Core e In Instruction n (Tie ier r One) ) in in Mathematics s The contents of this topic area were developed under a grant from the US Department of Education, #H323A120009. However, the contents do not necessarily represent the policy of the US Department of Education, and you should not assume endorsement by the Federal Government. Project Officer, Jennifer Coffey.
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The In Instructional Core Principle #1: Increases in student learning occur only as a consequence of improvements in the level of content, teachers’ knowledge and skill, and student engagement. Principle #2: If you change one element of the instructional core, you have to change the other two. Richard Elmore, Ph.D., Harvard Graduate School of Education
General Shif ifts in in Mathematics s Focus: focus strongly where the standards focus Co Coherence: think across grades, and link to major topics in each grade Rig igor: in major topics, pursue with equal intensity – conceptual understanding, – procedural skill and fluency, and – applications
Shift One: Focus
Traditional U.S .S. Approach K 12 Number and Operations Measurement and Geometry Algebra and Functions Statistics and Probability
Focusing Attention Wit ithin Number and Operations Expressions and Operations and Algebraic Thinking → → Equations Algebra Number and Operations — Base Ten → The Number → System Number and Operations — → Fractions K 1 2 3 4 5 6 7 8 High School
HS S Alg lgebra a Resources s as s an n Example e Achieve th the Core Grade Two Overview (http:// //www.achievethecore.org/) Dana Center HS Year at t a Gla lance (Y (YAG) Dana Center Se Sequenced Unit its (http:// //www.utdanacenter.org/) /) (R (Reference Documents)
Shift Two: : C Coherence
Th The Why : : Sh Shift Two Coherence ac across grades, an and lin link to maj ajor topics wit ithin grades (th thinking in interdependently) • Ca Carefu full lly connect th the lea learn rnin ing with ithin in and across grades so so that students can build ild new understandin ing onto foundatio ions buil ilt in in previ vious years. • Be Begin in to count t on so soli lid conceptual l understandin ing of core content and buil ild on it. it. Each standard is is not a new event, but t an ext xtensio ion of f previ vious lea learn rnin ing.
Coherence: : Thin ink Across Grades Fraction example: “ The coh oherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). Th The teachin ing of of fr fractions must be ackn acknowle ledged as as crit criticall lly im important an and im improved before an an in incr crease in in stu tudent ach achie ievement in in alg algebra can an be expected. ” Final Report of the National Mathematics Advisory Panel (2008, p. 18)
In Instructional l Content t (F (Focus) and (C (Coherence- th thin inkin ing in interdependentl tly across grade le levels ls) Potential l St Startin ting Poin int • Le Learning Targets ali lign wit ithin grade-levels and across grade-levels focus* 1 • Id Identify fy major clu lusters, supporting clu lusters, and additional clu lusters • Summative assessments are selected pri rior to teaching
TRANSITION: : Swit itching Gears Your Turn
Reminder fr from the In Instit itute: Technical Problems and Adaptive Chall llenges Sharon Dalo loz Parks, in in the book Leadership Can be Taught (2005)notes that: “ technical pro roblems (even though they y may y be complex) can be solv lved wit ith knowle ledge and procedures alr lready in in hand. . In In contrast, , adaptiv ive challe ch lenges requir ire new le learnin ing, g, in innovation, and new patterns of behavio ior.
The Pra ractice of Adaptive Leadership: T Tools and Tactics for r Changing Your Org rganization and the World (2009) by y Heif ifetz, , Grashow, , and Lin insky “ But t auth thoritie ies cannot t solv lve an adaptiv tive chall llenge by y is issuin ing a dir irectiv tive or brin ingin ing togeth ther a group of f exp xperts ts, , because th the solu luti tions to th the adaptiv ive proble lems lie lie in in th the new attit titudes, , competencies, and coordination of f th the people with ith th the proble lem its itself lf ”(p 73).
