•What are number bonds? Partitioning •Creating number triads using•Number bonds to make 10 number bonds and partitioning •What is partitioning? •How are number triads useful? •How can we partition different •Finding out about reverse numbers in different ways? calculationsNumber Bonds Number triadsFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). Number bonds Partitioning numbersNumber triads The Dyscovery Programme |Eleanor Machin 51
• Our number system Building large • Why do we need• Place value: numbers zero? Thousands, Hundreds, • Using blocks to build • What does it mean Tens and Units large numbers when there is a zero in a number? Base 10 • Fitting digits together to make large Zero as a numbers place-holderFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). What is Base 10 How to build large numbersWhat is zero and how do we use it? The Dyscovery Programme |Eleanor Machin 52
•What is trading and why do Carrying •What does 'borrowing' we need it? mean?•How to trade different units •What does 'carrying' mean? •When would we need to •When would we need to borrow a number? Trading carry a number? •Subtraction with borrowing •Addition with carrying BorrowingFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). An example or your own definition of trading An example or your own definition of carrying An example or your own definition of borrowing The Dyscovery Programme |Eleanor Machin 53
•What is multiplication? Patterns in •Multiplication myths•What different words multiplication •Relationship to addition relate to multiplication? •Revisiting the number •Repeat adding triad Introduction to •Arrays Multiplication •Multiplication Square Properties of multiplicationFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). Write down as many words for multiplication as you can remember Show some different ways of representing the sum \"3 x 4\" How does multiplication link to addition? The Dyscovery Programme |Eleanor Machin 54
• Simple patterns for Tricks • More patterns 9, 10 and 11 x tables • Making it easier! • \"Fingers\" method• Investigating other for 9x table Multiplication patterns Square • \"Bridge\" method for Investigating 6, 7 and 8 x table patternsFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). What patterns did you find? Which methods were easiest and hardest? How can you work on your most difficult times table? The Dyscovery Programme |Eleanor Machin 55
•What is division? How division •How division is linked to•What words mean works multiplication division? •Using the multiplication •Division as repeat square to solve division Introduction to subtraction problems Division •Division as grouping •\"Bus stop\" method Division and multiplicationFor each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). Write as many 'division' words as you can remember Show 20 divided by 5 in as many ways as you can Show how division relates to multiplication The Dyscovery Programme |Eleanor Machin 56
•What do we mean by a Parts of •What happens when we coninuous number line? numbers multiply fractions•How does this relate to •What happens when we the number line we •Identifying fractions on divide fractions made in session 2? the number line •What different fractions Working with Revisiting the are there and how do we fractions number line write them?For each section, write or draw your own thoughts and ideas (use the back of this sheet if you need more space). What is a continuous number line and why is it important? What is a fraction? Show something interesting about multiplying or dividing with fractions The Dyscovery Programme |Eleanor Machin 57
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Dot Enumeration Test SheetName:Answers:Item Circle your answer √ or x …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….1 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….2 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….3 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….4 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….5 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….6 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….7 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….8 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….9 The Dyscovery Programme |Eleanor Machin 65
…..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….10 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….11 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….12 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….13 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….14 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….15 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….16 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….17 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….18 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….19 …..1…....2…....3.......4..…..5…....6…....7…....8…....9…....10….20 The Dyscovery Programme |Eleanor Machin 66
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Number Triads: Reversing Calculations The Dyscovery Programme |Eleanor Machin 71
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Thousands Hundreds Tens Units
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Eleanor Machin – Year 1 Final ReportThousands Hundreds Tens Units Thousands Hundreds Tens Units The Dyscovery Programme |Eleanor Machin 82
