This blog is dedicated to The Multisensory Math Approach

 This blog is from the Multisensory Math 1 course.  The course is offered through The Atlantic Seaboard Dyslexia Education Center in Rockville MD.  

The course has been offered since 2003 but it has been through many updates.  We try to refilm the course every two years.  The pandemic made that difficult.  We last filmed it in 2019 and that version of the course has proven to be extremely popular.  

The course is based on multisensory strategies from decades of instruction for students with language based learning challenges. The strategies also come from recommendations from the research in learning and the brain as well as those from the What Works Clearinghouse and general best practices in mathematics.  

You can find workshops and classes in this approach at the www.asdec.org website and also at www.multisensorymath.online   

This approach is now used in many independent and public schools. It adapts to any curriculum.  The strategies in the course are appropriate for all but essential for some.  

You can also find us at Marilyn's Multisensory Math on Facebook.  

Coming to the End of the Course

We are coming to the end of our Multisensory Math Course.  Our final office hours video conference is about to occur.  I will answer questions, reiterate our core principles and strategies and encourage you to practice the language which supports meaning and  memory.  Those of us on the math team sincerely hope you have gained new skills and knowledge from your work with us over the past two months.  Try to summarize in your own mind the key components and strategies from this course and above all pay attention to your instructional language. 

Remember that even when students are not identified as having learning challenges, there will be those in your classes, those whom you are called to support or tutor, who do have learning differences and will benefit from the strategies in this approach.

View of the videos will end at midnight Sunday September 15th.  This blog will be available until Wednesday.  Dropbox access will end with the closing of the ASDEC access portal for this course section. 

Please continue to do good work.  Support all learners with concept based instruction which is delivered in a manner appropriate to their learning needs.  Be of good cheer and remember that there is no reason for most people to be weak in math skills.  Your students will appreciate what you do for them.   Help them succeed. 

Post # 40 More on Working with Older Students

The biggest bang for your instructional buck” in terms of time can be found in concentrating on core components and skills that build toward algebraic reasoning and computational/ procedural fluency with comprehension.  I am now couching some of these “big” ideas as the Super Powers of Math. 

With older students you need to shore up fluency by targeting specific facts for a period of time and using them in all work.  That said, students need to have an understanding of numeracy patterns, place value and expended form, the distributive property and fraction concepts such as the “Magic One.” 

You can work with older students on multiple levels at the same time if you provide linkages.  For example, regrouping from the whole to the part can be introduced by reviewing regrouping within whole number operations for addition and subtraction.  The student who needs to work on integers needs to see how these operations are connected to numeracy patterns.  The student who does not understand 3+4=7 will not understand why 3-7=-4.  The student who needs to understand FOIL or the box method for polynomial multiplication needs to see the connections to multi-digit arithmetic, the distributive property and expanded form. 

When I work with older students I always provide connections to lower level skills and in my summer programs for middle and high school students we see good results by reviewing those super powers.  Students continually remark that they have previously never understood fraction concepts and I find that on the post assessments, scores rise when I invest time in reviewing multiplication/division concepts and procedures and fractions.  Students who do not understand fractions will not fully understand the applications using decimals and percent.  Truly reviewing supporting concepts…not only procedures…can produce gains for the student.  It takes only a few minutes a day to shore up the foundations of math concepts but the payback is big for both students and teachers.

Post # 39 Questions and THAT Fraction Page

I loved the questions posed in our recent video conference and wanted to post the fraction image from a recent presentation.

Remember, the language of instruction for fractions is comparable to the language of place value instruction.  In terms of conceptual horizons in math, fractions are incredibly important. 

Many teachers tell me that they rely almost exclusively on procedural steps when teaching multiplication, division and the use of/creation of common denominators.  Yet, we know that students who use manipulatives retain information longer and perform better on follow up tests including standards based assessments.  Know the meaning behind the math allows students to construct mental math supports for their computations and applications. 

The discussion we had about fraction division led us to this image from my IDA presentation this year:  5 ÷ ¾ with a remainder of 2/3 …meaning two of the three fourths needed to create another “group.” 

The picture shows a colored image of “creating groups of the quantity ¾” and then a number line version showing division as repeated subtraction.  Remember that students with LD many not have sufficient fine motor skills to navigate small number lines so using those as the exclusive representation may not yield substantial results. 

For students with LD, “blow it up.”  Create large representations which can use whole body/ gross motor movements.  Use near point references at each student’s desk or small group table.  Students enjoy “coloring” fraction solutions but students who only color fractional parts are not reported to retain those concepts so include precise mathematically accurate language as demonstrated in class.  

