Post # 18 Manipulatives: Efficient & Effective for the Concept Being Taught

You will note that I continually state that the purpose of using 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 illustrate/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.  It is like learning the entire multiplication tables yet again which plays to the special needs students' weaknesses.  

The special educator can introduce the concept of long division using Unifix cubes.  By choosing a quantity such as sixteen and asking the student to make groups of various quantities ( 3, 4, 7,) the  can experience one of the meanings of division including with "left overs" or remainders.  Then the student can learn the division algorithm- the goal of the lesson- using friendly numbers.  The special education teacher can easily create a worksheet using one of the tools mentioned above.  The worksheet might use only a single  times table throughout but include problems with varying levels of complexity.  An example might be 30 ÷ 5, then 32 ÷ 5, then 34 ÷ 5.  Eventually the algorithm might be extended to 350 ÷ 5 and 360 ÷ 5.  The student would encounter the procedures in a meaningful way and learn the steps to the traditional algorithm incrementally, sequentially and meaningfully.   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?  Are you focusing on procedural fluency or torturing a student with automatic recall of facts while trying to teach sequential steps.   

Remember to separate the concept from the computations and build fluency while maintaining skills.  Applications do involve complexity, but students should be moved from using know facts to incorporating new ones in ways that build fluency and competence and do not lead to frustration. 

Post # 17 Work, Labor and Multisensory Math

When we think of labor, we think of a job, a life's work.  It is said that work is what you do not want to do.  For many of us, our daily work is pure joy; for others, not so much.  My own father worked for 37 years as a public servant.  His duties included ordering supplies for the White House and inspecting the midcentury repairs on the Capitol Dome.  He was sometimes called in the middle of the night to initiate a repair or rectify a problem.  He received little overt recognition, save for the occasional White House Christmas card.

I think about the word 'labor' itself and how much effort it takes some of our students to master those pesky math facts. I remember one older elementary student began to cry when he said his father was going to make him start practicing with flash cards again.  His tearful response was, "I’m not going to do it!"  Clearly he had experienced frustration and feelings of ineptitude.  The NCTM asks teachers to help students develop staying power and tenacity in problem solving.  For my students with challenges, I call it academic stamina.

Word retrieval is so very difficult for many of our most fragile learners.  I love that term “fragile.”  I love it because for some students repeated experiences of failure can doom the capacity to risk trying yet again.   One participant used it recently in a communication with me.  It so aptly describes the emotional state of so many students who struggle day after day and are humiliated day after day by those timed drills. They constitute too many facts required at one time for recall.  

There are ways to make learning math facts less tedious, some methods can be almost fun.  Certainly we can offer the facts of the day for use, and then use them consistently to reinforce memory.  We need to choose fewer facts and practice them with efficiency. We must practice fewer facts at a time to develop fluency over time.

Let us use a more humane method for developing fluency.  Let us practice fewer facts and practice them to mastery in a multitude of ways. Allow students to create their own near point references to use during class rather than giving them a “multiplication chart” already filled out…which constitutes just a less efficient calculator.  Let us keep them in the forefront of our activities and keep some manipulative or representation close by in case our word retrieval fails us...at that moment, or across several moments, until it does not fail us at all. Let us help children understand that though learning math facts is important, a skill that takes dedication and practice...as Red says to Rover in the comic strip- "the multiplication tables won't stay up all night with you when you are running a fever." 

By choosing the facts we will use during a lesson, practicing those facts early in the lesson and then employing them throughout the lesson, we build multiple experiences to support memory.  This is one focus of my approach to the Steeves Lesson Plan.  It is one thing I will look for in assigned projects.  Let’s hold that though for a video conference. 

Post # 16 Neuroplasticity and Multisensory Math

Neuroplasticity, what a word. It represents the brain’s ability to reshape itself and organize itself to react to what the body needs and does. In Judith Willis’ book, Research-Based Strategies to Ignite Student Learning, the scientist turned educator describes how the brain changes in order to maximize its capabilities in relation to its volume.  The brain can add gray matter and reinforce neural connections when demand requires it to do so.  Thus an action repeated over and over again, leads to additional neural connections and gray matter dedication in response to activity.  She cites as examples the area dedicated to use of the fine motor skills of the left hand in a violinist’s brain and the strengthened areas of the visual cortex in the brain of a juggler.  Most importantly, she cites research suggesting that the more sensory areas used in learning a task, the stronger the neural connections, the associations for learning, memory and fluency.  

