Some Solutions to Problems ABC

I have not completed the tabulation of peer evaluation for all groups. Do not panic. They will come in after I am done going through all your problem-solving journal which is more urgent as I need to return these to you by the end of this week.

The journals were graded as well kept (all problems are solved with some attempts at reflection), fairly well kept (all problems are solved but the element of reflection is lacking), adequate (many problems are solved but the documentation but communication of methods is wanting) and needing improvement (either incomplete or many solutions are only partial). A top grade journal would be like the examplar I gave during the course, for most of the problems. Journaling forming part of the participation and attendance score. Attendance and contributing significant ideas form the rest of this score.

Problems Modified for K

Vasandy has a suggestion on how to modify the apples problem for younger learners. Giving kids apples arrange in an AB pattern e.g. alternating rows of 3 and four apples and asking them to tell the number of a tray can be a platform to repeated addition which forms the basis for multiplication in later grade levels. Vicky also gave suggestions on how this problem can be made accessible to kindergarten kids.

The course has ended but I hope some of you will continue to share teaching insights on your respective blogs.

Lesson on Friday and Saturday

We solve two problems - the toothpick problem and the brick wall problem.

On Saturday we discussed the lesson study style lesson plan based on the hotdog problem. We also did Monkey Puzzle before we all setlled down to complete the final problem-solving test.

The Conversation Time on Saturday revolves the cotroversial issue of nurture and nature - Do children take to problem solcing like fish to water? and Do children need help in completing the problem-solving task?

The peer rating for Group 11 is 25 and for Group 12 is 27. The score is derived by removing the highest and lowest rating such that 10 scores remain. These are added up to give a score. A higher rating indicates your peers valued your insights more.

The ratings for Group 10 = 20. Group 8 = 24. Group 7 = 28.

Preliminary Assigment is graded based on Peer Evaluation + Quality of Slides + Individual Notes.

Lesson on Thursday

Listen to the Song which is based on this 1825 nursery rhyme by Sesame Street. There was an earlier version from 1730 but that was a nine version, not seven. It dates back to 1650 B.C. when the Egyptians ruled that part of the world.

Solve the problem to find how many cats and kittens were there?

A video solution is available. My brother posted it on my Facebook.

We also solved the logic problem - three containers contain all yellow cubes, all green cubes abd a mixture of green and yellow cubes. The thing is the labels on the three containers were all mixed up and each container is labelled wrongly. The container labelled Yellow does not contain only Yellow cubes. Find the minimum number of cubes that need to be drawn to ascertain which containers contain what.

After the break we had three conversations and solved a problem on newspaper.

Lesson on Wednesday

Why things float in water (or any fluid,for that matter)? You see when things are in water, the things experience a push. This push is upwards. It is called upthrust, a kind of force.

The size of this push depends on how much water is displaced by the thing. That is why a marble sinks and a heavier soccer ball floats.

We also solve the problem of looking for names of animal that has the same number of vowel as there are consonants.

We had two groups chatting about assessment things during Conversation Time.

I also posed a problem - how to compare the performance of the groups based on the ratings that you give the groups.

For example - Group 3 (Liana's) scored 3,3,3,3,3,3,3,3,3,3,3,2,3,3,3,3,x,3 (x means comment without rating) and Group 4 (Noorlinda's) scored 3,2,3,3,1,2,x,3,2,2,2,2,2,2,x,x,2). How can we determine which team receives a higher rating. Come up with thee different approaches. Do all approaches result in the same outcome?

Test @ Home

This is an optional task. This is the take-home test which will be taken into consideration in the event that you did not do well on the Final Problem Solving Test. Each task is valued at 10 points.

Your submission is graded as Exceeding Expectations (good solutions in all tasks), Meeting Expectations (ggod solution in at least one task) or Approaching Expectations (some attempt at the problem s but without success).

Problem A

Jon has a set of cards, numbered 1 to 100.
He removed some cards with consecutive numbers and found that the total of these numbers are 50.

