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Nanci Smith, Ph.D., ASCD Faculty Member | nanci.smith.faculty@ascd.org| www.ascd.org© 2017] by Nanci N. Smith.

Best Practices for Elementary Math Instruction

WHICH ONE DOES NOT BELONG AND WHY?

8 4 3 2

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BEST PRACTICES

STUDENT ENGAGEMENT?

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Differentiated instruction is a teaching philosophy based on the premise that teachers should adapt instruction to student differences. Rather than marching students through the curriculum lockstep, teachers should modify their instruction to meet students’ varying readiness levels, learning preferences, and interests. Therefore, the teacher proactively plans a variety of ways to ‘get at’ and express learning.

- Carol Tomlinson

Acknowledges that students have different motivational factors. Interest differentiation can be connecting students’ interests with content, but also by giving students voice and choice.

Interest

Activities can be designed to access different ways for making sense. Some structures will be more natural for learning than others for students, and this can change based on topic and circumstances.

Learning Profile

ReadinessAcknowledges a student’s entry point into learning. Factors are prior knowledge, speed of learning new concepts, independence and home factors.

HOW DO YOU DIFFERENTIATE?

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oInterest

oLearning Profile

oReadiness

LP

Respectful

WHAT’S THE POINT?

Readiness

Growth

Appropriate Challenge

Interest Learning Profile

Motivation Efficiency

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What does READINESS mean?It is the student’s entry point relative to a

particular understanding or skill.C.A.Tomlinson, 1999

READINESS

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A FEW ROUTES TO READINESS DIFFERENTIATION

Varied texts by reading levelVaried supplementary materialsVaried scaffoldingTiered tasks and procedures Flexible time useSmall group instructionHomework optionsTiered or scaffolded assessmentCompactingNegotiated criteria for qualityVaried graphic organizers

12

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MAKE CARD GAMES!

Triplets

41

6416

=

Explain how you found your answer

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6416

=

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Select the activity organizer

•concept

•generalization

Essential to building

a framework of

understanding

Think about your students/use assessments

• readiness range

• interests

• learning profile

• talents

skillsreadingthinkinginformation

Create an activity that is

• interesting

• high level

• causes students to use

key skill(s) to understand

a key ideaChart the complexity of the activity

High skill/

Complexity

Low skill/

complexityClone the activity along the ladder as

needed to ensure challenge and success for your students, in

• materials – basic to advanced

• form of expression – from familiar to unfamiliar

• from personal experience to removed from personal experience

• equalizer

Match task to student based on student profile and task requirements

1

3

5

2

4

6

DEVELOPING A TIERED ACTIVITY

Task 1: Find a way to count & show how many people are in

our class today. How did you get your answer?Task 2: Find a way to show how many people are in our class.

How many are absent today? How many are here today? How do you know?

Task 3: Find a way to show how many boys are in our class

today. How many boys are absent today? How many girls are here today? How many girls are absent today? Prove you are right.

KINDERGARTEN COUNTING

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ADDING FRACTIONS

Green GroupUse Cuisinaire rods or fraction circles to model simple fraction addition problems. Begin with common denominators and work up to denominators with common factors such as 3 and 6.

Explain the pitfalls and hurrahs of adding fractions by making a picture book.

Blue GroupManipulatives such as Cuisinairerods and fraction circles will be available as a resource for the group. Students use factor trees and lists of multiples to find common denominators. Using this approach, pairs and triplets of fractions are rewritten using common denominators. End by adding several different problems of increasing challenge and length.

Suzie says that adding fractions is like a game: you just need to know the rules. Write game instructions explaining the rules of adding fractions.

Red GroupUse Venn diagrams to model LCMs (least common multiple). Explain how this process can be used to find common denominators. Use the method on more challenging addition problems.

Write a manual on how to add fractions. It must include why a common denominator is needed, and at least three ways to find it.

All Groups play “Guess the Quadrilateral” in pairs, triads or quads.1. One person thinks or chooses a quadrilateral.2. The remaining members of the group (or partner) ask yes / no questions about the properties of the quadrilateral.3. The group tries to guess the quadrilateral based on the answers to the question in less than ___ questions. (You can vary the number allowed.)

o Average Group:o Play the game as stated above.

o Struggling Group:o A list of quadrilaterals and basic properties is provided for reference. The degree

of completeness of the list can be determined by the teacher.

o Advanced Group:o After guessing the quadrilateral, the group sketches the quadrilateral and lists all

properties that belong to it.

