Chapter 7

To me it makes complete sense that calculators and technology are  part of the mathematics education today. According to the NCTM, technology is an essential tool for both learning and teaching mathematics (Van De Walle et. al., 2010). It is important that children learn how to make mental computations. However, I agree with Van De Walle, Karp and Bay-Wiliams (2010) that students should learn when to use mental mathematics,when to use estimation, when to do a problem with paper and pencil and when to use a calculator. In Sweden during my school days (which was long ago…) we were allowed to use a calculator in the younger grade levels. For me it was great because I felt less anxious and could concentrate on solving the problem rather than getting stuck with the computations. Although, I wish that there would have been a greater emphasis on mental computations just for the sake of practice and confidence. My 15 year old daughter is worried that my 10 year old son is using the calculator too often when doing his math homework. It is interesting to observe that he is more interested in the challenge of solving the problem than just the computation and he uses a lot of common sense. I do not think that we should be concerned with the use of calculators but instead see it as a tool to help improve learning. As stated by Van De Walle et. al. (2010) calculators that are used considerately and meaningfully can enhance the learning of mathematics.  The danger would be if the children do not have a conceptual understanding of math and this can happen with or without the help of calculators or other technology. According to Van De Walle et. al. calculators should be used when the activity is not to compute but computation is involved in the problem solving, generating and analyzing patterns or accommodating students with special needs. I agree! If the calculators are used in a variety of ways in lower grade levels such as counting by ones, pressing the equal button and looking for patterns (Van De Walle et. al., 2010). We are living in a technological society that is rapidly changing. Today’s children are used to computers and electronic games. They will be the ones who will create new technology in the future and have a career that is not even known of today. Teachers have to learn more about mathematical software and how it can enhance learning in the math class. As stated by Van De Walle et. al., 2010 a mathematical software tool is like a physical manipulative; it does not teach but gives the user of a well-designed tool an electronic “thinker toy” with which to explore mathematical ideas. There are programs that provide “counters”, base-ten blocks (ones, tens, and hundreds model) and so forth. Instructional software is designed for student interaction and functions in the same way as a textbook or a tutor (Van De Walle et. al., 2010). It is important to follow certain guidelines when selecting and using software and, again as with most things, there should be a balance, combine software activities with off-computer activities (Van De Walle et. al., 2010). This is big business and teachers need to be able to scrutinize and select good software programs to ensure, that indeed, the program develops conceptual knowledge and follows the objectives of learning. It was interesting to go through the teachers resources on page 124 with my son. We found some really interesting math games that helped him visualize, for example, the concept of fractions.

In conclusion, I think technology is great and we need to keep up with it but BALANCE with instruction and support is crucial.

Reference

Van De Walle, J., Karp, K. & Bay-Williams, J. (2010).  Elementary & middle school mathematics. Teaching developmentally (7^th ed.). Boston, MA: Allyn and Bacon   

Session Five

Lesson 17:

Looking at division of fractions in a visual way.

I think this is an excellent way to introduce fractions for children using pictures and models so that they can visualize and see what is actually happening with the different parts; this will enhance their conceptual understanding of math. I wish I had had this exposure when I was young.  We had an in-depth discussion about why ¾ divided by ½ is still ½? Another discussions was that it should be 1 1/2. Thinking about it would 3/4 divided by 1/2 be 1.5 and not half. It is important to visualize this in order to understand.  

Does	1	÷	1	really equal 3 ?
	2		6

You can change a question like "What is 20 divided by 5?" into "How many 5s fit into 20?"
In the same way our fraction question can become:

1	÷	1		How many	1	in	1	?
2		6			6		2

Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"?

How many

in

?

Answer: 3

So now you can see that		1	÷	1	= 3		really does makes sense!
		2		6

The above example, retrieved on August 29, 2011 from  http://www.mathsisfun.com/fractions_division.html.

Another example was the following question: What is 1/5 divided by ½? To do this we divided the model into equal 10 pieces. Thus, 2/10 divided by 5/10 (because we know that half of ten is 5). The answer is 2/5.

