Probability Clothes Line

The Resource:

This resource consists of a piece of string, tiny wooden pegs and a number of small shirt-flashcards, each containing a "probability word".



How to use this resource:

Divide the flashcards evenly among the children participating in this activity.

Ask the children to come hang their shirts out on the line in order, from the probability word which signifies "least like to occur" to most likely.

Go through each of the words and discuss the reasoning that justifies their choices.

Also ask the students to suggest events that have the corresponding probability of occurring in relation to each of the words. E.g. likely - that it will rain tomorrow.

Alternative to using the pre-written shirts, ask children to write their own probability words on blank shirts instead.

Mathematical focus/outcome:

Strand: Chance and Data

Outcomes QSA (2004):
1.1 Students use everyday language when commenting on aspects of chance in practical activities and familiar events.

2.1 Students identify and classify familiar events according to the likelihood of occurrence.

What is Algebra?

(Click on image to enlarge)

This graphic represents the concepts which, as a result of my reflection following week 12's workshop, I believe are integral to the teaching and learning of algebraic thinking in the early years. Each cog represents a key concept highlighted in QSA (2004). The choice of interconnected cogs is to demonstrate that knowledge and understanding in any one of these areas can affect the knowledge and understandings of another. It is important to note that the size of each of the cogs is not representative of the importance of these concepts. Furthermore, the boxes and arrows show the “central ideas promoted in the national algebra standard for young children” (NCTM, 2000 cited in Taylor-Cox, 2003, p. 15) and how each of these ideas relate to the concepts highlighted in QSA (2004). The rest of this reflection will concentrate on expanding on these concepts and ideas, and identifying some implications for how to they should be taught and learnt in the early years.

Patterns

According to Economopoulos (1998) patterning activities have long been accepted in early childhood educational contexts as a part of the early mathematical curriculum. However, often little consideration is given to why patterning activities are valuable and how they actually contributes to children's development of mathematical understandings. Partly as a result of this lack of consideration, many early childhood teachers seem to focus on recognising patterns, creating patterns, and "what will come next?" type activities.

As was discussed in the workshop, understanding patterns is important mathematically in that they are used to form generalisations and predictions based on mathematical reasoning. We discussed that most children (and many adults) seem to have a natural interest in forming patterns, particularly repeating patterns, but that does not mean that they understand and can make generations about the rule by which a pattern is formed, or that they are able to use the pattern to predict missing variables.

Economopoulos (1998) suggests, in order to generalise and make predictions about patterns, children must "move from looking at a [repeating] pattern as a sequence of 'what comes next' to analysing the structure of a pattern...seeing that it is made up of repeating units" (p.231). Therefore questions such as "What part repeats?" or "What comes here?" (see the blog-entry on train tracks game) and "Why do you think it's that?" can be useful to extend children's learning about patterns. Also useful are activities involving the translation of patterns, describing patterns, identifying similarities and differences between patterns (Economopoulos, 1998). These help children develop their understanding of the function of patterns as a means of making predictions, and develop mathematical reasoning abilities to justify predictions they make.

The approach of Economopoulos (1998) is also supported by Papic (2007) who highlights that although young children are capable of developing understandings of complex patterns (beyond repeating patterns), merely exposuring children to patterning experiences does not guarentee that they will come to see patterns as structures that repeat or grow predictably. This ability to analyse the structure of patterns will help children develop other algebraic concepts such as functional thinking (Papic, 2007).

Equivilance and Equations:

In the week 12 workshop, we discussed the idea of balance or equivilance being a difficult concept to explain to young children. According to Warren, Benson and Green (2007) in a study of children's understanding of the equal sign, many students mistakenly thought it meant "here comes the answer" (p.151). Using resources such as a pan balance can help children develop this understanding of balance which is a important part of algebraic thinking. For example, if I had two identical objects in either side of the pan balance, both sides are equal, or balanced. However, if I take one object from one side, the two sides are no longer equal, or are unbalanced. If I then remove the same object from the other side, the two sides should again be balanced or equal to each other again.

This idea is linked to the development of mental computation skills which we discussed earlier on in this subject (see mental computation blog-post for more details).

Functions:

According to Warren, Benson and Green (2007), a function describes the relationship between one element of a set of data with another unique element of another set of data in that the value of the first element consistently changes into the other according to a particular rule.

