Session One

I was really nervous about this course because I have never felt confident with my math abilities or lack of them. Also, I am not working as a teacher, therefore I never practice doing math except with my son who is probably better than I am at math. He is currently in Primary 4. However, I acknowledge that math is CRUCIAL and always have encouraged my own children to be confident in math and told them that they can do it. It is all about attitude; I believe you can do anything with support. Dr. Yeap made the lessons very interesting and I experienced how to DO math and that there are a variety of ways to arrive at an answer. In addition, I learnt that it is important to justify and check the results. Does it make sense?

Lesson One:

Name problem: We counted letters in a certain way (in Dr. Yeap’s name) and then tried to figure out which letter is the 99^th.

B A N H A R

1, 2, 3, 4, 5, 6

11, 10, 9, 8, 7

     12,13,14,15,16  and so forth…and then we arrived to 99 which was letter N.

It was interesting to observe the pattern that was created and it made me realize that yes, math is a science. As stated by Van De Walle, Karp and Bay-Williams (2010) mathematics is a science of concepts and processes that have a pattern of regularity and logical order. There is actually a logical way to figure this out and there is a variety of methods that we looked into. We looked at three methods:

1. Look at the pattern (9 in which place?)

2. Multiples of 10 or multiples of another number?

3. Division method (minus first row 6, then divide by 5). However, with this method how do we know which way to go left or right???

Another interesting way could be to look at the difference vertically. What pattern do we see now?

Also, we discussed how numbers are used differently.

Cardinal numbers:

We are counting the number of things. For example, number of cookies, people, etc. It is the quantity of something that we are trying to establish.  We also use cardinal numbers when comparing.

Ordinal numbers:

This is when numbers are used to indicate position in space respective time such as 1^st, 2^nd, 3^rd etc. For example, the 22^nd of August is an ordinal number.

As teachers Dr. Yeap highlighted it is important to pay attention to the way we ask math questions. At times teachers make mistakes in designing ordinal number tasks.

For example:

Who is third in the race? (Ordinal number with respect to time). Please note that the teacher cannot ask this question when only showing the picture because we do not really know who will win.

However, when showing the picture the teacher can ask: Who is third from the finish line? (Ordinal number with respect to space).  

As teachers it is important (especially when assessing students) to know the difference of “rote counting” this is when the child just recites the numbers instead of “rational counting” when the child actually has an understanding of counting and saying a number for each item.

Nominal Numbers:

The number is just the label. For example, bus number 14.

Measurement Numbers:

This is the number used to measure. For example, I am 1.57 meters tall. It is important to use non-standard units before primary school and get the children to compare a variety of items.

Continuous numbers are are limitless and used for measurement– For example, 1cm, 1.5 cm, 1.72 cm etc.

Discrete numbers are used for counting – 1, 2, 3, etc.

Lesson Two:

Sound of numbers: Is it possible to actually hear how many items are in the can? Dr. Yeap showed us the first can and inside he placed two buttons. Then shook it to allow us hear what two buttons inside a can sounds like. Next he had buttons in a bottle. The bottle had definitely more buttons inside judging from the - noise it made when shaking the bottle but how many buttons are inside…is it possible to guess?…well not that easy.

This lesson was about what we can count and cannot count. It depends on the unit…teachers need to ensure that the items to count are perceived as the same type (or set). For example, buttons do not have to be the same color but at least the same type of buttons. You clearly cannot add 2km + 1kg because there are not the same units.

Lesson Three:

5+6+7: We were discussing various ways children may count. It is crucial that children develop the basic number facts and operations. As teachers we have to look at a variety of different strategies children may use and the complexity of the operations. I learnt the value of introducing the ten frames. It helps the child to think in fives and tens. Also, teachers should do ongoing assessments to find out how the child is counting in order to understand which level he/she is at the moment and what other strategies can be introduced to facilitate counting. For example, count on, count back, make and count groups and how numbers relate to each other. Has the child reach commutative property of addition?  This means that the child understand that 5+7=7+5.  The child has addition facts.