QUADRILATERAL REVIEW

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© Kay Brimijoin, 2003

THINK DOTS

Nanci Smith

Describehowyouwould Explainthedifference

solve orroll betweenaddingand

thedietodetermineyour multiplyingfractions,

ownfractions.

Compareandcontrast Createawordproblem

thesetwoproblems: thatcanbesolvedby

+

and (Orrollthefractiondieto

determineyourfractions.)

Describehowpeopleuse Modeltheproblem

fractionseveryday. ___+___.

Rollthefractiondieto

determinewhichfractions

toadd.

53

51+

21

31+

1511

52

31

=+

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Nanci Smith

Nanci Smith

Describehowyouwould Explainwhyyouneed

solve orroll acommondenominator

thedietodetermineyour whenaddingfractions,

ownfractions. Butnotwhenmultiplying.

Cancommondenominators

Compareandcontrast everbeusedwhendividing

thesetwoproblems: fractions?

Createaninterestingandchallengingwordproblem

Acarpet-layerhas2yards thatcanbesolvedby

ofcarpet.Heneeds4feet ___+____- ____.

ofcarpet.Whatfractionof Rollthefractiondieto

hiscarpetwillheuse?How determineyourfractions.

doyouknowyouarecorrect?

Diagramandexplainthesolutionto___+___+___.

Rollthefractiondieto

determineyourfractions.

911

73

132

++

71

73 and

21

31

++

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BUILD – A – SQUARE

o Build-a-square is based on the “Crazy” puzzles where 9 tiles are placed in a 3X3 square arrangement with all edges matching.

o Create 9 tiles with math problems and answers along the edges.

o The puzzle is designed so that the correct formation has all questions and answers matched on the edges.

o Tips: Design the answers for the edges first, then write the specific problems.

o Use more or less squares to tier.o Add distractors to outside edges and

“letter” pieces at the end.

m=3

b=6 -2/3

Nanci Smith

What does INTEREST mean?

Discovering interest is important;Creating interest is even

more important.

Inventing Better Schools, Schlechty

INTEREST

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Choices vs. Requiredcontent, process, product no student voice

groups, resources environment restricted resourcesRelevant vs. Irrelevantmeaningful impersonal

connected to learner out of contextdeep understanding only to pass a test

Engaging vs. Passiveemotional, energetic low interaction

hands on, learner input lecture seatworkEQUALS

Increased intrinsic IncreasedMOTIVATION APATHY & RESENTMENT

BRAIN RESEARCH SHOWS THAT…

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o Requires children to be aware of their own readiness, interests, and learning profiles.

o Students have choices provided by the teacher. (YOU are still in charge of crafting challenging opportunities for all kiddos – NO taking the easy way out!)

o Use choice across the curriculum: writing topics, content writing prompts, self-selected reading, contract menus, math problems, spelling words, product and assessment options, seating, group arrangement, ETC . . .

o GUARANTEES BUY-IN AND ENTHUSIASM FOR LEARNING!

o Research currently suggests that CHOICE should be offered 35% of the time!!

CHOICE – THE GREAT EQUALIZER

o Learning profile refers to how an individual learns best - most efficiently and effectively.

o Teachers and their students may differ in learning profile preferences.

DIFFERENTIATING USING LEARNING PROFILE

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LEFT BRAIN FUNCTIONSuses logic detail oriented facts rule words and language present and past math and science can comprehend knowing acknowledges order/pattern perception knows object name reality based forms strategies practical safe

RIGHT BRAIN FUNCTIONSuses feeling "big picture" oriented imagination rules symbols and images present and future philosophy & religion can "get it" (i.e. meaning) believes appreciates spatial perception knows object function fantasy based presents possibilities impetuous risk taking

RIGHT BRAIN VS. LEFT BRAIN

Do you see the dancer turning clockwise or counter-clockwise? If clockwise, then you use more of the right side of the brain and vice versa.

