The Game Show Problem

This is a problem which I have found stimulated my ideas about analysing probability.

The interesting part is justifying the decision.

Pretend you are on a gameshow. The gameshow host shows you three identical doors and tells you: "Behind one of these doors is a brand new car. If you pick the right door, the car is yours!" You think about it...and choose one of the doors. The host asks you, "Do you want to lock that in?" You reply, "Yes. Lock it in."

Then the host opens one of the two remaining doors that you did not choose. It is empty. "Phew," you thought, "I'm glad I didn't choose that one". Then comes the interesting part. The host asks you, "Do you want to change doors?"

What would you do? Why?

The Time (Clothes) Line

Teaching Focus:

To help children develop understandings about words used to describe periods of time.

How to play:

Prepare a number of time words, a piece of string and some pegs.

Divide the time words among the students.

Ask them to put the time words in order from the longest period of time to the shortest period of time.

When the line is assembled, discuss how many of each period of time fits into or makes up a larger period of time.

Alternatively, the children can brain-storm and come up with a list of their own time words.

The Fishing Game

This resource is used to create an authentic context for the learning of chance and data.

Distribute the fish into small ponds. Lay the fish out on the floor (fish side down)



Let the children take turns to go fishing. The fishing rod is made of a magnet stuck to a piece of string which is tied to a chopstick.



When the (small) group of children have had five turns each, ask them to assess the fish that they have caught.



Then, just like we did with the smarties in the week 11 workshop, discuss how they can represent the data. Here is an example of a picture-graph.



It is important to discuss what each graph tells you, for example: "What does this graph tell you?" That the purple fish (cuttle-fish) is the most common.
"If I am to catch another fish, what is it most likely to be? Why?"

Gradually the children may like to compare the types of fish which they have caught with another group. When the data pool gets large enough for the picture-graph to become less useful, prompt a discussion about some other ways which the children can display the information about their fishing expedition.

Concluding the EAB023 Journey: Some final thoughts

This concept map maps out the blog-entries which represent much of my reflections throughout the EAB023 learning journey. (Click on the image to enlarge)

The explorations of mathematical concepts which I have undertaken through this unit have been very rewarding for me as a future maths teacher. I have enjoyed the first-hand experience of being challenged to think mathematically, and use everyday language to explain and justify the reasoning I use to reach a solution. I also found this extremely helpful for me as a teacher as my aim is to stimulate children to engage their natural curiosity and use it to explore and develop systematic understandings of mathematical concepts (Australian Association of Mathematics Teachers and Early Childhood Australia, 2006).

In the future, I wish to engage in more professional development opportunities so that my own love of mathematics learning will not come to a close. Furthermore according to Cockburn (2008), on-going professional development is also important and valuable as it informs teaching practitioners to develop and improve teaching practices that are informed by empirical research. Many of the readings which I have accessed in order to inform my blog-posts have really taught me a lot of valuable things about mathematics teaching. I am more thankful than ever for the researchers who dedicate their careers to conducting good quality research to inform and improve teaching practices. Sadly, according to Cockburn (2008) the amount of educational research being applied to professional practice is still very limited. Perhaps it is because much of the teaching style recommended by research requires more time and effort than traditional approaches. For example, teaching mathematics without resorting to text-books and worksheets takes a lot more time and effort, both in the preparation stages and reflection stages of teaching and learning. However, it is so much more worthwhile.

Throughout this unit my ideas about mathematics learning have grown and I have realised that although quality resources do play a role in facilitating mathematical understandings, there are other factors which are even more critical when it comes to developing children’s understanding of mathematics. These include: how the resource is used, the sequence of mathematical concepts being taught/learnt, the teaching style adopted by the teacher (which determines the type of relationships that exist between the teacher and child, and between children and their peers), the identification of the connections that exist between different mathematical concepts, and the teachers’ ability to mathematise, recognise mathematising and identify common misconceptions. As early mathematical development and its benefit for future learning is so well documented through research (Godfrey, 2006), I believe it is of critical importance for teachers to adopt approaches that encourage children to see themselves as mathematicians and to develop a love for learning mathematics that stretches beyond getting the right answer. This has become an important part of mathematics learning for me as I have grown to change my perceptions about mathematics as being straight forward and unambiguous. This is evident as I grew to enjoy mathematical problems where there is more than one solution and more than one strategy can be used to solve it.