Functional thinking contributes to mathematics learning as it assists children to develop understandings of the inverse relationship between the operations of addition and subtraction; multiplication and division.

The sequence of learning argued for by Warren, Benson and Green (2007) is as follows.
1. Learn how to "follow a change rule";

2. Learn to "follow a backtracking";

3. Identify a change rule;

4. Identify a backtracking.

For more reflections on identifying a change rule, please read the blog-post entitled "Button Triangles"

For lesson ideas for qualitative functions please read "The train game" "Word riddles".

For lesson ideas for quantitative functions please read "function machines".

Measure the Lake

Here's an activity to get student's to come to terms with different aspects of measurement. It's called Measure the Lake.

The owner of a lake wants to turn his lake into an ice skating rink in Winter, when the lake freezes into solid ice. In order to begin the preparations in advance, he wanted to take some measurements of the lake during Summer time, while the lake is not yet frozen.



Can you suggest ideas for what measurements he needs in deciding:
1. The number of benches he needs to put down in the seating area;
2. How many barriers to put up around the lake so skaters have somewhere to hold onto when skating;
3. How many people to allow inside the skating rink so it does not get over crowded;
4. How many trucks to hire to carry away the snow that will need to be cleared away each morning;

Please also suggest ideas for what to use in order to obtain these measurements

Authentic contexts for learning chance and data

Week 11’s workshop was all about the strand of chance and data. The main activity surrounded the use of Smarties to create an authentic context to learn chance and data concepts.
I work at an OSHC. In my experience working with children in primary school, chance and data is often a dreaded strand of maths, notoriously well-known for useless worksheets such as the one I witnessed this afternoon when discussing with the children what they have been learning in maths:

After reading through this worksheet, I discussed its contents with the girl, a grade 4 student, who had completed it for homework just a few days ago. When I asked if she enjoyed doing this worksheet, she answered, “No, not particularly, it was easy, but it was boring. But then again maths is boring, so yeah. ” Then I asked her is there anything about it that she liked, and she optimistically pointed out that at least she got to make a choice about whether to colour the bars in colour or not, and she chose not to.

It was not just the lack of authentic context that annoyed me about the worksheet, but that fact that its design did not really highlight the usefulness of graphs. If you look at the worksheet in detail, it asks the student to graph the table of information provided at the top of the page, then use the graph to answer the following questions: Which was the city with the highest recorded temperature? Which was the city recorded the same temperature? How much hotter was Darwin than Hobart? Much of this information would have been just as easy to answer without constructing the graph, let alone the last question which would have been easier to calculate without the graph. I was astonished and annoyed by how chance and data, a topic which lends itself so easily to authentic investigations based on real-life contexts or other hands-on learning experiences, could be treated in this way. In the same afternoon at work, while playing connect four with another child in grade 1, an authentic context arose for her to record some real data, and she did so on her own accord.

This grade 1 girl started to tally the number of wins each of us achieved as a result of playing a series of games of Connect Four. In actual fact, we did not play as many games as was shown on this piece of paper. The score started off as:

She accurately recorded the results and made statements about them, “You’ve won four games and I’ve only won one”. “If you win one more game then I’ll put an across mark and you would have won five games.”

After playing four more games the score table read:

She said, “If I win one more game we’ll be tied.” We talked about where she got the idea of doing up a table from. She said that she picked it up from watching other girls keep score when they play Connect Four.

Then the call for afternoon tea was announced and she quickly scribbled in the remaining tally-marks. When I asked her what she was doing she said, “I’m just mucking around, look we both won lots of games”.

If we had more time, we could have explored different ways of graphically representing the information she collected or made up with post-its, bundle sticks, stickers and so forth. I am confident that she would have enjoyed that experience and gained a lot more from it than the girl in grade 4 seemed to have from completing her homework. This led me to think deeper. I wonder, what does actually constitute an authentic context for learning chance and data? Furthermore, is having an authentic context the crucial deciding factor in determining if this strand of maths is taught well or not?