This is an example of a ten frame:

The prerequisite to meaningful counting are:

1. The ability to classify and sort.

2. The ability to rote count in sequence.

3. One to one correspondence.

4. The last number that is uttered represents the number in the group (the cardinal number).

Once again, it is important to do ongoing assessments to see if the child appreciates the conservation of numbers (that numbers do not change by moving the items around). Does the child make a ten strategy (beyond addition of ten)? Or does the child not see the value to grouping the numbers and is just counting them? Counting them is the most primitive way of counting It is important to look at what the child does and does not have an understanding of.

 Lesson Four:

Spelling Card Trick:

Here we were to arrange nine playing cards face down.

Tap the first card and say “o”, the next say “n”, the next say “e” (spelling “one”)and the fourth card you turn over and it is the Ace! (or one card)

1

O        N      E    

                                               

Next spell t…w…o…starting with the 5^th card. Then you will show the playing card number two and so on.

1

2

                                       T     W     O     

You have nine cards from Ace to 9 that you have arranged in this particular order face down.

6

5

4

1

9

3

8

2

7

Dr. Yeap asked us to arrange the cards using a variety of methods.

1. Acting it out. However, you had to rely on your memory. Do not use memory to learn because we forget easily. I agree!!!

2. Lining the cards up on the table.

3. Writing the cards on a piece of paper.

This is how it should be arranged from top to bottom:

Lesson Five:

Number Tiles Puzzles:

In this lesson we had to arrange five numbers so that they add to the same total, both vertically and horizontally. For example 1, 2, 3, 4, 5.
                        3
                  5    1    2

                        4

 

With this arrangement the total number is 8.

With 3 in the middle the total number is 9.

With 5 in the middle the total number is 10.

We also tried using number 2, 3, 4, 5, 6.

1 in the middle the total number is 8.

3 in the middle the total number is 9.

5 in the middle the total number is 10.

2 in the middle the total number is 11.

4 in the middle the total number is 12.

6 in the middle the total number is 13.

Observe the pattern! We also recognize that two even numbers can never equal an odd number. Two odd numbers always equal an even number.

We discussed how to observe a vertical pattern versus a horizontal pattern.

Numbers                    Total

1, 2, 3, 4, 5                  8          Look at the pattern 1+2+5=8, then 2+ 3+4 = 9

                                    9

                                    10

_____________________

2, 3, 4, 5, 6                  11        Look at the pattern 2+3+6=11, then 3+4+5=12

                                    12

                                    13

_____________________

3, 4, 5, 6, 7                  14       

                                    15

                                    16

_____________________

How about?

72, 73, 74, 75, 76       

IMPORTANT! A pattern is a sequence that has a term (elements that make up a pattern) and a rule and both have to be stated in order for the student to solve the pattern. It could be a repeating pattern or a growing pattern. The teacher must state the terms and rules for the students in order for them to solve the pattern.

Thank you Dr. Yeap! I wish I had this hands-on math experience when I was younger… it would have made a world of difference to me. Exhausting but fun!!!!!! 

It Is All About Math!

“For math to make sense to young children, it needs to relate to everyday events happening in their lives.” (MacDonald, 2007).  This made me reflect about math in my surroundings; I am working in the admissions office at an International School in Singapore. I meet an average of 4 families a day, that is 4 x 5 days = 20 families per week, 20 x 4.4 weeks = 88 families per month, and 88 x 12 months = 1,056 families per year. Also, when I take families for a tour around the campus, which is 37 acres (converted to hectares it is 14.9), I found out with the use of a pedometer that I take about 18,000 steps in one day, 18,000 x 5 = 90,000 steps in one week....I could go on and on. It made me truly realize that math is all around us and I do not need to go to the gym to get my exercise and boy, do I meet a lot of people! It is important to connect  math to children's reality in order to make it interesting. For example, at our school the younger grades keep track of how many days they have been in school then celebrate the 100th day and money raised for charities is tracked on a thermomenter. All making math a part of their lives.  

Reference:

MacDonald, S. (2007). Math in minutes, easy activities for children ages 4-8.
Beltsville, MD: Gryphon House

Pre-course reading

Curious About Problem Solving + Supportive Environment – Teacher Directed = ?