Lastly, the value of reflection as a tool for evaluating one’s own learning is an important component of mathematics learning which I have come to appreciate more throughout this unit (Griffiths, 2000). I have found that creating this blog required me to analyse my own learning at a much deeper level than I would have, had I simply participated in the workshops alone. Similarly, the workshops provided me with the stimulus to think more deeply about mathematical concepts. I see now that the two go hand-in-hand. Writing is a way of documenting thoughts tangibly, and I see how having a tangible copy of the process of my learning journey will assist me as a future teacher. In my future as a teacher of mathematics, I hope I will be just as motivated to keep reflecting on my learning.

References:

Australian Association of Mathematics Teachers & Early Childhood Australia. (2006). Position paper on early childhood mathematics. Retrieved 18 Oct, from http://www.aamt.edu.au/content/download/722/19512/file/earlymaths.pdf

Cockburn, A. D. (2008). How can research be used to inform and improve mathematics teaching practice? Journal of Mathematics Teacher Education, 11, 343-347.

Godfrey, R. (2006). Early mathematics development and later achievement: further evidence. Mathematics Education Research Journal, 18(1), 27-46.

Griffiths, V. (2000). The reflective dimension in teacher education. International Journal of Educational Research. 33, 539-555.

Robot Navigation Games

Here are some ideas for teaching mathematical outcomes in the location, direction and movement topic in the strand of space:

S1.2 Students follow and give simple directions to move through familiar environments and located and place objects in those environments.

S2.2 Students interpret and create simple maps, plans and grids to follow and give directions, and to locate or arrange places or objects.

The learning sequence:

Step 1: Discuss the directional terms such as forwards, backwards, left and right, to assess children's understandings of those terms.




Set 2: Set up a grid using hula-hoops. Place an object, e.g. teddy-bear in one of the hoops, then ask a child-volunteer to be a robot, who can only move by recieving directions from their robot master.




The robot master can be you (the teacher) to begin with. The robot master's role is to give directions to the robot, so that the robot can navigate through the hula-hoop grid in order to rescue the teddy-bear. The directions should include position language such as "Move forward one step. Take one step the right. Then move forward two steps. Then take one step the left." and so forth.

You can introduce other complications to make the game more difficult, e.g. mark some of the hoops as being "lava hoops" which the robot cannot pass through. The robot master must navigate the robot to the teddy-bear without going through the lava spots.



The next step in the learning sequence is to ask the children to map out the route which they have taken. It is a good idea to provide the children with pre-drawn grids so that they can concentrate on the mathematical aspects of mapping.




After the children have completed their maps ask them to use their maps to describe the route to their friends as they follow the directions either on the grid or on a blank grid/map.

It is important to remember that the concept of space concerns more than just shapes - it includes developing understandings about direction, location and movement too.

Robot Navigation the boardgame:




I turned this game into a boardgame to help children visualise what they are doing now that they have experienced moving in differnt directions. It was also designed as a resource to teach the children about more complex directional language, such as "anti-clockwise" and "clockwise".

Aim: For each player to navigate their robot to the pot of gold (symbolised by the yellow spot).

How to play:
Each player is given 4 position cards (yellow) and 4 step cards (green) at the beginning of the game.




Players can only move their robot by playing step cards and/or position cards, e.g. a player may play one “to the left” position card and one “1 step” step card to move a robot one step to the left.

changes to


Robots can also rotate using position cards, e.g. a quarter-turn clockwise, half-turn anti-clockwise etc. But rotation cards cannot be used in conjunction with a step card, i.e. if a player rotates his/her robot, they must wait till the next turn to move the robot steps in any direction.


changes to
(after playing "quater-turn anticlockwise" card)

Each player must replace the cards they play by picking up another card from the deck.

Red spots symbolise lava hot-spots which must be avoided.

If another robot is blocking the path a player planned to take, they cannot land on, or pass through that space.

If a robot cannot make a valid move, that player may choose to discard one position or step card and pick up a new one.

The Train Tracks Game

Who is it designed for?

Children who are beginning to develop understandings of repeating patterns as structures with a core that repeats and that these structure can be analysed and used to predict missing elements in that pattern.

The train tracks game is designed to move teachers and students away from the question of "what comes next?" and on to "what comes here?" type questions (Economopoulos, 1998). I came up with the idea of creating this resource after the tute in week 12 and considering the question "when would you encounter a situation which required you to discern a missing element in a pattern?" I thought, "maybe if the missing element was covered up by something." That was how I came up with the train tracks idea.