Nisbet, Langrall and Mooney (2007) conducted a study which raised the research question "How do students knowledge of real-life contexts affected their ability to analyse some data provided to them?" The result of the study suggest that when primary-aged students were given sets of data related to a topic area which they had special knowledge and interest in, they used their understandings of the real-life context to "rationalise their data or their interpretations, in taking a critical stance towards the data, and in ways that were not necessarily productive or pertinent in addressing the task at hand." (p.16) According to Nisbet, Langrall and Mooney (2007), providing opportunities for learners to integrate contexual knowledge and statistical information through investigating real-life data is important. However, teachers need to be aware that learners can be just as easily distracted by their contextual knowledge, which can lead them to disengage with the mathematical problem or task. This hypothesis seems to be supported by English and Watters (2005) who found in their study that children who were participants in their study used their informal knowledge to relate to and identify important problem information, but at times became "absorbed in applying their informal knowledge" (p.72).

Therefore it is clear that providing data which is taken from real-life contexts or may be of interest to children is not enough. As teachers we need to crtically reflect on our lesson designs so as to assist students to use their informal knowledge to critically evaluate data where it is appropritate, as well as develop statictical literacy in considering the data itself and how it applies to the problem (English & Watters, 2005). Perhaps activities where students take part in collecting the data, as well as manipulating it to solve problems which they pose for themselves would be appropriate (see fishing game blog-post).

References:

English, L. D. & Watters, J. J. (2005). Mathematical modelling in the early school years. Mathematical Education Research Journal. 16(3), 58-79.

Nisbet, S. Langrall, C., & Mooney, E. (2007). The role of context in students' analysis of data. Australian Primary Mathematics Classroom, 12(1), 16-22.

Measurement

The concept of measurement covers a variety of attributes including, length, mass, area, volume, time, temperature, weight and capacity. The following reflection will discuss important principles and common misconceptions held by young children (and some adults) in relation to the attributes covered in the early years of mathematical learning, these are – length, area, volume, mass, and time; key concepts/misconceptions regarding the relationships between perimeter and area, volume and mass, as well as length and volume as well as suggest the general sequence of learning in regards to principles of measurement.

Length, Area and Volume

According to Currie, Mitchelmore and Outhred (2006), five important principles in measuring length, area, or volume are:

1. Congruent units must be used when measuring an individual object


2. The attribute of the measuring object has to be the same as the attribute of the object being measured to have


3. The transitivity principle – the units used to measure two different objects need to be the same (or converted to being the same)in order to compare the two objects


4. There is a relationship between the size of the units used to measure length, area and the number of units required to make the measurement (i.e. the smaller the unit, the more of that unit is needed to make the measurement. And in reverse, the bigger the unit, the less of that unit is needed)
   
5. Iteration principle –there is to be no gaps and no overlaps between each unit used to measure an object

The result of a study conducted by Currie, Mitchelmore and Outhred (2006) highlight some misconceptions held by the participants of this study in relation to the measurement of length, area and volume, in that “young children appear to have a much poorer understanding of the need for identical units that leave no gaps and no overlaps than teachers often presume and they may indeed have no clear concept of what they are actually measuring” (Currie, Mitchelmore and Outhred, 2006, p. 383). This is why it is important to begin learning of measurement by identifying attributes for measurement, e.g. by discussing what is meant by the attributes of length, perimeter, area and so forth (see the sequence of learning).

Currie, Mitchelmore and Outhred (2006) suggests that mathematical learning experiences being implemented in early years classrooms should focus on helping children develop the skills to justify reasons for adhering to measurement principles, (e.g. no gaps no overlaps) as well as make children aware of errors that can possibly result from misapplying measurement principles. An idea I have to facilitate this is to give children opportunities to choose the appropriate instruments to use to take different measurements, rather than always being told what to use.

Furthermore, Kribes-Zaleta and D’Lynn (2003) shows how everyday play situations often give rise to measurement-related problem posing by the children or teacher. In the vignette of children discussing how long their snake was, Kribes-Zaleta and D’Lynn described how the children identified a problem in how they were comparing the lengths of two snakes drawn by different children in that although one snake was clearly longer, it was only “25-long”, whereas a snake drawn by another girl was “28-long” but was clearly shorter. The author highlighted that although the children were not yet aware of the reason why their data is the way it was (the fact that they had used different units to measure length) the fact that they recognised that a problem existed created a context for learning. To me, this reminds me to observe not only the misconceptions that the children hold when attempting measuring tasks, but also what they can and do notice when provided with a little scaffolding. It also reminds me to seize opportunities for learning that arises from play because they can and do occur all the time.

