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Presenters:Dr. Umesh NagarkatteLavoizier St. JeanHerbert OdunukweKay LashleyDepartment of MathematicsMedgar Evers College, CUNY1638 Bedford Ave., Brooklyn, NY 1122511/19/2011

Acknowledgements -1. U. S. Department of Education – Minority Science EngineeringImprovement Program (MSEIP) two grants – Institutional andCooperative, 2010-20132. The Singapore Model Method for Learning Mathematics - Ministry ofEducation, 20093. What the United States Can Learn From Singapore’s World-ClassMathematics System (and what Singapore can learn from the UnitedStates): An Exploratory Study - American Institutes for Research®prepared for: U.S. Department of Education Policy and ProgramStudies Service (PPSS), 2005(Note: 2 contains the recommendations made in this monograph)11/19/2011

Background -• Singapore Mathematics and Science consistently ranks firstin the world in the Trends in International Mathematics andScience (TIMSS) studies. Currently it is second, and U.S. 26 th .• Medgar Evers College received two MSEIP grants – oneInstitutional and the other Cooperative with QCC for 2010-2013. The main activity is to implement Theory ofConstraints (TOC) to increase retention and graduationrates. We are implementing TOC for the last nine years withthe support of five federal grants.• One of the numerous activities this time is to adapt theSingapore Model Method (a TOC implementation) toCollege Level.• Starting with Basic Skills, we are revamping our mathcourses. Will involve the Singapore model and TOC thinking11/19/2011tools. (handouts)

Peferred Features of the Singapore MathematicsSystem11/19/2011SingaporeFramework Logical, National,develops topicsin-depth, alternateTextbooksThin, Fewer topics. Lesswords. Build deepunderstanding throughmulti-step problems,Concrete to visual toabstract. Illustrationsdemonstrating howabstract concepts can beused for differentperspectives.U.S.No national, NCTMlacks logical structure, noalternateHeavy, Too many topics.Limited to definitions andformulas, developingstudents’ mechanicalability to apply concepts.Real- world illustrationsindicate relevancy – butdo not show how to applyconcepts to solve thoseproblems.

11/19/2011Mathematics Framework• The Singapore Model method is deeply rooted in anunderlying Mathematics principles of effectiveproblem solving methods that is represented in apentagonal framework.• Mathematical Problem Solving which is the core ofthe pentagonal framework is central to mathematicslearning. There are five interrelated componentsassociated with the framework and they are listed as:• Concepts• Skills• Attitudes• Metacognition• Processes

MetacognitionMetacognition, is defined here as awareness of or ability tocontrol one’s thinking process. The following strategiesmay be use to develop metacognitive awareness of studentsand enrich their metacognitive experience:-• Expose student to problem–solving skills, thinking skills and heuristics insolving problems.• Guide students to use appropriate strategies and methods in solving problems.• Provide students with planning and evaluation before and after solving aproblem.• Encourage students to seek alternative ways of solving the same problem• Create conducive environment for students to discuss appropriateness andreasonableness of answers.11/19/2011

AttitudesAttitudes here refers to the affective aspects ofmathematics learning such as:• Beliefs about mathematics and its usefulness• Interest and enjoyment in learning mathematics.• Appreciation of the beauty and power ofmathematics• Building confidence in mathematics• Perseverance in solving a problem11/19/2011

Part-Whole ModelAddition and Subtraction• The part-whole model also known as the part-partwholemodel is a quantitative relationship between awhole and two parts.wholepartpart• The pictorial model shows that the whole is the sum oftwo parts. That is: part + part = whole.• Furthermore, to find a part, the other part can besubtracted from the whole. That is: whole – part = part11/19/2011

Comparison ModelAddition and Subtraction• The comparison model is a quantitative relationshipamong three quantities: larger quantity, smallerquantity 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 quantityLarger quantitySmaller quantitydifference11/19/2011

Model Method and Concepts pfFraction, Ratios and Percentages• Part-Whole Method- Fraction• Comparison Method- Fraction• Part-Whole Method- Ratio• Comparison Method-Ratio• Part-Whole Method-Percentage• Comparison Method-Percentage11/19/2011

Example Problem #1-FractionPart-Whole Method• There are 125 students in a class. 2/5 of them are girls. How many girlsare 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- FractionComparison Method• There are ¾ as many Republicans as Democrats at theWhite House Ball. If there are 80 Democrats, howmany Republicans are there in attendance?• ?RepublicansDemocrats804 units = 801 unit= 80/4=203 units= 20 x 3 = 6011/19/2011There are 60 Republicans

