As many of you have noticed, this blog has not had an entry for some time. I have another blog just for my multisensory math classes. It is where participants in my distance classes can interact, ask questions, comment on the class and generally interact with each other and with me.

I will try to post more frequently in the near future. Lots of things have been going on in the world of Multisensory Math. I have given workshops in several states and am scheduled for several more presentations in the near future. I seem to be traversing the country right now.

You can see two Multisensory Math Presentations at the NCTM Regional Conference in Richmond VA in November 2014. I will also be helping to moderate a panel discussion about differentiation and reaching all learners. Recent presentations have included full day seminars in Ohio and Seattle WA. I will be giving presentations at the LDA National Conference in Chicago in February and in VA later in the spring. Look for a new Multisensory Math website in the near future as well. I am working on various projects as quickly as I can.

If you are accessing this blog as part of the ASDEC Multisensory Math Class, you are in the wrong place! You should be accessing: asdecmultisensorymathonline.blogspot.com. It is there that you will find postings related directly to the course.

## Wednesday, October 22, 2014

## Friday, April 18, 2014

### Change: Tick Tock and Too Many Words

In the field of multisensory education there are some things that
remain the same. The use of concrete objects and tactile if not gross
motor strategies to anchor the student in the learning are fundamental.
These strategies address the conceptual component as well as the
attentional component of learning. This is one reason that the
multisensory strategies work well with language based learning
disabilities as well as related learning differences such as ADD/ADHD.
They do not interfere with traditional learners and in fact enhance the
learning for all students. They provide experiences which are memorable
through multiple learning channels.

Some things are open to interpretation and adjustment. In the field of education, we sometimes practice certain strategies which seem to become imbedded in our practice without question. They become tradition and though anecdotal evidence may suggest efficacy, in fact they are not as efficient or as effective as sometimes newer ideas. Take for example the "mad minute." Though this may work for and encourage some students, the timed stressful activity is a detriment to those who have processing or retrieval difficulties. It is not that children do not need to practice math facts, it is just that some methods of practice may not be the best for some children.

When I formatted the new math manual, in fact as I worked on it over the years, I jettisoned some strategies that were based on a verbal approach to teaching facts. As I reviewed the research on how the brain processes mathematics, it became clear to me that using too many words and convoluted stories to teach basic facts might be less useful than other visual and numeracy based strategies being suggested by researchers such as Dehaene and Butterworth. Over the years, the mFRI studies continued to suggest automatic recognition of small quantities and building larger quantity awareness on the construction and deconstruction of smaller quantities. Thus I jettisoned the traditional O-G based strategies based on language.

As I have updated the Multisensory Math Manual, the approach of this program has become multifaceted. I am attempting to combine the best of what we have from the traditional multisensory approach and the best of what we can learn from the research in mathematics. Please review the addition and subtraction chart in your manual. You will find no verbal strategies which ask children to use working memory to traverse and link known facts to others. Directionality is an issue for many alternative learners. Back and forth addition and subtraction based on words is not easy for many of our students. Instead, the student is encouraged to create the mental imagery for the construction and deconstruction of quantity based on numeracy patterns. With these he can continually draw on that mental imagery to solve even more complex problems.

I have demonstrated this in our Skype sessions with subtraction across place value. The What Works Clearinghouse suggests that multiple representations be used for concepts and strategies. This is one reason I use craft sticks, Unifix cubes, base ten place value blocks, tally marks, dice etc. All of these can be used to create mental imagery to support computation. Repeated exposure to numeracy patterns can form the basis of more complex calculations. Experience with these patterns will support memory and extensions to larger more complex applications.

Many new text book series are beginning to employ this idea of pattern recognition and numeracy in developing number sense. Older strategies such as go to the ten and counting on may still be used. They are based on skills and visual imagery. The number line and concrete manipulatives are only tools in helping students develop full numeracy awareness. We use all the tools available to us but for those students who have language based differences we need to be careful of using strategies based on too many words or associations of patterns based on words. We need to remember that quantity does not have a single color or shape. Over reliance on a single manipulative or rigid verbal strategy actually may limit a student. Multiple representations are key and memorable patterns based on visual stimuli and concrete constructions are only the beginning.

