I never did very well in math – I could never seem to persuade the teacher that I hadn’t meant my answers literally. ~Calvin Trillin
Manipulatives are proving to be effective tools for teaching certain math concepts to all students (Berkas, 2007). Students learn best through active experiences rather than lectures or passive listening. Visual images stick with people longer and better than abstract or language based concepts. (Stover, ret. 2011) Manipulatives are objects that can be touched and moved by students to introduce or reinforce a math concept. Since 1940 to the present day, the National Council of Teachers of Mathematics (NCTM) has encouraged all grade levels to use manipulatives in daily math instruction. Many concepts in math are abstract. Manipulatives are useful in helping students move from the concrete to abstract level. (Hartshorn, 1990) Students are being encouraged to use manipulatives to demonstrate understanding in representational and abstract stages (Berkas, 2007). They are beneficial for students who have been diagnosed with a learning disability. Manipulatives provide additional reinforcement of basic math skills and enables them to complete math problems independently. (Schreiner, 2010)
Once a child is familiar with a manipulative it should be made available to use at any time to help them think, reason and solve problems. After a while, they begin to develop a great toolbox of ideas for solving problems. (Rudnicki, 2010)
Manipulatives show positive impacts when combined with:
- virtual manipulative software
- reflective practices
- cooperative learning
- activities that are exploratory and deductive in their approach
Types and Uses
Unifix Cubes (colored linking cubes)
- learn measurement using nonstandard units
- visualize concept of area and geometric shapes
- percent, probability and fractions
- teach equivalence
- number concepts/ strategies for addition and subtraction
- create visuals showing ratio, proportion, fractions
- understand odd/even numbers
- create patterns
Playing cards, dominoes and number cubes
- illustrate decimals or whole numbers
- mental math games (addition, subtraction and/or multiplication war)
- understand probability and logical reasoning
Pattern blocks and tangrams
- understanding of spatial sense and geometry
- create shapes with certain number of sides, angles, complex patterns
- solving word problems
Measuring cups, volume containers, rulers, scales, and clocks
- concept of measurement
(Stover, ret. 2011)
Research has shown that long-term use of manipulatives is more effective than short-term use (Hartshorn, 1990).
“Some state and local systems have mandated the implementation of manipulatives through policy, law, or curriculum documents. Some have also provided funding.” (Hartshorn, 1990)
Manipulatives help hold the attention of the inattentive learner (Stover, ret. 2011).
Using manipulatives in math help students to understand and analyze concepts with a concrete purpose.
In terms of mental and physical activities, manipulatives allow students to use their mind and parts of their body to solve problems. To show students how much a gallon is, start with cups of water, then pour those cups to make pints, then quarts to the eventual gallon. This is both a physical and mental process that show students the meaning of a gallon.
Often times in the fields of math and science we work with a group of people to solve a problem. Using manipulatives in groups allow students to work together and to listen to the ideas of their peers to come to a conclusion.
Universal Design for Learning
I. Provide Multiple Means of Representation
1. Provide options for perception
- Options that customize the display of information
- Options that provide alternatives for visual information
2. Provide options for language and symbols
- Options that illustrate key concepts non-linguistically
II. Provide Multiple Means of Action and Expression
5. Provide options for expressive skills and fluency
- Options in the tools for composition and problem solving
III. Provide Multiple Means of Engagement
7. Provide options for recruiting interest
- Options that enhance relevance, value and authenticity
8. Provide options for sustaining effort and persistence
- Options that foster collaboration and communication
Berkas, N. & Pattison, C. (2007). “Manipulatives: more than a special education intervention.” National Council of Teacher of Mathematics.
Hartshorn, R. & Boren, S. (1990). “Experiential learning of mathematics: using manipulatives. ERIC Digest
Rudnicki, A. (2010). “Why use manipulatives to teach math.” http://www.ehow.com/print/about_664188_use-manipulatives-teach-math_.html.
Schreiner, E. (2010). “How to use manipulatives to teach math to ld students.” http://www.ehow.com/print/how_5991197_use-teach-math-ld-student.html.
Stover, E. (ret. 2011). “About the use of manipulatives in math.” http://www.ehow.com/print/about_5048097_use-manipulatives-math.html.
Stover, E. (ret. 2011). “How to use math manipulatives in the classroom.” http://www.ehow.com/print/how_5031920_use-math-manipulatives-classroom.html.
For most English Language Learners, content literacy (using reading, writing, speaking, and listening to gain new knowledge) depends on the proficiency in the language of the text, the majority of which is English (Carrier, 2006).