Or

2. Two Fifths divided by One tenths = Four tenths divided by One tenth = 4 divided by One = 4.

Lesson 18:

We were looking at a homework example for a primary three student. Question: Valentino has ½ liter of orange juice but he needs 5/7 liter of orange juice.

Well, in this case we used a model. We know that 7/7 is a whole and Valentino has half In order to divide this easily we cut the model into 14 equal parts.

Therefore, 10/14 minus 7/14 = 3/14.

It was interesting to learn that this type of question (when used to assess students) would be considered at application level for a primary three student but for a primary five student it is at comprehension level. The primary 5 student would have come across this question more frequently in math class. Bloom’s Taxonomy provides an important framework for teachers to use to assess students’ cognition. In primary school the focus is on the following three levels of Bloom’s Taxonomy:

1. Knowledge level

2. Comprehension level

3. Application level 

We discussed the significance of giving students time. If you wait long enough you will find interesting ways that students can come up with to solve the problem. Teachers should not be too eager to say, “Yes, that’s right.” Instead say, “Are you sure?” Maybe there are other ways to arrive at the solution…I agree! Students learn right or wrong from external stimulus. It is important not to give artificial signals (for example, body language, answers, tone of voice) for students to depend on. Try to be as natural in all situations. This made me reflect on the impact on a student’s self-confidence when a teachers gives feedback  to one student with  a comment of, “excellent” and another student gets a, “good”. How does the student feel who only received the comment, “good”?

We repeated the use of numbers and added that numbers can be used to represent proportion:

1. Ordinal numbers

2. Cardinal numbers

3. Measurement numbers

4. Nominal

5. Represent proportion (for example, 3/4 of a class)

Lesson 19:

Area

In this lesson we drew different sizes of squares on a dotted paper. It was interesting to see that we could come up with 8 different squares and discussed the size of the squares. How do we know that one is bigger than the other? Well, we counted the squares inside. It is important to start with non-standard units (how many squares per unit) instead of actual measuring of the square’s cm in the lower primary levels when children are comparing sizes. 

Lesson 20:

Area of the polygons related to dots.

Polygon means a closed figure.  We learnt that given a polygon on a grid of equal-distance points George Pick provides a simple formula for calculating the area of the polygon. The formula is A= i = b divide by 2 – 1 (retrieved August 29, 2011 from http://www.cut-the-knot.org/ctk/Pick.shtml). 

Thank you Dr. Yeap for yet a very interesting and engaging lesson! I have more confidence with math.

Session Four

Lesson 14:

Mind Reading Game (using only digits from 1 to 9).

This was another “magic trick” by Dr. Yeap. Well, actually there is a logical solution to the problem again…convincing me that math is actually a science and not magic. 
The "mind reader" will ask the person to think of a two digit number. Then follow three steps:
1. The two digit number, for example 72
2. Then add the two digits 7+2=9
3. Then difference between the first number 72 and the second number 9 which is 63. 
The person will call out the first number in the tens place and think about the other number in the ones place. Then the person will do the three step computation. The mind reader will actually get the same result.  

Column A will tell you the number that was called out and Column B is the number that Dr. Yeap figured out (which was indeed the same number).

Column A: Column B:

8 3 9 4 6 5

72 27 81 36 54 45

 

There were a variety of methods for the “mind reader” to figure out the person’s number.

1. Multiply by 10 then minus the 1^st number that the person says. For example, 8, I then think 80 – 8 = 72.

2. If the person say 5, you think 51 (5+1=6) then 51-6=45 (4+5 must add up to 9) or think 57 (5+7=12) then 57-12=45.

3. We see that it is multiples by 9.

4. Person says 8 then I will go back one number 7 (7 + 2 to make 9) so number is 72.

I thought this was an excellent introduction to Algebra. Very interactive and it made sense eventually.

For example, 3 (call it x) is in the tens value and 2 (call it y) in the ones value.

(30x +2y)=32

(3x +2y)=5

32 -5=27

There is still some hope even for someone like me!

Lesson 15:

Subtraction 37 – 19 = ?