We must ultimately move students on to the abstract level and gently encourage memory and retrieval through successful repeated practice which does not discourage or lead to despair. The NCTM is emphasizing perseverance in problem solving. It is emphasizing the great glow that students get when they accomplish something challenging. Student should be challenged, especially our gifted one. This is important but we need to ask ourselves what exactly those challenges should be. Some of our gifted students are gifted in ways that are not tested by a ticking clock or their ability to remember stories about how to get to a target sum. They can be defeated before they get to the mathematical starting gate which is applications. To this end, the multisensory programs always emphasize teaching for success and mastery. Thus, we seek the best tools and strategies available in an ever changing educational landscape. The new NCTM publication, Principles to Actions, emphasizes concept based instruction and seeks to summarize where we are today. The executive summary is available on line.

Some things are open to interpretation and adjustment. In the field of education, we sometimes practice certain strategies which seem to become imbedded in our practice without question. They become tradition and though anecdotal evidence may suggest efficacy, in fact they are not as efficient or as effective as sometimes newer ideas. Take for example the "mad minute." Though this may work for and encourage some students, the timed stressful activity is a detriment to those who have processing or retrieval difficulties. It is not that children do not need to practice math facts, it is just that some methods of practice may not be the best for some children.

When I formatted the new math manual, in fact as I worked on it over the years, I jettisoned some strategies that were based on a verbal approach to teaching facts. As I reviewed the research on how the brain processes mathematics, it became clear to me that using too many words and convoluted stories to teach basic facts might be less useful than other visual and numeracy based strategies being suggested by researchers such as Dehaene and Butterworth. Over the years, the mFRI studies continued to suggest automatic recognition of small quantities and building larger quantity awareness on the construction and deconstruction of smaller quantities. Thus I jettisoned the traditional O-G based strategies based on language.

As I have updated the Multisensory Math Manual, the approach of this program has become multifaceted. I am attempting to combine the best of what we have from the traditional multisensory approach and the best of what we can learn from the research in mathematics. Please review the addition and subtraction chart in your manual. You will find no verbal strategies which ask children to use working memory to traverse and link known facts to others. Directionality is an issue for many alternative learners. Back and forth addition and subtraction based on words is not easy for many of our students. Instead, the student is encouraged to create the mental imagery for the construction and deconstruction of quantity based on numeracy patterns. With these he can continually draw on that mental imagery to solve even more complex problems.

I have demonstrated this in our Skype sessions with subtraction across place value. The What Works Clearinghouse suggests that multiple representations be used for concepts and strategies. This is one reason I use craft sticks, Unifix cubes, base ten place value blocks, tally marks, dice etc. All of these can be used to create mental imagery to support computation. Repeated exposure to numeracy patterns can form the basis of more complex calculations. Experience with these patterns will support memory and extensions to larger more complex applications.

Many new text book series are beginning to employ this idea of pattern recognition and numeracy in developing number sense. Older strategies such as go to the ten and counting on may still be used. They are based on skills and visual imagery. The number line and concrete manipulatives are only tools in helping students develop full numeracy awareness. We use all the tools available to us but for those students who have language based differences we need to be careful of using strategies based on too many words or associations of patterns based on words. We need to remember that quantity does not have a single color or shape. Over reliance on a single manipulative or rigid verbal strategy actually may limit a student. Multiple representations are key and memorable patterns based on visual stimuli and concrete constructions are only the beginning.

