Number of Green Apples In One Tray

The picture has a 5,4,5,4 alternating row pattern. Therefore, one end (long side as seen) of the tray must be 5 while the other end (long side as seen) is 4. In this way, when the trays are stacked up in facing order, one on top of the other, the pattern of 5,4,5,4 will appear. In order to have one long end 5 and the other 4, the tray should take 4 rows of apples, thus giving 5+4+5+4 = 18. I rule out adding a further 5+4 as it will give a total of 27. 18 seems more likely as it is one and a half dozens. A common unit of measurement for quantity of groceries.

The content areas required include patterning, addition, shapes, area and estimation. To help children arrive at a solution, I will try the following:

1) divide the children to form groups of 3

2) provide each group with the picture

3) tell them that their task is to guess and find out how many apples are 

    there on one tray.

4) tell them to write down and/or draw any pattern your group can see.

5) materials like cardboard, crashed paper balls, writing materials, blu-

    tack, tapes and counters will be available should the groups want to

    explore the problem using concrete objects.

6) while the groups are working out the solution, I will circulate to observe and

     prompt appropriately.

Reflections

Session 1 : 22 August 2011

Doing Math but "not doing Math"

Thanks to Dr Yeap, I now have the opportunity to enjoy the process of maths rather than my schooling days (I am now 54 yrs old) of hating math because it is always about who has the right answers first. Those days, I never heard of pattern, making connections.... All I remembered was "just learn the steps by heart"; "recite your multiplcation tables"; "don't ask me why like that. You will get the right answers. Just follow steps. "
Lesson 1: Name Problem
Lesson 2: Sound of Numbers
Lesson 3: 5+6+7
Lesson 4: Arrange the cards
Lesson 5: Number Titles Puzzles

I enjoyed the above hands-on lessons very much and could feel my brain nuerons zapping to make connections, searching for patterns, count, adding, taking away and so on.

My key take-away is how Dr Yeap conducted the lessons, his materials used, his prompts and others. He uses very accurate instructions, prompts and queations which I as a teacher will borrow and use. Some of these are:
> "write down and make a list and then models how to do it".
> "look for a pattern in the ones place".
> "are your sure? Explain"
> "do you think so? Why? How come?"
> "You are correct. Tell us how you know".

My mindset for maths has now taken a new perspective. I now do not see Math as Math, but "Mathematics is an excellent vehicle for the development and improvement of  intellectual competence ... ". (MOE)

As a teacher in early childhood education, my philosophy has always been to give the children "opportunity" to explore and construct to learn.
Process is more important than the product.

I will definitely share the following information with my colleagues.

How Do You Use Numbers? (The Uses of Nymbers)

1) Rote Counting

2) Rational Counting

- cardical number (number of items counted)

- ordinal number (position in space and position in time)

- nominal number (number used to identify. e.g. bus number)

- measurement number (continuous qualtity and infinite quabtity)
- proportion ( from Session 5)

Know That The following Pre-requisite Skills Are Needed For  Counting 
- be able to classify

- be able to do rote counting

- understand and appreciate that the last number uttered is the number of item 

  being counted.

- have one-to-one correspondence

Other useful termologies/teacher tips

- number fact

- count all

- count on

- cummutative property of addition (5+7 is the same is 7+5)

- skip counting (2s. 5s, 10s ...)

- conservation of numbers (move them around but quantity remains)

- Initially, always use identical objects for counting, then move on to

  introduce change ot colour. You can only count things that are in the same 

  set. You cannot add oranges to apples, unless you change the unit to fruits.

- ten-frame cards which are cheap to produce, are useful for  training children 

  to see pattern and develop number sense.

Wow! I am more confident teaching children Maths already and am eager to use these activities.

Session 2 : 23 August 2011

"Subitize" was the word of the day. According to Dr Yeap, if you look at the thing and know the numbers without the need to count them, you can subitize.

I really enjoyed the "Holiday Games" . The "magic" with the die was really fun, but most importantly, the children would like "myself " in the class, observed and discovered the patterns and the numerical logic behind it.