What are these cards?

Write an explanation to show how you arrive at the solution.
Show that you engage in the look back stage of Polya's model.

Problem B

Find the largest rectangular box (by capacity) that can be made using a sheet of A5 paper.

Explain the steps that you take to make sure this is the largest capacity possible i.e. how do you know this is the largest possible value?

Provide an explanation if a container with a larger capacity floats better.

Problem C
The Door Bell Rang
Ma makes some freshly baked chocolate chip cookies, and her two kids sit down to eat the when ding dong! the doorbell rings! More kids arrive to share the cookies, but just when they sit down, ding dong! Finally, when there is only one cookie for each child, the doorbell rings again. Who is it? Grandma with a new tray of fresh baked cookies! And no one bakes cookies as good as Grandma's! Hutchins sneaks a bit of math into this funny tale.

In the original story, Ma baked 12 cookies. This is my version. Needs to have basic literacy in Singlish.

Cookies, cookies, there's chocolate chip and durian bits too.
One cookie left when put in twos.
Cookies, cookies, enough for you, you, you and me.
One cookie left when put in threes.
Cookies, cookies, good to eat with Milo and Kopi-O.
One cookie left when put in fours.
Cookies, cookies, come, come, die, die must try.
One cookie left when put in fives.
Cookies, cookies, baked from nenek's secret mix.
One cookie left when put in six.
Cookies, cookies, more sedap than the ones you bought in London.
No cookie left when put in sevens.

Give one, or better, two, method(s) to find the number of possible cookies in the given story. Originality is valued.

Footnotes (in case you are not literate in Singlish and local dialects): durian = a type of fruit, Milo = brand of a chocolate drink, Kopi-O = black coffee with sugar, die, die must try = definitely must give it a try, nenek = grandmother, sedap = delicious.

Note for Absentees

Students who are absent for any session or significant part of the session (more than an hour) need to complete make-up work and submit these with your Final Assignment. Indicate clearly on your Final Assignment that you are submitting the make-up work to earn credits for the hours you missed from the course. Letters and other documents to support your absebce must be submitted to Wendy (if not already submitted to the professor).

For Monday, the work should include the Solutions to the two problems as well as a summary of the definition of a problem, Polya's problem-solving model, and five examples of problem solving heuristics.

For Tuesday, the work should include the solutions to the problems. Make short notes about the important of affective factors in successful problem solving. Make a list of strengths and difficulties children face when solving the tangram problems - based on the Conservation Time held this evening.

For Wednesday, we had two groups that chat about assessment - what amazes them when they observe the kids and what the observation say about the child. Please find out the main points of this discusssion and make a summary. Your other task is to make notes on (1) the format of lesson-study style lesson plan and (2) explain why things float.

For Thursday and Friday, the make-up task is to solve the problems done in class.
For Saturday, the make-up is to complete an alternate test and taks based on Conversation Time.

You are also expected to complete tasks (short notes) based on the Conversation Time topics that you missed. You can do this by discussing it with a friend who was present, to read up relevant sections from the book, and to base it on your own experience.

Thursday: (1) Give five examples of tasks you expect K2 children to be able to solve? Use different subject areas. (2) Share two surprising things that you saw during the tangram task.

Friday: (1) Creativity & Brilliance in Problem Solving (2)A parent complained that you are wasting time doing the tamghram puxzzle in class. Write an e-mail to justify why you did the task in class. (3)Difficulties Students Have with Problem Solving.

Saturday: (1) Is problem solving taught or caught? Are some people just born to be good at this? (2) What Help Teacher Can Give Children During Problem Solving?

Lesson on Monday

Problem: Use a given number of sticks (12, 10, 9 and 6) to make four triangles. A triangle is made using three sticks with the vertices (corners) formed using the dough.

How many different figures comprising of four triangles can be formed using 9 sticks?

In this lesson, we discuss what makes a task a problem. We also learn to analyse a problem if it is suitable for a given age group.