I often relate the memorization of the times tables to learning the names of one hundred forty four countries and their capitals.   Given one in isolation, the student must retrieve the name of the other.  Sometimes the country.  Sometimes the capital.  It is a devious plot to weed out the inefficient memorizers.  We know that children who fail to learn their number facts are frequently coded as having a math learning disability which is really more of a language deficit.  They may be offered the accommodation of a calculator and all targeted practice at math facts discontinued.  The thinking is that they will master isolated facts as they use them, but this is not true when each practice set contains too many differing sets of facts.  A use of a fact here and the use of it there does not constitute repetitive practice.

Dr. Willis also relates a more urgent concern for those of us in the field of teaching mathematics.  Her description of childhood brain growth suggests that students learning their math facts prior to age eleven need ample time to master this skill and repeated practice of isolated patterns.  Her rationale for repeated practice of any skill is strongly supported though she does not specifically target math facts.  This time in a child’s life, when there is a great, as she calls it, “growth spurt, with increase gray matter and connections reaching a maximum density at about age 11” is when our students are mastering those pesky multiplication facts.   

“When children are between the ages of 6 and 12, their neurons grow more and more synapses that serve as new pathways for nerve signals.  This thickening of gray matter (the branching dendrites of the neurons and the synaptic connections they form) is accompanied by thickening in the brain’s white matter (fatty myelin sheaths that insulate the axons carrying information away from the neuron and making the nerve-signal transmissions faster and more efficient. As the brain becomes more efficient, the less-used circuits are pruned away, but the most frequently used connections become thicker, with more myelin coating making them more efficient (Guild,2004).”    

This may seem too technical for us to use as a basis for instruction but it is incredibly important to understand and indeed to use.  Dr. Willis speaks to the advantages of using simultaneous processing and the need to offer information and practice it in multiple modalities.  This provides for duplication and multiple connections which serve memory and retrieval.

Now to put this into context.  For three weeks during the summer I ran a math camp for struggling middle school students.  I used many of the same strategies with a private student for three more weeks.  I used one times table throughout.  I placed strings of beads in front of the students and we touched as we counted the groups of seven.  Each group, a different color, allowed us to look at three groups of seven and say twenty one.  Dehane’s research on numeracy provided the context.  The research supporting multisensory practice and simultaneous processing added more.  Even the most multiplication deficient students left counting by sevens and retrieving the facts.  Each student used the facts over and over again throughout the lessons.  We multiplied by seven.  We divided by seven.  We formed fractions and common denominators in groups of sevens.  We place seven on a times table and then multiples of seven.  We named factors and multiples.  We found the least common multiple and the greatest common factor and perfect squares and roots…all with sevens.  We used other number facts too, but when we explored new material, it was always with sevens.  Students had their strings available and could construct the facts as needed if required.

Post # 15 Early Thoughts and "Oh-h-h-"

An example occurred for me just the other week. I was working with a middle school student, a bright young lady, and she asked if we could work on decimals.  It quickly became apparent that she did not see decimals as fractions.  It is true that decimals can represent both rational and irrational numbers and so many teachers do not clearly make that distinction.  I am sure this child's teachers did all the correct representations using the hundred's grid and linking decimals to fraction concepts, but for this student it never really became clear.  She had never created decimals.  We do not cut pizza by decimals... or at least call those portions by decimal names. She needed some way to make the linkage in her mind. 

There it is again, that word "linkage."  For those of us trained in language it is a critical term.  This linking of new concepts to prior knowledge is so very important.  So for this young woman I knew the most effective teaching for this teachable moment, was to put things in her hands.  The entire concept took about twenty minutes.  

I first brought out the craft sticks, that beautiful bundle of one thousand and  along with hundreds, tens and ones.  We  placed them on a place value mat and spoke very quickly about the meaning of regrouping.  As we moved through the tracking activity outlined in your manual (Place Value) we whittled our quantity down to three.  I then asked her to show me two minus one half. 

Puzzled, as all students are, she asked me if she could break the stick.  "A woman's gotta do what a woman's gotta do," I said.  She did.  We renamed the new quantity two and two halves before "performing the indicated operation." 

At that point we transitioned to the base ten blocks, only this time after tracking I asked her to remove one tenth.  I had placed the clay unit cube among her units and I gave her the "handy dandy decimal fraction creator."  (Thank you Siena staff-who came up with the dental floss idea.)  She cut off her tenth, then another and another.  We renamed our quantities as we continued to cut that tiny flat into hundredths and that tiny hundredth into thousandths.  And when we cut the first tiny thousandth she said, "But it's so small!"  At that point I heard the response I most ardently await, "Oh....." 