Some Big Ideas :

• Thinking skills
• Problem solving skills
• Meta Cognition - manage information
• Looking for patterns
• Communication - justification / representation / reasoning
• Visualization
• Number sense
• Generalization

Session 3 : 24 August 2011

This evening, I felt creative and had sense of visualisation. I totally enjoyed it and would like my children to experience this achievement. Using 5 cubes, many different "toys" were constructed.

This task teaches conservation of number. No matter how they are arranged, the number of cubes remains the same.

Peggy shared with us "Lesson Study", "Mathematical Investigation" and "Recent PD Trends". These were comforting to me - our future generations should be in good hands. Yes. Learning is a continuum.

Session 4 : 25 August 2011

"Word Problems Disected" - Addition and Subtraction (Lesson 15)
There are three types of word problems, namely:

• Change
• Part/Whole
• Compare

All word problems have a situation, an initial amount, a change amount and a final quantity. This simple explanation meant a lot to me.

Children may be overwhelmed by the sentences in the problem. As a preschool teacher, I will address their fear and spur them to explore. Model them to do  one sentence at a time, using appropriate concrete manipulatives .

Example of Change Situation:

Tommy has 31 marbles. (concrete quantities)

He gave Peter 19 marbles.

How many marbles has he left?

Example of Change Situation:

Rashda had $37 in her bank. (continuous quantities)

She spent $19 on books.

How much money has she left?

Example of Part/Whole Situation:

There are 37 children in a class.

19 are boys. How manay girls are they?

Example of Comparison Situation:

I have 37 pens. I have 10 pens more than you.

How many do you have?

Lesson 16 : Equal Parts

I had to re-learn. Thanks to Dr. Yeap I will now avoid using the incorrect terms.

Key learning points:

• When you can make equal parts, you can name them
• The name is "one half " "one third" "one fourth".... (Not one upon four...)
• Fraction as part of a whole
• Fraction as part of a set
• Being equal doesn't need to be of the same shape
• Being equal need not be identical parts

Session 5 : 26 August 2011

It was Visualization Galore"
Lessons 17, 18, 19 & 20 had me on the edge of my chair visualizing.
Division of fractions and looking at area.

• How many fourths can you make from a piece of square paper?
• In one how many one fourths are there?
• One unit can cut into equal units. Once they are equal, we can rename it.

• Two methods of dividing fractions (i) model method and (ii) change to whole number method
• 5 Transformation of shape # reflect # rotate # translate (no change to the shape) #stretch #shear (will change the shape)

• Polygon is a closed figure  

Lesson 20

Pick's Theorem on calculating the area of a figure was discussed. His theory was interesting, but what another possible idea suggested in class was even more amazing.

Lesson 21

Graph Making (also covered during last session)
Revision was given to my priior knowledge that there are 5 common types of graph namely, Picture Graph, Bar Graph, Line Graph and Pie Graph.
What I did not know:

• Width of Bar Graphs must be equal (best used for category data)
• Histogram - the area counts and must be arranged in order. (best for continuous data)
• Line Graph is best used for plotting time concerns
• In pie graph the area of each sector makes the calculation. If comparing data, diameter of pie grpah must reflect the difference of toal being compared. 

Session 6 : 30 August 2011

This last session primary focused on Assessment. Dr. Yeap through his in-class task, showed us how assessment need not be paper and pencil test alone. Oral test/interviews as well as observation during classroom time are also assessment opportunities. One candid example was "Telling time". Why make the chidren draw hour hand, minutes hand, when one can simply ask them to TELL you the time.

Task at the MRT

A group task was given where we had to calculate the height between the basement level and the street level.
There were four flights of steps as seen in drawing below. 3 were of 16 steps each and one of 14 steps. The height of each step is 15cm.  Therefore 16 x 3 + 14 = 62 steps . 15 x 62 = 930cm.
Therefore, the height between the basement level and the street level is 930cm.