The class was introduced to Polya's problem-solving model.

Problem: How many apples are there on each tray? See previous post.

In discussing this problem, the idea of mathematical modelling was introduced. The term problem-solving heuristics were also explained with several examples.

MAT150 BSc05 Wheelock College / SEED Institute

The course on problem solving has begun. Please prepare an exercise book which will be your problem-solving journal - you can refer to this during the test, so take good notes. You will also submit this to be graded for the Participation component of the course. Bring your own crayons, glue, scissors etc if you are the kind who like to make your journal interesting. At least one of you is using your iPad to take notes - in this case you can submit your 'notebook' in the soft copy form. Each problem will be assessed for (a) the quality of solution (b) the quality of presentation of the solution.

Please complete your pre-course task. You will be given time on Monday to form your Conversation Time Groups and have a short meeting. All Groups must be ready to talk about the pre-course work from Tuesday onwards - Groups and Questions will be drawn randomly.

This course is on problem solving in numeracy with extension to other domains.

The required textbook is the same as EDU330. Please be familiar with the chapters on problem solving.

To whet your appetite, here is one based on the photograph of a grocery store in New York City. Look at the green apples. Problem: Explaining your method and stating your assumptions, find the number of apples on one tray. Suppose the tray is a square tray.

Also think about how you will help K2 children solve this problem. What can a teacher do to help the children arrive at a solution? What content areas are required in solving this problem?

Development of Numeracy in Early Childhood Education

The slides are available here:

Development of Numeracy in Early Childhood Education

View more presentations from Jimmy Keng

Reading Your Blogs

Photo Credit: Vasandy

Today I read two blogs. Vicky's pre-course blog entry was highlights key ideas discussed. It is comprehensive. Her course blog entries (which is all-in-one) gives a glimpse into what happens in class with slivers of her own experiences with those activities.

Vasandy's daily entries are interesting reads especially her observations of how her kids respond to tasks. It is important to combine authentic observation with formal learning in professional learning. Even realizing that a ruler does not start with zero and then figuring out how to use it to measure the steps is something that cannot be achieved if we do not include authetic learning experiences such as this. Measuring a line on a worksheet does not pose such a challenge.

A Maths Trail is when students go to different stations and carry out mathematics learning using the environment. The tasks at the MRT Station is an example of a task in a Maths Trail. To follow up on Vasandy's thoughts, what are the things that students can learn in a maths trail task compared to a paper-and-pencil task. Think about what additional learning that takes place during the practical-based tasks than if that was a paper-and-pencil task.

Anyway, why does a ruler not start with a zero? Janica offered a possible reason. See Comments.

Make Up Work for Absentees

If you are absent for a session, you are required to

(1) solve these problems and submit them - you may need to consult a classmate to find out the details; in cases where the Lesson is not a problem, to describe the main idea of the Lesson (you can find out the Lessons that we did in each of the six sessions);
(2) make short notes of three main learnig points from the lessons you missed (you can read the relevant chapters in the textbooks; in each session we did either whole numbers, fractions, measurements or graphs)

Please submit make-up work when you hand in your Final Assignments. If there is no make-up work for the sessions you were absent, you would be considered to have incomplete learning for this course.

Story Problem on 4 divided by 2/3

The class was asked to write a story problem that can be answered by calculating 4 divided by 2/3.

This does not count!
What is 4 divided by 2/3? (name withheld to protect the innocent)

This is correct but I was hoping for a context:
How many 2/3 are there in 4?

This is pretty close but I think you meant 2/3 cups
How many servings of 2/3 are there in 4 cups? (Nur Ain)
Several people gave responses like hers.

All you need to do is to the 2/3 the same unit as the 4.
I have four pints of ice-cream for a children's day party. If I serve 2/3 of a pint of ice-cream to each child, how many children can be served?
My first reaction is 2/3 pint - is that a lot?

Final Session

Today is the last session for EDU330.

