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Development Continum presentation The Developmental Continuum: From Theory to Practice Jamie Schmitz Head of Primary Westlake Academy The Developmental Continuum A developmental continuum is comprised of particular discipline’s expectations (what we and would like each child to understand and be able to do), arranged in developmental phases, with each phase building on the one before it. MAT H LEAR NING CON TINUUM FOR NUM BER IB/TEKS ALIGNMEN T PH ASE 1 PHASE 2 PHASE 3 PHASE 4 PH ASE 5 PHASE 6 PHASE 7 PHASE 8 Connect number names and numerals to the quantities they represent (through 20) using verbal and symbolic descrip tions Read and writ e whole numbers up to hundreds Read and writ e whole numbers up to hundreds or beyond Use plac e value to read, write, and describ e the valu e of whole numbers through 999,999 Use plac e value to read, write, and describ e the valu e of whole numbers through 999,999,999 Use plac e value to read, write, compare and order whole numbers through 999,999,999,999 Compare and order non-negative ratio nal numbers Compare and order integers and positiv e rational numbers Use sets of concrete objects to repre sent quantities giv en in verbal or written form (through 20) Compare and order whole numbers up to 99 (less than, greater than, or equal to) using sets of concrete objects and pictorial models Use plac e value to read, write, and describe the value of whole numbers to 999 Use plac e value to compare and order whole numbers through 9,999 Use plac e value to read, write, compare and order decimals involvi ng tenths and hundredths, including money, using concrete objects and pic torial models Use plac e value to read, write, compare and order decimals through thousandths Generate equivalent forms of r atio nal numbers including whole numbers, fractions and deci mals Convert between fractio ns, decim als, whole numbers, and percents mentall y, on paper, or with a calculator Name the ordin al positions in a sequence such as f irst, second, third, etc. Create sets of tens and ones using concrete objects to describe , compare, and order whole numbers Use plac e value to compare and order whole numbers to 999 and record comparisons using numbers and symbols Determine the value of a collection of coins and bills Use concrete objects and pictorial models to generate equivalent fractio ns Convert imprope r fractio ns to mix ed numbers and vice versa Use integers to represent real-life situatio ns Represent squares and square roots using geometric models Share a whole by separating it into two equal parts Use words and numbers to describe the values of individ ual coins such as penny, nickel, dim e and quarter and their rela tio nships Use concrete models to represent and name fractional parts of a whole object or a s et of objects (with denominators of 12 or less) Construct concrete models of fractions Model fractio n quantitie s greater than one using concrete objects and pictorial models Simplify fractio ns Write prime factorizatio ns using expo nents Represent multiplication and divisio n situatio ns involvi ng fractions and decimals with models, in cluding concrete obj ects, pictures, words, and numbers Explain why a given part is half the whole Share a whole by separating it into equal parts and use appropria te language to describe the parts such as three out of four equal parts Use concrete models to determine if a fractional part of a whole is clos er to 0, , or 1 Compare fractional parts of whole obje cts or sets of obje cts in a problem situatio n using concrete models Compare and order fractions using concrete objects and pic torial models Use models to relate decimals to fractio ns Identify factors of a posi tive integer, common factors, and the greatest common factor of a set of posi tive integers Use additio n, subtraction, multipl ication, and divisio n to solve problems involvi ng fractions and deci mals Model and create addition and subtraction problems in real situations with concrete objects Use appropria te language to describ e part of a set such as three out of the eight crayons are red Recall and apply basic addition and subtractio n facts (to 18) Use fractio n names and symbols to describ e parts of whole objects or sets of objects Relate decimals to fraction s that name tenths and hundredths using concrete objects and pictorial models Generate equivalent fractio ns Identify multiples of a po sitive integer and common multiples and the least common multip le of a set of pos itiv e integers Use models, such as concrete obje cts, pictorial models and number lin es, to add, subtract, multipl y, and divide integer s and connect the actio ns to algo rithms Solv e problems with guid ance that incorporate processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solutions for reasonableness Model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences Model additio n and subtraction of two-digi t numbers with obje cts, pictures, words and numbers Construct concrete models of equivalent fractions Use additio n and subtractio n to solv e probl ems involvi ng whole numbers Compare fractio ns Use additio