Post# 14 The Language Makes the Difference

Many participants in the distance classes remark that they love being able to rewind and listen to the language of demonstrations again and again.  To do this effective, all you need to do, is take the briefest of notes while you watch the videos.

At the bottom of the viewer, you will see an hour and minute reading/marking.  As you watch, if you see something you might like to see again, simply note the Day and Video title and the hour and minute marking.  This way, if you think you would like to review a part of a demonstration again you can do so.  You can also pause the video as you write down the specific language you like.

Indeed, this program and course is well known especially for its use of instructional language.  The language of place value quantity tracking and the demonstration on factors and multiples in multiplication are but two.  Another demonstration we will replicate in the video conferences is the language used for regrouping from the whole to the fractional part for operations with fractions.

One of the beauties of the video class is that you can pause, record, rewind and practice.  This is one feature the on-site live class does not have. (Also, if you see a place in a video that is not quite right, out of sync or some other glitch, please note the segment and minute and share that with me.  This is a newly edited version and the editing software updated in the middle of our production.  This caused several technical issues which will need tweaking.) 

Recently I was at a school consulting and we reviewed the language of division as it relates to story problems and problem solving.  The language was key to offering the students explicit instruction in ways to draw solutions and differentiate between the two possible meanings of division.  This same language was used to illustrate meaning in division of fractions which provided meaning and context.  

The concept-focused language paired with simultaneous processing - the "read the quantities with your hands" or the "touch and say"- is one of the central tenets of this course.  It is not only the use of the manipulatives which solidifies learning.  It is the "multisensory loop" as I call it: the hear it, see it, say it, touch it - all at the same time, creating multiple memory strands in the brain.  If the language is off, the thread may not endure. 

Post # 13 Readings for This Course

This information was sent by email but I also wanted to post it on the blog for some readers who are not taking the course. There are required readings for this course.  You will find them in the supplemental materials folder in the course DROPBOX.  They will be labeled: Required Readings.  You should begin with Subitizing and The Learning Brain.  Subitizing explains our reasons for approaching numeracy as a core deficit.  Subitizing is the automatic recognition of quantity and I would add, quantity relationships.  It is a crucial component of math instruction/education. The chapter from Blakemore and Frith explains some of the research on how the brain processes mathematics that you will find throughout the manual.  

The UDL Principles chart is something that should be downloaded along with Adler Fracts and the Pattern Block Activity and placed in your manual.  I have provided a section of the manual in the rear where you find all the power point presentation handouts.  It is also a place for you to put articles and references that you find relevant. Note the language used to describe the interactions between the student and content.  These are phrases we use when we talk the talk of education.  In order for us to be universally understood when we describe the methodology and impact of multisensory, we need to have a shared vocabulary. 

The articles from NCTM and The Learning Brain may be printed and placed in your binder too if you wish.  I like having them in color and full size so that I can make notes or underline passages.  Because they are in pdf format, you may also print them as 2 pages per sheet and double sided if you want.  That will save paper and bulk in the manual.

Once you understand numeracy, begin looking at Ten is the Magic Number and the Joyce Steeves articles.  You should also begin looking through the first sections of the binder and reading the material there.  I have made an effort to condense research and rationale for you so that you do not need to do a lot of research on the internet.

Be prepared to reflect on Subitizing, The Learning Brain, and Ten is the Magic Number for the first video conference sessions.  These are not the only required readings but they are the early ones. There are others which you will find in Dropbox. Most teachers also enjoy the articles Tying It All Together by Jennifer Suh. 

I would recommend that you also look at Fluency Without Fear by Jo Boaler.  It is readily available on the internet.  There is a particularly poignant paragraph about the abuse way many programs teach fluency.  We should pay attention.  

You should also become familiar with the Website: Achieve the Core: Math Focus By Grade and the content focus in math for each grade. This is a very valuable resource which is fairly new. 

Post # 12 The Mental Number Line

Some children have a great deal of trouble forming and using a mental number line.  Unless the whole body is used to experience magnitude changes, these children may have problems using the number line to solve problems and represent the relationships between numbers.  

I was working with two rising 5th graders recently.  We were working on fraction and decimal concepts.  For these two children the key was to experience the math concepts using multiple representations.  We constructed fraction problems with our rainbow fraction tiles.  We used pattern blocks to illustrate fraction operations.  We colored shapes to solve fraction problems and finally we used the number line.  Still, one of the children who has spatial deficits had to work at placing his fractions on the number line.  Lots of practice.  The use of alternative support strategies.  And then, more practice at large motor levels and at fine motor levels.  

