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1、The Mathematics Curriculum K12,A STANDARDS-BASED Search for CoherenceJohn A.DosseyIllinois State UniversityJeremy KilpatrickUniversity of Georgia,Curriculum,Not just a syllabus or textbookIncludesAimsContentMethodsAssessment proceduresDepends on ones positionPlanExperienceA linear path through a mul
2、ti-dimensional domain,What is a curriculum?,A plan of action directing the content and delivery of a program of mathematics learning for a specified group of studentsThis plan consists of an articulated set of statements that provide:general goals for the plana structure of content domains and cogni
3、tive processes to be developeda philosophy concerning the general conditions of learning expecteda listing of general standards and specific outcomes of learning related to levels of schooling germane to the population for which the curriculum is intended,The Importance of Mission,“A discussion of m
4、athematical education,and of ways and means of enhancing its value,must be approached first of all on the basis of a precise and comprehensive formulation of the valid aims and purposes of such education.Only on such basis can we approach intelligently the problems relating to the selection and orga
5、nization of material,the methods of teaching and the point of view which should govern instruction.”(The Reorganization of Mathematics in Secondary Education,1923),Source of Goals,Society/Workforce,Discipline,Psychology,Mathematics Education,Assessments,Most Curricula do not answer Who?,What?,Where?
6、,Why?,Situating a curriculum NCTM and PISA discuss:Personal Life,Workplace,Cultural and Societal,and Scientific and Technological.College Board Standards for College Success:Mathematics and Statistics:“College Board presents these standards as a guide to provide all students with the rigorous educat
7、ion that will prepare them for success in college,opportunity in the workplace,and effective participation in civic life.”,Recommendation 1,What is the purpose of the curriculum?Who is it to serve?What are the intended outcomes in:Broad brush strokes(overview)Unit level content and processes(teacher
8、s and materials)Specific concepts,principles,and skills(specialist and resource guide level),Previous Periods of Curricular Reform,Early 1900sCommissions.Late 1950slate 60s“New math”Late 1980s90sStandardsEarly 2000s Based on assessments,Today&Recent Past,NCLB and state“curriculum”reformsCollege Boar
9、ds Standards for College Success:Mathematics and Statistics(2009)NCTM Curriculum Focal Points(Pre-K8)(2006)&Focus in High School Mathematics:Reasoning and Sense Making(912)(2009)NGA&CCSSO release draft of College-and Career-Readiness Standards(2009),the first stage of the Common Core State Standards
10、 Initiative-a move toward a curriculumhttp:/www.corestandards.org/,Structuring a curriculum,How are the“levels”laid out?How is the“content”divided?How are the“processes”described and interwoven?How is knowledge itself related to intended learning(and to envisioned pedagogy)?Problems:ConnectionLinear
11、ityDifferentiationChoice,Providing Structure,Content Domains,CognitiveProcesses,Content Domains,Number&operationsAlgebra&functionsGeometry&transformationsMeasurementData analysisChanceBe careful to:Avoid straying too far from a concise list because that rapidly leads to a lack of focusMake sure ther
12、e are ample areas to connect!,E.g.,algebra is algebra?,Margaret Kendal and Kaye Stacey in The Future of the Teaching and Learning of Algebra:The 12th ICMI Study(Stacey et al.,2004):“Dont take your countrys curriculum and approach to teaching algebra for granted and dont assume all other educational
13、jurisdictions operate in a similar waythey conspicuously do not.”Striking differences inWho takes algebraWhether integrated or layered across yearsEmphasis put on generality and patternAttention to symbolism,formalism,and abstractionWhether approached through functions and multiple representationsRo
14、le played by technology,Interconnected Content Domains&Cognitive Processes,ContentDomains,Concepts,Relationships,Procedure,Facts,Recommendation 2,Keep the lists of content and processes focused and conciseDo not stray greatly from current models without strong justificationit just makes work for tea
15、chers and costs moneylittle real changeMake sure statements of goals,standards,and outcomes have clarity and reflect a range of cognitive processes,A Philosophy Of Pedagogy,Reasoning and Sense MakingTeaching should value and emphasize reflective reasoning(sense making)on the part of the student in o
16、rder to promote construction of new mathematical concepts based on previously learned concepts and new ideas.Teaching should help students see the need for justifying their work and help students to develop justification strategies.Problem Solving and Mathematical ModelingTeaching should engage stud
17、ents in solving open-ended problems,in groups and individually,to illustrate the need for mathematics and demonstrate the connectedness of mathematical ideas.Teaching with a problem-solving approach should encourage students to investigate mathematical models as a vehicle for analyzing problems,maki
18、ng sense of related content and structures,and finding problem solutions.,A Philosophy Of Pedagogy(cont.),Multiple PerspectivesTeaching should develop students capabilities to look at mathematical situations and outcomes from multiple perspectives,compare and critique methods,and reflect on the valu
19、e of each.Teaching mathematics should encourage students to employ appropriate tools and techniques such as computing,graphing,and other analytical methods.Mathematical AutonomyTeaching mathematics should support student development and ownership of mathematical ideas.Teaching mathematics should str
20、engthen students capability to illustrate and communicate their mathematical ideas,imbuing each student with the belief that they have the foundations and related capabilities required to succeed in mathematics.,Pedagogical Framework,While the instructional implementation is a partially separate mat
21、ter,a broad statement of the boundaries and shape of the program should be included.“What you learn reflects how you learned it.”Such a framework also reflects the approaches and conditions for achieving“depth of understanding.”,Coherence,Above alla curriculum must have coherence.Where coherence can
22、 be measured as the ratio of the number of outcomes that link to other outcomes when compared to the total number of potential outcome pairs.In addition,coherence relates to the overall consistency of expected depth of understanding and cognitive level consistency.,Recommendation 3:ATTEND to the Cha
23、llenges,Attend to the related teaching models Understand difficulty of making“real change”Strive for balance:content and processesGet buy-in of schools,teachers,parentsBe responsive to challenges of diversity ALL CAN DO IT and IT IS FOR ALL!Give it time to grow and prosperProvide for systematic revision,