Chapter 1: Teaching Mathematics in the Era of the NCTM standards

According to Van De Walle, Karp and Bay-Williams (2010) mathematics education has undergone steady changes for the last two decades in both the content of school mathematics and teaching methodology due to various sources including knowledge of research. There are two important factors behind these changes; the professional leadership of the National Council of Teacher of Mathematics (NCTM) and the political pressure due to U.S. students not performing as well compared to international studies (Van De Walle et al., 2010).  I agree with Van De Walle et al. (2010) that although high expectations for students are important, testing alone is not appropriate to improve students’ learning. Teachers need to support students to reach their fullest potential in math by focusing on mathematical thinking and reasoning (Van De Walle et al., 2010).   Also, I believe teachers need to look at some approaches to math assessment that would give a more comprehensive understanding of student accomplishment and growth in math concepts and processes. In addition, teachers need to know the research-based best practices in teaching of mathematics for students and how to use technology in order to support and enhance students' learning in mathematics. 

I was born and raised in Sweden and remember vividly how I dreaded math in school. This could have been my way of coping with academic stress and pressure of performance rather than my math learning skills and potential. I think that in order for children to learn mathematics we have to spark children’s natural curiosities and interests. It should be taught in a supportive and fun way. Even for me, I am able to learn best when I am in a comfortable supportive environment; where I am free to make a mistake knowing I will not be laughed at. It was interesting to read and learn more about the Principles and Standards for School Mathematics released by NCTM. Problem solving has become an important point in the mathematics curriculum (Van De Walle et. al., 2010). I think it is important for students to enjoy math and to discover the many ways there are to arrive at the solution.

The six principles are as follows (Van De Walle et. al., 2010):

• Equity                        

• Curriculum

• Teaching

• Learning

• Assessment

• Technology

According to Van De Walle et. al. (2010) the principles clarify that excellence in mathematics education is much more than listing content objectives.

The Five Content Standards are as follows (Van De Walle et. al., 2010):

• Number and Operations

•  Algebra

•  Geometry

• Measurement

• Data Analysis and Probability.

As stated by Van De Walle et. al. (2010) instead of using different sets of mathematical topics for each grade band, the authors agreed on a common set of five content standards throughout the grades and each content standard has a small set of goals applicable to all grade bands.

The Five Process Standards are as follows (Van De Walle et. al., 2010):

• Problem Solving

• Reasoning and Proof

• Communication

• Connection

• Representation

These five processes direct the methods of doing all mathematics and it is an integral component of all mathematics and teaching (Van De Walle et. al., 2010).

Among the ideas in Mathematics Teaching Today are six shifts in the classroom environment to allow students to develop mathematical understanding (Van De Walle et. al., 2010).  The six shifts are as follows (Van De Walle et. al., 2010):

•  Communities that offer an equal opportunity to all students.

•  A balanced focus on conceptual understanding as well as on procedural fluency.

•  Active student engagement in problem solving, reasoning, communication, making connections, and using multiple representations.

•  Well-equipped learning centers in which technology is used to enhance understanding.

•  Mathematics authority that lies within the power of sound reasoning and mathematical integrity (NCTM as cited in Van De Walle et. al., 2010).

Since the late 1960s the United States has collected data on how students are doing in mathematics through the National Assessment of Educational Progress (NAEP) (Van De Walle et. al., 2010). The data provide important information for policy makers and educators to measure the overall improvement of U.S. students over time and it examines both national and state level trends (Van De Walle et. al., 2010). I think that it is important to have some form of assessment over time in order to know what needs to be changed in a system and what works best for children. However, my concern is that a standardized test alone may not show an accurate picture of the child’s mathematical knowledge and understanding.

