Presenters:

Dr. Umesh Nagarkatte

Lavoizier St. Jean

Herbert Odunukwe

Kay Lashley

11/19/2011

Department of Mathematics

Medgar Evers College, CUNY

1638 Bedford Ave., Brooklyn, NY 11225

Acknowledgements -

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1. U. S. Department of Education – Minority Science Engineering

Improvement Program (MSEIP) two grants – Institutional and

Cooperative, 2010-2013

2. The Singapore Model Method for Learning Mathematics - Ministry of

Education, 2009

3. What the United States Can Learn From Singapore’s World-Class

Mathematics System (and what Singapore can learn from the United

States): An Exploratory Study - American Institutes for Research®

prepared for: U.S. Department of Education Policy and Program

Studies Service (PPSS), 2005

(Note: 2 contains the recommendations made in this monograph)

Background - Singapore Mathematics and Science consistently ranks first

in the world in the Trends in International Mathematics and Science (TIMSS) studies. Currently it is second, and U.S. 26th.

Medgar Evers College received two MSEIP grants – one Institutional and the other Cooperative with QCC for 2010-2013. The main activity is to implement Theory of Constraints (TOC) to increase retention and graduation rates. We are implementing TOC for the last nine years with the support of five federal grants.

One of the numerous activities this time is to adapt the Singapore Model Method (a TOC implementation) to College Level.

Starting with Basic Skills, we are revamping our math courses. Will involve the Singapore model and TOC thinking tools. (handouts)

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Peferred Features of the Singapore Mathematics

System

Singapore U.S.

Framework Logical, National, develops topics in-depth, alternate

No national, NCTM lacks logical structure, no alternate

Textbooks Thin, Fewer topics. Less words. Build deep understanding through multi-step problems, Concrete to visual to abstract. Illustrations demonstrating how abstract concepts can be used for different perspectives.

Heavy, Too many topics. Limited to definitions and formulas, developing students’ mechanical ability to apply concepts. Real- world illustrations indicate relevancy – but do not show how to apply concepts to solve those problems.

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Peferred Features of the Singapore Mathematics

System (contd.) Singapore U.S.

Assessments Questions on high-stake grade 6 Primary School leaving test more challenging than

Questions on grade 8 National Assessment of Education Progress or state assessments in 8 states

School performance - Value added contribution – growth of student outcomes.

Average Yearly Progress measure in NCLB does not have any such measure.

Teachers Required to pass stringent examination to enter teacher ed. program. Students are paid teachers’ salary. 100 hours annual prof. development

Lowest SAT scores. Take fewer math courses than average students. Test items on PRAXIS I and II lower than Singapore 6th grade test items.

Singapore Mathematics The Singapore Model Method for learning mathematics

was developed by a team of curriculum specialists in the 80’s by Singapore Ministry of Education.

The Model Method was an innovative way in teaching and

learning of mathematics; specifically design to reduce difficulties that students are faced solving word problems.

The Model Method entails incorporating pictorial models

to represent mathematical quantities and relationships.

The Model Method has undergone changes over the years since the 80’s to meet today’s challenges and has been integrated with algebra methods from primary to early college algebra.

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Mathematics Framework The Singapore Model method is deeply rooted in an

underlying Mathematics principles of effective problem solving methods that is represented in a pentagonal framework.

Mathematical Problem Solving which is the core of the pentagonal framework is central to mathematics learning. There are five interrelated components associated with the framework and they are listed as:

Concepts

Skills

Attitudes

Metacognition

Processes 11/19/2011

Singapore Mathematics Framework

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Concepts and Skills Mathematical concepts

• Covers numerical, algebraic, geometrical, statistical, probabilistic and analytical concepts.

• Students will develop and explore deep mathematic ideas by use of manipulatives.

Mathematics Skills

• Covers procedural skills for numerical calculation, algebraic manipulation, spatial visualization, data analysis, measurement, use of mathematical tools and estimation.