Fir irst Two Shif ifts: F Focus and Coherence • Tea eacher: How mig ight I I tak akes steps to mas aster th the math content knowledge for th the ar area of f focus I I am am teaching? (T (Technical Problem an and/or Adaptive Challenge- es especiall lly coherence ac across grades) • Dis istrict Lea eadership: What steps mig ight we e take to ad address th the al alig ignment an and coherence of f math cu curriculum, , text xtbooks, an and materials wit ith th the Id Idaho Core Standards? (T (Technical Problem)
Shift Three: : R Rig igor
Rig igor: In In majo jor topics, pursue wit ith equal in intensity Conceptual understanding (addressing student mis (a isconceptions) Procedural skil ill and fl fluency Applications (D (Depth of f Knowledge- DOK Levels 1-4) 4)
Rig igor: Pursue wit ith equal in intensity Th The challenge of “rigor” is fin findin ing th the bala lance betw tween conceptual l understandin ing, procedural flu fluency, and appli licatio ion. Many of f us s wil ill l need to lea learn rn new knowle ledge, str trategie ies, and sk skil ills ls in in order r to fin find th the bala lance. We have organized the implementation of “rigor” into th three areas: : D Deliv livery ry of f In Instr tructio ion, , In Instructio ional Activ tivit itie ies, , and Materia ials ls.
Several modes of f Deli livery of f In Instruction wil ill be needed for r conceptual knowledge, procedural skil ill, and application • St Structure e and d pacing options considered d (poss (p ssible e adaptive e chall llenge) • Routines, , procedu dures, , and d in instructional group uping • Questioning f Knowledge g and d Depth h of
TRANSITION: : Swit itchin ing Gears
In Instructional l Activ ivities s • Formative e ass ssessment t practices gu id ide decisions s to o adju just t and d dif ifferentiate • St Stud udent t th thin ink-aloud uds • Mathematical l Practices are e uti tilized d
TRANSITION: : Swit itchin ing Gears
Overarching Habits of f Min ind: Persistence and Effort “The Secret to Raising Smart Kids” in Scientific American Mind: By Carol S. Dweck; Dec2007/Jan2008, Vol. 18 Issue 6, pp. 36-43 More than three decades of research shows that a focus on effort — not on intelligence or ability — is key to success in school and in life World Economic Forum, Panel Discussion, December 2013: ANDREAS SCHLEICHER, THE OECD'S SPECIAL ADVISER ON EDUCATION POLICY, ON THE RESULTS OF THE PISA TEST (PROGRAMME FOR INTERNATIONAL STUDENT ASSESSMENT) NOTES: "IN SHANGHAI OVER 30% OF STUDENTS CAN CONCEPTUALIZE, GENERALIZE AND USE ADVANCED MATH IN CREATIVE WAYS.” “ HE ARGUES THAT EAST ASIA'S RESULT IS THANKS TO A BELIEF IN THE VALUE OF HARD WORK AND PERSISTENCE RATHER THAN INHERENT ABILITY.”
Grouping of f Mathematical Practices Reasoning and Explaining 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others Modeling and Using Tools 4. Model with mathematics 5. Use appropriate tools strategically Seeing Structure and Generalizing 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning Overarching Habits of Mind of a Productive Mathematical Thinker 1. Make sense of problems and persevere in solving them 6. Attend to precision Adapted from (McCallum, 2011) 26
TRANSITION: : Swit itchin ing Gears
M Materials s • Ali lign wit ith in instructional content for grade- le level focus • Vis isual and graphic depictions of f problems • Embed assessment it items to detect (d (dia iagnosis) ) and address common stu tudent misconceptions* 6
Mathematics: Understanding the Score (2008) ; Office for Standards in Education, Children’s Services and Skills kills (Ofsted), p. . 6-in in the U.K .K. • “Satisfactory lessons were […] ch characterised by th the teacher doing most t of f th the talk lking, g, emphasising rule les and procedures rath ther th than concepts or lin links with ith oth ther parts of f mathematics.” • “In the most effective lessons […] teachers listened to pupils carefully and observ rved th their work th throughout th the le lesson. . They aim imed to id identify fy any potential mis isconceptions or barriers to understanding key concepts, and responded accordingly.”
Question: D Detecting a Mis isconception is a 2 + b 2 = c 2 2 ? In In whic ich of f th these rig ight t tr tria iangles is b c A B a a c b a c C D b b c a a b E F c c b a
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