CURRICULUM EVALUATION

Citation and Skill Focus  Charles, R. I., et al. (1999). Math, Teacher’s Edition, Vol 2. New York: Scott Foresman-Addison Wesley. (Note: This is the second volume of the 1 st grade edition.)  Reading and writing numerals above 10

Strategy 1. Look in the table of contents and identify the likely lessons where the skill is being taught. 2. Look for a skills trace or scope and sequence that addresses where skills are taught for more information. 3. Limit or expand scope based on how many lessons you find.

Strategy cont. 4. Review lessons, focusing on issues addressed in Chapter 2 and on the instructional guidelines in the relevant chapter of the text. Strategies explicit and generalizable? Teaching procedures of high quality? (modeling, adequate examples, scaffolding) Sequence appropriate? Examples, practice and review adequate? Assessment aligned and frequent?

Table of Contents

Skills Trace First introduced IntroducedDevelopPractice/ Apply Review Make and write numbers to 19 Grade K , , 271 Count, add, and write groups of 10 Grade 1259 S60, , 271, , 420, 532 Write numbers to 60 as tens and extras Grade , , , 356, 420

Expand or Limit?  3-5 lessons…  Lessons 7-1, 7-2, 7-3.

Strategies?  Make and write numbers to 19  Use double ten frame. Give each student 19 counters.  “Have children count out 13 counters and place them on Workmat 3. How many ten frames did you fill? How many counters are left over? What number shows 10 and 3?”  Explicit?  Not bad. Strategy is replicable and steps are clear.  Generalizable?  Appropriate at conceptual stage. No focus on teaching reading and writing teen numbers without ten frame.

Strategies? Count and add groups of 10. Write the decade number. “For each number, ask children the following questions: How many counters are in each ten frame? How many ten frames are there? How many counters are there all together?” Explicit? Ok for teaching tens place value skill. Again, no model. Generalizable? Again, appropriate at conceptual stage, but not taken to the next level. They only say “Guide children to see that an easy way to add 20 and 10 would be to think: 2 tens and 1 ten is 3 tens, or thirty.”

Strategies?  Write numbers to 60 as tens and extras. “Explain that 2 groups of ten and 6 extras make 26. Help children verify by counting each object in the picture.” Then students have 3 more opportunities to count tens and extras to achieve a total. Explicit? Not very. (better would be…”first I count groups of tens, next…) Generalizable? Not very. Students are still learning in context of counting rather than focusing on place value and column alignment.

Teaching Procedures? For all skills Almost no modeling. Usually only one example. Teacher assistance is not gradually faded. It appears to go from high support to no support. Error correction is available, but not extensive. For example, in “Make and Write Numbers to 19” Observation: Children may have trouble counting out given numbers of counters. How to Help: Count aloud with the children as they touch or move the counters.

Sequencing General Guidelines  Preskills are taught before they are needed in strategies.  Easy skills are taught before more difficult ones.  Strategies and information that is likely to be confused are spaced or separated.

Sequencing  Make and write numbers to 19 (recommended from Stein text)  Preskills Reading: Read numerals between 0 and 10, rational counting of 2 groups. Writing: Read teen numbers accurately and fluently (i.e., 5 teen numbers at a rate of approximately 1 per sec.)  Sequence of instruction similar to reading teen numbers. Introduce first irregular teen about 2 days after regular teens; Then introduce at a rate of about 1 per day if students are successful.

Sequencing  Make and write numbers to 19 (from AW text)  Preskills taught:  Read and write numbers to 12  Addition  Sequence  All teens numbers introduced at once

 Sequencing  Count and add groups of 10. Write the decade number.  Preskill (recommended from Stein text): Count by 1s to 100; Skip count by 10s to 100  From AW text: Not clear where or if these preskills are addressed

Sequencing  Write numbers to 60 as tens and extras.  Preskill (recommended from Stein text): Read teens numbers accurately and fluently (5 teens numbers in 8 seconds); Count by 1s to 100; Skip count by 10s to 100; tens place value facts.  From AW text:  Teen numbers introduced, but not to fluency  Not clear whether Count bys and Skip counting were introduced.  Tens place value facts in previous lesson, but not to mastery.