Post # 11 The Decimal Place Value System

The early days of the class focus on what we believe is the core deficit in math disabilities, numeracy.  The automatic recognition of quantity and quantity relationships is incredibly important.   Using manipulatives and simultaneous processing as a strategy, we help children create multi-modal memories to support applications.  Early addition and subtraction skills rely on these representations and as we extend them across place value we help students to prepare for more complex computations.  In reading we would say the child moves from learning to read to reading to learn.  In math we build the tools for later calculations and problem solving by building fluency with number awareness and facts. 

One early activity I like to use with young students is building  place value concepts without counting.  Ask pairs of students to build tally marks with craft sticks on a place value mat.  One student builds the tallies and quickly passes two sets of tallies to a partner who bundles them into a ten.  We time them to see how many tens they can build in two minutes.  On day two we bundle more tens.  On day three we make tallies with ten bundles and build one hundred with our tens.  They quickly learn how our system of quantity bundling builds our awareness of the place value system.  And, they learn the purpose of our place value system is to be able to recognize quantities without counting.  It is a higher order of subitizing.   

When we get to place value as a concept, we must utilize the quantity awareness built in early subitizing activities and apply it across greater magnitudes.  Thus, 3 + 2 = 5 becomes 30 + 20 = 50.    On day two of the course, we explored the place value system first with craft sticks and then transitioned to base ten blocks.  You may notice that the third section of the course after lunch is quite short.  It was at this point that the class went into the hall and built the place value system to the ten million rod.  

Using 10 cm x 10 cm boxes from the craft store which are the identical size of the one thousand cube, we build the ten thousand rod and outlined the hundred thousand flat.  Then using yellow cord from the dollar store knotted at every meter, we constructed the outlines of the one million cube.  We held the cord out at one meter lengths and outlined the ten million rod.  Children often do this in class and in a previous post I included some pictures from the summer program in which the children constructed quantities from one to twelve thousand. 

One thing I have been emphasizing lately is the use of the term "proper number" and "improper number."  Think of proper numbers that conform to the rules of place value, no more than nine in any place.  Think of improper numbers as those which need to be "simplified." They are numbers that occur as the result of an operation such as adding seven and eight...or which we create in by decomposing  or regrouping in order to subtract.  If we begin talking to students about improper numbers when they are learning place value concepts, we are setting the stage for fractions ahead of time.  We are also emphasizing the order and symmetry of the place value system.  It really is quite elegant.  

Finally, I want to add that for many students the use of the craft sticks is a game changer.  I noted this summer in my program that older elementary students who were practicing subtraction with decomposing a ten did not understand regrouping as "unbundling" the ten.  As they sought to subtract they would try to reach into the bag of sticks for a "ten" because they were used to trading...the ten rod for unit cubes.  Even for older students, that task of bundling and unbundling must come before we transition to the place value blocks which are indeed more efficient for many concepts.  

Post # 10 Subitizing

You will see a strong emphasis on numeracy in this course.  In fact number sense is a critical skill.  Older students who receive an emphasis on counting strategies often do not develop the number sense they need for later operations including work with integers in algebra.  

The multisensory math approach basis many of its practices and strategies on the foundation skill of subitizing.  It is at the heart of estimation, and general math reasoning.  In this course we apply the concept of subitizing in many different areas and on many levels.  

As a reading for this course, you are asked to read Douglas Clements' article on subitizing and learn the two levels of subitizing he articulates:  perceptual and conceptual.  

Occasionally we find other resources which may support these ideas.  I recently came across this website from Australia which lists the research supporting the teaching of subitizing.  It is always good to be able to cite sources and produce evidence to support practice.  

The website also  lists several activities for the classroom or individual session.  It is worth a look.  

https://subitizing.weebly.com/exploring-the-research.html

Post # 9 Video Conferences

Your Video Conference Schedule choices will be posted this week.  You will sign up for two video conferences.  You will have three choices from which to choose for each conference.  

We are on Eastern US time.  One of the choices will be a Saturday afternoon.  The other two will be evening choices, usually from 7:30 pm to 9:30 pm.  They will not be scheduled in the same week so that we can accommodate vacation schedules and religious observances. 

We use video conference software from Zoom which is free.  You can download the software in advance- a free account- or wait until the day of your conference.  When you sign in to the meeting, you will be prompted to download the software is necessary.  

The invitations with the links for the conferences are generally sent the day before each conference.  Check your email.  

If you miss a conference, you can view the recording of the first one of each set.  That said, if you are taking the course for credit, this is part of your class participation grade.  