"Thank you Dr. Yeap". What I feared of math has been erased and I am finding and seeing more math in the world around me. Hope to have you "solve my problems" in the "Developing problem solving skills" module.



Bsc 05. EDU 330 Elementary Mathematics

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

Diagram 1 below summarized the various principles and standards that govern the teaching of mathematics today.



Diagram 1



NCTM Standards

Excellence in mathematics education involves more than content objectives. The National Council of Teachers of Mathematics' (NCTM) release and updates of Principles and Standards for School Mathematics is being used as the guide to reform mathematics education throughout the world. The six principles are: Equity, Curriculum, Teaching, Learning, Assessment and Technology. These standards should be intertwined with the mathematics programme.

5 Content Standards

There are five content standards. These are Number and Operations, Algebra, Geometry, Measurement as well as Data Analysis and Proficiency. Four grade bands have also been created and each has its set goals and expectations. The four grade-bands are: pre-K-2, 3-5, 6-8 and 9-12. E.g. For the pre-K-2 grade band, under the content of number and operations, they must count with understanding and recognise "how many" as compared to grade-band 3-5 where they must know place values.

5 Process Standards

Following the five content standards, there are five process standards which are integral components of mathematical learning and teaching. These are Problem Solving, Reasoning and Proof, Communication, Connection and Representation. To teach in accordance with these process standards will enable us to meet the learning objectives set in the grade-bands of the content standards. Problem Solving: solve the problem rather than using a method to get the answer. Reasoning and Proof: use logical thinking to decide if the answer makes sense. Communication: be able to describe and explain the idea. Connection: know that part and part makes a whole, and that fractions are also part of a whole, like in percentages.  Representation: the use of charts, symbols, graphs, manipulatives and diagrams to express mathematical ideas.

7 Teaching Standards

These teaching standards enables the teachers to resource and evolve the classroom environment, teaching tools and strategies to support the principles and standards.
The teaching standards are Knowledge of Mathematics and General Pedagogy, Knowledge of Student Mathematical Learning, Worthwhile Mathematical Tasks, Learning Environment, Discourse, Reflection on Student Learning and Reflection on Teaching Practice.

Becoming a Teacher of Mathematics
As summarized in the diagram, a teacher of mathematics must enjoy challenges. One must have profound, felxible and adaptive knowledge of mathematics content. The teacher must model persistence to work out the problem with investigations. The positive attitude of the teacher towards mathematics will influence the student to enjoy mathematics. Always be ready to change and embrace new invented strategies anytime. Take on a reflective disposition. Contine to learn content and methodology of teaching mathematics.



Diagram 2

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

The elaborations presented in the book brings me to view mathematics in a different light. To me, mathematics has always been finding anwers to a mathematical problem and that many of the disciplines can be used in real live, especially in counting. According to J.A.Van De Walle, K.S. Karp & J.M.B.Willaims (2010), "Doing mathematics means generating strategies for solving problems, applying those approaches, seeing if they lead to solutions, and checking to see if your answers make sense..... mathematics is a science of concepts and processes that have a pattern of regularity and logical order....Finding and exploring this regularity or order, and then making sense of it, is what doing mathematics is all about."

Asking simple questions like "What is the pattern ? to 6+7 is the same as 5+8 and 4+9" easily throws up concept that -1 +1 to each set of numbers.

I agree to the statement that "Studying relationships .... helps students understand what they are doing, therefore increasing their accuracy and retention."  

The Language of Doing Mathematics

I tried using some of the verbs listed and could immediately see the children engaging themselves thinking rather than working out the answers. I will definitely work towards using these verbs in my mathematics programme.

explore         justify               construct             develop
investigate    represent           verify                  describe
conjecture    formulate           explain                use
solve            discover             predict

The Setting for Doing Mathematics

Create the spirit of inquiry, trust and expectations. The productive classroom culture for mathematics should include:

• Ideas expressed have the potential to contribute to learning and warrant respect and response.
• Respect the need to understand their own method and recognize that there are other methods to the solutions.
• Mistakes give opportunity to examine errors in reasoning, thereby raising level of analysis.