n and subtractio n to solv e problems involvi ng fractions and decimals Use divisio n to find unit rates and ratios in propo rtio nal rela tionships such as speed, density, pric e, recipe s and student- teacher ratio Select or develop an appropriate problem-solving strategy including drawing a picture, look ing for a pattern, systematic guessing and checking, or acting it out in order to solve a problem Learn and apply basic additio n facts (sums up to 18) using concrete models Determine the value of a collection of coins up to one dollar Model additio n and subtraction using pictures, words and numbers Add and subtract decimals to the hundredths place using concrete objects and pic torial model s Use additio n and subtractio n to solv e problems involvi ng whole numbers and decimals Use multipl icatio n and div ision of whole numbers to solve proble ms Simplify numerical expressio ns involvi ng order of ope ratio ns and expo nents Use tools such as real objects, manip ulatives and technolog y to solve problems Use a proble m-solving model, with guidance as needed, that in corporates understanding the problem, making a plan, carrying out the plan, and evalu ating the solu tio n for reasonableness Describe how the cent symbol, dollar symbol, and the decimal point are used to name the value of a collectio n of coins Select additio n or subtractio n and use the operatio n to solve problems involvi ng whole numbers through 999 Model factors and products using arrays and area models Use multipl icatio n to solv e problems involvi ng whole numbers Estima te and round to approxim ate reasonable result s Select and use appropriate ope rations to solve problems and ju stify the selections Communicate mathematical ideas using obje cts, words, pictures, numbers, and technolog y Use tools such as real objects, manip ulatives, and technolog y to solve problems Model, create and des crib e multiplication situatio ns in which equivalent sets of concrete objects are joi ned or separ ated Learn and apply multiplic atio n facts through 12s using concrete models and objects Represent multiplication and divisio n situatio ns in pic ture, word, and number form Use div isio n to solve problems involvi ng whole numbers (two-digit div isors and three-digit divid ends), including interpreting the remainder within a given context Use order of op eratio ns to simplify whole number expressions in problem solving situatio ns Solv e problems that incorporate the processes of understanding the proble m, making a plan, carrying out the plan, and evalu ating the solu tion for reasonableness Relate everyday language to mathematical language and symbols Explain and record observations using objects, words, pictures, numbers, and technolog y Select additio n or subtractio n to solve problems using two-digit number s, whether or not regrouping is necessary Solv e and record multipl icatio n problems (up to two digit times one digit ) Recall and apply multiplicatio n facts through 12 x 12 Identify common factors of a s et of whole numbers Use ratio s to des cribe propor tio nal situatio ns Use logical reasoning to ju stify thinking Use logical reasoning Use fractio ns in real-life situation s Select or develop an appropriate problem- solving plan or strategy including drawing a picture, looki ng for a pattern, systematic guessing and checking, or a cting it out in order to solve a proble m Use models to solv e divisio n proble ms and use number sentences to record the solutions Use multipl icatio n to solv e problems (two digits times two dig its without technolog y) Estima te solutio ns to determin e reasonable result s using strategies such as rounding Represent ratios and percents with concrete models, fractions and decim als Use tools to solve problems Use logical reasoning Use log ical reasoning to justify thinking Round whole numbers to the nearest ten or hundred to estim ate reason able result s Use div isio n to solve problems (one-dig it divisors and three-dig it dividends without technolog y) Explain and record observations Use ratio s to make predictions in proportional situatio ns Use strategies, in cluding rounding, to estim ate solutio ns to additio n and subtractio n problems Round whole numbers to the nearest ten, hundred, or thousand to estimate reasonable result s Use fractio ns and decimals interchange ably Simplify fractions Develop strategie s for memorizing addition, subtractio n, multiplic atio n and division number facts Use strategies including rounding and compatib le numbers to estim ate solutio ns to multipl icatio n and div isio n problems Add and subtract fractions with rela ted denominators Solv e problems that incorporate the processes of understanding the proble m, making a plan, carrying out the plan, and evalu ating the solu tion for reasonableness Solv e problems that incorporate the processes of understanding the proble m, making a plan, carrying out the plan, and evalu ating the solu tion for reasonableness Solv e problems that incorpor ate the processes of understanding the proble m, making a plan, carrying out the plan, and evaluating the solu tion for reasonableness Solv e problems that