Sentence Walls/ Frames
Sentence walls/ frames are similar to word walls that are prominently displayed in the classroom. They provide a visual display of well-formed phrases and sentences, allowing students to communicate in classroom discussions about content. Sentence walls/ frames provide the language necessary for talking and writing about a given topic. It allows English Language Learners to become familiar with vocabulary and sentence structures. (Carrier, 2006)
- available for immediate use
- expands use of language (labeling and simple sentences to complex and grammatically correct statements)
- demonstrate knowledge of new concepts (assessment)
- helpful for struggling native English speakers in constructing well-formed sentences in their writing
- may create a challenge to develop questions and statements for sentence wall/ frame in advance
Developing Sentence Frames/ Walls
- Write sentences that express the target language function.
- Replace target vocabulary with blanks.
- Create a word bank of the words.
|Expected Outcomes||Simple Sentences||Comparative Sentences||Complex Comparative Sentences|
|Sentence frame with vocabulary underlined||Oranges are sweet.Lemons are sour.||Oranges and lemons are both fruit, but oranges are sweet, and lemons are sour.||The main difference between oranges and lemons is oranges are sweet, while lemons are sour.|
|Sentence frame with vocabulary removed||___ are ___.||___ and ___ are both ___, but ___ are ___, and ___ are ___.||___ and ___ is ___ are ___, while ___ are ___.|
|Example taken from: Donnelly, W.B. & Roe, C.J. (2010). Using sentence frames to develop academic vocabulary for English learners. The Reading Teacher, 64(2), pp. 134|
The best way for students to learn about sentence walls is through modeling and plenty of examples for when and how to use sentence frames/ walls. Modeling helps the student learn proper pronunciation and intonation. (Carrier, 2006)
- Listen to teacher say sentence.
- Student says sentence with teacher.
- Student says sentence to teacher.
- Student says sentence to a peer.
The main purpose is oral practice, but should equally be used for written practice. (Donnelly, 2010)
This a great opportunity for classroom teachers to share sentence walls with their English Language Education teachers. This can further enhance the child’s language development.
Connections to Multiple Intelligence
By using sentence frames/ walls students are learning to use language. They are able to express their ideas about the concepts being taught. As an end result, it is hoped that the child can remember and in turn show what they have learned.
This may be a stretch, but hear me out.
Language has certain structures and patterns. As Donnelly’s article shows, you need to think about and target the language function (compare/contrast, problem/solution, etc.). Each of these has a certain pattern and structure to follow. Once a child knows the pattern of a compare/contrast statement, they will be able to independently develop their own statement down the road.
Connections to Universal Design for Learning
Provide Multiple Means of Representation
1. Provide options for perception
• Options that customize the display of information
• Options that provide alternatives for visual information
2. Provide options for language and symbols
• Options that clarify syntax and structure
3. Provide options for comprehension
• Options that highlight critical features, big ideas, and relationships
Provide Multiple Means of Action and Expression
4. Provide options for expressive skills and fluency
• Options in the scaffolds for practice and performance
5. Provide options for executive functions
• Options that support planning and strategy development
• Options that facilitate managing information and resources
• Options that enhance capacity for monitoring progress
Provide Multiple Means of Engagement
8. Provide options for sustaining effort and persistence
• Options that foster collaboration and communication
Let’s Put It All Together
This clearly shows how sentence frames/ walls are linked to the UDL principles. Sentence frames display information in a visual format, showing students the proper language structure, which can show relationships in language, such as a problem and solution statement. Teachers will scaffold instruction to the point where students will become less dependent on this tool. Sentence frames help students plan and manage their ideas and how to express those ideas. It is also a handy assessment tool for teachers to monitor student progress in written and oral language. It also allows opportunities for the student, classroom teacher and the English Language Educator to work together.
Carrier, K.A. & Tatum, A.W. (2006). Creating sentence walls to help English-language learners develop content literacy. The Reading Teacher, 60(3), pp. 285-288.
Donnelly, W.B. & Roe, C.J. (2010). Using sentence frames to develop academic vocabulary for English learners. The Reading Teacher, 64(2), pp. 131-136.
Here is a great story frame to use with your students to learn about story elements.
Here is a Cinderella story frame.
Look sentence frames can be used in math!
“Babies are born with the instinct to speak, the way spiders are born with the instinct to spin webs. You don’t need to train babies to speak; they just do. But reading is different.” — Steven Pinker
Phonemic awareness is the ability to examine language independently of meaning and to manipulate its sounds (Griffith, 1992).