We must ultimately move students on to the abstract level and gently encourage memory and retrieval through successful repeated practice which does not discourage or lead to despair. The NCTM is emphasizing perseverance in problem solving. It is emphasizing the great glow that students get when they accomplish something challenging. Student should be challenged, especially our gifted one. This is important but we need to ask ourselves what exactly those challenges should be. Some of our gifted students are gifted in ways that are not tested by a ticking clock or their ability to remember stories about how to get to a target sum. They can be defeated before they get to the mathematical starting gate which is applications. To this end, the multisensory programs always emphasize teaching for success and mastery. Thus, we seek the best tools and strategies available in an ever changing educational landscape. The new NCTM publication, Principles to Actions, emphasizes concept based instruction and seeks to summarize where we are today. The executive summary is available on line.

## Friday, April 11, 2014

### Linkages

A word we use in the language field is "linkage." This is the connection between things like the sound and symbol correspondence. In math it might mean the connection between the quantity, the numeral and the name.

Numeracy is such an essential component of all mathematics and it must be addressed if it is a deficit for any student. Quantity awareness allows a student to fluently calculate, to estimate and apply. It has a spatial component when thought of along a number line. This can be an important component which lends itself to gross motor activities for the learning disabled child or the alternative learner.

One aspect of numeracy that goes beyond subitizing is pattern recognition. Think of place value and recognizing of four and four hundred are related. Subitizing allows us to recognize quantity but pattern recognition allows us to apply it at higher levels.

The What Works Clearinghouse suggests that children need to see multiple representations of math concepts. Absolutely, I agree. Keep in mind though that children need to have those representations linked so that the broad concept makes sense. Think fractions, decimals and percent! If a child comprehends 1/2 and taking one half of a quantity, he should also be able to link that to multiplication by 0.5 and taking 50% of a quantity. The careful teacher makes sure to revisit previously taught concepts and connect the dots and not just teach each new topic as if it exists as a set of procedures unto itself.

I routinely encounter teachers who attend my workshops and say, "I wish I had been taught this way." I believe it is because I stress a concept based approach and as we would say in the field of dyslexia, an approach that is incremental, sequential, cumulative and thorough with practice to mastery. To too many people though that sounds like procedures. It is not. The concept in math is the central piece for understanding. Without, applications are hit and miss. Procedural knowledge may get a student through one high stakes test but may be lost over time and not lend itself to applications such as problem solving. Procedures do not lead to deep thinking about mathematics.

The NCTM has a new publication which is really worth a look. Check out Principles to Actions on the NCTM website: http://www.nctm.org/principlestoactions/

Numeracy is such an essential component of all mathematics and it must be addressed if it is a deficit for any student. Quantity awareness allows a student to fluently calculate, to estimate and apply. It has a spatial component when thought of along a number line. This can be an important component which lends itself to gross motor activities for the learning disabled child or the alternative learner.

One aspect of numeracy that goes beyond subitizing is pattern recognition. Think of place value and recognizing of four and four hundred are related. Subitizing allows us to recognize quantity but pattern recognition allows us to apply it at higher levels.

The What Works Clearinghouse suggests that children need to see multiple representations of math concepts. Absolutely, I agree. Keep in mind though that children need to have those representations linked so that the broad concept makes sense. Think fractions, decimals and percent! If a child comprehends 1/2 and taking one half of a quantity, he should also be able to link that to multiplication by 0.5 and taking 50% of a quantity. The careful teacher makes sure to revisit previously taught concepts and connect the dots and not just teach each new topic as if it exists as a set of procedures unto itself.

I routinely encounter teachers who attend my workshops and say, "I wish I had been taught this way." I believe it is because I stress a concept based approach and as we would say in the field of dyslexia, an approach that is incremental, sequential, cumulative and thorough with practice to mastery. To too many people though that sounds like procedures. It is not. The concept in math is the central piece for understanding. Without, applications are hit and miss. Procedural knowledge may get a student through one high stakes test but may be lost over time and not lend itself to applications such as problem solving. Procedures do not lead to deep thinking about mathematics.