Time

Something else which the workshop activities reminded me of the how abstract the concept of time is. No wonder young children sometimes have trouble understanding some of the words used to describe time. When you have only been alive for three, it is difficult to contemplate how long a year is, as it is one third of your whole life so far. What would be even more difficult is appreciating how long a decade is, as it is a period of time which is just outside your experience. Nevertheless most children have some understanding of concepts of time even prior to starting school. It is important to help children clarify the relationships between the different words used to describe different periods of time. It is important to discuss concepts such as, the smaller the units of time, the more of that unit is needed to measure a period of time; whereas the larger the unit, the less number of units is needed to measure that same period of time, just like measuring other attributes such as length, area, volume or mass. The activity of The Time Clothes-Line can be helpful in discussing the relationships that exist between different words used to signify different periods of time.

Another aspect of time is related to how to tell what time it is on a watch or clock, which is something that affects our everyday lives. According to Krech (2000) the one-handed clock is a useful resource for addressing the problem that many children face in not being able to tell the time on an analogue clock when the minute hand is pointing to anything other than 12 or 6.

Sequence of learning:
The general sequence of learning in regards to developing understanding of measurement concepts specific to each of these attributes are:

Shape Story Video

The main idea/teaching focus this resource can be used for is to provide children with an interesting context for describing and identifying attributes of shapes.

They can conduct real-life investigations, researching questions such as "What do the shapes look like from differnt perspectives?" "Which shapes can roll?" "Which shapes can slide?" "What constitute a shape?"

This video can be watched a few times. The first time you may watch it the whole way through and discuss specific bits that stood out to the students. The second time, you may like to watch a small segment and conduct an investigation (with a bucket of the shapes) before watching the second part.

Disclaimers:
Please take note to mention that the 2D shapes shown in these images are not actually 2D because they have a thickness.

Also with younger children do not worry about naming shapes so much - you may like to call the shapes by their first names or just pick a shape out of a pile to talk about it. Even a simple activity such as getting the young childrern to identify "Which shape is Shaphia?" out of a pile of unpainted geometric shapes could be a good learning experience to get children to identify and describe attributes. "Why do you think she would be that shape?"

If I were to produce it again, I would probably do it on an animation program so that the shapes move as they talk, and I will try to write a story that is not as corny as this one. I would probably think twice about the bit about fat shapes and flat shapes too as it does contribute to the misconception concerning 2D shapes.

Balancing Teddies

The Problem:

(Click on the photographs to enlarge them)

If one yellow teddy balances with two green teddies...




And one green teddy and one yellow teddy together, balances with six red teddies...





Then what balances with two red teddies?

The Solution:

Well if one yellow teddy equals two green teddies; and one green teddy and one yellow teddy together equals six red teddies, then you can replace the yellow teddy with two green teddies. So when you add these two green teddies to the one green teddy that is already on the balance pan, then three green teddies must equal six red teddies.




Once the teddies are put in this arrangement, then it is easy to see that one green teddy is the same as two red teddies.



So what balances with two red teddies?


One green teddy!

What did I learn from doing this activity?

I learnt that to solve this problem, I had to apply algebraic thinking in that I had to figure our how the two facts presented in the problem related to each other - I had to identify the relationship between the two rules. Once I saw the connection, I saw that the yellow teddy on the scale could be replaced with two green teddies, the rest of it was easy to solve visually. From this activity I learnt that algebraic thinking is not limited to manipulating numbers and abstract symbols, but that in some instances visual and even concrete representations can present easier solutions. This activity taught me that using visual and concrete reprentations does not necessitate a trial and error strategy or approach. I also learnt that being able to reflect on the process of how I reached my solution, explain my thinking processes to someone else, then documenting it visually (in a way that made sense) were all difficult to do because it required a much deeper appreciation of the mathematical reasoning in order to justify my solution.

If I were to provide opportunities for children to investigate this activity, I would provide the relevant materials and set the scales up similar to the picture above. I would also allow time for the children to manipulate the material, contemplate their own reasoning before asking them to share how they got their solution and justify their processes.

Week 9 Workshop I - Tangrams

What are tangrams?
Tangrams are a type of ancient Chinese geometric puzzle which contains seven pieces, made up of two large triangles, one medium triangle, two small triangles, one square and one rhomboid (or parallelogram) (Bohning & Althouse, 1997). These pieces can be assembled in different ways so as to make shapes such as...

A square...