• Skills proficiency and procedural skills are important but should not be over emphasized over mathematical principles.

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Processes Mathematics Process:-

Reasoning:-Ability to analyze math situation and construct logical arguments.

Communication:-Concise use of mathematical language to express

math ideas and argument.

Connections:-Ability to see and make linkages among math ideas and

other subjects.

Thinking Skills: Skills that can be used in a thinking process such as

comparing, analyzing parts, sequencing, classifying etc.

Heuristics:-Strategies student use to approach problem where there are no

obvious solution.

Applications & modeling:-Process of formulating and improving a

mathematical model to represent and solve real world problem

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Metacognition Metacognition, is defined here as awareness of or ability to

control one’s thinking process. The following strategies may be use to develop metacognitive awareness of students

and enrich their metacognitive experience:- Expose student to problem–solving skills, thinking skills and heuristics in

solving problems.

Guide students to use appropriate strategies and methods in solving problems.

Provide students with planning and evaluation before and after solving a problem.

Encourage students to seek alternative ways of solving the same problem

Create conducive environment for students to discuss appropriateness and reasonableness of answers.

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Attitudes Attitudes here refers to the affective aspects of

mathematics learning such as:

Beliefs about mathematics and its usefulness

Interest and enjoyment in learning mathematics.

Appreciation of the beauty and power of mathematics

Building confidence in mathematics

Perseverance in solving a problem

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Comparison to the US- simplicity in wording

USA-Mathematics Proficiency

Singapore-Mathematics Framework

Conceptual Understanding

Procedural Fluency

Strategic Competence

Adaptive Reasoning

Productive Disposition

Concepts

Skills

Processes

Metacognition

Attitudes

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Model Method and Concepts of the Four Operations

The Singapore primary mathematics curriculum places great emphasis on the quantitative aspect of how students learn the concepts of numbers and the four basic operations: addition, subtraction, multiplication and division. The key feature of the model method is illustrated in rectangular bars which are pictorial representations of the models use of helping students learn the mechanics involved in solving mathematics word problems.

This concrete-pictorial-abstract approach is depicted by the part-whole and comparison models.

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Part-Whole Model Addition and Subtraction The part-whole model also known as the part-part-

whole model is a quantitative relationship between a whole and two parts.

The pictorial model shows that the whole is the sum of two parts. That is: part + part = whole.

Furthermore, to find a part, the other part can be subtracted from the whole. That is: whole – part = part

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part part

whole

Comparison Model Addition and Subtraction The comparison model is a quantitative relationship

among three quantities: larger quantity, smaller quantity and the difference. That is,

Larger quantity- smaller quantity = difference

Students can also find one quantity:

Smaller quantity + difference = larger quantity

Larger quantity – difference = smaller quantity

Larger quantity

Smaller quantity difference

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Part-Whole Model Multiplication and Division This model displays a whole divided into a number of

equal parts. That is, a whole, one part and the number of parts.

One part x number of parts = whole

Whole ÷ number of parts = one part

Whole ÷ one part = number of parts

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whole

part

Comparison Model Multiplication and Division Two quantities are compared such that one quantity is a

multiple of the other. Moreover, larger quantity, smaller quantity and the multiple. The model simply displays:

Larger quantity ÷ smaller quantity = multiple

Smaller quantity x multiple = larger quantity

Larger quantity ÷ multiple = smaller quantity

Larger quantity

Smaller quantity 11/19/2011

Example 1. Part-whole- Addition and Subtraction

Malik and Miguel brought tickets to the school dance. Malik brought 30 tickets while Miguel brought 20 tickets, how many tickets did they bought altogether?

Malik and Miguel brought 50 tickets for the school dance. If Malik brought 30 tickets, how many did Miguel brought?

Malik Miguel

?

30 20

Malik Miguel

50

30 ?

30 + 20 = 50

50 tickets were brought

for the dance.