We discussed a variety of ways to subtract and how to teach the concept of place value. This is important for children to effectively calculate. They need to be able to count in tens and ones. Explain to children that each place has a value, for example, 37 (3 is in the tens place hence 30 but we don’t write it out). We also discussed that children should explore different strategies. However, the teacher needs to bring it back and anchor the lesson to introduce the new idea to effectively help them to count.

Word problems:

It was interesting to discuss the necessity to introduce a variety of different word problems. Thus, elicit thinking and problem solving skills. The problems should have different structures such as: Join Problems, change situation (before and after), part-part-whole situation and compare. We discussed Zoltan Dienes who has emphasized the importance of variation in mathematics (vary the unknown). Also, a math problem can be introduced at any grade level just knowing which approach to use (Concrete, Pictorial or Abstract). Teachers need to be aware of their language when discussing math. For example, do not say when there is the word “more” in word problems always use addition and when there is the word “less” then you should say take away. This may cause confusion for children when looking at the word problem such as:  Valentino has 12 cookies and Sandra has 8 cookies. How many more cookies does Valentino have than Sandra?

Lesson 16:

Equal Parts

The square is divided into 4 equal parts:

¾ is colored blue. We discussed that the parts are equal and you can name them. Also, it is important to start by introducing the name, three fourths instead of writing the fraction ¾. We also talked about the importance that we can count only the same sets of things (no difference in nouns). In addition, be aware of our language; do not say 2 upon 5 or 2 out of 5 (as this may confuse the child and interfere with his/her conceptual understanding).

How to do two fifths plus one half? It has to belong to the same set.

Well, cut it in half…now you can do it.

Always introduce informally then move to formal math.

One piece is one fourth.

Now if you cut each piece into halves they will become one eighths. Thus one fourth is equal to two eighths.  

Being equal does not mean being identical. As long as the area is the same it will represent the same amount.

It is interesting to see that a math exercise can be done at any grade level using the CPA approach. 

Concrete Approach: Children have a piece of paper and cut it out.

Pictorial Approach: Use drawings to explain the concept.

Abstract Approach: Use the formula to find out the area of the triangle. Multiply the base by the height, and then divide by 2.

Lesson 17:

Dividing Fractions

Division have two meanings; sharing and grouping.

12 divided by 4 equals 3

 

How many fourths are there?

¾ divided ¼ = 3 (simply put, how many fourths are there in three fourths…)

12 divided by 4 =3 (simply put, how many fourths there are in 12…)

So ¾ divided by ¼ = 3

Thank you Dr. Yeap!

Session Three

Lesson Study:

Peggy started the lesson by asking us if we knew anything about “Lesson Study.” Well, just by looking at the name we can figure out that it is to observe and critically analyze a lesson. Interesting!

The objectives for today’s session:

1.      Examine two case studies using lesson study.

2.      Identify factors of good mathematics teaching and learning for numeracy development in early childhood.

3.      Understand lesson study,

I liked the interactive and engaging approach Peggy used to share the knowledge and information about lesson study. We started by watching the first lesson (case study one) and critically looking at the lesson and teaching pedagogy. The lesson study is about finding out what factors support or  hinder  the research theme. We wrote down one item each on a post-it on what was good and also, what could be improved with the lesson. Then categorized the comments into:

1.      Sitting arrangement.

2.      Level of engagement.

3.      Use of materials.

4.      Flow and sequence of the lesson.

5.      Classroom management.

6.      Communication. – Teacher to student, student to teacher.

7.      Questioning

8.      Attitudes/Disposition

9.      Differentiated learning.

The first lesson (case study) focused on “more than or less than” for Kindergarten students. There were discussions on how to improve and make it more challenging for the high ability children (differentiated design of the task.) It was interesting to see how Peggy brought the task to a higher level.