Group Orientation

independent/self orientationgroup/peer orientation

adult orientationcombination

Learning Environmentquiet/noisewarm/coolstill/mobile

flexible/fixed“busy”/”spare”

Cognitive Style

Creative/conformingEssence/facts

Expressive/controlledNonlinear/linear

Inductive/deductivePeople-oriented/task or Object oriented

Concrete/abstractCollaboration/competitionInterpersonal/introspective

Easily distracted/long Attention spanGroup achievement/personal achievement

Oral/visual/kinestheticReflective/action-oriented

Intelligence Preferenceanalyticpracticalcreative

verbal/linguisticlogical/mathematical

spatial/visualbodily/kinestheticmusical/rhythmic

interpersonalintrapersonal

naturalistexistential

Gender

&

Culture

LEARNING PROFILE FACTORS

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Visual• Take numerous detailed notes

• Tend to sit in the front • Are usually neat and clean

• Often close their eyes to visualize or remember something

• Find something to watch if they are bored • Like to see what they are learning

• Benefit from illustrations and presentations that use color

• Are attracted to written or spoken language rich in imagery

• Prefer stimuli to be isolated from auditory and kinesthetic distraction

http://www.usd.edu/trio/tut/ts/styleres.html

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Auditory• Sit where they can hear but needn't pay attention

to what is happening in front • May not coordinate colors or clothes, but can

explain why they are wearing what they are wearing

• Hum or talk to themselves or others when bored • Acquire knowledge by reading aloud

• Remember by verbalizing lessons to themselves (if they don't they have difficulty reading maps or diagrams or handling

conceptual assignments like mathematics). http://www.usd.edu/trio/tut/ts/styleres.html

Kinesthetic• Need to be active and take frequent breaks • Speak with their hands and with gestures

• Remember what was done, but have difficulty recalling what was said or seen

• Find reasons to tinker or move when bored • Rely on what they can directly experience or perform

• Activities such as cooking, construction, engineering and art help them perceive and learn

• Enjoy field trips and tasks that involve manipulating materials • Sit near the door or someplace else where they can easily get up

and move around • Are uncomfortable in classrooms where they lack opportunities for

hands-on experience • Communicate by touching and appreciate physically expressed

encouragement, such as a pat on the back http://www.usd.edu/trio/tut/ts/styleres.html

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MODALITY TASK PROMPTSVisual Auditory Kinesthetic

Pictures Speeches Matching games

Graphic Organizers Discussions Modeling

Color coding Infomercials or PSAs “Becoming” the task

Posters Creating Question Lists Hands-on tasks / touching

Charts / Graphs Read Alouds “Peg Board” yarn game

Videos Books / Instructions on tape Gestures and Motions

Detailed Notes Self Talk (Whispies) Motion

Visualizing Tape Recording Answers Drama / Skits

Making Books Interviews Charades

To Do Lists Lectures / Tone & Inflection Manipulatives

Written Directions Spoken Directions Modeled Directions

o Visual:o Make a poster to show the place value in a number. Roll two

(three) dice and form a number. Record your number and show at least two different models for the place value of your number. Answer: How many (hundreds), tens and ones?

o Auditory:o Play “Guess My Number” with a partner. On your game sheet,

write down a number, and how many tens and ones are in your number. Tell the tens and ones to your partner who will guess your number. Now trade roles.

o Kinesthetic:o Build numbers using two different models. Record your

models and your numbers.

PLACE VALUE

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BEYOND IQ

o Robert Sternberg

use of intellect rather than quantity of it

intelligence as problem-solving

analytical

practical

creativeTriarchic Theory

Linear – Schoolhouse Smart - SequentialANALYTICALShow the parts of _________ and how they work.Explain why _______ works the way it does.Diagram how __________ affects __________________.Identify the key parts of _____________________.Present a step-by-step approach to _________________.

Streetsmart – Contextual – Focus on UsePRACTICALDemonstrate how someone uses ________ in their life or work.Show how we could apply _____ to solve this real life problem ____.Based on your own experience, explain how _____ can be used.Here’s a problem at school, ________. Using your knowledge of ______________, develop a plan to address the problem.