50 – 30 = 20

Miguel brought 20 tickets 11/19/2011

Example 2. Comparison Model-Addition and Subtraction Kael has 40 marbles. Her sister Kate has 10 less, how many marbles

does Kate have?

Kael has 40 marbles. Her sister Kate has 30, how much less Kate has than Kael?

Kael-40

Kate- ?

10

Kael-40

Kate-30

?

40 – 10 = 30

Kate has 30 marbles

40 – 30 = 10

Kate has 10 marbles less

than Kael 11/19/2011

Example 3. Part-whole- Multiplication and Division Andrew brought 6 hot dogs at $2 each. How much did the hot

dogs cost?

Andrew brought 6 hot dogs for $ 12. How much did each cost?

?

2

12

?

6 x 2 = 12

The cost of the hot dogs was $12

12 ÷ 6 = 2

Andrew paid $2 for each hot dog.

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Example 4. Comparison Model-Multiplication and Division Andy has 4 birds. He has 3 times as many fish as birds. How many

fish does Andy have?

Andy has 12 fish. He has 3 times as many fish as birds. How many birds does Andy have?

?

4

12

?

Birds

Fish

Birds

Fish

3 x 4 = 12

Andy has 12 fish.

12 ÷ 3 = 4

Andy has 4 birds. 11/19/2011

Model Method and Concepts pf Fraction, Ratios and Percentages Part-Whole Method- Fraction

Comparison Method- Fraction

Part-Whole Method- Ratio

Comparison Method-Ratio

Part-Whole Method-Percentage

Comparison Method-Percentage

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Example Problem #1-Fraction Part-Whole Method There are 125 students in a class. 2/5 of them are girls. How many girls

are in the class?

?

125

The fraction 2/5 means 2 units out of 5 units. To find the value of girls (2 units), students find 1 unit:

5 units = 125

1 unit = 125/5= 25

2 units = 2 x 25 = 50

There are 50 girls in the class.

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Example # 2- Fraction Comparison Method There are ¾ as many Republicans as Democrats at the

White House Ball. If there are 80 Democrats, how many Republicans are there in attendance?

?

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Republicans

Democrats

80

4 units = 80

1 unit= 80/4=20

3 units= 20 x 3 = 60

There are 60 Republicans

Example #1-Ratio Comparison Method A recipe requires 3 ingredients A, B, C in the volume

ratio 2:3:4. If 6 pints of ingredient B are required, how many pints of ingredients A and C are required?

A

B

C

3 units = 6 pints

1 unit= 6/3 = 2 pints

2 units =2 x2 = 4 pints

4 units = 4 x 2 =8 pints

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Example # 2-Ratio Comparison Method If 24 ounces of a certain liquid fills ¼ of a pail, how

many ounces of the same liquid will fill 1/3 of the pail?

96

1/4 unit = 24 ounces

1 unit =24/(1/4) = 96 ounces

1/3 unit = 96 x 1/3 = 32 ounces

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24

Example #3-Ratio Part-Whole Method A settlement of $600 was to be divided in the ratio 1:2:3

between 3 brothers. How much money did the third brother received?

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600 ?

6 units = $600

1 unit = 600/6 =$100

3 units = 100 x 3 = $300

Example #1- Percentages Part -Whole Method 15 is 2.5% of what number?

15

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2.5

?

2.5 units = 15

1 unit = 15/2.5 = 6

100 units = 100 x 6 = 600

Example #2- Percentages Part- Whole Method If 20% of a number is 14, what is 80% of that number?

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20

80

?

20 units =14

1 unit = 14/20 = 0.7

100 units = 100 x 0.7 =70

? 70

100 units = 70

1 unit = 70/100 =0.7

80 units = 0.7 x 80 = 56

14

Example # 3-Percentages Comparison Method 40% 0f 30 equal 20% of what number?

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40

30

100 units = 30

1 unit = 30/100=0.3

40 units = 40 x 0.3 =12

?