You do not need to "bring" anything to the video conferences, though printing the card stock fraction tiles from the dropbox would help you to follow the language for fractions in video conference 2.  

The members of your math team share these presentations.  You will receive an email introducing members of the team and the conference schedule.  When you get it, respond as quickly as possible.  Do not wait until the week of a conference to sign up!  

We look forward to "meeting" each of you face to face via Zoom. 

Post # 8 All the Usual Suspects

All the usual suspects for a distance class.  There may be glitches and frustrations as we all learn how the pieces fit together.  If find something that is not quite right with the videos, please email me with the segment name and exact time.  We may need to tweak the editing.  And, yes, it is an enormous amount of information and that is one reason I have deliberately reserved assignments until people find their feet.  You have lots of time to view the videos and you should take it in.  One of the advantages of the distance classes is called "rewind."  The instructional language is so precise in many activities that you will need to pause the video to take notes, or process the research, or simply to practice the instructional language.  In the live class this is not possible.  There are advantages to each. You should, however, practice with the class, use the manipulatives and learn by doing.

I will be posting several blogs each week.  Some will be new.  Some will be the best of previous class blog postings, recognized by class participants as having been especially meaningful. Your class will add contributions, ideas for me to consider from your comments and questions.   You may and should post comments for me and for your classmates to see.  Do not despair if your comments are not posted right away. I have to preview and approve them.  This waits for email access and I check fairly frequently unless I am on the road presenting.  Read the blog postings in order.  React.  Print or note.  They represent content, instruction and musings about this methodology in practice.

Here is an example:  Picture two 5th graders, twins.  One struggles with numeracy.  The other does not.  Which one used the partial quotient method of division to state that twenty three divided by seven gave a result of eleven remainder two?  And, how did that student reason through the solution?  The answer is that the challenged student got it correct and the twin with no disabilities got is wrong.  Why?  She followed procedural as opposed to conceptual instruction.  The strings with wings allowed the numeracy challenged student to automatize the seven times table over the course of a week and a half... through usage and visual associations. The sibling who had learned only the procedure could not “see” the effects on quantity.  Her “estimation” was wrong.

And then there is the language.  It is always the language.  How many groups of seven can you make if you start with twenty three?   For many concepts, I believe it is more about the language than the math.  And off we go.

Post # 7 Support for Older Learners

For the teacher working with older students who lack foundation skills, this course should offer ideas for building skills while addressing grade level content. This can be done with even a few minutes dedicated to skill building each day.  More importantly, it can be done by focusing on fewer facts at a time and then using or practicing them to automaticity. 

I reference the NCTM (National Council of Teachers of Mathematics) position paper on interventions.  In this document the organization stressed the need to invest time in hitting the “big picture” concepts rather than relegating an older student to a lower level curriculum.  Later in the video conferences I will refer to aspects of this as the “super powers of math.” 

In a recent presentation I gave at a university STEM conference, I outlined two type of intervention: the intensity of a summer ‘boot camp” or what I call “infusion for inclusion.”  In a summer session, an interventionist can attempt to lay a foundation for math processing by trying to address the core concepts in a hierarchy of math concepts outlined as foundation skills.  In an “infusion” model, an instructor would attempt to direct targeted practice and focus on smaller aspects of these same concepts but on a daily basis in warm ups, practice pages or homework.  Neither is ideal, but core concepts and foundation skills must be addressed if students are to understand the math they are currently studying.

First, you must determine if students have a sense of numeracy.  Are they still counting on fingers?  Are they unable to perform basic calculations without using a calculator?  Do they lack a sense of what operations mean?  You will need to assess your students to find what they know and don’t know. 

In the first day of my summer math camps for adolescents, we do not use pencil and paper.  There are no worksheets, no paper homework, no quizzes.  Students begin by solving problems with manipulatives.  I use the craft sticks, base ten blocks, fraction manipulatives, pattern blocks…anything that can model basic operations.  I like to begin with subtraction with regrouping using the craft sticks and a place value mat.  Sometimes, depending on the age of the students, I include regrouping from the whole to the part so they would need to break the stick to solve for a fractional amount.  They may work independently or in pairs.  In every case, the students go home and say that this is not like any math class they have attended.  They like the nature of the hands on work. 

I usually quickly find that many students do not understand what they have been doing on paper.  They “borrow” and “carry,”  “flip and multiply,” create lattices or divide with partial quotients, but they have poor understanding of what they are doing on paper because it was taught only at the abstract level with few or no quantity representations.  Taking them back to the concrete level is enlightening for many of them.  It is also fun.