Our other interesting question related to the traditional division algorithm and why the “FIRST NUMBER FOR DIVISION”  box might be a two digit number.  We linked the algorithm to the array model in which the dividend must align to the divisor in terms of place value.  If the divisor is smaller than the first digit of the dividend, the “tens place digit” would need to be deconstructed into units for proper alignment.  It is why we do not use base ten blocks to divide a quantity greater than one hundred by say “9”.  The hundreds flat would need to be deconstructed into single unit cubes to create “groups of 9.”  This would be an inefficient use of the manipulatives which we avoid.  It is not that it cannot be done, but it is cumbersome and time consuming. 

So, if we are dividing 27 by 9 and we start with the representation of 27 as two ten rods and seven unit cubes, we would need to “unbundle” or deconstruct the two ten rods into single cubes to align them with our divisor of nine.  Not a good use of time or manipulatives.  If you need to do that once to demonstrate the concept, fine; but don’t over use larger numbers for demonstrations. A better manipulative for this concept would be Unifix cubes and making groups of nine. Remember manipulatives are used to illustrate the concept, then you move on to other representations and finally calculations at the abstract level.

Post # 38 Multiplication, Division & Gozinto

“Many of my students struggle with organization, and I believe this is a major barrier to them retaining concepts and procedures. My algebra student, for example, consistently mixes up rules for adding terms versus multiplying terms, and my 6th grade student continues to forget the different rules for decimal operations. I have asked them both to make "Trouble Area" pages, and we've also done compare and contrast sheets, but it still has not been effective in helping them continue to add to it other difficult concepts, easily refer back to it, or ultimately master the material.”

This is a very good question.  It is a question that has a multitude of answers though.  Answers depend on a diagnostic prescriptive observation and approach.

When we think of applying rules to calculations such as the example above in algebra, we must consider the roots of this problem which go back many years.  The problem can be remediated through language, through simultaneous processing and guiding the student through appropriate language for an internal monologue.  The student who is confusing multiplication and addition with like terms and variable may not have fully understood the meaning of the distributive property with whole numbers. This begins in fourth grade with multi-digit multiplication.  It is knowing that we arrive at partial products which then must be added to attain the complete product.   Ex.  7(32) can the thought of as 7(30 + 2).  7(30) must be linked to 7(3) in the student’s mind and then adapted for place value.  Then the student must understand that he is multiplying “parts” of the quantity 32 separately and to get the total product he must add the partial products. Thus 7*30 is 210 + 7*2 which equals 14.  The total product is 210 + 14 or 224.  

Taking the student back to the box method of multiplication with simple whole numbers is a good way to make that linkage.  Then, after several one digit by two digit applications AND one digit by three digit applications, all using the box, the instructor can adapt to using x(x+3).  The teacher can use the same box method again but demonstrate how and why the distributive property works.  One of the problems with using FOIL with students who have never performed the distributive property with whole numbers is that they do not understand the place value implications of what they are doing.  It is not linked to anything the KNOW and so it is like a new skill coming out of nowhere…a skill which involves directionality challenges and rules based on words. 

Another challenge is that students do the calculations in their heads without language to support it.  They see 2 and 3 but do not register the operational sign.  Think back to “touch/say the sign, follow the line…”   This begins at a very basic level.  Getting students to note the operational sign is basic.  They must say the operation in order to perform the correct one. 

Another tactic is to say the sign of a term as its “first name.”  Thus the term 3x in 5(4-3x) is read 5 times (4 minus 3x); but for distribution it must be read as 5 times 4 and 5 times (-3x).  Its first name is “negative.”   This requires explicit instruction and practice.

Think what happens when we multiply (x-3)(x² +2x -1).  With FOIL the partial products are: x³ + 2x² –x –3x² -6x +3.  Now the student must add like terms that are not adjacent and recognize exponents as indicating place values.  With the box method, whose values are explicit on the diagonals. 

Now you apply language strategies to help with processing: On the outside we MULTIPLY but inside – the partial products- we ADD!  Chant it with rhythm, touch if need be, practice combining terms on the diagonals. 

The most difficult procedure in this unit is adding and subtracting polynomials. One method is circles and diamonds and squares “Oh My.” Circles and diamonds and squares!   Code like terms by surrounding them with shapes,  then add or subtract as indicated.  If they are written as two parenthetical expressions, use chanting again as a sub-skill:  (…..) + (….) Addition, remove (parentheses) and add.   (….) – (….) Subtraction, distribute (the negative) and add.

Again, explicit modeling and observed practice with sub-skills is important.  Only then can one re-integrate the sub-skills into multi-step equations.   Reduce the language.  Chant repetitive guiding language in simultaneous processing activities.  Add questioning to slow the student down.  Ask the student to reason aloud while processing until the error is remediated. 

Often at this level, the problems can be myriad: directionality, impulsivity, lack of internal self-monitoring, lack of internal monologue as moderator, lack of really processing the operation or operational sign, not touching the numbers or terms with a pencil point- (I call it air calculation), font that is too small to assist in visual processing and sustaining visual attention, gaps in conceptual or procedural knowledge….etc.