The NCTM has a new publication which is really worth a look. Check out Principles to Actions on the NCTM website: http://www.nctm.org/principlestoactions/

## Friday, March 28, 2014

### Thinking Vertically About Teaching Math

**For remediation with older students, one needs to think of skills introduced but not mastered. Begin to think about concepts which bridge multiple operations and levels. Two examples are "Regrouping" and "Place Value." With older students who have been taught procedures without concepts this is a terrific place to begin.**

__Thinking Vertically:__
Start by using manipulatives to model whole number
operations with the place value mat you were given. Ask students to
"prove by construction" answers to basic problems without
regrouping. With severe students I would recommend the craft sticks
because as I have said, the student may need to physically bundle and

*unbundle*quantities. If the student needs only to reinforce the concept, base ten blocks may be used. After regrouping is introduced, practiced and mastered, you can move the student to fraction concepts.
Creation of fractions with your fraction circles is a first
step. Students may keep one circle uncut to remind them of how many
pieces it takes to make a "whole." Other fraction pieces may be
used to add and subtract like fractions. If the solution is an improper
fraction, the students quickly see that laying the whole circle on top
"simplifies" the fraction to a mixed number.

The next step is helping them understand that we may
"regroup" from the one's place value to the fraction place value by
moving "one" to the fraction place value and representing it as the
required circle "cut" into the required size pieces. The whole
can never be in the fraction place value unless it has been "cut"
into its required number of pieces, thus creating an improper fraction in the
fraction place value.

The student learns that we may get a "sum" which
is improper in any place value by the operation of addition. We then
"simplify" the quantity to its proper form. We may need to
create an improper quantity in ANY place value in order to subtract. This
is a fundamental concept for both whole number operations and fractions.

The older student feels validated in that he or she is
working at higher levels of math, but is also beginning to understand the
fundamental math concepts which form the foundations of higher level skills.

### What Comes Before & What comes After

**What Comes Before & What Comes After**

In thinking about the math courses, I wanted to ask each of
you to consider thinking about one concept such as multiplication or
fractions. Begin to think about how that concept appears at various
levels of instruction. What would be the earliest exposure a student
might have? What vocabulary is essential for the child to comprehend the
concept? How could a child experience the concept, practice the concept
and demonstrate proficiency at an early level?

Then, I would like you to jump ahead several levels and
years. How is this concept applied at higher levels of math? How
does the early vocabulary continue to be important in concept formation and
application? How does one expand this concept to extremely abstract
levels?

As a primary grade teacher, we need to understand how what
we do at basic levels forms the foundation of what is to come. As a
secondary teacher, we need to understand the basic concept instruction and
vocabulary so that we may go back to fill in gaps for those who need remedial
instruction.

You might also choose a concept such as division or
multiplication. Try to spend a few moments considering the various levels
and applications

## Sunday, February 23, 2014

### Gross Motor Activities

Those of us who work in multisensory education know the importance of using large motor muscles as part of daily instruction. Using large movement helps to reinforce directionality and sequence. It provides a direct path to the brain in a way that fine motor movements do not. Think of the muscle memory involved in navigating a dark bedroom at night. You know where everything is without the aid of eyesight. You have tactile memory and spatial awareness.

Now consider mimicking the direction and sequence of multi-digit multiplication operations, or numeral formation. These activities allow students the opportunity to get up, flood the brain with oxygen and move. Consider Geometry Simon Says, or even the direction and spatial movements associated with transformations of function.

Dr. Joyce Steeves believed that students needed the opportunity to get up during a lesson each day. It might mean tossing a weighted object during skip counting or forming numerals in the air, but they do need to get up and not just to offer samples of homework worked the night before.

I observed a wonderful lesson in North Carolina in which the 4th grade teacher in an independent school, used dry erase pens to write problems for the students to solve on various desk surfaces around the room. They were given a single sheet of paper quartered for completion. They could move to the various desks to copy and solve, work collaboratively etc. They were given a fixed amount of time and had to pass the papers in before exiting for the next class. There was a seat time discussion of the problems before dismissal so there was an orderly transition to the next class, but the students were well regulated and engaged...up and actively involved in the math lesson.

