Considerations around Math Instruction for Students with Disabilities1
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Chapter Two: Place Value
If anybody doesn’t have the book, there are extra’s in the cabinet in Room 170
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Outcomes for Session Four
Participants will understand the scope and sequence of the development of number sense
Participants will deepen their understanding of place value in the Hindu-Arabic number system
Participants survey the various models for Least Restrictive Environment to treat number sense issues
Participants will understand the foundation of a systematic lesson plan for math (30 minute lesson plan)
Participants will explore instructional techniques for the mastery and application of skills for number sense
Place Value
Place Value: Brahmi Numbers
The Hindu-Arabric number system that we use today can be traced back to the Brahmi numerals (India) 3rd Century BC
What number is missing from the system?
numbers 1,2,3 are the number of stokes to equal the number (same as modern Japanese number system
number 4 is marked by the four point in the cross
historians are not sure about the origin of the rest of the numbers; some think that they were distinct marks that were combined like a system of cursive writing
Talking about the history of the current number system, helps to understand the evolution of the number 0 that is used today to denote place value. Remember the number 0 is not a number that exists on our internal number line. This was invented to makes number writing efficient. During this time, each group of tens had it own symbol. There was a different symbol for 20, 30, 40, etc., thus making the number system more cumbersome.
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1944
1999
MDCCCCLXXXXVIIII
MCMXCIX
MIM
In the Roman Number system, place value was denoted by a symbol. Zero was not used to denote place value. Letters were used to denote value. Symbols are placed in order of value, starting with the largest value first. There are no standard rules to writing in Roman Numbers. For the number 1999 there are three possible ways. This was confusing and was gradually replaced by the Arabic numerals.
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Place Value: Arabic Numerals
Arabic numbers were introduced to Europe in the 10th century by Arabs of North Africa. There are many numbers systems in the Arab world. These are the numbers systems introduced from North Africa.
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Irrational Number
Digit comes from Latin (digita meaning fingers)
10 Symbols of the base 10 number system
0123456789
a number is one or more digits
The Arabic number system used a Zero, which eventually was developed by Babylonian mathematics but it was typically just a blank space, then became wedges, to eventually the symbol we used today 0. The Babylonian system was base-60 and still used today in telling time.
The Zero is both a number and a digit. A number is a mathematical object used to count. Numbers include zero, negative numbers, rational numbers, irrational numbers, real and complex numbers.
The Zero also is a digit in the a number system. In the base 10 system you have the digits 0-9. In a base 2 system (binary) you have the digits 0 and 1).
The Zero made writing numbers much easier and efficient. Less digits were required, especially for large numbers. It became the preferred method in Europe over the Roman Number system.
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Place Value: Positional Systems and Fraud?
Positional systems or place values systems with the use of an absolute 0 were controversial. One could simply put a number before or after the string of digits thus causing fraud. This is why when writing checks we must write out the words as well as using the digits to represent the amount.
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themselves.
The location of the digits in relationship to each other, helps us to determine the value
80 + 4
The number 84 has two digits. The 8 and the 4. The placement of the digits tells us it value.
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Place Value: Language
The language of place value can be confusing for students with any type of language processing difficulty. What does ten thousands really mean? This abstract concept can lead to problems with place value understanding.
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Solve this problem…
Express 5 hours and 20 minutes as 4 hours and ______ minutes.
Once a student is use to base 10 systems, we throw strange system at them like telling time. This can create confusion for students.
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Common Errors for Place Value
Confusion when to use rules to group objects, trade objects or compute
These are pieces of evidence to indicate an error in place value understanding, which is a type of number sense error.
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Common Errors for Place Value
Content gaps and language barriers
These are pieces of evidence to indicate an error in place value understanding, which is a type of number sense error.
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Common Errors for Place Value
Content gaps and language barriers
These are pieces of evidence to indicate an error in place value understanding, which is a type of number sense error.
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Misapplies the procedure for “counting on”
Counts this as 10, 20, 30, 1, 2
Instead of Counts this as 10, 20, 30, 31, 32
These are pieces of evidence to indicate an error in place value understanding, which is a type of number sense error.
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Does not identify multi-digit numbers in terms of place value
reads the number correctly
but…
3 + 2
instead of
30 + 2
When asked the values of the numbers the child says “3” and “2” instead of “30” and “2”
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Does not understand that place determines value
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Misapplies the rule for reading numbers from left to right
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eighteen
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Common Errors for Place Value
Student orders numbers based on the value of the digits, instead of the place value.