Peggy asked us to explore cubes and construct a different structure with 5 cubes. We used both our creativity and thinking skills. I agree that math emphasizes visualization in the cubes. The cubes also help children learn the idea of conservation of numbers (no matter how the cubesare arranged, the number stays the same.) This was the introduction to the second case study which was a lesson for K1 students. The research theme: Explore different options to articulate their thinking and apply what they have learnt in new situations? The Focus: Constructing different structures/models with five cubes. We followed the same procedure as the first case study.  

It made me reflect that yes this is an excellent approach for teachers to  learn more about others, themselves and also, observe how children learn. Lesson study is the recent trend in professional development. Traditionally, teachers went for workshops, lectures and conferences.

However, this approach is: Teacher driven/initiated, Job-embedded (relevant to teachers’ area of work) and authentic involving their own students. Collaborating with fellow colleagues within school/center and beyond. It is a professional development process/tool that teachers are engaged in to systematically examine lesson plans.

Lesson Plan Study Flow:

Identify Research Theme (which is aligned with center’s vision and mission).

Plan Lesson

Research Lesson

Post Lesson

Lesson Plan Revision

Back to Plan Lesson

From Peggy I could really see the benefits from "lesson study" for both teachers, children, and the school. For example, collaboration, changes of mindset, scaffolding techniques, allwo teachers to study children's level of understanding and or misconceptions.

Thank you Peggy! Great job keeping all of us actively engaged.

Session Two

Lesson 6:

Take 1 or 2 Game (two players).

In this game each person takes turns drawing one or two sticks, the last person drawing one or two sticks is the winner. We discussed what could be a bad number to end up with which would indicate that the other person will win the game.

Here are some thoughts:

Take                Bad Numbers

1,2                   3

                        6

                        9

1, 2, 3              4…

We observed the pattern to find the logic and saw multiples of three when taking 1 and 2 sticks. We made predictions. Also, we looked at the number bonds while playing.

It was easier to find out who the winner would be once we counted a smaller number of sticks. We look at the sticks and know exactly what number is without counting. It is called subitize.

Teachers have to differentiate the learning within the class. For example, we discussed that this activity is good for adding and counting numbers. However, if you have a high ability student in your class who already know how to do this, you need to challenge this student.

The teacher should ask him/herself the following questions:

1.      What is that I want them to learn?

2.      What if they cannot do the tasks (intervention)?

3.      What if they already know it?

Also, we discussed probability. The numerical value to a chance happening. .. could be either low or high. According to Dr. Yeap this is not taught in the primary schools in Singapore.

Lesson 7:

Making Largest Even Number.

Dr. Yeap had 4 spinners (with numbers 0 to 9).

The goal was to make an even number as big as possible. Dr. Yeap spun each spinner once and then we had to make a choice where to place the number in the empty boxes.

6

2

0

Dr. Yeap asked us that if we were using cards would it change our decision on where to place the numbers in the boxes.

One explanation was that if the number does not go back in the card stack it will never show up again. Hence, there may be a difference in our decision because of this knowledge.

Holiday Games in Primary One

We watched a video of Dr. Yeap teaching a primary one class. We were asked to assess the children in the following areas:

1.      The skills needed for addition.

2.      Can the students see a pattern?

3.      Can the students communicate their ideas?

Dr. Yeap used two big dice, holding them together and asked the students to figure out which numbers are hidden.  

The students were engaged and interested. This was like a magic trick for them. How could Dr. Yeap figure out the sum of the two numbers hidden on the dice when putting them together? They were problem solving, reasoning, communicating and connecting ideas.  One child actually communicated clearly that the die has six numbers and thus we can figure out what number is missing. Great job for a Primary One student!

One student tried to do the trick; by looking at all the sides that appeared he could find out the total sum of the two sides that were hidden. He said, “I think it is 8.” Let us see if there is a logical pattern to solve this problem.

Dice                 Total

1+6                  7          =

3+5                  6          =

2+5                  7          =

4+2                  8          =

5+4                  5          =         

Is there something special about the numbers on the opposite side of the hidden side? What happens when we add the three numbers? We then see that it all equals to 14.

The dice is usually a rounded cube, with each of its six faces showing a different number from one to six. The opposite numbers on the dice add up to seven.