Now consider mimicking the direction and sequence of multi-digit multiplication operations, or numeral formation. These activities allow students the opportunity to get up, flood the brain with oxygen and move. Consider Geometry Simon Says, or even the direction and spatial movements associated with transformations of function.

Dr. Joyce Steeves believed that students needed the opportunity to get up during a lesson each day. It might mean tossing a weighted object during skip counting or forming numerals in the air, but they do need to get up and not just to offer samples of homework worked the night before.

I observed a wonderful lesson in North Carolina in which the 4th grade teacher in an independent school, used dry erase pens to write problems for the students to solve on various desk surfaces around the room. They were given a single sheet of paper quartered for completion. They could move to the various desks to copy and solve, work collaboratively etc. They were given a fixed amount of time and had to pass the papers in before exiting for the next class. There was a seat time discussion of the problems before dismissal so there was an orderly transition to the next class, but the students were well regulated and engaged...up and actively involved in the math lesson.

## Friday, February 21, 2014

### It's the Language

As I return from my latest two day workshop, I am continually
impressed with the need to address instructional language. When I
completed my teaching degrees, I received no instruction in identifying
and dealing with learning differences. It was assumed that if a child
were not in special education he or she would be able to learn in my
class. That was many years ago. Within the first three years of my
teaching, all teachers were required to take a special education class
or some in house training in special needs. That is when I first
received training in multisensory methods as part of a public school O-G
based program.

Now teachers are expected to teach inclusion classes and sometimes work with another teacher to support special needs students in blended classrooms. There is lots of talk about differentiation. What I find in the field is that teachers are not always given sufficient professional development to feel confident in addressing the myriad learning differences that can occur in the same class.

One of the simplest vehicles for addressing the needs of all students is clear, precise, concept oriented language with a rate of speech that does not race of some student's heads. A well articulated lesson delivered at a moderate rate of speech is more apt to reach a greater number of students.

Yes, there are times when a faster rate of speech, emotion and excitement should infuse the classroom with urgency. These are moment which inspire students and engage them in questioning and analysis. However, when a new concept is being introduced or sequential directions are given, the rate of speech should allow for processing speed deficits and be delivered in such a way to meet the needs of all students.

Now teachers are expected to teach inclusion classes and sometimes work with another teacher to support special needs students in blended classrooms. There is lots of talk about differentiation. What I find in the field is that teachers are not always given sufficient professional development to feel confident in addressing the myriad learning differences that can occur in the same class.

One of the simplest vehicles for addressing the needs of all students is clear, precise, concept oriented language with a rate of speech that does not race of some student's heads. A well articulated lesson delivered at a moderate rate of speech is more apt to reach a greater number of students.

Yes, there are times when a faster rate of speech, emotion and excitement should infuse the classroom with urgency. These are moment which inspire students and engage them in questioning and analysis. However, when a new concept is being introduced or sequential directions are given, the rate of speech should allow for processing speed deficits and be delivered in such a way to meet the needs of all students.

## Sunday, February 16, 2014

### Additional Resources and Intern Comments

I receive many questions from educators about the use of
manipulatives in the classroom. This is one reason that a sample
professional development contract involving me often includes
demonstration lessons. In these sessions, I teach the students in front
of their teachers to show how manipulatives will be received by the
students and how best to use them. I have taught in both public and
private schools, large classes and small. In very large classes, I
limit the time spent and the number of activities with manipulatives to
make sure that students use them efficiently. I also know that the more
you use manipulatvies, the more students get used to them and learn the
behavior rules associated with them. They get acclimated so to speak.

I would like to relate a discussion I had with an intern after the Skype session yesterday. This particular intern is at the end of her practicum. She has been using multisensory math methods in her classroom for two years. Basically her reaction includes the following observations:

Additional resources mentioned in a Skype session with my MSM II class yesterday include:

Don't forget LearnZillion for video ideas regarding the Common Core State Standards and Hippocampus.org for a free, open source algebra course with videos acceptable for student use at home. Hippocampus has other resources available as well and it is well worth a look.

