With Christmas approaching I always take time to reflect on some of my favorite things. One thing I have come to love is snow shoeing. Last year my parents retired to the Colorado Rockies, this was our first Christmas ever spend in the mountains. Go figure-right. I have lived in Colorado most of my life and have never spent Christmas in the mountains. As a family we went show shoeing. We took the dog-mine being Italian Greyhounds-had a grand time.

I'm donating a gift card from one of my favorite stores-Target. I need a 12 step program.  I love their $1 bins. They are perfect for everything from dressing my dogs to filling my treasure chest. I have to work at staying away. I get most of everything I need from Target. Have fun and enjoy the shopping trip.

I have a group of struggling learners who have had a hard time showing growth on Nonsense Word Fluency from Dibels. I created a set of cards that I can give students that have them sound out the nonsense word and then blend it back together again. They are perfect for reading centers, RTI Intervention work or struggling readers who need extra practice with nonsense words. This has do wonders to bring student scored up from an average of 40 sounds and no whole words to 15 whole words read correctly. This item is at my store for 50% off. Click on the picture.



Enjoy you Blog Hop! Have a great and safe holiday!




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This month I'm hooking up with Lisa from PAWSitively Teaching for a Pick 3 from Pinterest. The items I have picked out are ones that I can see my students doing over the next three weeks. Click on each picture to take a closer look on how these teachers created each Pinterest.


I work with preschool students and students who need to practice counting and one to one correspondence whenever I can build it in. Making snowmen would be perfect to tackle both of those. I could even see them cutting them out to work on fine motor skills. Who knew by creating snowmen that I could cram all those skills into them.

What do the tiers mean?

Description of Critical Elements in a 3-Tier RtI Model The following table outlines the essential features of a three-tier model of RtI including suggested ranges of frequency and duration of screening, interventions and progress monitoring. This is intended as guidance as they determine the various components of their RtI model.
Elements	Tier 1 Core Curriculum and Instruction	Tier 2 Supplemental Instruction	Tier 3 Increased Levels of Supplemental Instruction
Size of instructional group	Whole class grouping	Small group instruction (3-5 students)	Individualized or small group instruction (1-2 students)
Mastery requirements of content	Relative to the cut points identified on criterion screening measures and continued growth as demonstrated by progress monitoring	Relative to the cut points identified on criterion screening measures and continued growth as demonstrated by progress monitoring	Relative to the student’s level of performance and continued growth as demonstrated by progress monitoring
Frequency of progress monitoring	Screening measures three times per year  (DIBELS, AIMSWeb, iReady)	Varies, but no less than once every two weeks	Varies, but more continuous and no less than once a week
Frequency of intervention provided	Per school schedule	Varies, but no less than three times per week for a minimum of 20-30 minutes per session	Varies, but more frequently than Tier 2 for a minimum of 30 minutes per session
Duration of intervention	School year	9-30 weeks	A minimum of 15-20 weeks

What are the Benefits of RtI?

• RtI ensures a shared approach is used in addressing students’ diverse needs.
• Parents are a very important part of the process.
• RtI eliminates the “wait to fail” situation, because students get help promptly within the general education setting.
• The RtI approach may help reduce the number of students referred for special education services while increasing the number of students who are successful within regular education.
• RtI helps to identify the root cause of achievement problems.
• RtI’s use of progress monitoring provides more instructionally relevant information than traditional assessments

How Parents/Guardians can support at Home:

• Reading is Fundamental (These tips have been adapted from Reading is Fundamental (www.rif.org)
• Invite your child to read with you every day.
• When reading a book where the print is large, point word by word as you read.
• Read your child’s favorite book over and over again.
• Read many stories with rhyming words and repeated lines.
• Discuss new words and ideas.
• Stop and ask about the pictures and what is happening in the story. Encourage your child to predict.
• Read from a variety of materials including fairy tales, poems, informational books, magazines and even comic strips.
• Let your children see you reading for pleasure in your spare time.
• Take your child to the library. Explore an area of interest together
• Scout for things your child might like to read. Use your child’s interests and hobbies as starting points.