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because 6 and 9 are bigger than 1 and 2
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Lacks understanding of positional systems
Struggle with trading groups (when you reach 10 units then trade for a 10 bar)
Poor understanding of place value structure
make sure it isn’t due to the following
Poor number sense (instant recognition of number, one-to-one correspondence, counting forwards and backwards, skip counting , poor cardinality)
Place Value: Conceptual Errors
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4000700208
The child is asked to write out a set of numbers. This child express is each digit as a separate Cardinal value rather than multiplying each by its place value. (4 x 1000). For example, he writes “50” instead of “ 5” to indicate the number of tens in “57”. This child lacks a conceptual understanding of place value in that his notation is not corrected to any sense of a positional system.
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They are equal
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215
They are equal
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Can you figure out why he is making these mistakes? There could be a number of things going on however the most prominent one is that he is reading numbers both right to left and left to right. In question number one he thinks that 13 and 31 are the same number. This is a strong indication that this child lacks an understanding of place value.
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Scope and Sequence of Number Sense Development
In a group of 2-4 develop a scope and sequence of Number Sense Development based on what we have studies in class so far.
Write one skill on a separate post-it using a bold marker.
Arrange the post-it in order from the most basic to the most advanced skill on the wall.
After all groups have completed the task, then look at each others scope and sequence.
This basic activity is designed to get a specific sequence to the development of skills that must be mastered in order to ensure success in all other mathematic domains. While there is no agreed upon specific scope and sequence, based upon the work of your group’s would there be anything that you might change from your group’s scope and sequence? Lead a discussion around the various Scopes and sequence is of skill development. Proceed to the next task where they will sequence the CCSS sequence of number sense including place value.
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CCSS and Number Sense
The common core state standards provide a very specific scope and sequence of skill development around numbers and the base 10 system.
At each table is an envelope we have a summary of various skills to be mastered at each grade level related to number sense.
Rearrange the skills from kindergarten through high school. We will check your answers in the next upcoming slides.
A copy of this activity is located at the end of the PowerPoint.
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Kindergarten
Counting and cardinality (know number names and count sequence, count the number of objects, compare numbers)
Number and operations and base 10 (work with numbers in 11-19 to gain foundations for place value)
First grade
Number in operations in base 10 (extend the counting sequence, understand place value, uses place value understanding and properties of operation to add and subtract)
Second grade
Number in operations in base 10 (extend the counting sequence, understand place value, uses place value understanding and properties of operation to add and subtract)
Third grade
Number in operations and base 10 (uses place value understanding and properties of operation to perform multi-digit arithmetic)
Number in operations-fractions (develop understanding of fractions as numbers)
Fourth grade
Number in operations in based 10 (generalize place value understanding for multi-digit whole numbers, use place value understanding and properties of operations to perform multi-digit arithmetic)
Number in operations-fractions(extend understanding of fraction equivalents and ordering, build fractions from unit fractions by applying and extending previous understandings of operations on the whole numbers, understand a small notations for fractions, and compare decimal fractions)
Fifth grade
Number in operations and base 10 (understand the place value system, perform operations with multi-digit whole numbers and with decimals to hundredths)
Number and operations-fractions (use a equivalent fractions as a strategy to add and subtract fractions, apply and extend previous understandings of multiplication and division to multiply and divide fractions)
Sixth grade
Ratios and proportional relationships (understand ratio concepts and use ratio reasoning to solve problems)
The number system (apply and extend previous understanding of multiplication and division to divide fractions by fractions, compute fluently with multi-digit numbers and find common factors and multiples, apply in extend previous understandings of numbers to the system of rational numbers)
Seventh grade
Ratios and proportional relationships (analyze proportional relationships and use them to solve real-world and mathematical problems)
The number system (apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide all numbers)
Eighth grade
The number system (I know that there are numbers that are not rational, an approximate them by rational numbers)
High school
The real number system (extend the properties of exponent stew rational exponents, use properties of rational and irrational numbers)
Quantities (reason quantitatively and use units to solve problems)
The complex number system (performs arithmetic operations with complex numbers, represent complex numbers and their operations on the complex plane, use complex numbers and polynomial identities and equations
This slide is a animated. Each click will reveal the different grade level expectations. Go through each one and the groups can self check to see if they got it in the correct order.
Provide a hand out that contains a combination of preschool and school aged skills for a scope and sequence of number sense. This will be used to develop intervention for students struggling with this skill.