I would like to relate a discussion I had with an intern after the Skype session yesterday. This particular intern is at the end of her practicum. She has been using multisensory math methods in her classroom for two years. Basically her reaction includes the following observations:

- The use of manipulatives is time consuming and can be messy, BUT after using them she feels that she has done much less reteaching. In other words, the concepts are retained more thoroughly.
- She has already seen growth in her assessment
scores. She uses both the
*CMAT*and the*Woodcock Johnson*because many of her students are funded. She teaches in a school for students with learning differences and must answer to local schools as she assures them that IEP goals are being met. Her administration is thrilled with student progress and growth in skills as well as comments. - She also believes that it is important to go back in the CRA instructional sequence and link the abstract to the concrete. This is one thing we recommend in the class when students with variable memory seem to have lost what they had previously mastered. A return to the concrete can be a good way to review and cement gains.
- Difficulties she has encounter are in using the complete lesson plan. It takes practice. However, she does like thinking through Joyce Steeves' lesson plan because it reminds her to get all the strands of math in over time.

Additional resources mentioned in a Skype session with my MSM II class yesterday include:

__by John A Van de Walle, Karen Karp and Jennifer Bay-Williams. It is a little pricey but can be rented or purchased for Kindle. I have the eighth edition which runs $149 new but closer to $100 used. It rents for as little as $46. The earlier editions range from $2 to $32 used in paperback.__**Elementary and Middle School Mathematics: Teaching Developmentally,**__Grades K-5 by Sherry Parrish - This is available for rent and purchase, paper back (used and new around $40-50) New, it comes with a DVD__**Number Talks: Helping Children Build Mental Math and Computation Strategies**__, by Theresa R. Fitzgerald, Billed as the #1 homework helper, it runs anywhere from $5 -$8, new and used.__**The Math Dictionary for Kids****Teaching Student-Centered Mathematics Developmentally Appropriate Instruction for Grades Pre-K-2,**again by John A Van de Walle et al. It is from Pearson Publishing. Various editions run $15-$50 on Amazon. It is a series and includes various levels so teachers could purchase the one which is appropriate for the level they teach.Don't forget LearnZillion for video ideas regarding the Common Core State Standards and Hippocampus.org for a free, open source algebra course with videos acceptable for student use at home. Hippocampus has other resources available as well and it is well worth a look.

## Saturday, February 15, 2014

### Evidence

Fractions- Look at the work of the Rational Number Project. You can find a compilation of the work from the University of Minnesota, find lesson plans, download fraction manipulatives and read of the success the project has had in public schools.

In the Multisensory Math Program at ASDEC we use a similar concept based approach. Students create fraction concept cards for the student notebook. They choose their manipulatives and assemble card stock graphic organizers which they keep for reference. The illustrate key vocabulary and concepts and all operations. Reaction from teachers using them has been extremely enthusiastic. "The children love them."

There is evidence to support using circular models first and then transitioning to other models. Teachers need to solidify concepts before teaching children to simply "push numbers around." So many teachers tell me that they wish they had been taught this way. You do not need commercial manipulatives to teach fractions either. You can print fraction circles on cardstock. You can use simple construction paper squares. Folded paper can come later after students fully comprehend the concepts and remember: fract- is a Latin root that means to break into parts. I suggest that students need to cut or break something to really understand the linkage.

In addition, students need to comprehend that the "fractions of one" is a place value concept. This should be fully understood before decimals are taught. Students will benefit from using manipulatives to add, subtract and simply fractions and mixed numbers. Then, students can move on to the abstract level of experience using only numbers.

In the Multisensory Math Program at ASDEC we use a similar concept based approach. Students create fraction concept cards for the student notebook. They choose their manipulatives and assemble card stock graphic organizers which they keep for reference. The illustrate key vocabulary and concepts and all operations. Reaction from teachers using them has been extremely enthusiastic. "The children love them."