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Eleanor Machin – Year 1 Final Report SESSION EVALUATION FORMSession Number…………………………………………..Criteria 0 score descriptor 1 score descriptor 2 score descriptor 3 score descriptor Score Reason forPrerequisite given score/Additionalskills Clearly identifies comments prerequisite skills andMath Does not identify Identifies prerequisite Identifies prerequisite provides sufficient warm-vocabulary prerequisite skills or skills, but provides skills, however, offers up activities to engage provide warm-up warm-up activities limited warm-up students’ background activities. unrelated to target activities to engage skills. topic. students’ background skills. Does not identify Identifies key Identifies and directly Identifies and directly vocabulary words. vocabulary but at the embeds (e.g., embeds (e.g., underlines) beginning of the underlines) key key vocabulary within chapter or unit rather vocabulary. Does not lesson. Also, offers than individual provide opportunities opportunities for students lessons. for students to use to use vocabulary. vocabulary. The Dyscovery Programme |Eleanor Machin Research Documents 89
Eleanor Machin – Year 1 Final ReportExplicit Does not provide Provides limited Provides opportunities Systematically providesinstruction opportunities for opportunities for for teacher-led opportunities for teacher- teacher-led teacher-led instruction, but limited led instruction. Offers instruction. instruction. directions on how to explicit directions on how Instructional teach the target topic. to directly teach the target activities use a topic. discovery learning approach only.Instructional Does not provide Provides poorly Provides clear teaching Provides clear andexamples teaching examples. selected teaching examples, however, sufficient number of examples. Examples examples are limited in teaching examples. Also, are irrelevant to number (i.e., only two examples are at least as student practice. examples). complex as student practice.Math Does not Incorporates math Incorporates math Appropriately models for student models for teacher and incorporates modelsmodels/frame incorporate math use only. Does not student use. Provides throughout the lessons incorporate models limited directions on (e.g., C-R-A approach).works models. during teacher-led how teachers should Provides clear directions instruction. use models during on how to teach with the teacher-led instruction. models.Practice Does not provide Provides Provides opportunities Provides sufficientopportunities practice and review opportunities for for math verbalizations opportunities for mathand opportunities. math verbalizations but are limited in verbalizations, includingcumulative but those offered are number. Includes occasions for group andreview limited in structure opportunities for individual responses. Also, (e.g., peer discourse). written exercises, but includes written exercises The Dyscovery Programme |Eleanor Machin Research Documents 90
Eleanor Machin – Year 1 Final Report Includes limited does not allow for and discrimination opportunities for written exercises. discrimination practice. practice.Academic Does not provide Includes only brief Provides hints to Provides hints tofeedback academic feedback hints for anticipating anticipate student anticipate student errors or correction student errors and errors and and misconceptions. procedures. misconceptions. misconceptions. Provides procedures for Procedures for academic and corrective correcting errors and feedback. Also, offers re-teaching are limited. reteaching strategies.Formative Does not provide Provides end-of-unit Provides opportunities Provides frequentfeedback assessments or assessments only. to check for student opportunities to check forloops opportunities to understanding and student understanding check for student identifies an acceptable and identifies an understanding. criterion for moving on acceptable criterion for with instruction. moving on with instruction. Also, it has procedures to link test results.Taken from Doabler et al (2012) Evaluating Three Elementary Mathematics Programs for Presence of Eight Research- Based Instructional Design Principles,Learning Disabilities Quarterly, 35(4), p200-211 The Dyscovery Programme |Eleanor Machin Research Documents 91
Eleanor Machin – Year 1 Final Report MATERIALS EVALUATION FORMTeacher Materials (The Programme Booklet: Programme Rationale, Overview, Session Plans, General Layout)Criteria Score 0 Descriptor Score 1 Descriptor Score 2 Descriptor Score 3 Descriptor Score Reason for Given score/AdditionalTeacher materials Neither curriculum commentssupport teachers to development norbetter meet the lesson design Curriculum maps, unit Most curriculum maps, Curriculum maps,learning needs of the guidance are lessons, and/ or unit lessons, and/ or unit lessons, and/ orstudent. provided. individual lessons are individual lessons are easy individual lessons are difficult to understand. to understand and easy to understandTeacher materials No strategies or implement; are clearly and implement; areprovide effective activities are Few strategies and written and present clearly written andstrategies and activities provided activities support the accurate concepts. present accurateto support the development of concepts.development of Teacher materials do academic English for all Some effective strategiesacademic English for all not support teachers’ students, especially and activities support the Effective strategiesstudents, especially understanding. English Learners development of academic and activitiesEnglish Learners English for all students, support theTeacher materials Few teacher materials especially English Learners development ofsupport teachers’ support teachers’ academic English forunderstanding of understanding of Some teacher materials all students,central concepts and central concepts. support teachers’ especially Englishtools of understanding of central Learners.inquiry/research. concepts and tools of inquiry/research. Teacher materials support teachers’ understanding of central concepts and tools of inquiry/research. The Dyscovery Programme |Eleanor Machin Research Documents 92