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Models for LRE for math specially designed instruction in number sense
Mild number sense disability
Usually has instant recognition of number, carnality, and counting ability.
Struggles with place value.
-Adaptation of general education materials (Manipulatives ) -Alternative presentations and demonstration of concepts by GEN Ed teacher (Universal design for learning ) -Additional instruction and practice in general education in the area of place value (tutor, peer assistance, small group) -Special education teacher provides consult
-Adaptation of general education materials (Manipulatives ) -Alternative presentations and demonstration of concepts by GEN Ed teacher (Universal design for learning ) -Pull out for 10 to 15 minutes to the back of the room for direct instruction in place value by special education teacher
-Co-teaching the lessons, with special education teacher providing alternative presentation of the concepts to whole group or small group -Special education teacher suggesting alternative presentation and demonstration of learning techniques -Special education teacher providing small group or one on one instruction in the back of the room
Go over each one of these possible intervention options for a student with a mild number sense disability. After going over these options ask the group to comment. Are you doing any of these options with your students? Is there something else that you are doing with your mild students?
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Moderate number sense disability
Knows the name of numbers, has rote counting and 1 to 1 correspondence
Struggles with instant number recognition, inconsistent cardinality, and struggles with place value to the number 10
Option One
Option Two
Option Three
-Pull out for math instruction -Alternative approach to curriculum -Analyze each lesson to consider if it meets an essential goal for the grade level -Conduct lesson using alternative presentation and demonstration -Eliminate lessons that do not meet an essential goal for grade level -Include daily systematic routines to develop automaticity in basic number sense skills
-Pull out for math instruction -Rearrange the curriculum so that one concept is developed at a time to mastery level -Include daily systematic routines to develop automaticity in basic number sense skills
-Co teaching with pull out to the back of the room for certain topics -Allow student to participate in whole group instruction to be exposed to the concept -Special education teacher provides suggestions on alternative presentation and demonstration --Include daily systematic routines to develop automaticity in basic number sense skills
Go over each one of these possible intervention options for a student with a mild number sense disability. After going over these options ask the group to comment. Are you doing any of these options with your students? Is there something else that you are doing with your mild students?
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Has basic number knowledge to five
Has no instant number recognition, cardinality, magnitude of number, and place value concepts
Option One
Option Two
-Alternative core curriculum designed for students with adaptive functioning concerns
-Pull out for math instruction -Rearrange the curriculum so that one concept is developed at a time to mastery level -Include daily systematic routines to develop automaticity in basic number sense skills
Go over each one of these possible intervention options for a student with a mild number sense disability. After going over these options ask the group to comment. Are you doing any of these options with your students? Is there something else that you are doing with your mild students?
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3rd Grade
Struggles with instant recognition of 6-9 on 10 frame
Doesn’t understand the base 10 system with trading 10 ones for 1 ten
Poor number formation through handwriting (reverses 2 and 7)
This is Sam. Sam is in third grade and has a specific math learning disability in math, that is rooted in weak number sense and recall of math concepts.
We are going to quickly look at a weeks worth of Everyday Math lessons for Third Grade. Keep Sam in mind when we look over these lessons. What do you think he will be successful with? What is he going to struggle with? What accommodations does he need? What should his instruction look like?
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In this lesson the students explore numbers that you might find in a museum. From this discovery based exercise they are to conclude that there are a variety of a type of numbers and numbers have many purposes. Then the students will work in their journal doing number sequences. Then they explore the rooms for evidence of numbers. The differentiation activity is to play a game called “Number Squeeze”.
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While a majority of the lesson is to focus on building rituals and routine, this lesson deals with the number grid. Patterns are explored through Query type of activities. Then students follow up previous lessons by doing number sequence sheets and then solve number grip puzzles. The differentiation activity has the student solving the number grid puzzles by covering up portions of a number grid with “sticky notes”
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In this lesson the students are learning how to use their reference materials to look up information. Then the students play an addition fact games called “Less Than You!”. The differentiation task as student’s scooping up a handful of pennies, counting and then comparing quantity with each other.
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In this lesson students review telling time, measurement and basic shapes. The purpose is to review the use of tools in mathematical problem solving. The differentiation task has students making a paper plate clock.
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This lesson focuses on how to record and display data. It also introduced the concept of a math box. The differentiation has the students practicing making tally marks.
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What is he going to struggle with?
What accommodations does he need?
What should his specially designed instruction look like?
Use these guiding questions to lead a discussion regarding Sam. Also refer to the models of LRE for supporting a child, What model might be most appropriate for him?
