Standard for MathaticsQuia

1、domain iii: mathematics competency 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 mathematic

2、s equity, curriculum, teaching, learning, assessment and technology (2000). 1. equity. excellence in mathematics education requires equity high expectations and strong support for all students. 2. curriculum. a curriculum must be coherent, focused on important mathematics, and well articulated acros

3、s grades. 3. teaching. 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 previo

4、us knowledge. 5. assessment. assessment 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 student

5、s learning. principles of mathematics 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

6、 cards, etc.). use a variety of cues to check 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 mat

7、hematics (cont.) use instructional approaches that will ensure comprehension and mastery of skills and concepts, such as cooperative learning, graphic organizers, use of manipulatives, etc avoid excessive paper-and-pencil drill that serves merely as busy work rather than as a meaningful practice exp

8、erience. link new instructional knowledge to present knowledge. 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 read 20% of what we hear 30% of what we see 5

9、0% of what we both hear and see 70% of what we discuss with others 80% of what we experience personally 95% of what we teach to someone else william glasser how do children learn mathematics? research indicates that children must develop higher level thinking abilities in order to interpret certain

10、mathematical concepts. because the development of some of these abilities is so natural, 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

11、level of abstraction they can deal with the concept. basically there are three levels 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 be

12、ans into each of four margarine tubs and finding the total number of beans illustrates that 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.

13、 example: 4 x 3 = 12. standard for mathematics the standards for mathematics are divided into two sections content and process. content standards process standards standard for mathematics content standards 1. number and operations. these components include the concept of number, and fraction, and b

14、asic computation. 2. algebra. this component includes elementary algebraic reasoning involving ing patterns and sets of numbers 3. geometry. this encompasses the study of geometric shapes and spatial reasoning, 4. measurement. this component includes the units of measurements (standard and metric) a

15、nd the process for measurement in general. 5. data analysis and probability. these two components cover the collection, analysis, and display of mathematics information. standard for mathematics content standards number and operations. pre-k-1st grade. draw, read, and write values to 99. count objec

16、ts by grouping tens. compare and order numbers to 99 using models/pictures. identify coins and value (pennies. nickels and dimes) 2nd grade. 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 a

17、nd 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 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

18、 hundredths. compare and order decimals using models to hundredths. round numbers to ten and hundred. compare fractions using pictures and patterns. standard for mathematics content standards numbers and operations (cont): (addition and subtraction) pre-k-1st grade. use objects to act out and subtra

19、ction stories. addition and subtraction to ten. act out or use objects to describe addition or subtraction situations including: comparing, missing parts, how many left. represent addition and subtraction situations with a number sentence. 2nd grade. estimate sums and differences to 99. select corre

20、ct 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. use subtraction to solve problems with minuends through 99. basic fact recall addition and subtraction to 18, 3 5 sec

21、onds average recall per fact. 3rd grade. estimate sums and differences to 9,999. use addition to solve problems with numbers through 9,999 4th grade. use addition and subtraction of decimals to solve problems (tenths and hundredths). standard for mathematics content standards numbers and operations

22、(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 problem. 2nd grade. use a multiplication or division number sentence to describe a modeled situation. dr

23、aw 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 division. 4th grade. estimate products of 2 digit by 2 digit factors. use multiplication and divisio

24、n 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 for mathematics content standards algebra. pre-k-1st grade. identify and extend pattern us

25、ing 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: addition and subtraction with zero, addends orders does not matter (commutative pr

26、operty). 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, multiplication commutative and distributive. use patterns to solve problems. st

27、andard for mathematics content 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 real objects by shape: cube, cone, sphere, cylinder. const

28、ruct 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: quadrilaterals, parallelograms, rectangle, rhombus, squ

29、are, trapezoid, pentagons, hexagons, octagons. classify 3 dimension figures and faces: cube, sphere, cone, cylinder, prisms, pyramids standard for mathematics content standards measurement. pre-k-1st grade. compare length and weight. read a calendar. tell time to hour and half-hour. 2nd grade. estim

30、ate 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. temperature 4th grade. find areas of rectangles.