Once we establish these quantity representations at the concrete and pictorial level, we can quickly make the links to what they have been doing in the abstract.  Some students will progress in a matter of a day or so.  Others will need more time. 

  

The second key to reaching the students beyond the concrete representations is using explicit language and "friendly" or easy numbers.  Whenever you teach something new, make your language as precise as you can and use easily accessible numbers so the students can focus on the meaning behind the math.  If they must struggle to retrieve number facts in the middle of some sequential algorithm, you have lost them before you start.  Using easy numbers for instruction does not harm any student and it leaves the teacher the possibility of differentiating and adding rigor for those students who need it.  Using “friendly” numbers for instruction and differentiated numbers for practice and applications allows all boats to rise when the curriculum tide comes in.

Thus, one strategy is to present numbers students will use but early in the lesson.  The warm up is perfect.  Lead students to fill in PART of a times table chart rather than giving them one already filled out.  They create their own near point reference.  I like to use the seven times table and the perfect squares. Then USE those numbers in class for all work that day and for several days so that the facts become friendlier and friendlier.  With the seven times table and the perfect squares, you can teach multiplication, division, simplifying fractions, mental math, the distributive property, multi-digit arithmetic and simplifying radicals.  Food for thought.

Finally, I will add that one common deficit among older students is in numeracy- composition and decomposition patterns- for the digits from six through nine.  Early on, students are taught to "count on" to add and to "count back" to subtract.  They are not given sufficient practice in these numeracy patterns and therefore have great difficulty in addition and especially subtraction across a ten.  Provide small incremental practice with these patterns in warm-ups or brief practice sets.  Use no more than one or two patterns at a time and use the patterns across place value.  If you subtract for example 13-7, decompose the 7 as 3 and 4.  Lead students to subtract 3 to get to 10 and then subtract the 4.  Then use the same pattern to subtract 7 from 43, 63, 93 and then to subtract 17 from 83.  Demonstrate math reasoning but with the same pattern in different contexts.  With some students you could even take the pattern to 3-7=-4.  

Post # 6 Getting the Most From This Course

You are not in isolation.  Ask your own children or spouse to help you with your homework.  Practice with real “students.” Get out those manipulatives and ask your students or even your spouse/significant other to play the student while you practice the language and the hands on applications.

Participants take this class for different reasons.  Some are classroom teachers who want to expand their skill set for reaching all types of learners especially those who are challenged by learning differences, learning English as a second language, poverty or previous poor instruction.  Some of you are home educators who want alternative methods for sculpting lessons.  Some are supporting professionals such as academic language therapists, tutors or learning center administrators.  Each of you has specific goals and ideas of what you want to gain from this course. 

The course outlines the research supporting a multisensory, conceptual approach to teaching mathematics.  It was originally geared toward those working with students who have challenges with language based learning disabilities, but it has come to incorporate so much more.  Students struggle for many reasons and we need to be cognizant of all the ways learning differences may occur.  I have endeavored to address this in the manual and in the course. 

In this posting I would like to address one more difference among participants.  It is the level of math taught. 

A primary or elementary school teacher will immediately be struck by the basic level of addressing numeracy detailed in the early videos.  I have worked very hard to enumerate creative ways to build this foundation skill.  I have extended the concept of numeracy to automatic pattern recognition in the form of place value. 

Building on that, we apply our numeracy skills to operations and fractions and even integer operations.  Each skill builds on the previous one and links concepts and applications.  The language is often the unifying principle. 

Some of the participants in this class have an additional unique challenge.  They are dealing with older students who lack foundation skills, and yet they are tasked with teaching upper grade level content.  These might be educators in special education, resource room teachers, educators in independent schools for students with learning disabilities, and even ELL teachers who work with older students who have missed crucial foundation skills and have the impact of learning a new language. 

These reasons all present unique challenges for the educator. I will try to address these in several blog postings this week.  The blog entries will be important for all of you because early childhood educators will need to know the importance of using appropriate language and providing linkages for teachers who follow them.  Upper elementary, middle school and high school teachers need to know the language and methods necessary for building skills even while they address on grade level content.

 On your final exam, you will be asked to comment on blog postings that have specifically resonated with you.  Keep that in mind this week as you check in with the blog. Don’t fall too far behind in reading the blog.  I sometimes have participants who wait until the last week and find it impossible to catch up before access is closed.