Segmenting sounds is important because children need to blend sounds when using letter-sound correspondences to read new words. Phonemic awareness is a powerful predictor of later reading achievement (Griffith, 1992). Segmenting isolates the sounds in spoken words by pronouncing each one in order (Ellis, 1997). Learning to differentiate sounds in words is confusing for some students. Differentiating sounds is an auditory process, but with the scaffolding of a kinesthetic method such as sound boxes help students manipulate those sounds (McCarthy, 2008).
Phoneme Awareness Activity Guidelines
- Identify and focus on a phoneme awareness task.
- Use phoneme sounds (/ /), not letter names. Remember: some sounds are represented by two or more letters (/sh/).
- Teach the following sequence: consonant vowel pattern (CV), vowel consonants (VC), and the consonant vowel consonant pattern (CVC). (Ellis, 1997)
As a rule, most activities should not exceed 15-20 minutes. Words used should be chosen from the curriculum being taught. (Ellis, 1997)
A great segmenting activity is the use of Elkonin boxes or sound boxes. A card is presented to the child. On top is a picture of a simple word. (Ellis, 1997) The picture helps students remember the word being stretched, allowing for independent refocus (McCarthy, 2008). Below the picture is a series of boxes, one for each sound (phoneme). The teacher articulates the word phoneme-by-phoneme, while pushing a counter into each box for each phoneme. Eventually, the child will “say it and “move it” independently. Overtime, the picture and boxes are eliminated. Students will be able to segment the words without visuals. (Ellis, 1997).
Elkonin boxes help the students hear the sequence of phonemes in words. They verbally stretch the word’s sounds while pushing a counter into boxes, one for each sound heard. (McCarthy, 2008) This method helps them think of the order of spoken sounds in words (Griffith, 1992).
When you first teach this process start with three phonemes and a continuous initial consonant sound (flow of breath is not constricted, /s/, /f/, /m/). Students are able to use one continuous breath to stretch the words. Once successful, move to initial stop consonants (/d/, /k/, /t/), then use words that segment phonemes with blends at the initial and/or final position (/bl/, /mp/). (McCarthy, 2008)
Teaching Sound Segmentation
- Explicit modeling of stretching the word out into phonemes. Have students repeat after you.
- When students can stretch the words explain and model sound boxes.
- Students practice stretching words and using sound boxes simultaneously until they achieve independence. (McCarthy, 2008)
Research links some reading failure to insufficiently developed phoneme awareness skills. These skills should be a part of the daily kindergarten and first grade reading program. This can easily be done through phonemic awareness training activities. It should be a natural extension of the lesson. (Ellis, 1997)
Positive results in one study show students who received training in phonemic awareness outperformed the students in the control group (NRP). Studies have also shown phonemic awareness training is a better indicator than measures such as IQ. Poor readers who enter first grade phonemically unaware are likely to remain poor readers at the end of fourth grade. (Griffith, 1992)
It is essential for student at risk for reading difficulties to receive training in phonemic awareness (Ellis, 1997).
Students with learning disabilities have difficulty learning how to relate lettsers to sound. It benefits all students to receive explicit instruction in phonemic awareness. Teachers need to use a variety of research based techniques to help children learn to read. Teachers should provide explicit instruction, lots of modelling of strategies, guided practice, and independent practice. Students should be given the opportunity to practice these skills.
Developing Phonemic Awareness
- Expose students to literature that plays with sounds in language (rhyming, alliteration).
- Provide extensive writing experiences (invented spelling)
- Provide explicit instruction in sound segmentation. (Griffith, 1992)
Phonemic Awareness Blog, http://papahere.com/2009/03/10/phonemic-awareness/
Ellis, E.S. (1997). How now brown cow: phoneme awareness activities. Reading Rockets. http://www.reading rocket.org/article/388?theme=print
Griffith, P.L. & Olson, M.W. (1992). “Phonemic awareness helps beginning readers break the code.” Reading Teacher, 45(7), 516-522.
McCarthy, P.A. “Using sound boxes systematically to develop phonemic awareness.” Reading Teacher, 62(4), 346-349.
National Institute of Child Health and Human Development. (2000). Report of the National Reading Panel. Teaching children to read: An evidence-based assessment of the scientific research literature on reading and its implications for reading instruction (NIH Publication No. 00-4769). Washington, DC: U.S. Government Printing Office.