A Few Remediation Strategies: 

·       “when in doubt, write it out”

·       Font too small- blow it up, use dry erase and super-size it.

·       Color code

·       Touch and say – the pencil point focuses the eye, the voice slows down the speed, the simultaneous interaction of senses is more apt to catch errors.

·       Chant the rules

·       Note the sign  

·       Link to previous skills, real life or whole number operations then apply to the abstract.

·       Visual dictionaries with examples to codify rules – Major concept sheets rather than single operation sheets.

·       Consistency in concepts, vocabulary and many operational procedures from early skills to algebra- Using similar formats and linking them in instruction

Post # 37 Dyscalculia

If a student can recognize patterns, respond to multisensory instruction, and continue to successfully apply mathematics concepts, the student’s primary disability may be language based and not true dyscalculia.  This is where we must look for primary causes of deficits and not simply an inability to learn math facts by traditional methods and in traditional time frames.  Yet, more and more I see this as a diagnosis and an excuse to simply offer accommodations.  Difficulty learning math facts is NOT an excuse to stop teaching them. Oh, and I believe in calculators.  I also believe in teaching children how and when to use them.  They are valuable tools especially as students move toward and into algebra. 

One of my cooperating teachers at a school where I consult observed a middle school teacher simply telling a student to use a calculator for something as simple as 4 x 6.  The teacher did not stop to help the student reason through it or develop a strategy which he could use again.  When this happens we rob our students of the ability to reason mathematically and abrogate our responsibility to teach.   Much of higher math requires students to express whether a solution is reasonable or not and then explain why.  This is real life math and when we do not help students develop these skills we do not help them become proficient enough for the skills they will need to work today's jobs and manage their own personal finances.  

We must begin to look more deeply at the root causes of deficits in math and simply offer a label as an excuse to use accommodations.  We all use technology for complex calculations but they should be used sparingly as we develop the skills necessary to survive in real life

Post # 36 Additional Diagnostic Procedures

If a student can recognize patterns, respond to multisensory instruction, and continue to successfully apply mathematics concepts, the student’s primary disability may be language based and not true dyscalculia.  This is where we must look for primary causes of deficits and not simply an inability to learn math facts by traditional methods and in traditional time frames.  Yet, more and more I see this as a diagnosis and an excuse to simply offer accommodations.  Difficulty learning math facts is NOT an excuse to stop teaching them. Oh, and I believe in calculators.  I also believe in teaching children how and when to use them.  They are valuable tools especially as students move toward and into algebra. 

In a differentiation model, think about where it might be appropriate to guide the student in reasoning, offer practice in specific facts for developing automaticity,  and where it would be appropriate to offer technology assistance for independent work even while the student is building fluency.  

One of my cooperating teachers at a school where I consult observed a middle school teacher simply telling a student to use a calculator for something as simple as 3 x 6.  The teacher did not stop to help the student reason through it or develop a strategy which he could use again.  When this happens we rob our students of the ability to reason mathematically and abrogate our responsibility to teach.   Much of higher math requires students to express whether a solution is reasonable or not and then explain why.  This is real life math and when we do not help students develop these skills we do not help them become proficient enough for the skills they will need to work today's jobs and manage their own personal finances.  

We must begin to look more deeply at the root causes of deficits in math and simply offer a label as an excuse to use accommodations. We all use technology for complex calculations but they should be used appropriately and sparingly as we develop the skills necessary to survive in real life

Post # 35 Linkages

We teach calendars.  We teach the days and weeks, the months and years.  Elementary school is fixed in what day it is until the day dissolves into minutes and hours.  We teach the sequencing and that one day follows the next with regularity.

For some reason we fail to teach that years evolve into decades and decades into centuries.  We fail to hang events on the larger tree of time.  I regularly ask social studies teachers to use time lines.  Like our number lines stretching from 0 to 1 and 0 to 10 and 10 to 100, history teachers should teach number lines in centuries.

I regularly see students who think that the Renaissance  was just a little while before the Great Depression.  So we in our extensions can help students across the curriculum by teaching number lines in larger ...to the extreme...representations.  For middle school and high school students, I like to draw a number line in centuries and hang the major events of modern world history like ornaments dangling like ornaments from some odd branch.  I draw linkages between events and decades, between major periods our history and out time.

Students like to see the march of time with pictures linked to the centuries.  When we draw a number line from 0 to 2000 and paste up a picture of knights and castles or the Mona Lisa, we help them to see history and numbers as immense and linear, sequential, building and growing from number to number, year to year, century to century up to today.