Post # 5 Assignments and Readings

Assignments:   There are no small incremental homework types of assignments for the distance class.  You may have a quiz before a video conference, but no specific assignment to turn in until the end.  There are some larger ones though that will help you to internalize the methodology as you go along. The project at the end of the course ties it all together. The broad scope and sequence information is in the back of your binder in your Resources section.  It will tell you which readings will enhance specific content in the videos. 

First:  Really get to know the early chapters of your manual.  Read the front matter to the early chapters.  Try to get up to and through the lesson plan.  You should get to know the resources in the chapter on students who learn differently because that chapter offers justification for the methodology.  The resources here are the keys to my phrase "appropriate for all but essential for some."  It is not enough for you to hear that rate of speech is an issue.  You must know why some children look as if they are inattentive, really are not paying attention, or why some children just can't seem to memorize the times tables.  Knowing the “why” allows you to create strategies which will reach all students. 

Recently I was working with a fourth grade boy who is extremely bright but has very poor sense of numeracy and almost no ability to retrieve words.  He understands math concepts but has extreme difficulty with fluency for facts.  He has directionality problems and needs lots of gross motor activities to cement meaning.  We spent this week's lesson creating base ten models at the thousand's level.  He compared the quantity 14 to the quantity 14,000 with concrete manipulatives.  ( I use 10cm x 10cm x 10 cm gift boxes from the craft store.) He sat in the middle of what would be 100,000 unit cubes and then he was asked to decode large numbers.  Everything I do with this child is diagnostic and specific to his needs.  The wonderful thing though, is that I have done the same activities in a classroom with many children working at the same time. 

So, assignment number one, is become familiar with your manual, especially the first five sections,  and watch the first day's video segments.  By days 2 and 3 of the videos we are emphasizing place value and operations. As we move into the "Lesson Plan" you will have an opportunity to discuss possibilities in our first Video conference session.  Try making your own "Name That Quantity" Powerpoint presentation appropriate for the level you teach.  

Requirements are outlined in the reference section of the course at the end of your binder.  You may elect to take the course for CEU clock hours, Credit, or Graduate Credit.

Reading Summaries-available Resources found in Dropbox Turn in one single .doc or docx file with all reading summaries 

	Your Name Title, Author (Each summary & reaction not to exceed ½ page typed)         Brief Summary –content 4-5 sentences or bullet points         Brief Reaction – Liked/ Disliked?  Will/Won’t Use (How)-How it may help or inform your teaching. Title, Author of Reading 2 Brief Summary Brief Reaction Continue to next reading using the same format.

All Readings  are Available in Dropbox  They may be printed for your own reference and placed in the back section of the binder. This way you can annotate, high light or take notes. 

Group # 1 to be completed before your first Video Conference:

·     Excerpts: The Learning Brain, 

·     Subitizing: What Is It? Why We Teach It?

·    Contributions of Joyce Steeves: Using an O-G Approach to Teach Mathematics- Place in the Lesson Plan Section

·   The Initial Lesson-Steeves, place  in The Lesson Plan Section of the binder

·     Linking the Language, Ebbers,  place in the Language of Math Section

·      Ten is The Magic Number, place in References

Group  # 2 to be completed before your second Video Conference

·      Tying It All Together, place in References

Review and document

·   UDL Principles Graphic Chart- DO NOT Summarize- Simple Reaction is sufficient. Place in the General Theory and Background of Teaching Section of the binder 

· Achieve the Core – Math Standards by Grade   https://achievethecore.org/category/774/mathematics-focus-by-grade-level

NOTICE:  Graduate Credit Candidates

If you have signed up for graduate credits from Trinity University for this class, you must notify your instructor as soon as possible.  You may not wait until the course is underway to make that decision.  

Post # 4 Pacing Guide

You must get through 30 Plus hours of video to complete this course.  This does not take into account any desire to pause, review, rewind and re-watch.  Distance classes sometimes can be problematic in that some participants do not know how to approach a loose schedule which they may partially create themselves.  One of the participants from the last class asked how to approach viewing the "daily" segments; a good question. 

The videos are arranged by "days" of instruction from the live class.  This means that each group of four segments was a single day of the on-site class.  The segments are usually about 1.5 hours each.  This comes out to about three hours before lunch including a break and three hours after lunch including an afternoon break. We break when there is a natural place in the instruction. 

For this class you must complete ½ of a Math “Day” per week.  That would be about 3 hours of video per week. 

Add to that your readings and review of blog postings; and you would have about four to five hours per week.  When you have a 2 hour video conference session scheduled, you will have that as well, but you only have two of those.  Leave time to practice the instructional language and modeling with manipulatives.  Carve out about 4 to 5 hours per week and you should have time to cover the material and get through everything. You will then have one week to complete your project and exam.