Lesson 22 Writing a story problem for 4 divided by 2/3. I am glad we see several suitable responses. I randomly selected a few responses.

There are 4 cakes. Each child receives 2/3 of the cake. How many children are there? (Joycelyn)

Inger has 4 pizzas and she needs to cut it into two thirds. How many two thirds can she get out of it? (Ylva)

Lesson 23 How Big is a Foot is a good story to introduce the concept of standard unit in length. One activity I can do in class is to ask students to use the king's foot to measure the size of the bed that fits each of the children in the class. Or for homework, let them bring the 'foot' home and find the dimensions of the beds that fit each member of the kid's family.

Lesson 24 MRT Station. The class was sent to find the height between Level 1 and Basement 1 of the station.

Lesson 25 was to make a container to fill 15 beans.

Lesson 26 was to make a graph to show how we get to school.

We ended with a caterpillar story to remind us that shortcuts can 'damage' a child, easpecially in early childhood education.

The course has ended but learning continues ...

Multiplication in Hort Park

These are photographs taken at Singapore's Hort Park.


Each flower has 5 petals. How many petals are there in 2 flowers? 4 flowers? 5 flowers? This is a 'rate' situation that can be modelled using multiplication
Another is the 'array' situations. How many pots of plants are there in the photograph. In an array, the number of objects in each row is equal.

Pick's Theorem

At the top of this blog is a link to one of the many, many sites on Pick's Theorem. There is an applet there that you can use - it gives the area automatically, so no need to calculate the area yourself.

There is also a proof of the theorem. It is a little involved so be warned that it is not a walk in the park.

For those of you mathematical types, there are other things on this site. Enjoy.

Session 5

Lesson 17 continues which is looking at division of fractions in a visual way. Then we did Lesson 18, which is actually the homework of the children of one of the participants.
We went on to Lesson 19 on area. Today we did some work on measurement.

See photo.

Lesson 20 revealed an interesting pattern! Who would think that the area of the polygon is realted to the number of dots inside it. And the relationship that someone came up with (add the number of dots minus three) is awesome!

George Pick would have been impressed. The link to Pick's Theorem is available above.

We had been focusing on whole numbers and fractions the last few sessions.

We also used graphs to document the number of 'pegs' for good ideas. That is Lesson 21.

Selamat Hari Raya to all. Final session for EDU330 on 31st August. But the learning journey continues ...

Addition & Subtraction Situations

There are three basic situations for addition and subtraction - part-whole, change and comparison situations.

In Change Situation, there is an initial quantity, a change in quantity and a final quantity. In Part-Whole Situation, there are at least two parts and the whole. In Comparison Situation, there are two quantities that are compared.

Session 4

Are the four parts equal?
We started with Lesson 14 The Mind Reading Game - the class came up with at least four different ways to figure out the final difference. We explained how the trick works using base ten materials and algebra.

Lesson 15 was on Subtraction 37 - 19. We wrote different word problems based on this.

After dinner, we did several lessons on Fractions. Lesson 16 was on Equal Parts. Lesson 17 was on Dividing Fractions.

See you tomorrow.

Session 3

Photo Credit: Raudha

I hope you see how maths emphasizes visualization in the cubes lesson and tangram homework.

The cubes lesson is to help children learn the idea of conservation of number. That is, no matter how the cubes are arranged, the number remains the same.

By the way did you watch the two kids in Jasmine's blog playing a variant of the sticks game that you played in class (Session 2)? I enjoy reading many of your blogs.

If you were absent, please complete the three tasks and do a write-up on lesson study and the two case study done in this lesson. You may read your classmates' blogs to find out more.

Session 2

We have covered key ideas in Chapter 8 to 14. Tomorrow we will do some lessons in kindergarten and lower primary and explore ways to develop key components of mathematical thinking.