Using Technology

Introducing calculators must be done together with how mathematics operations work, so that students do not use it simply for quick answers.

Experiences to maximize learning opportunities

• Constructivist Theory: Knowledge cannot be poured into a learner. The student has to construct knowledge by reflective thought and active thinking in order to make connections and contruct knowledge.
•  Sociocultural Theory: Learning is dependant on the student's social interactions with other learners within and beyond the classroom.

This knowledge of learning theories will influence the teaching strategies of teachers. I never thought of applying these knowledge into my teaching strategies for mathematics until now, and will surely do so. Some of these applications as elaborated in the book (Elementry & Middle School Mathematics Teaching Developmentally) will include:

# Provide Opportunities to Talk about Mathematics: Encourage students to talk about their strategies. Hear them reflect and interact with peers.
# Build Opportunities for Reflective Thought: Use problem-solving approach and have the students explain and justify their solutions.
# Encourage Multiple Approaches:  Take simple counting off a table of dots arrange in 5s. Once the solution is revealed, have children explain their strategy. Some will be counting in 1s, others in 2s while yet others by 5s. Student's reflective thinking on these strategies will already construct new knowledge.
# Treat Errors as Opportunites for Learning: Asking the student to regroup and use another strategy will help the student realize the misapplication of prior knowledge and thus accommodate a new learning.
# Scaffold New Content: Challenging the student to the next level of learning by providing structure, content and assistance.
# Honor Diversity: Value the uniqueness of each individual. Establish a classroom culture where ideas are valued and respected.

Understanding Mathematics Defined

Understanding is a measure of the quality and quantity of connections that a new idea has with the existing ideas. The greater the number of connections to a network of ideas, the better the understanding. Thus the two ends of the continuum of understanding: Relational Understanding (high end) and Instrumental Understanding (low end).

Mathematics Proficiency

• Conceptual understanding and procedural understanding is the basic knowledge about relationships or foundational ideas of a topic.

• Five Strands of Mathematical Proficiency:

        1) conceptual understanding - comprehension of mathematical concepts, 
            operations and relations
        2) procedural fluency - flexible, accurate, efficient and
            appropriate execution of procedures
        3) strategic competence - ability to formulate, represent, and  
            solve mathematics problems
        4) adaptive reasoning - able to think logically, reflect, explain and justify
        5) productive disposition - have the habit to see mathematics
            as sensible, useful, and worthwhile, coupled with a belief   
            in diligence and one's own efficacy.

     Add all these up and you get the mathematical intertwined strands of 
     proficiency as illustrated in Diagram 3 below.





Five Strands of Mathematical Proficiency

Diagram 3



Implications for Teaching Mathematics

Instead of asking "Does she know it?" I will replace with "How does she understand it? What ideas does she connect it?" This way I will be able to see the different ideas and understanding of each student. Example, for additions, instead of counting counters, or using traditional algorithm of lining up and adding the ones and then the tens, with the understanding of place-value concept, I will also show them the invented approach. (add 10s to the root number and count foward the ones).

Benefits of a Relational Understanding

Effective learning of new concepts and procedures, less to remember, increased retention and recall, enhanced problem-solving abilities and improved attitudes and beliefs are the reasons why relational understanding is so important and beneficial.

Models and Manipulatives

Educators are promoting the use of concrete and hands-on methods to teach mathematics. This reading impacts upon me the effective use of manipulatives to teach concept by letting the student think, reflect, show and explain, to solve the problem, and not asking them to mimic the teacher's procedure to reach an answer. Manipulatives could include countable objects, rods, strips and squares, sticks, dot grid, integers as well as calculators.

Conclusion - My Calling

Plan my programme to focus on creating opportunities for students to develop their own network of blue dots. Include a comprehensive treatment of mathematics. Design my instructions to elicit prior knowledge, reflect the social and cultural background, challenge them to think critically and creatively. Continue to use manipulative models and introduce use of technology.