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1、Domain III:MathematicsCompetency 016(Mathematics Instruction),The teacher understands how children learn mathematical skills and uses this knowledge to plan,organize,and implement instruction and assess learning.,Principles of Mathematics,The NCTM identified six principles for school mathematicsequi
2、ty,curriculum,teaching,learning,assessment and technology(2000).1.Equity.Excellence in mathematics education requires equityhigh expectations and strong support for all students.2.Curriculum.A curriculum must be coherent,focused on important mathematics,and well articulated across grades.3.Teaching.
3、Effective mathematics teaching requires understanding of what Students know and need to learn,and then challenging and supporting them to learn it well.4.Learning.Students must learn mathematics with understanding,actively building new knowledge from experience and previous knowledge.5.Assessment.As
4、sessment should support the learn of important mathematics concepts,and furnish useful information to both teachers and Students.6.Technology.Technology is essential in teaching and learning mathematics it influences the teaching of mathematics and enhances students learning.,Principles of Mathemati
5、cs,Use various materials to teach skills and concepts.Use different instructional techniques.Have students explain skills and concepts.Allow students to see their progress(e.g.,records of performance).Teach the language of mathematics(e.g.,word walls,flash cards,etc.).Use a variety of cues to check
6、for understanding(e.g.,thumbs up,color coded cards,happy face cards,etc.).Include concrete,representational,and abstract activities.Avoid over reliance on workbooks for dictating curriculum and providing practice opportunities.,Principles of Mathematics,(cont.)Use instructional approaches that will
7、ensure comprehension and mastery of skills and concepts,such as cooperative learning,graphic organizers,use of manipulatives,etcAvoid excessive paper-and-pencil drill that serves merely as busy work rather than as a meaningful practice experience.Link new instructional knowledge to present knowledge
8、.Show students-and have students explain-how mathematics is part of daily living.Teach mathematical skills and concepts within a problem-solving context.,Remember.We learn:,10%of what we read20%of what we hear30%of what we see50%of what we both hear and see70%of what we discuss with others80%of what
9、 we experience personally95%of what we teach to someone elseWilliam Glasser,How Do Children Learn Mathematics?,Research indicates that children must develop higher level thinking abilities in order to interpret certain mathematical concepts.Because the development of some of these abilities is so na
10、tural,teachers often fail to consider that children in certain stages of development may not have acquired them.To be an effective teacher you must know when children can be introduced to a given concept and at what level of abstraction they can deal with the concept.Basically there are three levels
11、 at which concepts can be represented:concrete,pictorial,and symbolic.,Levels of math development,Concrete Level.A concept can be represented by the appropriate manipulation of objects.Example:Placing three beans into each of four margarine tubs and finding the total number of beans illustrates that
12、 4 X 3=12.Pictorial Level.A concept can be represented by appropriate pictures.Example:A picture of four groups of three to illustrate 4 X 3=12.Symbolic Level.A concept can be represented by symbols.Example:4 X 3=12.,Standard for Mathematics,The standards for mathematics are divided into two section
13、s content and process.Content standardsProcess standards,Standard for MathematicsContent Standards,1.Number and operations.These components include the concept of number,and fraction,and basic computation.2.Algebra.This component includes elementary algebraic reasoning involving ing patterns and set
14、s of numbers3.Geometry.This encompasses the Study of geometric shapes and spatial reasoning,4.Measurement.This component includes the units of measurements(standard and metric)and the process for measurement in general.5.Data analysis and probability.These two components cover the collection,analysi
15、s,and display of mathematics Information.,Standard for MathematicsContent Standards,Number and operations.Pre-K-1st Grade.Draw,read,and write values to 99.Count objects by grouping tens.Compare and order numbers to 99 using models/pictures.Identify coins and value(pennies.nickels and dimes)2nd Grade
16、.Draw,read,and write values to 999.Compare and order numbers to 999.Count values of all coins.Recognize fractions using models.Skip Count.Recognize odd and even numbers.3rd Grade.Draw,read,and write values to 99,999.Compare and order numbers to 99,999.Show bills and coins to equal a given number.4th
17、 Grade.Read,write,and determine place value of numbers up to one million.Draw pictures of numbers to millions.Draw,read,and write values to hundredths.Compare and order decimals using models to hundredths.Round numbers to ten and hundred.Compare fractions using pictures and patterns.,Standard for Ma
18、thematicsContent Standards,Numbers and Operations(cont):(Addition and Subtraction)Pre-K-1st Grade.Use objects to act out and subtraction stories.Addition and subtraction to ten.Act out or use objects to describe addition or subtraction situations including:comparing,missing parts,how many left.Repre
19、sent addition and subtraction situations with a number sentence.2nd Grade.Estimate sums and differences to 99.Select correct operation and solve real life problems involving addition and subtraction.Use addition to solve problems to 999.Addition and subtraction with money:cents to 99,dollars to 999.