                                                                                              

I would suggest viewing an entire segment (1.5 hours) at one sitting.  Pause the videos during play to take notes, replay instructional language, or gain a deeper understanding of a concept. You can practice strategies or techniques with a family member or friend.  Try explaining a concept to someone else and taking that person through the CRA instructional sequence.  

Remember:  Practice Makes Permanent.  The important thing to remember is that the more you watch at once, the less you will retain. And, the most complex parts of the course are at the end!  

The one thing you should NOT do is get behind and try to binge watch near the end of the course.  One participant in an on-site class remarked that there was so much information in the course that it was “like trying to get a drink from a fire hose.” If you are a teacher returning to the classroom, plan on doing more earlier so that you will not be pressed to complete the course in time for school to open.

Remember that one of the most unique aspects of this course is the attention to instructional language and the other is simultaneous processing in which the student uses as many sensory modalities as possible. Take note of this when watching and taking notes.  Do the activities along with the class participants.  If there is a moment to practice something such as Quantity Tracking...practice it along with the class. 

Post # 3 Manipulatives and Materials

You will want to supplement the manipulatives you were sent with your manual.  Many schools have some of these supplies available.  If not, there are only a few I consider indispensable.  I actually do not use excessive numbers of commercial manipulatives.  Sime things you have around the house can work just fine for teaching many math concepts.  That said, there are a few which are easy to use and tremendously effective. To get you started though, I have placed a new folder in your class Dropbox with printable manipulatives.  These will include base ten materials, pattern blocks, place value mats and some color fraction tiles I created on MS Word.  I can’t create printable craft sticks, but when I see something I can make for you to access on line, I will do so. Here are some of the things I consider to be especially useful. 

• Place Value Sticks The most useful place value manipulative for young children is craft sticks.  You can purchase 2,000 craft sticks at any craft store for about $8 per thousand. Wholesale from a company such as www.darice.com, you can find them for about $5 per thousand.   You can then purchase rubber bands or elastic hair bands from discount stores such as Dollar Tree, the Family Dollar Store or even on-line beauty supply shops.  I also purchase the extra large rubber bands used for file folders.  I get them at office supply stores.  You do not need many, but you will need several to hold together the thousand bundle.  You should bundle sets of ten to make one hundred.  Then bundle ten of those to make one thousand.  It is important for the children to see each individual stick which makes up the thousand. I also use these to teach multiplication, factors and multiples, prime factorization and fractions as part of the place value system.

• Base Ten Place Value Blocks:  
• You will also need access to a set of Base Ten Place Value Blocks.  In your welcome letter you were given a web based source for printing cardstock versions but you will ultimately want a set to use.  If you travel, you might want to purchase Quiet Shape versions out of foam.  For classroom use the plastics ones are the norm.  Note: EAI Education sells small packets of the individual pieces which can make them inexpensive. I also like the Quiet Shape Base Ten Blocks if you travel to tutor because they weigh almost nothing.

• Unifix Cubes are also one of the most useful manipulatives.  We use them for everything from numeracy, place value and multiplication/division to linear functions in algebra.  If you are in a school, you more than likely have access to these.  If you are not in a school, you can find a set of 100 for about $10. Available from teaching supply stores, on-line from various suppliers, and in bulk from companies such as www.eai.education.com or www.didax.com.

•    Pattern Blocks - A small set came with your manual.  You do not need these right away.  They will be used in the fraction unit.  They may be purchased at any educational supply store or on- line.  Alternatively, you can download a free version to print on cardstock

• Colored Fraction Tiles: plexiglass rainbow colored fraction tiles or if you tutor, the Quiet Shape Magnetic Fraction tiles- the rectangular ones.  These are inexpensive at about $5 a set.  I recommend having two sets so that you can create improper fractions. Alternately, I will place a printable version in Dropbox for you. You can print and laminate for personal use. I created the template using Microsoft Word and the insert table function. It is possible to subdivide the “cells” in the table to create these.

These are essential manipulatives.  I have given you cardstock versions of the fraction circle cards and you may download larger versions with other fraction pieces at the Rational Number Project website listed in your manual.  This file is available in Dropbox.  This study attributes greater understanding of fraction concepts and higher test scores on fraction assessments to the increased use of manipulatives especially the fraction circles. 

Have a complete set ready when you get to the fraction activities on day four and five. You may also want to have a bit of yellow children's modeling clay and a dental floss holder with floss on it.  You will understand this later.  A little mystery is good. 

I look forward to hearing from all of you.