There is evidence to support using circular models first and then transitioning to other models. Teachers need to solidify concepts before teaching children to simply "push numbers around." So many teachers tell me that they wish they had been taught this way. You do not need commercial manipulatives to teach fractions either. You can print fraction circles on cardstock. You can use simple construction paper squares. Folded paper can come later after students fully comprehend the concepts and remember: fract- is a Latin root that means to break into parts. I suggest that students need to cut or break something to really understand the linkage.

In addition, students need to comprehend that the "fractions of one" is a place value concept. This should be fully understood before decimals are taught. Students will benefit from using manipulatives to add, subtract and simply fractions and mixed numbers. Then, students can move on to the abstract level of experience using only numbers.

## Friday, February 14, 2014

### I Hate Fractions!

I think I have never met a student who claims to be proficient in fraction concepts and operations. More often than not, I am met with the title of this post. Why do you think that so many students feel inadequate when it comes to fractions.

I met with a student today who loved long division but would not even talk about fractions. "Could I use a calculator?" she asked. My students do not generally use calculators in sessions. I assure success by using student friendly numbers to teach concepts. We use calculators to check answers, to teach concepts such as transformations of functions or in cases where it simply makes sense not to struggle with complex solutions to function applications. We need a more exact answer.

So why is it that so many students do not feel confident with fractions? We all need to ask ourselves this question. What is it that we are doing right and what is it that we are not quite conveying? So much of higher math depends on an understanding of fractions. We need to consider what the research is telling us (circular models are better introduction) and there is a need to move from the concrete to the abstract for many students. They need to understand the concepts before they begin to push numbers around in operations they do not understand. In my workshops and courses, I keep hearing the following statement, "I wish I had been taught this way." Or, "I know how to tell them to get an answer. I just can't tell them why it works."

I met with a student today who loved long division but would not even talk about fractions. "Could I use a calculator?" she asked. My students do not generally use calculators in sessions. I assure success by using student friendly numbers to teach concepts. We use calculators to check answers, to teach concepts such as transformations of functions or in cases where it simply makes sense not to struggle with complex solutions to function applications. We need a more exact answer.

So why is it that so many students do not feel confident with fractions? We all need to ask ourselves this question. What is it that we are doing right and what is it that we are not quite conveying? So much of higher math depends on an understanding of fractions. We need to consider what the research is telling us (circular models are better introduction) and there is a need to move from the concrete to the abstract for many students. They need to understand the concepts before they begin to push numbers around in operations they do not understand. In my workshops and courses, I keep hearing the following statement, "I wish I had been taught this way." Or, "I know how to tell them to get an answer. I just can't tell them why it works."

## Sunday, February 9, 2014

### Foundation Concepts

We all look at modern curricula, textbooks, scope and sequence documents etc. and feel a bit overwhelmed. The Common Core State Standards initiative is one solution. It doesn't tell anyone how to teach or insist on specific content. It merely gives us a common set of skills with which students should be proficient at developmentally appropriate levels, or grades.

Even the Common Core can seem daunting if you dig too deeply at first. I would advise checking out the summary document on the website itself. You will find that the essential skills for each grade level are listed in focus documents that are incredibly simple to comprehend.

The Common Core Standards for math evolved from the NCTM Focal Points. For years these focal points formed a terrific guide for what needed to be mastered. They are based on recent research in how the mind processes math. This research has, I believe, fundamentally changed the way we teach mathematics.

We no longer believe that simply having procedural fluency is enough. Students today must understand what they are doing. I see this almost everyday in working with students at in algebra. some of my private clients attend schools which use ancient textbooks, heavy in language and complicated drills but little focus in the underlying concepts.

When we look at the evidence and milestones that students must achieve, consider investing additional time in teaching numeracy, construction and deconstruction of quantity for quantities up to and including ten, and of course, place value. Place value and the concepts underlying multiplication, division and fractions are essential. They simply must be done thoroughly. As important though is the way these concepts are introduced and developed. They cannot be rushed or taught procedurally for some test. They must be firmly placed and soundly developed to a level beyond familiarity.