What should parents do if they believe their child is struggling?

• Contact your child’s teacher
• Request a parent/teacher conference
• Access the parent portal and other daily means of communication
• Review your child’s work to see if there is progress
• Talk with your child to ensure they know you are supporting them at home as well as in school

Be sure to stop by my Teachers pay Teacher store for great RtI progress monitoring tools during the 2-day Cyber Sale all items 28% off. My students love  my RTI: Nonsense Word Fluency Activities. Its a great way to open guided reading so students to practice going from the individual sounds to the whole word. 

What is Guided Reading?

Moving school districts means that this year I have had to create new ways for my students
to work with the vocabulary. One thing with Storytown I have come to love is the Intervention series vocabulary matches the grade level vocabulary. The teachers have also requested that I run one week ahead of them. This is nice because students work through the vocabulary with me before they have to do it in class. I can help students understand with word with examples but also they can practice how to apply the word using the story as a background. I also have to make sure that I give real life or real world examples of the word—in most cases these are words that they will come across in middle school but will never use with writing or in a conversation. (Which doesn't help them with text access.)

• Tally-Ho!-Display the vocabulary word card. Add a tally mark beside the word each time a student uses the vocabulary word in conversation.
• May I Have Your Autograph?-Display an enlarged print vocabulary word card. Allow students to autograph the card each time they use the vocabulary word in conversation.
• Vocabulary String-Scan and print the cover of each book from which vocabulary words are pulled. Attach a kite “tail” of string, yarn, ribbon, etc. to the cover. Print out vocabulary word cards and glue each card on the tail. Allow student to write their name on a clothespin and clip it to the word card each time they use the word in conversation.
• Don’t Lose Your Marbles!-Display the vocabulary word on the outside of a small jar. Add a marble to the jar each time a student uses the vocabulary word in conversation.
• Stick With It!-Display the vocabulary word card. Each time a student uses the vocabulary word in conversation, have him add a small sticker to the card.
• Just Scrolling Along!-On a computer that is visible to students, set the screensaver to scrolling text and type in the vocabulary words and/or definitions. Set the screensaver to come on after 5 minutes or so.
• Rock On!-Make a vocabulary jar by gluing a vocabulary word card to the outside of a jar. Allow students to drop a rock in the jar each time they use the vocabulary word in conversation.
• Vocabulary Vine-Make a crepe paper vine to wind across the walls in your classroom. Cut out leaves and write a vocabulary word on each leaf. Attach the leaves to your vine and watch students’ vocabulary grow!
• Word Wizard-Purchase clear name badges. Write the vocabulary word on a card. Slide the card into the badge holder. Allow a student to be the “Word Wizard”. He will wear the vocabulary word and should use the word throughout the day.
• Chain, Chain, Chain!-Cut out construction paper chain links. Each time a student uses a vocabulary word in conversation, have him write the vocabulary word on the link and add it to the chain. The chain will hang straight down from the ceiling. Display this poem:

• Vocabulary Pop!-Set a large jar on the counter. Each time a student uses a vocabulary word, drop a small handful of popcorn kernels in the jar. When the jar is full, have a popcorn party.
• Movin’ On!-Take a piece of yarn the width of your classroom and hang it above you. The yarn should start on one side of the room and stretch across it horizontally to attach to the other side of the classroom. On the far side, attach a blown up balloon. (Before blowing up the balloon, slip a piece of paper inside with the treat to be given written on it—class homework pass, 5 minutes extra recess, etc.) Attach a sign that says “Our vocabulary is moving on!” with a gym clip. Each time a student uses a vocabulary word, move the sign a bit toward the balloon. When the sign reaches the balloon, pop the balloon and read the prize. You can then start over with a new balloon and a new secret prize.

Here some ideas that I have used with my students that are more aligned with their learning styles. Most of these I have students do at home for practice. This gives them the chance to pick how they want to practice the words. It works great with spelling as well.

