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.