Today we completed our discussion on what constitute mathematics. The class came up with Problem Solving & Thinking with five key components - Generalization (patterns, relationship, connections), Visualization, Communication (language, representation, reasoning, justification), Number Sense and Metacognition.

Lesson 6 was on Take 1 or 2 Game where we decide bad numbers are multiples of three. There was some attempt to generalize to when the rule allows taking one, two or three sticks. Lesson 7 was a favourite - Making Largest Even Number.

Lesson 8 was on Long Division. The photograph shows Lesson 9.

Lesson 10 was a word problem in response to a question raised.

Make-up work for absentees. Please solve all the problems we discussed and prepare a set of short notes for three key ideas discussed in class today.

Session 1

We had the first meeting today.

We did tasks in Lesson 1 (Name Problem), Lesson 2 (Sound of a Number), Lesson 3 (Jun-Jun, Lasene & Siti), Lesson 4 (Spelling Card Trick), Lesson 5 (Arrange Five Numbers).

If you are absent from the session
(1) solve these problems and submit them - you may need to consult a classmate to find out the details;
(2) explain the diferent uses of numbers - ordinal, nominal, cardinal and measurement numbers;
(3) describe in order of increasing complexity different strategies to do 5 + 7; and
(4) four pre-requisites to counting.

If you were present, you must have learnt, minimally, all the above. You may also recall two 'mistakes' that teachers make in designing ordinal number tasks and patterning tasks.

Hopefully, you will review Chapter 2 and Chapter 8.

Do not forget to write your daily reflections - even if it is just one paragraph.

For your class participation, I have given out pegs. Please trade them for a 'good idea token' when you have ten of the pegs.

See you tomorrow. Do not be late if you want to do the Quiz.

I randomly went to a few blogs and saw that Ain had done up an entry where she shared a bit of what she learnt with other teachers who were not with us, and reminded those of us who were there with what we learnt.

Session 1 has as a goal to give teachers a correct mindset about mathematics and teaching. When I read more blogs I will know if this goal has been achieved and to what extent.

Information on Quiz

During the course you will be given some quizes for you to demonstrate your learning of the course materials. These are 15-minute tasks to be done between 1800 and 1815. You must hand in your responses by 1815 for it to be graded. If you are late for classes, you will have to forgo the opportunity to attemp the quiz.

Some tasks are content based. Others are based on pedagogy.

Sample Quiz Items are available at http://mmepdpm.pbworks.com . Please request access the first time you log in. This site requires a password (created by yourself).

Here are the sample tasks for those who have difficuly accessing the site.

Content Type Task
Sally added 8 and 7 by thinking that 8 is 3 and 5. Thus, 3 and 7 is ten. Hence, 8 + 7 = 5 + 10 = 15.

Use Sally's Method to add 6 and 9.

Pedagogy Type Task
Miss Lim uses three types of materials to teach numbers to twenty (e.g. 18) - ice-cream sticks, 1-cent and 10-cent coins and base ten blocks.

Are these three unique materials or they can replace each other? Give your reason.

Welcome BSc 05 to EDU330 August 2011

Updated on 22 August 2011

Welcome BSc05. This is an update before we start today.
Please bring a new notebook / exercise book (any size) for use in this course.

You must have received your course information by now.

There is a pre-course task that has to be completed and the submission is via a blog. That has to be done before the course starts. Hence the 20 August 2011 deadline. Most of you have sent me the url for your own blog. I can see everyone's blog except Maslinda's.

However, throughout the course, there are also reflections that has to be done and these are, again, though the blog. The deadline to complete all reflections is 1 September 2011.

In the meantime, it is good to get your blog up and running - send me the link and your name (the one you prefer me to call you as well as your official name).

Many of you have sent me the url for your blogs and I have linked them to this blog. Please complete the pre-course reading and task by the deadline 20 August.

See you in class. Watch this space for all forms of communication related to the course.