• Backwards Words- Write your words forwards, then backwards.
• Silly sentences -Use all your words in ten sentences.
• Picture words – Draw a picture and write your words in the picture.
• Words without Vowels – Write your words replacing all vowels with a line.
• Words without Consonants – Write your words replacing all consonants with a line.
• Story words – Write a short story using all your words.
• Scrambled words –Write your words, then write them again with the letters mixed up.
• Ransom words – Write your words by cutting out letters in a newspaper or magazine and glue them on a paper.
• Pyramid Words – Write your words adding or subtracting one letter at a time. The result will be a pyramid shape of words.
• Words-in-words – Write your word and then write at least 2 words made from each.
• Good Clean Words –Write your words in shaving cream on a counter or some other surface that can be cleaned safely.
• Etch-A-Word – Use an Etch- A-Sketch to write your words.
• Secret Agent Words – Number the alphabet from 1 to 26, then convert your words to a number code.
• Popsicles – Make words using popsicle sticks.
• Newspaper Words – Search a newspaper page from top to bottom, circling each letter of a word as you find it.
• Silly String – With a long length of string, “write” words using the string to shape the letters.
• Backwriting – Using your finger, draw each letter on a partners’ back, having the partner say the word when completed.
• Choo-Choo Words – Write the entire list end-to-end as one long word, using different colors of crayon or ink for different words.
• Other Handed – If you are right-handed, write with your left, or vice versa.
• Cheer your words – Pretend you are a cheerleader and call out your words! Sometimes you’ll yell, sometimes you’ll whisper.
• Reversed words – Write your words in ABC order - backwards!
• ABC order- Write your words in alphabetical order.
• Puzzle words – Use a blank puzzle form. Write your words on the form, making sure that the words cross over the pieces. Then cut them out (color if you wish) and put them in a baggie with your name on it.
• Pasta Words – Write your words by arranging alphabet pasta or Alphabits.
• Sound Words – Use a tape recorder and record your words and their spelling. Then listen to your tape, checking to see that you spelled all the words correctly.
• 3D words – Use modeling clay rolled thinly to make your words. Bring a note if done at home.
• Dirty Words – Write your words in mud or sand.

Have a great week with your family. I hope these ideas help are we are looking for new ways to spice up vocabulary work.

Its difficult to meet the readers needs of all students. For some its too easy, others to way to hard, and for many its just right. Using guided reading to move students takes time--time that I don't have when students are struggling in the classroom. My reading program just doesn't cut it when it comes to meeting the needs of my readers who are three or four years behind.

Like many my daily schedule is already jam-packed, and it's challenging to add one more thing. Here are three ideas that won’t take a lot of extra time and will support your striving readers as they move toward reading independently and comprehending increasingly complex texts.

1. Read aloud texts that may be challenging, providing all students the opportunity to boost their comprehension and engage in collaborative conversations.

Reading aloud is such an integral part of my daily instruction that my students and I keep a read-aloud tally (pictured below), using one tally mark for each read-aloud experience. The essential literacy practice of reading aloud is also endorsed by the authors of the Common Core's ELA Standards in Appendix A (p. 27). I do the grade level reading and grade vocabulary works as a read aloud and ask higher order thinking questions.

 2. Guide readers individually and in small groups. To make the most of the time you will spend guiding readers one-on-one or in small groups, it is wise to pinpoint their strengths and areas of need. The best way to do this is by using a reading assessment that includes a running record. This will enable you to know your students' instructional level and, as important, whether your readers are struggling with decoding, fluency, vocabulary or comprehension. Armed with this information, meet with students as often as possible to prompt and coach them to apply decoding strategies for figuring out unknown words and comprehension strategies to better understand the text. If your reading program provides leveled texts, these will work well for guiding readers in small group. I spend three days a week working as a guided reading group, where we read instructional level text. I always have a written assignment on these days to push them to use the resource (spelling, sighting text).

3. I have a couple of students that this is not enough for them to access the material. So I use Boardmaker and Google Images to highlight the key part of the text in a picture based sentence strip. I tape the strip on the the story, so that the students have the original text but can read the sentence that I added to understand the text.