Example #1-RatioComparison Method• A recipe requires 3 ingredients A, B, C in the volumeratio 2:3:4. If 6 pints of ingredient B are required, howmany pints of ingredients A and C are required?• A• B• C3 units = 6 pints1 unit= 6/3 = 2 pints2 units =2 x2 = 4 pints4 units = 4 x 2 =8 pints11/19/2011

Example # 2-RatioComparison Method• If 24 ounces of a certain liquid fills ¼ of a pail, howmany ounces of the same liquid will fill 1/3 of the pail?2496• 1/4 unit = 24 ounces• 1 unit =24/(1/4) = 96 ounces• 1/3 unit = 96 x 1/3 = 32 ounces11/19/2011

Example #3-RatioPart-Whole Method• A settlement of $600 was to be divided in the ratio 1:2:3between 3 brothers. How much money did the thirdbrother received?600 ?6 units = $6001 unit = 600/6 =$1003 units = 100 x 3 = $30011/19/2011

Example #1- PercentagesPart -Whole Method• 15 is 2.5% of what number?152.5?2.5 units = 151 unit = 15/2.5 = 6100 units = 100 x 6 = 60011/19/2011

Example #2- PercentagesPart- Whole Method• If 20% of a number is 14, what is 80% of that number?142020 units =141 unit = 14/20 = 0.7100 units = 100 x 0.7 =70?8011/19/2011? 70100 units = 701 unit = 70/100 =0.780 units = 0.7 x 80 = 56

Example # 3-PercentagesComparison Method• 40% 0f 30 equal 20% of what number?4020?100 units = 301 unit = 30/100=0.340 units = 40 x 0.3 =123011/19/201112 ?20 units = 121 unit =12/20 = 0.6100 units = 0.6 x 100 = 60

Model Method & ProblemSolving• Model method is a synthetic-analytic process thatcan be used to express and solve structurallycomplex word problems:-This Method entailsdrawing a pictorial model and finding the so called“1-unit” from unitary method. Students will learnhow to identify known and unknown quantitiesand relationships among each - syntheticapproach, then logical steps are developed for thesolution of the problem – analytic approach.1/1/201211/19/2011

Example Problem #1: 3 out of 7 students in an Algebra coursepass a class quiz. How many students did not pass in a classof 91 students?Solution:Here a pictorial bar model is used to represent the whole, (total number ofstudents in the class). The whole is 7 units of which 3 units representsnumber of students passing the course.Therefore 4/7 of the whole students did not pass the course.7 units = 91 students1 unit = 91/7 = 13 students4 units will be = 13 x 491= 52 students failed the quiz.1/1/201211/19/20111unit

Example problem #2. A store gives 10% discount to allstudents off the original cost of any item. If an additional 15%is taken off the discounted price, how much is the originalprice if a student purchases an item for $306?Solution:1 st 10% store discount represent 90% of original price .(100% – 10% = 90%)2 nd Discount price represent 85% of the first discount price of 90% = 0.85 x0.90 = 0.765Therefore the final price represent 76.5% of the original price.We need to find the unit price per percent.1% represent $306/.765 = $4.00Original price (100%) = 4 x 100= $400.00Here a pictorial bar model is used to represent the whole, (original price ofthe item). The whole is 100%.Original Price?$3061 unit $4.0076.5% 100%1/1/201290%11/19/2011

Model Method and AlgebraSingapore Model Method also offers alternative approach tosolving algebra word problems. Students mostly encounterdifficulty manipulating algebraic skills necessary in solvingword problems; therefore the need to develop newstrategies is more urgent. Here we will explain how we usethe Model method in solving problems in Algebra courses atMedgar Evers College, CUNY. We will explore the use of theUnitary 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.1/1/201211/19/2011

Part-Whole Model in Problem solving• One major objective adopting Singapore method is toenable students to develop problem solving strategies. TheSingapore model method uses construction of pictorialmodel to solve part-whole and comparison type questions.• Part-Whole ModelThe part-whole model shows relationship between thewhole, f and its component parts, m and n.mn1/1/201211/19/2011fThe equation will be: f = m + n

Part-Whole Model in Problem solving• In another Part-whole model, the whole is divided into anumber of equal parts. The pictorial model is shown as:a a a a aL• The equation of this relationship is given below as:L = 5a(b) The Comparison Model• Comparison model shows relationship between twoquantities when they are compared.11/19/2011

Comparison Model cont’d• The pictorial model is shown below:axyPbd• The pictorial model shows the quantity y is more than thequantity x and their difference is d.• That is: d = y – x• And P = a + b• Still in Comparison model, we may express one quantity asa multiple of the other1/1/201211/19/2011