CREATIVE Innovator – Outside the Box – What If - Improver

Find a new way to show _____________.Use unusual materials to explain ________________.Use humor to show ____________________.Explain (show) a new and better way to ____________.Make connections between _____ and _____ to help us understand ____________.Become a ____ and use your “new” perspectives to help us think about ____________.

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o Bulletso Listso Stepso Worksheetso Tableso Venn Diagramso Timelineso Sequential Itemso Flow Chartso Compare and Contrast

• Findtheerror• Evaluating• SortingandClassifying• Appealingtologic• CritiqueandCriticize• ExplainingDifficultProblemstoothers

• MakingInferencesandDerivingConclusions

• PunsandSubtleties

ANALYTICAL

• Chart

• Graphic organizer

• Timeline

• Venn diagram

• T-chart

• Patterns

• sequencing

• Classifying• Definitions• Cause and effect• Code• Graph• Database• Blueprints• Newspaper• Fact file

ANALYTICAL

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• Working your way out of a problem• Notes to Self (what questions to ask myself, how to make sense of for

myself)• Here is a problem, explain what happened• Analogies• Draw real world examples• Advising and convincing others (Advice columns)• Hands-on Activities• Taking things apart and fixing them• Understanding and Respecting others / Friendships / Resolving

Conflicts• Putting things into Practice• Adapting to New Situations

PRACTICAL

• Explaining how things can be used

• Developing a plan to address a problem

• Help classmates understand

• Scenarios

• Role plays

• WebQuest

• Job shadowing

• Dialogs

• Newscasts

• Letters to the editor

• Flyers

• Demonstrations

• Experiments

• Surveys

• Field trips

• Petitions

• “Cheat sheets”

• Lesson plans

PRACTICAL

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• Figure out a way to explain• Idiot’s Guide To… (Book for Dummies)• How to represent• Make your own interpretation• Pictures or news bulletins to describe• Designing new things• Alternative solutions and methods• Thinking in pictures and images• Noticing things other people tend to ignore• Suppose something was changed… What would happen if?• Acting and Role playing• Inventing

CREATIVE

• Become a … and use your new perspectives to help us think about…

• Use humor to show…

• Explain or show a new and better way to…

• Figure out a way to explain…

• Pictures, picture books, doodles and icons

• Songs

• Riddles

• Mime or charades (think vocabulary!)

• Play

• Bumper stickersSmith, 2008

CREATIVE

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SOME VERBS THAT MIGHT HELP

Analytical Practical Creative

AnalyzeJudge

CritiqueCompareContrastEvaluateDiagram IdentifyExplainAssess

Present a step-by-step approach

ImplementApply

UseDemonstrate

TeachPut into practice

ConvinceShow how

EmployRelate to experience or

worldMake practical

InventDiscoverImagineSuppose

DesignPredict

PromoteEncourage

DevelopWhat if you (were)…

Find a new wayUse unusual materials

Analytical Task

Practical Task

Creative Task

Make a number chart that shows all ways you can think of to show 5.

Find as many things as you can at school and at home that have something to do with 5. Share what you find with us so we can see and understand what you did.

Write and/or recite a riddle poem about 5 that helps us understand the number in many, unusual, and interesting ways.

UNDERSTANDING NUMBER

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Analytic Task

Practical Task

Creative Task

Make a chart that shows all ways you can think of to use order of operations to equal 18.

A friend is convinced that order of operations do not matter in math. Think of as many ways to convince your friend that without using them, you won’t necessarily get the correct answers! Give lots of examples.

Write a book of riddles that involve order of operations. Show the solution and pictures on the page that follows each riddle.

Nanci Smith

ORDER OF OPERATIONS

Fairness is noteveryone getting the same

thing.It is everyone getting

what they need.

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BURNING QUESTIONS?

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EVALUATE THE EXPERIENCE

Below is a link to ASCD’s Professional Learning Evaluation. We encourage all participants to complete the online evaluation at the

conclusion of the workshop. All responses will be anonymously reported to ASCD.

www.ascd.org/ascdpleval

Session PIN = NNS3

Thank you for taking the time to honestly evaluate the program. The results we receive help us to improve the quality of services we

provide.