I hope these ideas help you in the classroom. I've been playing with student access and have created a couple of adaptive books for a student who is working on her shapes and colors. Enjoy them free below.

Color Adaptive Book

Shape Adaptive Book

As parent/teacher conferences approach (or in my case later this week), something to keep in mind to think about changing up ideas to support struggling readers while talking with parents and to try as students are brought to an RTI team. Sometimes rethinking the basics is all students need.

• The teacher's knowledge matters: knowing which skills to teach and when, teaching reading skills in balanced reading programs.
• Classroom organization matters: access to books and writing materials, classroom routines, community reading, "just right" reading, "on your own" reading.
• Reading choices matter: levels of difficulty, genre, topics, cultural representation, task difficulty and achievement.
• Explicit instruction in phonemic awareness and phonics matters: effective word study instruction, assessment, building decoding fluency.
• Explicit and strategic instruction in comprehension matters.
• Response to reading matters: types, contexts, purposes and assessing reader response.
• Assessment matters: frequency, context and type.
• The amount of text that children read matters.
• Fluency matters: correct words per minute, tone, phrasing.

A person's ability to use and understand numbers:
• knowing their relative values,
• how to use them to make judgments,
• how to use them in flexible ways when adding, subtracting, multiplying or dividing
• how to develop useful strategies when counting, measuring or estimating.

What is number sense?

The term "number sense" is a relatively new one in mathematics education. It is difficult to define precisely, but broadly speaking, it refers to "a well-organized conceptual framework of number information that enables
a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms". The National Council of Teachers identified five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers and quantities. These skills are considered important because they contribute to general intuitions about numbers and lay the foundation for more advanced skills.

Researchers have linked good number sense with skills observed in students proficient in the following mathematical activities:

• mental calculation
• computational estimation 
• judging the relative magnitude of numbers 
• recognizing part-whole relationships and place value concepts and;
• problem solving 

How does number sense begin?

An intuitive sense of number begins at a very early age. Children as young as two years of age can confidently identify one, two or three objects before they can actually count with understanding. Piaget called this ability to instantaneously recognize the number of objects in a small group 'subitizing'. As mental powers develop, usually by about the age of four, groups of four can be recognized without counting. It is thought that the maximum number for subitizing, even for most adults, is five. This skill appears to be based on the mind's ability to form stable mental images of patterns and associate them with a number. Therefore, it may be possible to recognize more than five objects if they are arranged in a particular way or practice and memorization takes place. A simple example of this is six dots arranged in two rows of three, as on dice or playing cards. Because this image is familiar, six can be instantly recognized when presented this way.

Usually, when presented with more than five objects, other mental strategies must be utilized. For example, we might see a group of six objects as two groups of three. Each group of three is instantly recognized, then very quickly (virtually unconsciously) combined to make six. In this strategy no actual counting of objects is involved, but rather a part-part-whole relationship and rapid mental addition is used. That is, there is an understanding that a number (in this case six) can be composed of smaller parts, together with the knowledge that 'three plus three makes six'. This type of mathematical thinking has already begun by the time children begin school and should be nurtured because it lays the foundation for understanding operations and in developing valuable mental calculation strategies.

What teaching strategies promote early number sense?

Learning to count with understanding is a crucial number skill, but other skills, such as perceiving subgroups, need to develop alongside counting to provide a firm foundation for number sense. By simply presenting objects (such as stamps on a flashcard) in various arrangements, different mental strategies can be prompted. For example, showing six stamps in a cluster of four and a pair prompts the combination of 'four and two makes six'. If the four is not subitised, it may be seen as 'two and two and two makes six'. This arrangement is obviously a little more complex than two groups of three. So different arrangements will prompt different strategies, and these strategies will vary from person to person.

• verbalizing a strategy brings the strategy to a conscious level and allows the person to learn about their own thinking;
• it provides other children with the opportunity to pick up new strategies;
• the teacher can assess the type of thinking being used and adjust the type of arrangement, level of difficulty or speed of presentation accordingly.