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30 Min Math Lesson: Systematic
We build houses based on blueprints. A well planned out description of how to build the house. House builders must have blueprints. Specially designed instruction also needs blueprints. This is referred to as systematic instruction or well though out sequence of instruction and well established routines that focus on understanding, mastery, fluency and application of learning.
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30 Minute Lesson Plan: Systems
The 30 minute lesson plan is adapted from the work of Susan Hall in the area of reading. Since most specially designed instruction occurs in blocks of 30 minutes, the 30 minute lesson plan is designed to maximize the short about of time to get the most out of instruction and is designed for mastery, understanding, fluency and application.
Systematic: Each step is done the same way every day, building one skill onto the next. Use a time to make sure you are not taking too long on one step. Make sure all materials are ready before the lesson for seamless transition between the skills.
Students who receive their instruction in a systematic way, are higher achievers.
This can be adapted for 15 min, 45 min, 60 min and 90 min lesson plans based on needs.
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Application (10 min)
Fluency of previous over learned skill (5 min)
The 30 Minute Lesson plan
State your goal/purpose
Review Previous Lessons
Accelerated version of previous skills designed to activate their background knowledge
Might be fluency drills
Pass out the one page 30 min lesson plan. Go over each section.
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3. New Lesson
1. Sometimes the same lesson must be repeated in this section because they didn’t get it the first time ( or first 3 times)
2. Use proven strategies for teaching new skills
a. CRA
b. Visualization
c. Multi-Sensory
d. Discovery
4. Practice
1. I do- Model the practice they need to do
2. Do it together
3. Gradually turn over the work to them with supervision
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5. Fluency Practice
1. I do- Model the practice they need to do
2. Do it together
3. Gradually turn over the work to them with supervision
It is very important that only previously taught skills are worked on for fluency. The purpose is rapid retrieval of already learned skills. These should be rapid fire drills with the purpose of increasing speed along with accuracy.
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The 30 Minute Lesson plan
5. Problem Solving: The whole purpose of doing the skill is to eventually solve problems. This can be an instructional task or a simple story problem.
1. Teach strategies to address word problems
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All have number sense issues and weak place value concepts
Following the Mathematics Navigator Scope and Sequence supplemented with Hands on Standards manipulatives and application
Complete daily “Instant Number Recognition” tasks to provide repetition and rapid recall
All students participate in the large group lesson with accommodations, does a modified math box (some boxes are eliminated and others added), participates in the differentiated Everyday Math lesson
Pulled to the back of the room for the second half of the Math Block
Thoughts
Written out in images instead of words for the students,
Connected to an IEP goal
Connected to a standard
Thoughts
This is based on previous lessons to make sure they are building concepts from one lesson to the next
In the previous lesson they built visual representations of models of numbers
Thoughts
In this new lesson, we are introducing a new concept
Always use visuals and manipulatives to help establish mental images that students can refer to later
This comes directly from Mathematics Navigator “Beginning Place Value” page 13
Thoughts
Follow the “I do” , “We do”, and “You do” procedures
Will need to develop your own materials as the curriculum and the supplements do not necessarily have enough examples and repetition needed for a student with a disability
Thoughts
Always have a fluency practice stage.
In this case we are practicing a skill that they have learned and should understand. If they have not fully developed this understanding then go back to a skill that they know.
We only develop fluency of a skill that we have developed conceptual knowledge of
In this lesson we are going to use Power Point to work on Instant recognition of two digit numbers (show the base ten, then the students use their knowledge of the ten frame to recognize the numbers)
Switch to the Instant Recognition of two digit numbers ppt to demonstrate a fluency drill.
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Thoughts
Always have a practical application component to every lesson. This can be an instructional task, story problem, or a project.
Teach strategies for solving problems
This problem comes from HOS page 43
Instruction for Number Sense
Project Based Learning or Instructional Tasks and Application of learning
Systematic Instruction and Application Learning
systematic instruction for the acquisition of skills
project work (problems, dilemma, instructional tasks, projects) for the application of skills acquired earlier
Special Education is often criticized for only focusing on skills development and not purposely transferring the skills to application. Both must occur! At the elementary, middle, and high school levels, there are some parts of the curriculum in which students are necessarily dependent on the teacher, and others in which students can work more independently. In particular, two aspects of the curriculum provide for students’ learning needs:
1.    systematic instruction for the acquisition of skills 2.    project work for the application of skills acquired earlier
This is why the “30 min” lesson always ends with an Instructional Task that applies the skills learned.