Even the Common Core can seem daunting if you dig too deeply at first. I would advise checking out the summary document on the website itself. You will find that the essential skills for each grade level are listed in focus documents that are incredibly simple to comprehend.

The Common Core Standards for math evolved from the NCTM Focal Points. For years these focal points formed a terrific guide for what needed to be mastered. They are based on recent research in how the mind processes math. This research has, I believe, fundamentally changed the way we teach mathematics.

We no longer believe that simply having procedural fluency is enough. Students today must understand what they are doing. I see this almost everyday in working with students at in algebra. some of my private clients attend schools which use ancient textbooks, heavy in language and complicated drills but little focus in the underlying concepts.

When we look at the evidence and milestones that students must achieve, consider investing additional time in teaching numeracy, construction and deconstruction of quantity for quantities up to and including ten, and of course, place value. Place value and the concepts underlying multiplication, division and fractions are essential. They simply must be done thoroughly. As important though is the way these concepts are introduced and developed. They cannot be rushed or taught procedurally for some test. They must be firmly placed and soundly developed to a level beyond familiarity.

## Tuesday, January 28, 2014

### Manipulatives: Must Be Efficient & Effective For What Is Being Taught

I continually state that the use of manipulatives is to teach concepts. Manipulatives are used to give the students a hands-on experience, one that is memorable and helps them interact with different representations. This is a core principle of UDL (Universal Design for Learning).

The goal of using manipulatives is to illustrate a concept and then get rid of them. Students should seldom perform calculations with manipulatives unless it is skill building and aids in memory. For example, using manipulatives to illlustrate/see calculations of large quantities using craft sticks and a place value mat is extremely useful...for a while. It reinforces our place value system and allows them to physically experience regrouping and renaming-a concept that is one of our "continuous threads." Once the student begins to recognize the concept and has formed a mental representation of the procedure involved, we would want to move the student to the representation and abstract levels.

Picturing groups of quantities can certainly explicate the meaning of multiplication and division. They can help automatize select facts. They can illustrate the concepts of multiplication and division easily. They should lead to the use of specific fact families-and for LD students a very few- which are practiced and applied to the automatic level.

An inefficient use of manipulatives would be using counters to solve successive problems beyond the child's fact base knowledge system. This is where the general education teacher and the special educator may part ways in using a book or set curriculum. The published curriculum assumes that the child using the textbook has attained certain skill levels. The special needs student may not have the skills required to use the worksheets and practice pages associated with a specific concept.

This is not to say that a special education student cannot be taught higher level concepts. It only means that, as the What Works Clearinghouse suggests, struggling students practice math facts daily and as I say, use

*THOSE*facts in their activities. The teacher may make up a worksheet...yes, in your spare time of course...to fit the needs of the struggling student. Using a computer program such as Math Type, or the equation editor in MS

Word, the teacher can create a simple worksheet with fewer problems on a page, ample white space, and a restricted set of number facts which can be practiced to complex levels.

Take for example, long division. The typical text book would ask that the student work with a single digit divisor and two digit dividends until all multiplication facts have been worked through the division algorithm. Then, as the student approaches multi-digit dividends, the student is expected to have mastered the times table facts. This would preclude the special needs student from doing the activities. The special education teacher can easily create a worksheet using one of the tools mentioned above. The worksheet might use only one times table throughout but include problems with varying levels of complexity. This would prevent a student from simply using counters to solve problems by hand repeatedly practicing unrelated and isolated facts independently of each other, and leading to frustration without serving to build any mastery of any fact families.

As always, look at what you are teaching. Decide what your goal is. Are you teaching a concept or practicing applications? Applications do involve complexity, but student should be moved from using know facts to incorporating new ones in ways that build fluency and competence and do not lead to frustration.

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