How lovely it would be to see the days of the week, linked to the weeks of a month, linked to one year and that one full year linked to some tiny point on a larger number line, one with historical events told in pictures.   I'd like to see the fall of Rome, feudalism, the Renaissance, the Age of Enlightenment, the Industrial Revolution....locomotives, early automobiles, rockets.  All, on a gigantic number line.

We should team teach more. 

Post # 34 Fewer Facts At A Time to Develop Fluency Over Time

One of the strategies for inclusion classes and special education settings is to practice math facts as per the suggestions of the What Works Clearinghouse Practice Guide for working with students who struggle with math.  Research suggests about ten minutes per day in math fact practice.  This does not mean the dreaded Mad Minute which taxes word retrieval and causes anxiety among those with learning deficits. 

For this course, I suggest focusing on a limited set of facts at an age appropriate level and practicing them to automaticity.  This can include multisensory practice with manipulatives and or representations, games, patterns and matching to algorithms, any activity which reinforces the facts being practiced.  These targeted math facts become the basis for introducing new skills and concepts. 

In much the same way a reading program would use a controlled text leveled to facilitate success and focusing on adding new vocabulary incrementally, a math program that seeks to include all students in lessons on higher level concepts can include all students by using student friendly numbers for new introductions. 

One school with which I consult is trying this approach this year and reports greater self confidence in the middle grade elementary students.  The students do continue to add additional math facts in  ongoing incremental practice, but new skills are taught and immediately practiced with a core set of number facts to which the entire school has dedicated itself to teaching to automaticity.  At specific horizons, the math staff has chosen to focus on core concepts for each grade level as suggested in the Common Core State Standards.  Even the severely learning disabled students are working on grade level concepts.  They are just doing so using specific number facts. 

For example, utilizing the early times tables of two, three, five and nine; students can work through multiplication and division algorithms.  They do not rush at breakneck speed to master all times table facts before using them in applications.  Multiplication facts are targeted toward inclusion students in such a way that they can master chunks at a time and work within their mastered facts.  They go home saying to their siblings and peers, "Oh, I'm working on long division too." 

One of my current students has severe memory and word retrieval issues.  He is mastering numeracy patterns, and specific addition/subtraction facts even as he adds like fractions and the meaning of numerator and denominator.  He can accurately utilize the facts he has mastered to problem solve along with his class.  For the first time his conceptual awareness has surpassed that of his sister who has always been in the lead.  In the same lesson he might review the ordered pairs of ten, place value with regrouping and the seven times table.  He is a constant work in progress but he has made enormous progress since he has begun working in a more conceptual approach.  He is no longer relegated to levels of word retrieval before he can attempt something new.   He comes to his sessions asking to work on fractions.  What math teacher would not like to hear that? 

Addendum:  I am happy to report that the student discussed above took the SSAT and ISEE exams for entrance to independent middle schools.  He scored solidly in the "middle of the pack" as his parents said.  This is quite an accomplishment for a student who struggled as much as he did.  In addition to that, I want to report that my summer math camp for rising middle school students has had a similar result.  I use the strings with wings and teach the seven times table to automaticity. The students fill out portions of the times table chart daily:  the square numbers, the 2, 3, 5, and 9 times tables...and the sevens.  We use the seven times table for all calculations in class, multiplication, division, fraction operations- including simplifying and common denominators.  The students have no difficulty recalling seven as a factor or the products in the seven times table as multiples.  After the first several days, they do not even refer to their bead strings.  Post assessment scores are up as are confidence levels.

Post # 33 Along the Way: Thinking About Fractions

After our Video Conference sessions, I would like to plant some seeds for thought.  Think about concept based teaching.  How would you explain to a student the difference between 4 divided by 2 and 4 divided by 1/2.  How could you demonstrate the difference through drawings or representations? You have seen that we can often teach a concept itself with pictures that spark questions or address a concept creatively. 

As I work with older students who have been taught in a purely procedure driven manner, I find that they are as confused as many adults about the meaning behind this concept.  Even students who perform very well computationally may experience difficulty estimating and applying these concepts if they do not understand them.  Why is 3.5 divided by 0.125 a whole number answer?  How could we estimate a solution for 30.069 divided by 9?  How does a student know whether or not his answer is reasonable or not?  When the student reaches algebra, he should know. Before she gets there, we should model the thinking and the language of estimation. 

I find that the language of fraction division is among the most important we create.  We must strive to help students reason why ½ divided by ¼ is 2.  In division with fractions, we can ask how many pieces the size of “__” we can make if we start with a specific quantity…but sometimes it is not how many, but how much of.

How much of ¾ can we make from the quantity 3/8? Consider, ¾ divided by 3/8 is 2, but 3/8 divided by ¾ is ½.  Can you explain with simple, comprehensible language why that is so?  The language lies in the “how many of” vs “how much of” and we need to help students see the meaning behind the math if they are to reason mathematically.