20、Use subtraction to solve problems with minuends through 99.Basic Fact recall addition and subtraction to 18,3 5 seconds average recall per fact.3rd Grade.Estimate sums and differences to 9,999.Use addition to solve problems with numbers through 9,9994th Grade.Use addition and subtraction of decimals
21、 to solve problems(tenths and hundredths).,Standard for MathematicsContent Standards,Numbers and Operations(cont):(multiplication and division)Pre-K-1st Grade.Use a multiplication or division number sentence to describe a modeled situation.Draw a picture for a given multiplication or division word p
22、roblem.2nd Grade.Use a multiplication or division number sentence to describe a modeled situation.Draw a picture for a given multiplication or division word problem.3rd Grade.Multiply number with factors through 10.Select correct operation to solve real-life problems involving multiplication and div
23、ision.4th Grade.Estimate products of 2 digit by 2 digit factors.Use multiplication and division to solve problems:multiplication 3 digit by 2 digit,division one digit quotients with and without remainders.Basic fact recall,multiplication and division,3 to 5 second average recall per fact.,Standard f
24、or MathematicsContent Standards,Algebra.Pre-K-1st Grade.Identify and extend pattern using objects.Demonstrate the relation between addition and subtraction.2nd Grade.Identify and extend pattern.Determine missing elements.Write number families for addition and subtraction.Use operation properties:add
25、ition and subtraction with zero,addends orders does not matter(commutative property).3rd Grade.Find the relationship of number pairs(function)and extend the pattern.Write number families for multiplication and division.4th Grade.Use operation properties:multiplication and division by one,multiplicat
26、ion commutative and distributive.Use patterns to solve problems.,Standard for MathematicsContent Standards,Geometry.Pre-K-1st Grade.Construct squares,rectangles,triangles on geoboards and with pattern blocks.Locate interior and exterior points(locations)on plane figures.2nd Grade.Identify and sort r
27、eal objects by shape:cube,cone,sphere,cylinder.Construct congruent shapes on a geoboard and on dot paper.3rd Grade.Classify(sort)polygons.Identify figures having symmetry.Identify line of symmetry.Identify a figure congruent to a sample figure.4th Grade.List characteristics of polygons:quadrilateral
28、s,parallelograms,rectangle,rhombus,square,trapezoid,pentagons,hexagons,octagons.Classify 3 dimension figures and faces:cube,sphere,cone,cylinder,prisms,pyramids,Standard for MathematicsContent Standards,Measurement.Pre-K-1st Grade.Compare length and weight.Read a calendar.Tell time to hour and half-
29、hour.2nd Grade.Estimate and measure:customary inches/feet/yards/pounds,metric centimeters/meters/kilograms.Tell time to 5 minutes.3rd Grade.Estimate and measure:customary inches/feet/yards/miles/kilometers/ounces/pounds,metric centimeters/meters/grams/kilograms.Temperature4th Grade.Find areas of rec
30、tangles.Find perimeters.Estimate and measure capacity:customary cup/pint/quart/gallon,metric milliliter/liter.Solve problems involving elapse time.,Standard for MathematicsContent Standards,Data analysis and probability.Pre-K-1st Grade.Make real graphs.Identify events that are sure to happen,sure no
31、t to happen,or unsure of outcome.2nd Grade.Make and interpret picture graphs and bar graphs.Interpret and use charts.3rd Grade.Make picture and bar graphs where each cell represent multiple units.Use bar graphs to solve application and non routine problems.4th Grade.List possible outcomes in a given
32、 situation.Interpret line graphs.Plot points in a coordinate plane.Collect record and organize data into tables,charts,bar graphs and line graphs.,Standard for MathematicsProcess Standards,1.Problem solving.In this component students are guided to formulate and solve mathematics problems that can be
33、 used in real life situations.2.Reasoning and proof.These two components allow student opportunities to examine problems,find solutions,and justify them using logical and mathematics principles.3.Communication.This component teaches children to use precise and appropriate mathematics vocabulary to e
34、xplain processes and outcomes.4.Connections.This component emphasizes the importance of mathematics and the connection to other content areas and life in general.5.Representations.This component teaches how mathematics information can be presented in various ways numbers,letters,tables graphs and so