Comparison Model cont’dExample of pictorial model is shown below:yxThe equation relating this pictorial model is given by:y = 4xWe use these models to solve lots of problems in pre-algebraand algebra courses.1/1/201211/19/2011

Example problem #1. A couch cost 5 times as much as a rocking chair.Altogether they cost $702. How much will the rocking chair costThe algebraic method involves using x to represent therocking chair. The pictorial representation shows therocking chair with respect to the couch (y) which is fivetimes (5x) more in price.yx702• Using Comparison model, we can form an equation.• y =5x and 702 = x + y• Therefore, x + 5x = 702• x =$11711/19/2011

Example Problem. #2. An inspector found 30defective bolts during an inspection, it is 0.25% ofthe total number of bolts inspected, how many boltswere inspected.Let total number of bolts inspected represent 100%.However, we know that 30 inspected bolts were defectiverepresenting 0.25% of defective bolts. We need to find howxmany bolts were inspected.Here a bar is drawn to representthe 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 bolts11/19/20110%300.25%100%

Sample Question #3 High school graduating class is made upof 674 students. There are 298 more boys than girls. Howmany girls are in the class?• Algebraic method using a variable, x to represent number of girls. Asthere 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 theequation; x + (x + 298) = 674. The solution of the equation will be:• x = 188• We can draw a Comparison model to represent the situation and usingalgebra method,boysgirlsx+ 298x 2986741/1/201211/19/2011

Sample Problem #3 Variation 2Total number of boys and girls is 674. The number of boys canbe expressed as 674 – x.x+298674xFrom the above pictorial model, students will see thedifference between (674 – x) and x is 298.Therefore, the equation representing this situation is(674 – x) - x = 298The solution of the equation isx =188.11/19/2011

Example Problem #4Question: Smith, Jones, and Miller have decided to split profits fromtheir business, so that Smith gets three times as much as Jones,and Jones get twenty less than twice smith. How much will eachget if they are to share a profit of $2016?Solution:We use algebraic variable, x, to represent the unknown. Let x representJones share of profit. Then Smith gets 3x and Miller will gets 2x - 20Using Comparison model, we can form an equation:3x + x + (2x -20) = 2016The solution of the equation isx = 336.We can draw a Comparison model2016x2x - 20201/1/201211/19/2011

Conclusion• The Singapore Mathematics Model Framework focuses on using analternative approach to reduce inherent problems of algebraicmanipulation skills used in interpreting application word problems inarithmetic and algebra. The Model Method has been demonstrated inthis presentation to show how an alternative approach can be used insolving word problem in college pre-algebra and algebra. In theseexamples we can see that students can apply basic mathematics conceptsand skills in solving application word problems and developingmathematical thinking. The model method has important feature ofSingapore primary mathematics curriculum.• Problem representation involves data (quantities and quantitativerelationship) and question. Students understanding of the problemsituation, relationship between the known and unknowns enable themto solve the problem. This is schematically shown in the next slide.1/1/201211/19/2011

Conclusion cont’dWordproblemProblemcompletionsolution• In the algebraic method student formulate analgebraic equation to represent the problem situationand to connect the known and unknown quantities.Then solve and answer the question.WordproblemAlgebraicEquationsolution1/1/201211/19/2011

Conclusion cont’d• Using the model method and algebraic method we wereable to construct a pictorial model to help formulate analgebraic equation to solve the problem. This can be shownin the sketch shown below.WordproblemsolutionPictorialmodelAlgebraEquation1/1/201211/19/2011

Conclusion cont’d• We consciously made use of schemas such as the Partwholeand Comparison models which are building blocksfor mental structures and cognitive processes. In addition,we interpreted the learning of algebraic method withmodel method building schemas as well.• The Model Method recognizes metacognition which wasearlier defined as self regulation of learning; enhancingstudents problem solving abilities.• The Model Method adopted Polya’s 4-step solving processwhich involves:1/1/201211/19/2011

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 asolution or seeking alternative solutions.• You can see that the use of Part-whole andComparison models as pictorial representations,facilitates meaningful learning of the abstractconcept of the four operations, fraction, ratio &proportion, and percentages in pre-algebracourses. Students will be able to draw a pictorialmodel as a virtual representation of known andunknown quantities and their relationship.1/1/201211/19/2011

Conclusion cont’d• Moreover, these are extended to more complexword problems. The model method approachprovides students with enriching opportunityto engage in the construction andinterpretation of algebraic equation throughmeaningful and active learning.1/1/201211/19/2011