To begin with, early number activities are best done with movable objects such as counters, blocks and small toys. Most children will need the concrete experience of physically manipulating groups of objects into sub-groups and combining small groups to make a larger group. After these essential experiences more static materials such as 'dot cards' become very useful.

Dot cards are simply cards with dot stickers of a single color stuck on one side. (However, any markings can be used. Self-inking stamps are fast when making a lot of cards). The important factors in the design of the cards are the number of dots and the arrangement of these dots. The various combinations of these factors determine the mathematical structure of each card, and hence the types of number relations and mental strategies prompted by them.

Consider each of the following arrangements of dots before reading further. What mental strategies are likely to be prompted by each card? What order would you place them in according to level of difficulty?

Card A is the classic symmetrical dice and playing card arrangement of five and so is often instantly recognized without engaging other mental strategies. It is perhaps the easiest arrangement of five to deal with.

Card B presents clear sub-groups of two and three, each of which can be instantly recognized. With practice, the number fact of 'two and three makes five' can be recalled almost instantly.

Card C: A linear arrangement is the one most likely to prompt counting. However, many people will mentally separate the dots into groups of two and three, as in the previous card. Other strategies such as seeing two then counting '3, 4, 5' might also be used.
Card D could be called a random arrangement, though in reality it has been quite deliberately organized to prompt the mental activity of sub-grouping. There are a variety of ways to form the sub-groups, with no prompt in any particular direction, so this card could be considered to be the most difficult one in the set.
Card E shows another sub-group arrangement that encourages the use (or discovery) of the 'four and one makes five' number relation.

Obviously, using fewer than five dots would develop the most basic number sense skills, and using more than five dots would provide opportunities for more advanced strategies. However, it is probably not useful to use more than ten dots. (See the follow-on article focusing on developing a 'sense of ten' and 'place value readiness'). Cards such as these can be shown briefly to children, then the children asked how many dots they saw. The children should be asked to explain how they perceived the arrangement, and hence what strategies they employed.

What games can assist development of early number sense?

Games can be very useful for reinforcing and developing ideas and procedures previously introduced to children. Although a suggested age group is given for each of the following games, it is the children's level of experience that should determine the suitability of the game. Several demonstration games should be played, until the children become comfortable with the rules and procedures of the games.

Deal and Copy (4-5 years) 3-4 players

Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.

Variations/Extensions
Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added. Increase the number of dots on the cards.

Memory Match (5-7 years) 2 players

Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.

Variations/Extensions
Increase the number of pairs of cards used. Use a greater number of dots on the cards. Pair a dot card with a numeral card.

What's the Difference? (7-8 years) 2-4 players

Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. He/she then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (E.g. If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.

Variations/Extensions
Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters. Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends. Use dot cards with random arrangements of dots.

Number Sense plays into how well order students grasp onto the more difficult concepts such as rounding, place value, and learning the basic math facts. Look for more information to come. Have a great week!

Concrete - Representational - Abstract:  Sequence of Instruction

The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning. When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math skill and truly understand math concepts at the abstract level.

What is it?

• Each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, beans and bean sticks, pattern blocks).
• Students are provided many opportunities to practice and demonstrate mastery using concrete materials
• The math concept/skill is next modeled at the representational (semi-concrete) level which involves drawing pictures that represent the concrete objects previously used (e.g. tallies, dots, circles, stamps that imprint pictures for counting)
• Students are provided many opportunities to practice and demonstrate mastery by drawing solutions
• The math concept/skill is finally modeled at the abstract level (using only numbers and mathematical symbols)
• Students are provided many opportunities to practice and demonstrate mastery at the abstract level before moving to a new math concept/skill.

As a teacher moves through a concrete-to-representational-to-abstract sequence of instruction, the abstract numbers and/or symbols should be used in conjunction with the concrete materials and representational drawings (promotes association of abstract symbols with concrete & representational understanding)

What are the critical elements of this strategy?