31、 find perimeters. estimate and measure capacity: customary cup/pint/quart/gallon, metric milliliter/liter. solve problems involving elapse time. standard for mathematics content standards data analysis and probability. pre-k-1st grade. make real graphs. identify events that are sure to happen, sure

32、not 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

33、 in a given 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 mathematics process standards 1. problem solving. in this component students are guided to formulate and solve mathematics p

34、roblems that can be 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

35、mathematics vocabulary to explain 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 num

36、bers, letters, tables graphs and so on. standard for mathematics process 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

37、 to represent word problems: multiplication and 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

38、extra information and determining missing information. solve problems working backwards. standard for mathematics process 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

39、 understand mathematical content. formulate problems from everyday 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 mathema

40、tics process standards problem solving. (cont) the mathematics teacher 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

41、 to prepare students to use mathematics. standard for mathematics process standards 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 expl

42、ain their thinking. justify their answers and solution processes. use patterns and relationships to analyze mathematical situations. believe that mathematical make sense. standard for mathematics process standards communication. for one step problems, students can be asked the following questions as

43、 a way to discuss their work: what are you trying to find? which data in the story were needed 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 a

44、nd your answer? standard for mathematics process standards connections. students should learn how 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

45、should know numbers, algebraic computations, basic operations, etc, to read the thermometer and use this information in graphs. standard for mathematics process standards representations. students should learn the different ways to represent the numerical information they gathered from a problem by

46、using algorithms, numbers, tables, charts, graphs, etc. the texas essential knowledge and skills teks the texas essential knowledge and skills (teks) requires a well balanced curriculum beginning in kindergarten and continuing through grade12. (tea, 2006) it incorporates the principles and standards

47、 of the nctm and introduces these in a sequential manner that reflects the cognitive development of children. children in kindergarten begin exploring number concepts using concrete objects and mastering one to one correspondence. in first and second grades they continue exploring number concepts an

48、d begin studying basic computations skills. in third through fifth grades they continue expanding number concepts to include multiplication, division fractions, decimal representations, geometric principles and algebraic reasoning. mathematics curriculum for k-4 kindergarten whole number concepts an

49、d using, patterns and sorting to explore numbers, data and shapes first adding and subtracting whole numbers and organizing and analyzing data second comparing and ordering whole numbers, applying additional and subtraction, and using measurement processes third multiplying and dividing whole number

50、s connecting fraction symbols to fractional quantities and standardizing language and procedures in geometry and measurement fourth comparing and ordering fractions and decimals, applying multiplication mid division and developing ideas related to congruence and symmetry cognitive development and ma

51、thematics the cognitive development of children in pre- kindergarten (pre-k) through grade 4 represents a special challenge when attempting to learn the symbolic and abstract representations used in mathematics. piaget classified students in pre-k through grade 4 into two broad stages: preoperationa

52、l (2 - 7 years), and concrete operational (2 1 1 years) (cited in sperry smith, 2001) children in the preoperational stage of cognitive development (2 - 7 years, pre-k through grade 2) experience problems with at least two perceptual concepts - centration and conservation (sperry smith, 2001). cogni

53、tive development and mathematics centration four and five year old children focus attention on one characteristic of an object and ignore the others. they might notice that the objects are round, but fail to notice that they have different colors and texture. children at this age play with blocks in

54、 a very simplistic and systematic way-linear fashion. they do not become more creative because they are emphasizing one feature at time. based on cognitive development, children at this age generally experience problems developing and recognizing patterns in mathematics. cognitive development and ma

55、thematics conservation four and five years olds might not understand that changes in the appearance do not necessarily change characteristics of the object (conservation). this limitation can affect childrens ability to measure volume and to understand the value of money. that is, children might get

56、 confused when liquid is moved between containers of different shapes or when trying, to determine the value of a quarter versus five nickels. these perceptual limitations can affect childrens ability to understand measurement and the value of money. cognitive development and mathematics during the

57、concrete operational stage of cognitive development second to seventh grades children experience rapid growth in cognitive development. this stage is characterized by the ability to think logically about concrete objects or relationships. some of the accomplishments of students at this stage are as

58、follows: can form conclusions based oil reason rather than perception can arrange objects based oil characteristics (classification) can organize objects based oil multiple criteria (ordering or seriation) call understand that changes iii appearance do not necessarily affect the substance (conservat

59、ion) can conceptualize what would have happened if an action is reversed (reversibility) cognitive development and mathematics despite the cognitive growth of this stage teachers have to structure lessons to provide students with a concrete foundation to support their thinking. for example, a teache

60、r can introduce the concept of graphing by asking students to follow the growth of a plant for a period of time and document the growth using a iine graph or to compare how one plant grows versus another plant using a bar graph. teachers should also break down a task into manageable components with