In the first session, we will look at what cosntitutes mathematics and how children learn it - we will use content in whole numbers. The third session will be taken by a guest lecturer Peggy Foo who will share with you the idea of lesson study and examples of lessons in kindergaten used in lesson study. Fractions will be covered in Session 2 and Session 4.

Make Up Work for Absentees

for those who missed more than one hour of each day's class please complete make-up work as part of the requirement of the course.

i have posted points from various lessons and make suggestions as to what you can do. please email me if you are unsure.

please submit these pieces together with your final assignment.

Problem 10

Today we solved a problem - draw on a geoboard paper a polygon with four dots on the sides of the polygon. Many gave quadrilaterals. Students were encouraged to draw figures other than quadrilaterals.

After finding the area of a few figures, a conjecture was made - figures with fours dots on the sides and none inside has an area of one.

This led to a discussion on instructional model for teaching problem solving.

Growing Plants

The groups did the planning on Day 2. On Day 3, they put into action their plan. And see the results next Monday.

This was a partial discssion of a problem solve din class today - a problem with insufficient (and useless) information.

Alice, Bob and Cheryl form a line to buy tickets.
The shortest is not first an the tallest is not last.
The boy is not the oldest and the youngest is a girl.
The oldest is not first in the line.

What are their positions in the line?

If there is no definite answer, are you able to work out all the possibilities?

Make-Up Work for Lesson 2

If you are absent for part of or the entire of Lesson 2, please complete Lesson 1 Reflection and submit it asap.

Also submit the solutions to the five problems done in class. make a list of problem-solving heuristics and illustrate three of them with examples. Submit this together with your Final Assignment.

One of the five problems is: Fold a square piece of paper in any way you like. With just one straight-line cut create each of the pieces shown in the second photo. The first photo shows your classmates solving the problem in class.

Shake Hands

Allan, Bella, Cheryl, David, Ernie and Farid shook hands with each other. Each person shook hands with everyone else once. Use three different methods to find the number of handshakes in the group.

Shake Hands, Say Good Evening and Other Problems

Three sticks are used to make a triangle this size. You need to make 4 such triangles - same size and also equilateral. How many sticks do you need? Can you use only six sticks?


There are five of you. Each says Good Evening to everyone else - once. How many Good Evenings are uttered?

What if instead of Good Evenings, each shake hands with everyone else once. How many handshakes are there?

What about if we do this with the whole class - 46 of us?

Did you like the Sound of Numbers Problem? One containers has 2 objects. The other one has an unknown number of objects. Shale both containers and listen to the sound. How many objects are there in the second container?

Structure

Use no more than 10 straws to build the tallest structure possible. It must be able to stand on its own. You cannot cut the straw. You can use masking tape if you wish.

Container

Make a container using newspaper to hold exactly one cup of green beans.

Problem Solving 1

Photo Source: Jalan Besar PCF Kindergarten, Singapore

Problem: Cut the bread (square) in half. In the photograph, the students have 2 triangles and 2 rectangles. Are there other ways to do this?

Further Question: Do you get more jam on the triangle or the rectangle bread?

Today is the first day of lesson and the focus is on experiencing problem solving in mathematics and other domains so that we can arrive at a definition of problem and hence problem solving.

Welcome to MAT150 Developing Problem-Solving Skills

Welcome to the course.

It is one week before the course starts. The course is about, well, what the title says - developing problem-solving skills.

More about the course later when we meet in class from 22nd Feb onwards.

First thing first - you have some preliminary reading and writing assignment to get you ready for the course. Please refer to your Course Outline.

There is no prescribed books you need to read but the one suggested is Frameworks for Thinking. It is available online. You can read any other books or even what people write on the internet. The product is My Dictionary of Problem-Solving Terms.

You are expected to submit during the first meeting this booklet that you compiled based on your reading. The idea is that you are familiar with terms common in talking about teaching thinking. Please submit one copy to me and have your own copy for reference. You are likely to edit your entries during the course when you get new insights in the problem-solving process.

Enjoy your reading and see you in class with a sharp pencil ... and mind.