• Use appropriate concrete objects to teach particular math concept/skill (see Concrete Level of Understanding/Understanding Manipulatives-Examples of manipulatives by math concept area). Teach concrete understanding first.
• Use appropriate drawing techniques or appropriate picture representations of concrete objects (see Representational Level of Understanding/Examples of drawing solutions by math concept area). Teach representational understanding second.
• Use appropriate strategies for assisting students to move to the abstract level of understanding for a particular math concept/skill (see Abstract Level of Understanding/Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers).

How do I implement the strategy?

When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding).

• Provide students many practice opportunities using concrete objects.
• When students demonstrate mastery of skill by using concrete objects, describe & model how to perform the skill by drawing or with pictures that represent concrete objects (representational level of understanding).
• Provide many practice opportunities where students draw their solutions or use pictures to problem-solve.
• When students demonstrate mastery drawing solutions, describe and model how to perform the skill using only numbers and math symbols (abstract level of understanding).
• Provide many opportunities for students to practice performing the skill using only numbers and symbols.

After students master performing the skill at the abstract level of understanding, ensure students maintain their skill level by providing periodic practice opportunities for the math skills.

How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems?

• Helps passive learner to make meaningful connections
• Teaches conceptual understanding by connecting concrete understanding to abstract math process
• By linking learning experiences from concrete-to-representational-to-abstract levels of understanding, the teacher provides a graduated framework for students to make meaningful connections.
• Blends conceptual and procedural understanding in structured way

Concrete

What is it?

The concrete level of understanding is the most basic level of mathematical understanding. It is also the most crucial level for developing conceptual understanding of math concepts/skills. Concrete learning occurs when students have ample opportunities to manipulate concrete objects to problem-solve. For students who have math learning problems, explicit teacher modeling of the use of specific concrete objects to solve specific math problems is needed.

Understanding manipulatives (concrete objects)

To use math manipulatives effectively, it is important that you understand several basic characteristics of different types of math manipulatives and how these specific characteristics impact students who have learning problems. As you read about the different types of manipulatives, click on the numbers beside each description to view pictures of these different types of manipulatives.

General types of math manipulatives:

Discrete - those materials that can be counted (e.g. cookies, children, counting blocks, toy cars, etc.).

Continuous - materials that are not used for counting but are used for measurement (e.g. ruler, measuring cup, weight scale, trundle wheel). See example - 1
Suggestions for using Discrete & Continuous materials with students who have learning problems:

Students who have learning problems need to have abundant experiences using discrete materials before they will benefit from the use of continuous materials. This is because discrete materials have defining characteristics that students can easily discriminate through sight and touch. As students master an understanding of specific readiness concepts for specific measurement concepts/skills through the use of discrete materials (e.g. counting skills), then continuous materials can be used.

Types of manipulatives used to teach the Base-10 System/place-value (Smith, 1997):

Proportional - show relationships by size (e.g. ten counting blocks grouped together is ten times the size of one counting block; a beanstick with ten beans glued to a popsicle stick is ten times bigger than one bean).

Non-linked proportional - single units are independent of each other, but can be "bundled together (e.g. popsicle sticks can be "bundled together in groups of 'tens' with rubber bands; individual unifix cubes can be attached in rows of ten unifix cubes each).

Linked proportional - comes in single units as well as "already bundled" tens units, hundreds units, & thousands units (e.g. base ten cubes/blocks; beans & beansticks).

Non-proportional - use units where size is not indicative of value while other characteristics indicate value (e.g. money, where one dime is worth ten times the value of one penny; poker chips where color indicates value of chip; an abacus where location of the row indicates value). A specified number of units representing one value are exchanged for one unit of greater value (e.g. ten pennies for one dime; ten white poker chips for one blue poker chip, ten beads in the first row of an abacus for one bead in the second row). See example - 1
Suggestions for using proportional and non-proportional manipulatives with students who have learning problems:

Students who have learning problems are more likely to learn place value when using proportional manipulatives because differences between ones units, tens units, & hundreds units are easy to see and feel. Due to the very nature of non-proportional manipulatives, students who have learning problems have more difficulty seeing and feeling the differences in unit values.