20

12 ?

20 units = 12

1 unit =12/20 = 0.6

100 units = 0.6 x 100 = 60

Model Method & Problem Solving Model method is a synthetic-analytic process that

can be used to express and solve structurally complex word problems:-This Method entails drawing a pictorial model and finding the so called “1-unit” from unitary method. Students will learn how to identify known and unknown quantities and relationships among each - synthetic approach, then logical steps are developed for the solution of the problem – analytic approach.

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Example Problem #1: 3 out of 7 students in an Algebra course pass a class quiz. How many students did not pass in a class of 91 students?

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Solution: Here a pictorial bar model is used to represent the whole, (total number of students in the class). The whole is 7 units of which 3 units represents number of students passing the course. Therefore 4/7 of the whole students did not pass the course. 7 units = 91 students 1 unit = 91/7 = 13 students 4 units will be = 13 x 4 = 52 students failed the quiz.

91

1unit

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Example problem #2. A store gives 10% discount to all students off the original cost of any item. If an additional 15% is taken off the discounted price, how much is the original price if a student purchases an item for $306?

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Solution: 1st 10% store discount represent 90% of original price .(100% – 10% = 90%) 2nd Discount price represent 85% of the first discount price of 90% = 0.85 x 0.90 = 0.765 Therefore the final price represent 76.5% of the original price. We need to find the unit price per percent. 1% represent $306/.765 = $4.00 Original price (100%) = 4 x 100 = $400.00 Here a pictorial bar model is used to represent the whole, (original price of the item). The whole is 100%.

Original Price?

1 unit

76.5% 100%

$306

11/19/2011 90% $4.00

Model Method and Algebra

Singapore Model Method also offers alternative approach to solving algebra word problems. Students mostly encounter difficulty manipulating algebraic skills necessary in solving word problems; therefore the need to develop new strategies is more urgent. Here we will explain how we use the Model method in solving problems in Algebra courses at Medgar Evers College, CUNY. We will explore the use of the Unitary model and the Comparison model to:

Process given information in the problem.

Identify known and unknown quantities

Understand relationships between those quantities.

Increase students reasoning and thinking skills.

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Part-Whole Model in Problem solving

One major objective adopting Singapore method is to enable students to develop problem solving strategies. The Singapore model method uses construction of pictorial model to solve part-whole and comparison type questions.

Part-Whole Model

The part-whole model shows relationship between the whole, f and its component parts, m and n.

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m n

f

The equation will be: f = m + n 11/19/2011

Part-Whole Model in Problem solving

In another Part-whole model, the whole is divided into a number of equal parts. The pictorial model is shown as:

The equation of this relationship is given below as: L = 5a (b) The Comparison Model Comparison model shows relationship between two

quantities when they are compared.

L

a a a a a

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Comparison Model cont’d The pictorial model is shown below:

The pictorial model shows the quantity y is more than the quantity x and their difference is d.

That is: d = y – x And P = a + b Still in Comparison model, we may express one quantity as

a multiple of the other

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x

a

b

P

y

d

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Comparison Model cont’d Example of pictorial model is shown below:

The equation relating this pictorial model is given by:

y = 4x

We use these models to solve lots of problems in pre-algebra and algebra courses.

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y

x

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Example problem #1. A couch cost 5 times as much as a rocking chair. Altogether they cost $702. How much will the rocking chair cost

The algebraic method involves using x to represent the rocking chair. The pictorial representation shows the rocking chair with respect to the couch (y) which is five times (5x) more in price.

Using Comparison model, we can form an equation.

y =5x and 702 = x + y

Therefore, x + 5x = 702

x =$117

x

702

y

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Example Problem. #2. An inspector found 30 defective bolts during an inspection, it is 0.25% of the total number of bolts inspected, how many bolts were inspected.

Let total number of bolts inspected represent 100%. However, we know that 30 inspected bolts were defective representing 0.25% of defective bolts. We need to find how many bolts were inspected.