Eleanor Machin – Year 1 Final ReportTeacher materials Teacher materials do Few materials support Some materials support Teacher materialssupport teachers in not support teachers teachers in connecting teachers in connecting support teachers inconnecting concepts in connecting concepts and differing concepts and differing connecting conceptsand differing concepts and perspectives to engage perspectives to engage and differingperspectives to engage differing perspectives learners in critical and learners in critical and perspectives tolearners in critical and to engage learners in creative thinking. creative thinking. engage learners increative thinking. critical and creative critical and creative thinking. Few materials support Some materials support thinking.Teacher materials teachers in planning teachers in planning andsupport teachers in Materials do not and adjusting adjusting instruction. Teacher materialsplanning and adjusting support teachers in instruction. support teachers ininstruction. planning and Teacher materials are Teacher materials are planning andTeacher materials are adjusting instruction. somewhat organized organized with easy to adjusting instruction.easy to use. with hard to read font read font and are Teacher materials are and provide difficult to correlated with student Teacher materials disorganized with follow correlations materials. are well organized hard to read font, with student materials. with easy to read providing little or no font and excellent correlations with correlation with student materials. student materials. The Dyscovery Programme |Eleanor Machin Research Documents 93
Eleanor Machin – Year 1 Final ReportStudent Materials (photocopiables)Criteria 0 Score Descriptor 1 Score Descriptor 2 Score Descriptor 3 Score Descriptor Score Reason for Given score/Additional commentsVisual and Visuals and Visuals and Manipulatives and High qualityManipulative manipulatives do not manipulatives are visuals are of good manipulatives andAttributes contribute. Distracts acceptable and quality and complements visuals contribute to from program content. aligned with program program effectiveness. the overallStudent content. effectiveness of thematerials The materials lack 70% of the materials program.provide appropriate use of font, 50% of the materials provide appropriate use Materials provideappropriate illustrations, and text provide appropriate of font, illustrations, and appropriate use of font,print, features, (e.g., use of font, text features, (e.g., illustrations, and textillustrations and illustrations, graphs, illustrations, and text illustrations, graphs, features, (e.g.,text features. tables). features, (e.g., tables). illustrations, graphs,Material is illustrations, graphs, tables).appropriate in Very little, if any, of the tables). Less than 80% of theterms of age, materials and activities 50% or less of the materials and activities 80% or more of thegender, culture are interesting, materials and are interesting and materials and activities engaging to a range of activities are engaging to a range of are interesting and diverse students. interesting and diverse students, engaging to a range of engaging to a range of promoting purposeful diverse students, diverse students, learning. promoting purposeful promoting purposeful learning. learning. The Dyscovery Programme |Eleanor Machin Research Documents 94
Eleanor Machin – Year 1 Final ReportAdaptability of There are few or no 50% of the program 70% of the program There are appropriatematerials appropriate provides appropriate provides appropriate accommodations for accommodations for accommodations for accommodations for various developmental various developmental various various developmental levels, providing levels, providing no developmental levels, levels, providing some effective strategies to strategies to activate providing very few effective strategies to activate background background knowledge strategies to activate activate background knowledge and ensure and ensure access to background knowledge and ensure access to content content knowledge. knowledge and access to content knowledge. ensure access to knowledge.Encouraging a There is no opportunity content knowledge. The development ofsense of number to develop ‘number- Most of the programme ‘number-sense’ is Only around 50% of enables learners to clearly integral to the sense’. the programme gives develop their ‘number- entire programme. the opportunity to sense’.Numeracy There are no obvious develop ‘number- Links to maths curriculacurriculum links between the sense’, There are a number of are clearly set out in(GCSE/Functional programme and maths links to maths curricula, every session.Skills, Basic curricula. There are few obvious but some are tenuous.Skills) coverage links to any maths curricula.Adapted from http://www.schools.utah.gov/CURR/imc/Rubrics/Language-Arts-Secondary/SECONDARY-COMMON-CORE-ELA-INSTRUCTIONAL-MATERIALS-.aspx The Dyscovery Programme |Eleanor Machin Research Documents 95
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