How do you play?

Form a simple repeating pattern with the train track cards (with the core containing no more than three elements).

Analyse the pattern with the children

"What kind of pattern is it?"

"How do you know that it is a repeating pattern?"

"Which part repeats?"

Form another simple repeating pattern but this time cover one of the tracks with the locomotive (as above).

Again, analyse the pattern with the children

"What shape is this locomotive standing on?"

"How did you work that out?"

*Note: When making patterns, make sure that the core repeats enough times for the pattern to be recognisable, e.g. A B C A B is not yet a repeating pattern.

Provide cards for a small group of children to make their own repeating patterns and cover one element of their pattern with the locomotive. Then they can invite each other to solve the missing element. It is important to give opportunities for the children to justify their answers by articulating their reasoning, including the rule that the pattern follows.

Alternative lesson idea:

The shapes on the cards alternate in colours and size. According to Taylor-Cox (2003), when teachers demonstrate repeating patterns with pattern blocks for example, the focus attribute seems always to be colour, which has meant that children often rely on colour to solve problems regarding repeating pattern. Children should be encouraged to construct patterns based on attributes other than colour. These train tracks allow this to occur.

Furthermore, the fact that the shapes alternate in colour and size also means that it can be used to teach how to analyse qualitative change. The train locomotive can be used to symbolise the function machine (Warren, Benson & Green, 2007) and I feel this is a helpful analogy to explain the idea of back-tracking.

Have a look at how this train changes the tracks after it goes over them.

Before & After






What would the train track look afterwards?


Or what about this train? Have a look to see what it does to the tracks after going over them.

Before & After

What could the track look like before if this is what it looked like afterwards?



After following the rule, and back-tracking, the following task would be to identify firstly the original rule, followed by the rule for back tracking (Warren & Cooper, 2005).

References:

Economopoulos, K. (1998). What comes next? the mathematics of pattern in kindergarten. Teaching Children Mathematics, 5(4), 230-233

Warren, E., & Cooper, T. (2005). Introducing functional thinking in year 2: a case study of early algebra teaching. Contemporary Issues in Early Childhood, 6(2), 150-162.

Who am I? (With shapes)

Teaching focus:

I love a good game of who am I. There are so many variations one can have. Here's one with shapes. The teaching focus is for students to focus on identifying and recognising attributes of shapes.

From our workshop discussions in week 8 and 9, we discussed how the more helpful learning sequence for shapes is to introduce 3D shapes before 2D shapes as 2D shapes are less tangible than 3D shapes. By learning about 3D shapes first, introducing 2D shapes becomes easier because you can explain how 2D shapes are used to dsecribe the faces of 3D shapes, thereby helping them to see the connection between the two seemingly different concepts.

How to play:

(Basic level)

• Display the 3D shapes on the floor in front of a small group of children
• Describe some attributes of a particular shape (e.g. This shape can slide but it cannot roll. It has six faces. None of them are curved. All the faces look exactly the same. It has 12 sides, all of which are the same length.)
• Have a child volunteer pick out the shape he or she thinks you are describing
• Ask the child to pick a shape. This can be from another set of shapes inside a box or opaque bag so that the child can have a look at the shape in order to describe it properly without anyone else noticing which shape it is.

(Medium level)

• Play the same game as before but have the child describe the shape by visualising it in his/her mind
• Alternatively, play a similar game with just two players
• Provide each player with one set of 3D shapes and a photograph of one of those shapes
• Have the students try to guess the shape on the other person's photograph by asking yes/no questions, E.g. Does your shape have six faces?

(Medium level)

• Play celebrity heads with shapes
• Stick a picture of a shape on the child volunteer's forehead (or a headband displaying the picture)
• Have the volunteer ask yes/no questions about their shape while the rest of the class answers them
• The aim of the game is for the volunteer to guess the shape by identifying it from a pile or saying the name of the shape (keep in mind naming the shape may take the focus away from describing attributes)

Function machines and hands on learning

In my readings I found this brilliant teaching idea for teaching functions to kindergartens by Willoughby (1997). I feel that this may even be a valuable learning experience for children in the older grades too. I would love to try this idea with children in an actual lesson. Please note that the function machine my graphic refers to is a large box big enough for a child to fit inside comfortably, decorated so that it looks like a machine, used for teaching functions. It must also contain a slot for the input and a slot for the output.




