Final Session

Lesson 22:

4 divided 2/3.

We discussed the importance of assessment. According to Richard Skemp there are three levels in the understanding of mathematics:

Procedural – Teachers may assess procedure

Conceptual – Teachers may assess conceptual knowledge

Conventional – Teachers may assess method

The lesson started out with a paper-and-pencil test where we first created story problems (do not use the term problem sum because this indicates “sum” as in addition) and then do the actual computation. It is important to assess to see how much the child has understood…maybe they can only do the computation but does not have a conceptual understanding. Also, teacher assesses in order to see what the child does not understand from which they may assist the child (what are the implications for teaching). Teachers should not rely only on paper and pencil test but also, do oral tests (interviews). If the child writes down the answer without any working but can explain how he or she arrived at the answer that means that the child understands. We also looked at how children conduct a count. For example, some count all, others may skip count, or count in twos, or in repeated addition etc. We should observe to see what strategies it is that they are using.  Do the children have good meta-cognition strategies to overcome their weaknesses?  It is important that the test is realistic in nature. Do not ask children to tell time by testing them with drawing the clock and time. Telling time means being able to “read” the clock; it does not mean drawing the hour hand and minute hand on the clock. Ask ourselves, “What are we actually assessing the children for? Equally important is that the results of assessment are reliable.  Equality in performance should attract the same score.

We also discussed that something per something is called rate and when there is a constant rate, we should apply multiplication, e.g. how many petals are there per flower?

Lesson 23:

“How big is a foot?” by Rolf Myller

 I found this story an excellent introduction to measurement. It was very entertaining. We discussed the right terminology for measurement. For example, 1 meter is actually compared to a wave length up to a certain light (frequency), which is more precise than a physical meter. Also, I learnt that I should say my mass is 52 kilograms and not my weight is 52 kilograms. This is because weight is a force that is pulling me down to earth and if I want to lose weight I should go to the moon…

I also learnt about how to introduce time to young children. It should be concrete (what to Jerome Bruner is called an “enactive representation") which means relating time to every day events. For example, now at three o’clock is our tea break. Young children learn about qualitative time (such as day, night, morning and afternoon). We also discussed distance being length and also location (displacement). As young children measure in non-standard units, we say it is ABOUT five sticks long.

Lesson 24:

MRT Station

Problem of the evening: We went down to the MRT station to measure the height Level 1 and Basement 1 of the station. There were 62 steps in total and each step was 14cm high. Thus, the total height is 62x14 = 868cm. The first thought of Group Members and I was…we know that each flight has 16 steps times 14cm and there are three flights with 16 steps and 1 flight with 14 steps. Let’s just multiply each flight and then the steps (14cm) but then that would have given us more steps which we think would be an inaccurate measurement. Math is all around us and I believe that math trails is an excellent way of learning mathematics rather than doing paper-and-pencil tasks. It is interactive, engaging and interesting. I think it is especially good for young children because math trails offer concrete learning experiences.  

Question: Why does a ruler not start with a zero?
Here are some thoughts...Zero is never shown becasue you do not need to measure nothing....Zero never appears on the ruler itself, however, the first point of measure could be considered as a zero. Often times the very left side of a ruler actually starts at a point that could be considered to be less than zero – so that it would take in account for the inaccuracies of the making of a ruler. Especially useful when measuring from corners of a wall for instance, but then you need to measure the start of the ruler to the start of the first long mark, which distance have to be included when using the very left side of the rule to measure.

Another answer found on Yahoo (retrieved on September 3, 2011 from http://answers.yahoo.com/question/index?qid=20090901213239AAxO4mT) Wooden rulers do not start at 0 because the cut of the wood alters the length of the first measure. Many carpenters usually measure from unit 2 to avoid the distortion in caused by the termination.
Lesson 25:

Make a container to fill 15 beans.

We did a practical assessment at the end of this lesson. We created a small box out of paper and the box should fit 15 kidney beans….Wow, we thought our box was tiny but still we could fit in about 100 beans inside…I learnt  that practical test can also be a formal assessment and tests may be conducted with individuals or in groups. 

Another interesting discussion we had was about volume and capacity. The capacity of the swimming pool is…and the volume of water needed for the pool is…However, in the U.S. they do not separate these two terms. “Volume and capacity are both term for measures of the size of three dimensional regions.” (Van De Walle, Karp and Bay-Williams, 2010, p. 380).

Lesson 26:

Make a graph to show how we get to school.

This was an interactive lesson on graphs and data collection. E.g. how we can make it interesting for children by a simple questions such as, “How we came to school” and then each person had to stick a colored square piece of paper…we found out that in our class most of the students came to school by MRT. I learn about bar graphs (the width is equal), histograms (area that counts), pie graph and line graph (used for time and not categorical data).  

In summary, this math course has been an invaluable experience for me. Mathematics is not just doing computations but it is all about thinking. "Reflection using prior knowledge, social interaction and solving problems in a variety of ways, among other strategies, are essential to learning and therefore becoming mathematically proficient (Van De Walle et. al, 2010, p. 25). This is what we have done throughout the math course and it worked! I have learnt so much and it made me realize that math can be fun!

"The essence of mathematics is not to make simple things complicated, but to make complicated things simple." S. Gudder (Retrieved on September 3, 2011 from http://www.quotegarden.com/math.html).

Thank you Dr. Yeap! It has been an interesting and entertaining learning journey!

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.