Examples of manipulatives (concrete objects)

Suggested manipulatives are listed according to math concept/skill area. Descriptions of manipulatives are provided as appropriate. A brief description of how each set of manipulatives may be used to teach the math concept/skill is provided at the bottom of the list for each math concept area. Picture examples of some of the manipulatives for each math concept area can be accessed by clicking on the numbers found underneath the title of each math concept area. This is not meant to be an exhaustive list, but this list does include a variety of common manipulatives. The list includes examples of "teacher-made" manipulatives as well "commercially-made" ones.

Counting/Basic Addition & Subtraction Pictures

Colored chips
Beans
Unifix cubes
Golf tees
Skittles or other candy pieces
Packaging popcorn
Popsicle sticks/tongue depressors

Description of use: Students can use these concrete materials to count, to add, and to subtract. Students can count by pointing to objects and counting aloud. Students can add by counting objects, putting them in one group and then counting the total. Students can subtract by removing objects from a group and then counting how many are left.

Place Value Pictures

Base 10 cubes/blocks
Beans and bean sticks
Popsicle sticks & rubber bands for bundling
Unifix cubes (individual cubes can be combined to represent "tens")
Place value mat (a piece of tag board or other surface that has columns representing the "ones," "tens," and "hundreds" place values)

Description of use: Students are first taught to represent 1-9 objects in the "ones" column. They are then taught to represent "10" by trading in ten single counting objects for one object that contains the ten counting objects on it (e.g. ten separate beans are traded in for one "beanstick" - a popsicle stick with ten beans glued on one side. Students then begin representing different values 1-99. At this point, students repeat the same trading process for "hundreds."

Multiplication/Division Pictures

Containers & counting objects (paper dessert plates & beans, paper or plastic cups and candy pieces, playing cards & chips, cutout tag board circles & golf tees, etc.). Containers represent the "groups" and counting objects represent the number of objects in each group. (e.g. 2 x 4 = 8: two containers with four counting objects on each container)
Counting objects arranged in arrays (arranged in rows and columns). Color-code the "outside" vertical column and horizontal row helps emphasize the multipliers
.
Positive & Negative Integers Pictures

Counting objects, one set light colored and one set dark colored (e.g. light & dark colored beans; yellow & blue counting chips; circles cut out of tag board with one side colored, etc.).

Description of use: Light colored objects represent positive integers and dark colored objects represent negative integers. When adding positive and negative integers, the student matches pairs of dark and light colored objects. The color and number of objects remaining represent the solution.

Fractions Pictures

Fraction pieces (circles, half-circles, quarter-circles, etc.)
Fraction strips (strips of tag board one foot in length and one inch wide, divided into wholes, ½'s, 1/3's, ¼'s, etc.
Fraction blocks or stacks. Blocks/cubes that represent fractional parts by proportion (e.g. a "1/2" block is twice the height as a "1/4" block).

Description of use: Teacher models how to compare fractional parts using one type of manipulative. Students then compare fractional parts. As students gain understanding of fractional parts and their relationships with a variety of manipulatives, teacher models and then students begin to add, subtract, multiply, and divide using fraction pieces.

Geometry Pictures

Geoboards (square platforms that have raised notches or rods that are formed in a array). Rubber bands or string can be used to form various shapes around the raised notches or rods.

Description of use: Concepts such as area and perimeter can be demonstrated by counting the number of notch or rod "units" inside the shape or around the perimeter of the shape.

Beginning Algebra Pictures

Containers (representing the variable of "unknown") and counting objects (representing integers) -e.g. paper dessert plates & beans, small clear plastic beverage cups 7 counting chips, playing cards & candy pieces, etc.
Description of use: The algebraic expression, "4x = 8," can be represented with four plates ("4x"). Eight beans can be distributed evenly among the four plates. The number of beans on one plate represent the solution ("x" = 2).

Suggestions for using manipulatives:

• Talk with your students about how manipulatives help to learn math.
• Set ground rules for using manipulatives.
• Develop a system for storing manipulatives.
• Allow time for your students to explore manipulatives before beginning instruction.
• Encourage students to learn names of the manipulatives they use.
• Provide students time to describe the manipulatives they use orally or in writing. Model this as appropriate.
• Introduce manipulatives to parents

Representational

What is it?