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Student follows instructions
Intrinsic motivation characterizes the work particularly
Teacher addresses student's deficiencies
Three Interventions
We are going to take a look at these three programs that are used in DPS and how they address number sense. These programs are not designed to meet the needs of all learners but provide supplement to the core curriculum for students who struggle. Each presents one lesson for a different topic on the scope and sequence. They may not provide the extra practice, instruct strategies appropriate for student with a disability, sufficient fluency practice or a sensible sequence of instruction for remediation.
Mathematics Navigator is grouped by subject which makes it easier to use for specially designed instruction. There is no grade level designation. The Beginning Place Value module would be appropriate as early as Kindergarten for the first lesson but probably most appropriate for 2nd grade and up.
Hands on Standards is loosely grouped by grade level. You will probably need multiple grade levels to see the entire scope and sequence of number sense skills. The manuals really just show you how to use the manipulatives. You have to create your own practice items.
Origomath is by grade levels. You will need to find the number sense skills at each grade level. Essentially you are rearranging the curriculum to be more like Mathematics Navigator. This does provide a lot of applicaion or instructional tasks that the other programs don’t provide.
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Hover mouse over Classroom
If possible go to school net to find these three curriculum. If not then use these slides to show how to find them in School Net. These slides are here for folks to refer to if they are struggling on how to find the.
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Click Search
School Net
23= 20 +3
Lesson 9 in “Beginning Place Value”
Developing Number Sense Using Place Value
1. Review: representing, counting, and recording tens (20, 30, 40, etc)
3. Practice using CRA
Sample Lesson: ORIGOmath
First Grade Unit 5 Working with Place Value and Relative Position
Using a two-digit Numeral Expander
tens
ones
fold
tape or laminate
This lesson uses the Numeral Expander. See how to make this manipulative. Pass out a copy of the Numeral Expander. Show them how to fold it.
Use this to demonstrate how a number is made with place and value. Show
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tens
ones
3
2
2
3
How many 10s? Write that number on the tape in the 10s box using a wet erase markers so you can use it over and over again. How many 1s? Write that number on the tape in the 1s box. Then fold the expander to reveal the number. You can do this reversed as well. Put the numbers in the boxes and have them build models with the base 10 blocks.
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Next time we meet… 2-5-13
Chapter 4-5 Teaching Students who struggle with addition and subtraction of whole numbers
Counting and cardinality (know number names and count sequence, count the number of objects, compare numbers)
Number and operations and base 10 (work with numbers in 11-19 to gain foundations for place value)
Number in operations in base 10 (extend the counting sequence, understand place value, uses place value understanding and properties of operation to add and subtract)
Number in operations in base 10 (extend the counting sequence, understand place value, uses place value understanding and properties of operation to add and subtract)
Number in operations and base 10 (uses place value understanding and properties of operation to perform multi-digit arithmetic)
Number in operations-fractions (develop understanding of fractions as numbers)
Number in operations in based 10 (generalize place value understanding for multi-digit whole numbers, use place value understanding and properties of operations to perform multi-digit arithmetic)
Number in operations-fractions(extend understanding of fraction equivalents and ordering, build fractions from unit fractions by applying and extending previous understandings of operations on the whole numbers, understand a small notations for fractions, and compare decimal fractions)
Number in operations and base 10 (understand the place value system, perform operations with multi-digit whole numbers and with decimals to hundredths)
Number and operations-fractions (use a equivalent fractions as a strategy to add and subtract fractions, apply and extend previous understandings of multiplication and division to multiply and divide fractions)
Ratios and proportional relationships (understand ratio concepts and use ratio reasoning to solve problems)
The number system (apply and extend previous understanding of multiplication and division to divide fractions by fractions, compute fluently with multi-digit numbers and find common factors and multiples, apply in extend previous understandings of numbers to the system of rational numbers)
Ratios and proportional relationships (analyze proportional relationships and use them to solve real-world and mathematical problems)
The number system (apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide all numbers)
The number system (I know that there are numbers that are not rational, an approximate them by rational numbers)
The real number system (extend the properties of exponent stew rational exponents, use properties of rational and irrational numbers)
Quantities (reason quantitatively and use units to solve problems)
The complex number system (performs arithmetic operations with complex numbers, represent complex numbers and their operations on the complex plane, use complex numbers and polynomial identities and equations
Common core state standards sorting activity: Cut out each section and put into an envelope for each table.
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