Practice and Review  Initially massed to solidify students knowledge  Opportunity to practice discriminating between similar skills or concepts.  Distributed  Revisited over time  Accumulated  Concepts that are initially taught separately are reviewed together  Varied  Concepts are applied to a range of applications to promote generalization

Practice and Review First introduced IntroducedDevelopPractice/ Apply Review Make and write numbers to 19 Grade K , , 271 Count, add, and write groups of 10 Grade 1259 S60, , 271, , 420, 532 Write numbers to 60 as tens and extras Grade , , , 356, 420

Practice and Review  Make and write numbers to 19  9 practice pages total 16 items for introduction Practice page: 8 items Reteaching page: 4 items Enrichment page: 2 items Problem Solving page: 5 items Practice Game: unlimited Stop and Practice: 7 items Mixed practice: 4 items

Practice and Review  Count and add groups of ten  13 pages total Introduction: 15 items Practice page: 11 items Reteaching page: 8 items Enrichment page: 7 items Problem Solving page: 5 items Mixed Practice: 1 item Cumulative Review: 1 item Skill Practice Bank: 6 items

Assessment and Instruction Link  Placement tests?  No  Recommendations for acceleration and remediation?  Yes, but limited “Another way to learn” section “Options for reaching all learners” Enrichment and problem solving pages Reteaching page  Assessments carefully aligned with instruction?  Not really. Assessment in “Close and Assess” section

Conclusions  Strategies are reasonably explicit, but not highly generalizable  Teaching procedures are not well-articulated, although there are some error correction procedures  Sequencing is weak. Preskills possibly taught, but not to mastery. New skills not sequenced well.  Practice and review needs work. Perhaps adequate initial practice, but not for discriminative, distributed, and varied.  Assessment needs to be more frequent and aligned.

Adaptations for Students in Special Education  Add more modeling of strategies  Provide more scaffolding (leading) on how to apply strategies  Assess preskills and make sure they are firmly taught.  Improve sequence of instruction. For example, introduce regular teens first, then irregular.  Add additional practice and review. Focus on appropriate discriminative practice. Distribute over time.  Plan for more frequent assessment opportunities. Align to instruction more closely.

ADDITION & SUBTRACTION

Precise Definitions What is addition? What is subtraction?

Precise Definitions  Addition  Combining of objects and counting the number of objects that results  Subtraction  Removing sets of objects and counting the number of objects that results Milgram, 2005

Math Vocabulary for Teachers  Addend, sum  Minuend, subtrahend, difference  Renaming or regrouping (rewriting a number as a greater and lesser unit (75 = 7 5 = 6 tend + 15 ones = )  Regrouping (uses manipulatives or counters)  Trading (current term – linked conceptually to place value)  Carrying and Borrowing (older terms – conceptually misleading)  Commutative, associative, identity, distributive properties (or laws or axioms)

Math vocabulary for students  Equal, equality  Add, plus, sum, how many  Subtract, take-away, minus, difference  Trade, regroup, rename, carry, borrow  Bigger bottom borrows  Commutative and Associative (order) property (rule) of addition and multiplication

Concrete  Representational  Abstract  Concrete: physical objects used to model mathematical concepts (Base 10 blocks, Cuisenare rods, fraction circles).  Representational (semi-concrete): pictures, marks, stamps used to represent objects.  Abstract: numbers, notation, symbols

Semi-concrete objects  Why does your text recommend the use of semi- concrete (representational) objects (lines)?

What to do?  Research is not clear about whether concrete objects are more effective for students with disabilities.  Neither objects nor images provide conceptual understanding.  How they are used by the teacher with what intent is likely more important.