At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level. It is appropriate for students to begin drawing solutions to problems as soon as they demonstrate they have mastered a particular math concept/skill at the concrete level. While not all students need to draw solutions to problems before moving from a concrete level of understanding to an abstract level of understanding, students who have learning problems in particular typically need practice solving problems through drawing. When they learn to draw solutions, students are provided an intermediate step where they begin transferring their concrete understanding toward an abstract level of understanding. When students learn to draw solutions, they gain the ability to solve problems independently. Through multiple independent problem-solving practice opportunities, students gain confidence as they experience success. Multiple practice opportunities also assist students to begin to "internalize" the particular problem-solving process. Additionally, students' concrete understanding of the concept/skill is reinforced because of the similarity of their drawings to the manipulatives they used previously at the concrete level.

Drawing is not a "crutch" for students that they will use forever. It simply provides students an effective way to practice problem solving independently until they develop fluency at the abstract level.

Examples of drawing solutions by math concept level

The following drawing examples are categorized by the type of drawings ("Lines, Tallies, & Circles," or "Circles/Boxes"). In each category there are a variety of examples demonstrating how to use these drawings to solve different types of computation problems. Click on the numbers below to view these examples.

Abstract

What is it?

A student who problem-solves at the abstract level, does so without the use of concrete objects or without drawing pictures. Understanding math concepts and performing math skills at the abstract level requires students to do this with numbers and math symbols only. Abstract understanding is often referred to as, "doing math in your head." Completing math problems where math problems are written and students solve these problems using paper and pencil is a common example of abstract level problem solving.

Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers

Students who are not successful solving problems at the abstract level may:

• Not understand the concept behind the skill

Suggestions:

Re-teach the concept/skill at the concrete level using appropriate concrete objects (see Concrete Level of Understanding).
Re-teach concept/skill at representational level and provide opportunities for student to practice concept/skill by drawing solutions (see Representational Level of Understanding).
Provide opportunities for students to use language to explain their solutions and how they got them (see instructional strategy Structured Language Experiences).

- Have difficulty with basic facts/memory problems

Suggestions:

• Regularly provide student with a variety of practice activities focusing on basic facts. Facilitate independent practice by encouraging students to draw solutions when needed (see the student practice strategies Instructional Games, Self-correcting Materials, Structured Cooperative Learning Groups, and Structured Peer Tutoring).
• Conduct regular one-minute timings and chart student performance. Set goals with student and frequently review chart with student to emphasize progress. Focus on particular fact families that are most problematic first, then slowly incorporate a variety of facts as the student demonstrates competence (see evaluation strategy Continuous Monitoring & Charting of Student Performance).
• Teach student regular patterns that occur throughout addition, subtraction, multiplication, & division facts (e.g. "doubles" in multiplication, 9's rule - add 10 & subtract one, etc.)
• Provide student a calculator or table when they are solving multiple-step problems.

Repeat procedural mistakes

Suggestions:

• Provide fewer #'s of problems per page.
• Provide fewer numbers of problems when assigning paper & pencil practice/homework.
• Provide ample space for student writing, cueing, & drawing.
• Provide problems that are already written on learning sheets rather than requiring students to copy problems from board or textbook.
• Provide structure: turn lined paper sideways to create straight columns; allow student to use dry-erase boards/lap chalkboards that allow mistakes to be wiped away cleanly; color cue symbols; for multi-step problems, draw color-cued lines that signal students where to write and what operation to use; provide boxes that represent where numerals should be placed; provide visual directional cues in a sample problem; provide a sample problem, completed step by step at top of learning sheet.
• Provide strategy cue cards that student can use to recall the correct procedure for solving problem.
• Provide a variety of practice activities that require modes of expression other than only writing

The Big Idea

Student learning & mastery greatly depends on the number of opportunities a student has to respond!! The more opportunities for successful practice that you provide (i.e. practice that doesn't negatively impact student learning characteristics), the more likely it is that your student will develop mastery of that skill.