presenters: dr. umesh nagarkatte lavoizier st. jean ... · pdf filepresenters: dr. umesh...
TRANSCRIPT
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 -
11/19/2011
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)
11/19/2011
11/19/2011
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.
11/19/2011
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.
11/19/2011
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
11/19/2011
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.
11/19/2011
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
11/19/2011
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.
11/19/2011
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
11/19/2011
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
11/19/2011
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.
11/19/2011
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
11/19/2011
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
11/19/2011
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
11/19/2011
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.
11/19/2011
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
11/19/2011
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.
11/19/2011
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?
?
11/19/2011
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
11/19/2011
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
11/19/2011
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?
11/19/2011
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
11/19/2011
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?
11/19/2011
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?
11/19/2011
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.
1/1/2012 11/19/2011
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?
1/1/2012
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
11/19/2011
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?
1/1/2012
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.
1/1/2012
11/19/2011
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.
1/1/2012
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
11/19/2011
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
1/1/2012
x
a
b
P
y
d
11/19/2011
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.
1/1/2012
y
x
11/19/2011
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
11/19/2011
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
11/19/2011
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,
1/1/2012
x+ 298
x 298
boys
girls 674
11/19/2011
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
11/19/2011
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
1/1/2012
x 2016
2x - 20 20
11/19/2011
Conclusion
1/1/2012
• 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.
11/19/2011
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.
1/1/2012
Word problem
solution Problem
completion
Algebraic Equation
solution Word
problem
11/19/2011
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.
1/1/2012
Pictorial model
Algebra Equation
solution Word problem
11/19/2011
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:
1/1/2012 11/19/2011
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.
1/1/2012 11/19/2011
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.
1/1/2012 11/19/2011