Sequencing Skills and Strategies: CRA Beginning Addition and Subtraction  Add single digits w/ concrete models  Demonstrate commutative property with concrete models  Add single digits w/ representational models  Addition fact memorization  Subtract single digits w/ concrete models  Subtraction fact memorization  Repeat above addition sequence with sums to 18  Repeat with subtraction with minuends to 18  Solve simple addition and subtraction word problems Concrete/ Conceptual Representational

Addition and Subtraction (CRA) Concrete  Representational  Abstract Number Lines / Hundreds Chart Counter/ Fingers / Place Value Models 

Using Concrete Manipulatives  Introduce manipulatives and explain their use.  Set ground rules and differentiate manipulatives from toys & games. Clarify how students should interact with manipulatives & each other.  Set up system for storing materials and familiarize students with it.  Clearly identify what conceptual information you are trying to teach.  Plan adequate time for their use. Adapted from Burns

Addition (CRA) Concrete  Representational  Abstract Addition the Slow Way = (DI Format 7.2 p.112) Missing Addends 4 + = 6 (DI Format 7.3 p.113) Addition the Fast Way* = (DI Format 7.4 p ) 8 2 8

Addition the “Slow Way”  What are the preskills?

Addition the “Slow Way”  What examples should one include?

Missing Addend Addition  Based on what rule?

Addition the “Fast Way”  When are the students ready for addition the fast way?  What potential pattern of errors might the students make?  How do you remedy this error?

Sequencing  When can you begin subtraction (concept)?  When can you start addition facts instruction?

Addition (CRA) Concrete  Representational  Abstract  Fact Memorization - Declarative

Subtraction  First Stage—conceptual and simple problems  Multi-digit stage—3 types of column subtraction 1. without “borrowing”, 2. simple borrowing problems, and 3. complex with multiple borrowing and/or zero

Subtraction (CRA) Concrete  Representational  Abstract Subtraction with Lines (DI Format 8.1 p ) = 2 I I I I I Missing subtrahend = 2

Introducing the Concept of Subtraction  How do students use the “crossing-out” strategy? 6 – 4 =  1) 2) 3)

Introducing the Concept of Subtraction  Example selection  Format 8.1: What is the difference between the examples in the structured worksheet and the less structured worksheet? Why?

Missing Subtrahend Problems  When do you teach them?

Subtraction (CRA) Concrete  Representational  Abstract  Fact Memorization - Declarative

CRA for Properties of Numbers Commutative Property a + b = b + a Associative Property (a + b) + c = a + (b + c) Identity Property Addition / Subtraction = 0 Concrete/ Conceptual Representational

Concrete  Representational  Abstract (CRA)  Pictorial representations to demonstrate the commutative property  Example from Russian 1 st grade text Milgram, 2005

Concrete  Representational  Abstract (CRA)  Pictorial representations to demonstrate commutative property Milgram, 2005

Concrete  Representational  Abstract (CRA)  Pictorial representations to demonstrate addition and subtraction as inverse operations Milgram, 2005

Getting from C to R to Abstract Operations with multi- digit numbers with place value models / base ten blocks Operations with multi- digit numbers with place value pictures Procedural Algorithms Concrete/ Conceptual Representational

Abstract: Procedural Strategies for Operations  New prerequisite skills  Facts – more on this later in the quarter  Place value (and renaming)  Reading & writing larger numbers

Teaching algorithms  Should students be taught traditional algorithms? Why?  How does conceptually understanding a procedural algorithm benefit students?

Precise Definitions  Addition and Subtraction  Combining or removing sets of objects and counting the number of objects that results  Rationale for addition and subtraction algorithms  Replaces the cumbersome adding on or counting backward for larger numbers Milgram, 2005

Concrete and Representational Models for Multidigit numbers - Place Value Models

Conceptual Understanding with Expanded Notation

Abstract Traditional algorithms for multidigit operations Addition  Column add with no renaming (no format)  Column add, 2 digit, renaming (format 7.6)  Column add 3+ digits, renaming (format 7.6 with modifications)

Multi-digit Addition without Renaming  When can these problems be introduced?  How?  Students read the problem  Teacher tells students that we start adding in the ones column and then the tens (Why?)  Students write the answer in each column

Multi-digit Addition with Renaming  What are the preskills?