A rectangle...



A triangle...



A trapezium...



A parallelogram...



And lots of other shapes e.g. birds, boats, etc.

What do you learn by working with tangrams?
According to Bohning and Althouse (1997), explorations with tangrams assist children’s skills development with “geometry vocabulary, shape identification, classification, discovering relationships between and among the pieces...recognising and appreciating geometry in their natural world.” (p.240).

In week 9’s workshop, through our explorations with tangrams I came to a deeper appreciation of how these skills may be developed. The use of geometry vocabulary was obvious as those of us present in our class tried desperately to assemble the shapes, in particular, the square shape. For example, in the spirit of collaboration, those of us who successfully assembled the square coached those of us who were less successful in our attempts, often by using location and direction terminology such as flip, slide and rotate. Furthermore, the names or the attributes of the shapes were also used to explain where they should go, for example, “Put the little two little triangles next to the square, but rotate the one on the right so that its longest side is on the edge”.

In terms discovering the relationships among the pieces, we discussed how each of the shapes was made up of certain numbers of the smaller triangle. For example, the square is made up of two of the smaller triangle, and the big triangle is made up of four smaller triangles. Therefore, in terms of area, the big triangle is actually equivalent to two squares. This discussion also led to the conclusion that using the tangram as a model can also be useful for discussing fractions, for example, if the total area of all of the shapes added together is understood to be the whole then the large triangle is one-quarter of the whole, which makes the medium triangle (which is one-half of the large triangle) one-eighth of the whole. This in turn makes the smaller triangle (which is one-half of the medium triangle) one-sixteenth of the whole, which also makes the square and the parallelogram (which both equal two small triangles) two-sixteenths or one-eighth of the whole.

Learning sequence of working with tangrams:

1. Placing shapes directly over the top of a diagram (of the same size) of the target shape with lines shown

2. Assembling the shapes by looking at a smaller-scaled example of the target shape with the lines drawn in

3. Assembling shapes by looking at an example of the target shape with some of the lines drawn in, e.g. Which shape can fit in here? How? Show me...

4. Assembling shapes without looking at an example at all (which effectively is the same as looking at an example without the lines drawn in) but with the correct-side-up identified

5. Assembling the shapes without looking at an example witht without the correct-side-up identified.

Week 8 Workshop - 2D Shapes

This week in our workshop we explored 2D shapes. Firstly, we constructed a 3D shape. We were given two pre-drawn nets of two 3D shapes which we cut out, constructed and sticky-taped together. Then we stuck the two 3D shapes together, and formed a triangular based pyramid. Afterwards, we reflected on what children would actually learn from such an activity if it were being done in a classroom. We discussed the fact that without further discussions about the 3D shape and its attributes, the activity would at best be providing children a chance to work on their fine motor skills and at worst be a complete waste of time.

This got me thinking - what is the purpose of constructing nets? It is supposed to help children understand and visualise the attributes of a 3D shape. Constructing 3D shapes using pre-drawn nets reduces the amount of visualisation and mathematical thinking children need to do, and can easily become a waste of time. Perhaps a better idea would be to get the children to construct their own nets. This may take more time, and may require children to persist through difficulties and multiple attempts. However, if the children are provided with appropriate scaffolding, they should come away with a deeper appreciation of the attributes of 3D shapes, than if they simply constructed a shape using a pre-drawn net.

Secondly, we discussed naming of common 2D shapes. This took up the majority of our lesson. We discussed terms such as: quadrilateral, polygon, rhombus, rectangle, square, trapezium, parallelogram...we discussed how some shapes fit into multiple categories, e.g. a square can also be classified as a quadrilateral, polygon, rhombus, parallelogram, a rectangle and a trapezium (according to the Australian definition of the word). We concluded by saying that what matters most about the teaching and learning of shapes in the early childhood years is developing children's abilities to describe and recognise attributes of shapes. This skill is more important than ascribing the right names to the shapes.