(Please click to enlarge this graphic)

As I reflected on this lesson plan by Willoughby (1997), I contrasted the potential learning described with Warren and Cooper's (2005) research findings, which stated that "the use of child volunteers to act as IN and OUT with respect to the function machine was a strong distractor in classroom 3." According to Willoughby (1997) the above-mentioned learning experience was one which he has carried out many times. Furthermore a photograph of a child remaining engaged while participating in this activity was included in the article. This led me to wonder, how could this be? Perhaps it was because of the fact that in the lesson conducted by Willoughby (1997), the child inside the box remained hidden to the rest of the class and therefore was less of a distraction. Or perhaps it could have been the fact that the child inside the box was in control of the transformation from recieving the input to the providing of the output. This was unlike the learning experience described by Warren and Cooper (2005) where several volunteers were responsible for only one aspect of the transformation, i.e. "Frank gives the green stick to Ned; Ned puts green stick in box; Researcher changes the green stick to red stick and gives this to Bonnie; Bonnie gives the red to Frank and the teacher records the change on the IN/OUT table." (p.157)

This reflection has reminded me that in designing learning experiences and assessments for children, simply having concrete examples, hands-on experiences and well-designed resources is not enough - aspects such as what the children are paying attention to must also be taken into consideration. As teachers we must ask ourselves, "What are the children learning?" and "How do I know he or she is learning it?"

References:

Willoughby, S. (1997). Functions from kindergarten through sixth grade. Teaching Children Mathematics, 3(6), 314-318.

Warren, E. & Cooper, T. (2005). Introducting functional thinking in year 2: a case study of early algebra teaching. Contemporary Issues in Early Childhood, 6(2), 150-160.

Word Riddles

The following two word riddles are a fantastic way to help children identify the rule which these qualitative changes (Taylor-Cox, 2006) follow. They can be very tricky especially when they are spoken instead of written.



First Riddle:
"I am going on a holiday.
I can take um...a tent.
I can take um...a my puppy.
But I can't take my book."

Invite the children to then ask more questions about what they are allowed to take on the holiday.

They might ask, "Can you take a computer?"
No, I can't take a computer.
Or another might ask, "Can you take me?"
No, I can't take you.
Or yet another might say, "Can you take um...a boat?"
Yes, I can take a boat.

Ask the children who think they have figured out the rule to test it by asking more questions and keep listening to others' suggestions in order to check the rule they are using is working consistently. It can go on for a long time and amuse many children (and adults alike) for a long time also. Discuss the rule at the very end, and invite the children to try it out with others in their family.

Or try this other riddle.

"What can fit through the little green door?
A dog can't fit through the little green door,
But a puppy can fit through the little green door.
A cat can't fit through the little green door,
But a kitten can fit through the little green door."

Again, invite the children to ask more questions about what can fit through the little green door.

Their first question may be, "Oh, this one is easy, can babies fit through the little green door?"
No, babies cannot fit through the little green door. Nor can ducklings, nor can baby chicks. But bunnies can fit through the little green door.

Surprised? Have you worked out the rules? Feel free to e-mail me/leave a comment if you'd like your answer confirmed.

Growing Patterns

What kind of pattern is it?

It is a growing pattern.

How many buttons do you think will be in the next row?

5.

How do you get five?

Because it will be one more than the row before it which has four buttons.

How many buttons do you think will be in the 10th row?

45.

How did you get that?
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

Is there another faster way of calculating it? Maybe doing up a table will help. Or if you can also try rearranging the shape of the triangle.

Row number Number of buttons
1 1
2 3
3 6
4 10
5 15

What I learnt at this point:

It was easy to recognise and describe the vertical rule, both by looking at the numbers in a table, or by working off the visual representation.You just take the previous number, and add on one number more than the number you added on to the last number to get the previous answer.

1+2=3, 3+3=6, 6+4=10, 10+5=15

But to recognise the horizontal pattern or see how the numbers in the second column related to the numbers in the first column was much more difficult.

Eventually I found the the rule was: take the row number, square it, add it to the row number, and divide the whole answer by two e.g. 5 x 5 = 25; 25+ 5 = 30; 30/2 = 15

As I reflect on the reasons why it is important to recognise the horizontal pattern and not just the vertical I realise itt is because the point of identifying patterns is to use it to make predictions; and being able to identify a horizontal pattern was necessary to efficiently predict the output when given the input, or the backtrack by figuring out the input from being given the output.

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.