35、 on.,Standard for MathematicsProcess Standards,Problem solving.Pre-K-1st Grade.Act out and draw pictures to represent addition and subtraction including:how many left,missing parts,comparing.2nd Grade.Draw a pictures.Use patterns.Act out and draw pictures to represent word problems:multiplication an
36、d division,non-routine problems.3rd Grade.Use strategies:make an organized list,make a table.Write a number sentence to describe word problems involving addition and subtraction,and multiplication and division.4th Grade.Solve word problems with extra information and determining missing information.S
37、olve problems working backwards.,Standard for MathematicsProcess Standards,Problem solving.(cont)In grades k-4,the study of mathematics should emphasize problem solving so that students can:Use problem solving approaches to investigate and understand mathematical content.Formulate problems from ever
38、yday mathematical situations.Develop and apply strategies to solve a wide variety of problems.Verify and interpret results with respect to the original problem.Acquire confidence in using mathematical meaningfully.,Standard for MathematicsProcess Standards,Problem solving.(cont)The mathematics teach
39、er understands and uses numbers,number systems and their structure,operations and algorithms,quantitative reasoning,am technology appropriate to teach the statewide curriculum(Texas Essential Knowledge and Skills TEKS)in order to prepare students to use mathematics.,Standard for MathematicsProcess S
40、tandards,Reasoning and proof.In grades k-4,the study of mathematics should emphasize reasoning so that students can:Draw logical conclusions about mathematics.Use models,known facts,properties,and relationships to explain their thinking.Justify their answers and solution processes.Use patterns and r
41、elationships to analyze mathematical situations.Believe that mathematical make sense.,Standard for MathematicsProcess Standards,Communication.For one step problems,students can be asked the following questions as a way to discuss their work:What are you trying to find?Which data in the story were ne
42、eded to find the solution?Were there unnecessary data?What action in the story suggested the operation you used to find the answer?Can you give the answer in a complete sentence?Have you checked your work and your answer?,Standard for MathematicsProcess Standards,Connections.Students should learn ho
43、w math is connected to the other content areas,such as,language arts,science,social studies,health,physical education,art,music,etc.For example,if they are studying“temperature changes,”they should know numbers,algebraic computations,basic operations,etc,to read the thermometer and use this informat
44、ion in graphs.,Standard for MathematicsProcess Standards,Representations.Students should learn the different ways to represent the numerical information they gathered from a problem by using algorithms,numbers,tables,charts,graphs,etc.,The Texas Essential Knowledge and Skills TEKS,The Texas Essentia
45、l knowledge and Skills(TEKS)requires a wellbalanced curriculum beginning in kindergarten and continuing through grade12.(TEA,2006)It incorporates the principles and standards of the NCTM and introduces these in a sequential manner that reflects the cognitive development of children.Children in kinde
46、rgarten begin exploring number concepts using concrete objects and mastering onetoone correspondence.In first and second grades they continue exploring number concepts and begin studying basic computations skills.In third through fifth grades they continue expanding number concepts to include multip
47、lication,division fractions,decimal representations,geometric principles and algebraic reasoning.,Mathematics curriculum for K-4,KindergartenWholenumber concepts and using,patterns and sorting to explore numbers,data and shapes FirstAdding and subtracting whole numbers and organizing and analyzing d
48、ataSecondComparing and ordering whole numbers,applying additional and subtraction,and using measurement processesThirdMultiplying and dividing whole numbers connecting fraction symbols to fractional quantities and standardizing language and procedures in geometry and measurementFourthComparing and o
49、rdering fractions and decimals,applying multiplication mid division and developing ideas related to congruence and symmetry,Cognitive Development and Mathematics,The cognitive development of children in Pre-Kindergarten(Pre-K)through grade 4 represents a special challenge when attempting to learn th
50、e symbolic and abstract representations used in mathematics.Piaget classified students in Pre-K through grade 4 into two broad stages:Preoperational(2-7 years),and Concrete Operational(21 1 years)(cited in Sperry Smith,2001)Children in the Preoperational stage of cognitive development(2-7 years,Pre-