Furthermore, one of the research findings of the study summarised in Hannibal (1999) showed that when the research participants were asked to identify which shapes are triangles from a number of stimuli presented, younger children in particular often drew upon "self-determined triangle-defining criteria" (p.355). It was also shown that this occurred more frequently when other shapes which were easily recognised to not be triangles such as a circle and a square were not present in the stimuli. According to Hannibal (1999) "teachers need to move beyond having children just label shapes to having them understand what defines a shape category" (p.356). In other words, it is important to help children identify the integral attributes of a shape and distinguish these from other non-intregral attributes such as size, ratio and orientation, in order to avoid the development misconceptions in regards to what constitutes shapes such as a triangle, a square, a rectangle, a pentagon, and so forth. Other implications for some teachers may include correcting some of their own misconceptions of what are integral and non-integral attributes of these shapes.

It has been suggested that the common usage of proto-typical shapes (in books, posters or worksheets on shapes) has contributed to the development of these misconceptions held by children (and perhaps even teachers). For example, if children are used to seeing equilateral triangles but are unfamiliar with seeing isosceles or scalene triangles, they are more likely to mistaken non-integral attributes (such as "three sides have to be equal") as integral attributes of a triangle. Therefore teachers should use more non-prototypical shapes in the classroom and including different examples of shapes, in order to show that size, orientation and ratio are not integral attributes.

Thirdly, in the workshop, we participated in an activity which focused on identifying and describing attributes of shapes. This involved creating different polygons out of pieces of paper. We were encouraged to form non-prototypical shapes. Each participant created as many shapes as they could in the given time, then we were asked to sort our shapes by one attribute. The photographs below show how my group sorted our shapes by the attributes of "number of sides".


Starting with shapes that have three sides, four sides, five sides, six sides and so forth...

Then we subcategorised each of our categories. For the shapes that have three sides we divided our shapes into triangles that contain a right-angle (right-angle triangles) and those that did not contain a right angle triangle. The pink line was used to show the subdivision.



We also subdivided our four sided shapes (quadrilaterals) into two groups. We created one group of quadrilaterals that have no right angles, a group of quadrilaterals that have four right angles. Then, we further subdivided the second group (of shapes with four right angles) into shapes that have four equal sides and those who do not.





We sub-divided the five-sided shapes (pentagons) into those that contained a reflex angle and those that do not.



With the activity, the main focus was on identifying and describing attributes of shapes, and justifying the reasoning behind how these shapes were sorted. Another research finding outlined in Hannibal (1999) suggested that when the children were more likely to make correct catagorisation decisions when asked to provide explainations of the reasoning for their decisions in the triangle sorting task. Justifying decisions requires one to reflect on one's actions, which also provides opportunities for self-correction.

From participating in this activity, I also see its potential to assist teachers in identifying misconceptions held by their students (for example if they experience difficulties in counting the number of sides of a shape) as well as develop children's conceptual understandings about shape that is foundational for future geometry learning. It is only by being able to identify and describe the attributes that they can develop a comprehensive understanding of why a square is also a rectangle - which is because that its attributes fulfils both the criterion of a rectangle (a quadrilateral with four right angles and therefore have two sets of parallel sides of equal length) and a square (a quadrilateral with four right angles and all four equal sides are of equal lengths).

There were ten in the bed...

Clarkson (2006) suggests that rhymes and folktales can be useful and creative resources for teaching mathematics. I have developed this resource as a response to the article.


Original Rhyme: There were ten in the bed and the little one said, “Roll over, roll over...


Teaching focus: Can be used for counting backwards or subtraction – each time a child rolls over, there’s one less


My variations:


Teaching focus: to help children understand the connections between the different operations by focusing on division by repeated subtraction (Reys et. al., 2006)


There were ten in the bed and the little one said, “Hey, two of you, roll over, roll over...”
How many times will the little one have to say “two of you roll over” for all ten to end up off the bed?



Teaching focus: to help children understand the connections between the different operations by focusing on multiplication by repeating addition (Reys et. al, 2006).


And then (after they have all rolled off)...There were none in the bed and the little one said, “Hey five of you, roll on the bed...”


How many are there now?


Child: Five


What if the little one said it again? “Hey, let all five of us roll on the bed...” how many would there be then?


Child: Ten


How did you know?


Child 1: Because I counted five more than five, so the answer is ten.


Did someone else have another way?


Child 2: Because I know that five plus five is ten.


Are there any more ways to working it out?


Child 3: Yes, five times two is ten.


References:
Clarkson, P. (2006). Rhyming and folk tales. Australian Primary Mathematics Classroom. 11(4), 18-22.

Reys, R. E. et. al. (2006). Helping children learn mathematics. Hoboken, NJ : John Wiley & Sons