Here a bar is drawn to represent

the data given.

0.25% represent 30 bolts

100% will represent x bolts.

So we can find x

Therefore x = 100/0.25 x 30

x=120,000 bolts

x

100% 0.25% 0%

30

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Sample Question #3 High school graduating class is made up of 674 students. There are 298 more boys than girls. How many girls are in the class?

Algebraic method using a variable, x to represent number of girls. As there are 298 more boys than girls, The number of boys will be x + 298. Total number of boys and girls will be

x + (x + 298)

Since the total number of students graduating is 674, we can write the equation; x + (x + 298) = 674. The solution of the equation will be:

x = 188

We can draw a Comparison model to represent the situation and using algebra method,

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x+ 298

x 298

boys

girls 674

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Sample Problem #3 Variation 2

Total number of boys and girls is 674. The number of boys can

be expressed as 674 – x.

From the above pictorial model, students will see the

difference between (674 – x) and x is 298.

Therefore, the equation representing this situation is

(674 – x) - x = 298

The solution of the equation is

x =188.

x+298

x 674

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Example Problem #4

Question: Smith, Jones, and Miller have decided to split profits from their business, so that Smith gets three times as much as Jones, and Jones get twenty less than twice smith. How much will each get if they are to share a profit of $2016?

Solution:

We use algebraic variable, x, to represent the unknown. Let x represent Jones share of profit. Then Smith gets 3x and Miller will gets 2x - 20 Using Comparison model, we can form an equation:

3x + x + (2x -20) = 2016

The solution of the equation is

x = 336.

We can draw a Comparison model

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x 2016

2x - 20 20

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Conclusion

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• The Singapore Mathematics Model Framework focuses on using an alternative approach to reduce inherent problems of algebraic manipulation skills used in interpreting application word problems in arithmetic and algebra. The Model Method has been demonstrated in this presentation to show how an alternative approach can be used in solving word problem in college pre-algebra and algebra. In these examples we can see that students can apply basic mathematics concepts and skills in solving application word problems and developing mathematical thinking. The model method has important feature of Singapore primary mathematics curriculum. • Problem representation involves data (quantities and quantitative relationship) and question. Students understanding of the problem situation, relationship between the known and unknowns enable them to solve the problem. This is schematically shown in the next slide.

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Conclusion cont’d

In the algebraic method student formulate an algebraic equation to represent the problem situation and to connect the known and unknown quantities. Then solve and answer the question.

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Word problem

solution Problem

completion

Algebraic Equation

solution Word

problem

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Conclusion cont’d Using the model method and algebraic method we were

able to construct a pictorial model to help formulate an algebraic equation to solve the problem. This can be shown in the sketch shown below.

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Pictorial model

Algebra Equation

solution Word problem

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Conclusion cont’d

We consciously made use of schemas such as the Part-whole and Comparison models which are building blocks for mental structures and cognitive processes. In addition, we interpreted the learning of algebraic method with model method building schemas as well.

The Model Method recognizes metacognition which was earlier defined as self regulation of learning; enhancing students problem solving abilities.

The Model Method adopted Polya’s 4-step solving process which involves:

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Conclusion cont’d

1. Understanding the problem:- Construct the information

2. Drawing a plan: – Draw a model

3. Carrying out the plan: – Carry out computation

4. Check solution or looking back:- Checking reasonability of a solution or seeking alternative solutions.

You can see that the use of Part-whole and Comparison models as pictorial representations, facilitates meaningful learning of the abstract concept of the four operations, fraction, ratio & proportion, and percentages in pre-algebra courses. Students will be able to draw a pictorial model as a virtual representation of known and unknown quantities and their relationship.

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Conclusion cont’d Moreover, these are extended to more complex

word problems. The model method approach provides students with enriching opportunity to engage in the construction and interpretation of algebraic equation through meaningful and active learning.

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