This was an excellent lesson for young students to practice addition. The students were actively looking for a pattern which is one of the “Big Idea” when learning math. Another “Big Idea” is to develop strong number sense. They were practicing number bonds. It is important to understand simple addition. Teachers talk about “number bonds” as pairs which make up each number. For example, the number bonds for seven are 3+4, 2+5, 1+6 and 0+7.

Lesson 8:

Long division.

It was interesting to learn that the Singapore Math Curriculum is a spiral curriculum. It means that students revisit topics at a higher level. It is important that students have a good foundation in math in order to be successful. Again, they need to fully understand the concept and not rely on memorization!

Also, students need to understand how they arrive to a solution not because the teacher told them to follow a certain formula or procedure. For example, for long division…why do we do it in a certain way? Are there other ways we could use to solve the division?

When doing mathematics we need to understand the following:

1.      Procedure.

2.      Conceptual understanding (meaning behind…).

3.      Instrumental understanding. For example, we know how to use a computer but if it breaks down we do not know what to do…

4.      Conventional understanding. For example, the society has agreed on certain conventions that we have to follow.

We discussed that some conventions are contradicting depending on which country you live in. For example, in the U.S. they use a comma in 51,000 but in Singapore we do not.

Again, please remember that we are teaching children, not teaching mathematics. Mathematics is only a tool. Children need to learn by self-correcting. Teachers should be there to encourage children to make decisions and to use their thinking skills. At times adults try to teach young children multiplication by memorizing the multiplication table which is not developmentally appropriate. This will not encourage conceptual understand of mathematics.

The Singapore math curriculum follows the CPA Approach:

Concrete

Pictorial

Abstract

When teaching multiplication for young children, start by counting groups. For example, five groups of twos and so forth.  It is helpful for the teacher to use a systematic arrangement of objects, in rows and columns, which is called an “array” or dot diagrams to help the children visualize the groupings.

It is crucial to teach children to figure out things for themselves. Do not teach them to be helpless. I agree!!!

It was interesting to discuss Howard Gardner’s multiple intelligences (linguistic intelligence, logical-mathematical intelligence, musical intelligence, bodily-kinesthetic intelligence, spatial intelligence, interpersonal intelligence, intrapersonal intelligence). Did you notice that there is no such thing as “memory intelligence?” This is not part of our brain structure to remember all these things…Hence, humans invented computers, thumb drives, etc. Invention comes out of necessity. Thank goodness for calculators and computers!!!

Intersting to know that the approach that is currently implemented in Singapore Schools:

TSLN (Thinking Schools Learning Nation) and subjects are vehicles.

TLLM (Teach Less Learn More).

The Singapore math has the fewest topics (thinnest syllabus) in the world. However, they go more in-depth. Could this be why Singaporean students score well on tests internationally?
We also discussed the importance of quality instruction. Children who have the opportunity with quality education teaching, do not need to spend hours doing tuition or be enrolled in extra commercial math programs. I agree!!!!

Lesson 9:

We were given the answer and should find the numbers in the subtraction question.

Three answers:

____     -    _____   =   3          (10-7=3)

____     -    _____   =   3          (12-9=3)

____     -    _____   =   3          (11-8=3)

Five answers:

____     -    _____   =   5          (10-5=5)

____     -    _____   =   5          (11-6=5)

____     -    _____   =   5          (12-7=5)

____     -    _____   =   5          (13-8=5)

____     -    _____   =   5          (14-9=5)

Can you see a pattern?

Lesson 10:

Word problem in response to a question raised.

John spent 1/4 of his money on a gift.

He spent 1/3 of the remainder on a book.

He was left with 36 dollars. How much money did John have at first?

This made me reflect about the advantages of breaking the word problem down to step 1, 2, 3 it is an easier and more failsafe approach.

So what constitutes mathematics?

Thinking Skills & Problem Solving
Number sense
Visualization
Generalization (patterns, relationship, connections)
Communication  (language, representation, reasoning, justification)
Metacognition (thinking about thinking)

Thank you Dr. Yeap for another engaging and interesting math session!!!