incorpor ate the processes of understanding the proble m, making a plan, carrying out the plan, and evalu ating the solu tion for reasonableness Zone of Proximal Development The “zone of proximal development” (ZPD) is that conceptual development that is just out of reach of a child’s current independent performance, but which s/he can reach with assistance. It is the area of development that that we need to be targeting, as it’s not completely out of reach, but also, not redundant. What Is Scaffolding? Scaffolding provides the support and structure necessary for students to learn new information or complete assigned tasks successfully. Scaffolding Why Scaffold Student Learning? Raises Student Achievement Provides Structure and Support Removes Surprises Keeps Students on Task Closes Achievement Gap Differentiates Instruction Increases Success-Extends Scaffolding-Examples Scaffolding-Examples Scaffolding-Examples How Does the Math Continuum Work? 1.Teachers will assess the students at regular intervals. Assessment will focus on evaluating understanding (Application, Explanation, Interpretation, Perspective, Empathy, and Self- Knowledge). How Does the Math Continuum Work? Using the assessment data, teachers will plot (on the developmental continuum ) what a student understands and is able to do. The next slide will illustrate this. PH ASE 4 PH ASE 5 Use place v alue to read, wr ite, and d escr ibe the valu e of who le number s through 999,999 X Use p lace valu e to read, wr ite, and describe th e valu e of wh ole numbers through 999,999,999 Use place v alue to co mpar e and orde r whol e numbers t hrough 9,999 X Use p lace valu e to read, wr ite, co mpar e and order d ecimals involving tenths and hundredths, inclu d ing mone y , using concr ete objects and pictor ial mod els Determin e th e valu e of a collection of coins and b ills X Use concr ete objec ts and pictor ia l mod els to gene r ate equiv alen t fr ac tions X Constru ct concr ete mod els of fr actions Mod el fr ac tion quan tities gre ater than one using conc rete ob jects and pictorial models X Compa re fr action al parts of who le ob jects or sets of ob jects in a prob lem s ituation using concr ete mod els Compa r e and order fr actions using concr ete objects and pictor ial mod els X Use fr ac tion names and symbol s to de scrib e parts of whol e ob jects or sets of objects Relate decimals to fr actions th at name tent hs and hundre d ths using conc rete objects and pictorial mode ls X Constru ct concr ete mod els of equivalent frac tions Use add ition and subtr action to solve problems invol v ing whol e numbers Mod el addi tion and subtraction using pictures, words and numbers X Add and subt r act decimals to th e hundredths pl ace usin g conc r ete ob jects and pi ctor ial mod els Select add ition or sub traction and use th e operation to solv e problems involving whol e numbers through 999 X Mod el f actors and products using arrays and ar ea mode ls Lear n and apply mu ltipli cation f acts through 12s using co n crete mod els and ob jects X Represent multiplication and division situ ations in pictur e, word, and number form Solve and record mu lt iplication problems (up to tw o di git times on e dig it) X Recall and apply mu ltiplication fa cts through 12 x 12 Use mode ls to solve d ivision problems and use number se n tences to record th e solu tions X Use mu ltiplication to solve problems (two di g its times two d igits wi thout technology) Round whol e number s to th e neares t ten or hundr ed to e stimat e reasona ble results X Use d ivision to solve problems (one -dig it divisors and three-di git div idends wi thout technology) Use strateg ies, in cluding rounding, to estima te solu tions to add it ion and subtraction problems X Round whol e number s to t he nearest ten , hundred, or thousand to es timate reasona ble results Develo p strategies for memori zing addi tion, subtraction , mu ltiplication and div ision num ber fac ts X Use strategies inclu d ing rounding and comp atible numbers to es tima te solu tions to m u ltiplication and div ision prob lems Solve prob lems that incorporat e th e processes of understan d ing the problem, making a plan, carrying ou t the p lan, and ev aluating th e solu tion for reasonab leness X Solve prob lems th at in corporat e the processes of under standing th e problem, making a plan, carrying ou t the plan, and evaluating the so lution for reasona b leness MAT H LEAR NING CON TINUUM FOR NUM BER IB/TEKS A LIGNMEN T How Does the Math Continuum Work? Teachers will design learning experiences and scaffold according to the student developmental clusters that have been identified through the use of the developmental continuum. How Does the Math Continuum Work? Teachers will review the developmental continuum periodically for each child, and make adjustments as necessary. In Summary: The use of Developmental Continua allows teachers to facilitate student learning in accordance with every child’s individual level of understanding. In essence, this is an individualized developmental map for each student, which will be vital in moving him or her forward at his or her own developmental pace.