The results of an International Mathematics and Science study where 41 nations participated in 1995 and 1996 showed that 11 countries have significantly higher scores than the United States (Singapore, Hong Kong, Japan, Chinese Taipei, Flemish Belgium, Netherlands, Latvia, Lithuania, Russian Federation, England, and Hungary) (Van De Walle et. al., 2010). One may wonder what these students do differently in order to have higher scores or is it that they are just better test takers. I agree with Van De Walle et. al. (2010) that it is important to focus on conceptual understanding and making connections to other mathematics strands rather than traditional emphasis on mathematical procedures. “We can predict that there will be work that requires interpreting complex data, designing algorithms to make predictions, and using the ability to approach new problems in a variety of ways.” (Van De Walle et. al., 2010).  Thus, it is my belief that teachers need to ensure that they support, guide and inspire students and prepare them for the demands of a changing technological society where new careers and knowledge are constantly being created. 

Chapter 2: Exploring What It Means to Know and Do Mathematics

What does it means to do mathematics? Doing mathematics means not just solving a problem but discovering the many ways there are to arrive at the solutions. I strongly believe that teachers need to challenge students forward along a path to solve a problem, and then go backward to find alternate methods to the solution. Let math become an adventure! As stated by Van De Walle et. al. (2010) mathematics is a science of concepts and processes that have a pattern of regularity and logical order. 

Also, it is important that students have a conducive and productive classroom environment where students are respecting each other’s ideas. Students should feel safe and supported while learning through trial and error to understand mathematical concepts. 

According to Van De Walle et. al. (2010) in the real world of problem solving there are no teachers with answers and no answer books – doing mathematics is about using justification as a means of determining if an answer is correct. Hence, a supportive environment is crucial where students are allowed to try a variety of ways without fear of making mistakes.

What does it mean to learn mathematics? As stated by Van De Walle et. al. (2010) we find the answers in current theory and research on how people learn, for example, constructivists theory – learners are not blank slates but creators of their own learning. People construct their own knowledge based on their prior knowledge (Van De Walle et. al., 2010). This made me reflect on the importance of differentiated learning in the classroom and to ensure each student is appropriately challenged. What would be the next level for this student and how can I as a teacher make it interesting. Equally important for learning is the social interactions in the classroom, students can reach the next level understanding with support (zone of proximal development, ZPD) and the culture within and beyond the classroom (Van De Walle et. al., 2010).  Students should be given opportunities to talk, reflect and be encouraged to think of multiple approaches when doing mathematics (Van De Walle et. al., 2010). I agree that we should honor diversity and each student’s ideas should be valued and included in classrooms discussion of the mathematics (Van De Walle et. al., 2010).

What does it mean to understand mathematics? Each child is unique and brings to the classroom different prior knowledge and understanding of math. According to Van De Walle et. al., (2010) we can think about understanding such as it exists along a continuum from a relational understanding – knowing what to do and why – to an instrumental understanding – doing without understanding. Also, it is important to know the difference between conceptual and procedural understanding. Conceptual understanding is knowledge about the relationships of a topic and procedural understanding is knowledge of the rules and procedures used in carrying out mathematical processes (Van De Walle et. al., 2010). Students need to have the conceptual understanding if math is going to be fun (Van De Walle et. al., 2010). I agree! However, while conceptual and procedural understanding is essential to being mathematically proficient, they are not sufficient, we have to take into consideration the five strands which are interrelated and interwoven with each other (NRC as cited in Van De Walle et. al., 2010):

• Conceptual understanding

• Procedural fluency

• Strategic competence

• Adaptive reasoning

• Productive disposition

Teachers have to be aware of how the student understands it and what ideas he or she connects it with (Van De Walle et. al., 2010). It is important to observe and assess the student to figure out what strategies he or she is using, which will then lead us to see how much the student really comprehend. 

The teacher needs to focus on supporting the students in their understanding so that they build more connections and are more likely to connect new ideas to the existing conceptual webs they have (Van De Walle et. al., 2010).  Thus, students have less to remember, increased retention and recall, enhanced problem-solving abilities and improved disposition towards mathematics (Van De Walle et. al., 2010).

If there are more ways for children to think about an idea the better chance they will develop a relational understanding (Van De Walle et. al., 2010).  To help students understand, teachers may use models and/or math manipulatives. I agree that teachers need to constantly think about a variety of creative strategies to reinforce student's understanding. It is when the student has grasped the concept that defines success!

Curious About Problem Solving + Supportive Environment – Teacher Directed = Students Who Love Math!

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