Multi-digit Addition with Renaming  Adding three single-digit numbers—Format 7.5  What are the example selection guidelines for these problems?

Multi-digit Addition with Renaming Format Students read the problem 2. Identify where to start adding (ones) 3. Add the ones and determine if they must rename 4. Use expanded notation to determine the number for the tens and ones column 5. Write the renamed number and ones number 6. Add the first two numbers in then tens, then add the next number to the sum 7. Write the tens number

Multi-digit Addition with Renaming Format 7.6 What is the common error? What should the teacher do?

Multi-digit Addition with Renaming Format 7.6 Example selection for Structured Board and Structured Worksheet? Example selection for Less Structured Worksheet?

3 or More Multi-digit Addends with Renaming  Why are these particularly difficult?

3 or More Multi-digit Addends with Renaming  How are the complex addition facts sequenced?

Abstract Traditional algorithms for multidigit operations Subtraction  Column subtraction with no renaming  Column subtraction, 2 digit, renaming (format 8.3)  Column subtraction complex renaming (format 8.5)

Subtraction without renaming  How taught?

Subtraction with Renaming  Preskills?

Subtraction with Renaming  Format 8.2—concept of regrouping (semi concrete)

Subtraction with Renaming  Format 8.3  Part A: What is the rule? Example selection?  Part B: Borrowing component skill

Subtraction with Renaming Format 8.3 C—computation summary 1. Read the problem 2. Determine if we must rename 3. Borrow the ten and put it with the ones 4. Subtract the ones column 5. Subtract the tens column

Subtraction with Renaming  Format 8.3  What types of problems should one include on less structured?

Subtraction with renaming  Renaming from tens  ¾ subtraction; ½ require renaming  ¼ addition  Renaming from 100s  Mostly subtraction ½ rename from 100s ¼ rename from 10s ¼ no renaming

Complex Renaming Problems  Problems requiring renaming more than once (without zeros)  Possible errors?

Complex Renaming Problems  Problems with zeros:  Strategy?  Preskill?  Format 8.5: A—structured board, B—structured worksheet, C—less-structured worksheet

Complex Renaming Problems  Format 8.5: C—less-structured worksheet  What are the example selection guidelines?

Diagnosis and Remediation 4 Steps  Diagnosis: Analyze pattern of errors; if necessary ask student to solve a problem “thinking aloud”  Determine type of pattern of errors (fact, component, or strategy)  Determine how to re-teach/remedy  Determine examples (problems)

Pattern of Errors--Facts  Most common  Pattern of errors—what do you look for?  How do you remedy missing the same fact?  How do you remedy inconsistent fact errors?

Pattern of Errors—Component Skill Example: “Carrying” the wrong number Remediation (Go back to teaching the component skill):  Reteach expanding notation for the total in the ones column  Practice examples can have a box for the carried number and ones number  Practice examples should include problems with and without renaming

Pattern of Errors—Strategies  Example: Not regrouping  Reteach: For all strategy errors reteach the format for that particular strategy  Examples: Structured board, structured worksheet, then less structured.  Then a worksheet similar to original

Other Algorithms for Addition  Add each column in any order  Adjust

Other Algorithms for Subtraction  Trade first then subtract

Alternative / Bypass Strategies  Number line for add and subtract  Finger Strategies  counting up  Calculators

Alternative / Bypass Strategies Touch Math

Factors to consider for alternative strategies…  Goals  Conceptual development v. procedural fluency and accuracy?  Age of the student  Instructional priorities  Other strategies tried  Ease of use  Availability