An example of this would be understanding that after 7 comes 8 then 9 and 10 (Doverborg, Elisabet and Samuelsson, 2000, p. 85).

Cardinal principle

The possession of the skill of knowing that the last listed number is in fact the number of objects present in the counted quantity (Doverborg, Elisabet       and Samuelsson, 2000, p. 85). Example is shown below.

Q What is the last listed number?

 A 9.

Q How many objects are there?

A 9(this is an example of cardinal principle being acquired).

Abstraction Principle

A concept of all appropriate objects of a well-defined capacity being counted irrespective of the kind of objects present (Doverborg, Elisabet and Samuelsson, 2000, p. 85).Example, a rectangular prism and a cylinder both being counted regardless of their differing shapes.

Order-irrelevance principle

When counting the number of objects in a set, the order in which they are counted is not important, but rather simply that all objects are counted.

Online programs such as Five Little Ducks and 1O Fat Sausages can assist the students in promoting the development concept. The audio, which is the songs help the children to listen and at the same time do the counting.

As, the children are able to mastering all the five principles in counting, the teacher can teacher the teach the children to the number word sequences.

1) Forward Number Word Sequence

   The number sequence is from ascending to descending order.

Q Looking at the number sequence above, what number would come after 5?

A 4

2) Backward Number Word Sequence

    The number sequence is from descending to ascending order.

Q Looking at the number sequence above, what number would come before 6?

A 5

However, if you ask the young children about the quantity of an object, they will confidently say "three flowers" without counting. Hence, Piaget called this ability to instantaneously recognise the number of objects in a small group as 'subitising'. 

Subitising is defined as the side recognition of a quantity. They are two different recognize forms of substising:

Perceptual subitizing - Distinguishing a number without using any other form of mathematical process, for example children seeing and recognising the number ‘4’ without any taught mathematical familiarity (Clements, 1999, p. 402).

Conceptual subitizing - “Recognizing the number pattern as a composite of parts and then together as a whole, for example an eight dot domino" (Clements, 1999, p. 402)- children will automatically say two rows of four dots as eight without counting.

Teacher can develop students' learning experience on substizing through simple activities in the classroom. Example of the activities are shown below.

This activity develop students' skill on substizing. Children have to recognize the number on the dice and on the pictures in order to complete the 'Humpty Dumpty puzzle'.

Q  What is the number on the Humpty Dumpty's hat?

A  4.

 Q  How did you know this?

A  There are 2 and 2 dots, which together equal 4.

As Clements (1999, p. 403) explained, subitizing along with tens frames can help with early addition for students to understand addition combinations. The game pictured below is the example of subtizing with the tens frames.

In this game students will substizing through out the rolling dice. When the dice shows the same number as appears on their koala's line, their can move the koala ahead. The first koala that can reach home is the winner.

As an adult that played those activities in the tutorial class-Humpty Dumpty puzzle and Koala race for example, I can see games that include dice, domino or playing card help the children to recognize and identify the quantity in numbers. Indirectly, it develops students' substising skill in counting.