Post # 28 What Role Skip Counting

I have had a recent reason to review the role of skip counting and its relationship to multiplication.  The standards suggest that in second grade, teachers begin skip counting by 2’s and 3’s to lay the foundation for the concept of multiplication.  And which one of us could not skip count by 5’s at an early age. The problem I see is that many teachers believe that one teaches or reviews multiplication facts  by skip counting. 

Let us examine first what multiplication is:  making many of the same quantity…like repeated addition of the same quantity. The goal of multiplication fluency though is automatic retrieval of an individual product in isolation.  Using skip counting to locate a specific product in response to a prompt by going through a string of numbers every time would be highly inefficient.  In many children it might also be inaccurate.

So what is the proper role of skip counting then?  It does help us to recognize patterns. It familiarizes us with the specific products in a set of products.  It trains the ear, so that if our retrieval fails us, we might self-correct because we know that a specific number is not part of that set.  With many of my students who have dyslexia, I will hear them say the algorithm and a product they retrieve rapidly, only to follow with “no, no….” and then the correct solution. 

I write this in response to some lesson plans I received as assignments.  More than one teacher wrote of skip counting by several different numbers in the counting section of the lesson plan.  This was not counting by 3’s that day and then using the facts in all work to provide repetitive practice as I often advocate.

No these teachers suggested skip counting first by 3’s, then 4’s, then 6’s, then 8’s…all in the same day, as if that would be reviewing multiplication facts.  I found this troubling on so many levels. 

For students with language based learning disabilities, multiplication is a major hurdle.  They have no visual or quantitative reference. This is one reason I invented the Strings with Wings.  I don’t make the strings for all times tables though we certainly could.  I usually reserve them for the upper times tables. 

There is another possible use for skip counting which I feel is necessary but must be judiciously used.  I have had older students with undiagnosed LD who are not permitted to use calculators on standardized tests. Without fluency support, these students would perform miserably if not simply give up and not try.  This is the “mark answer ‘b’ and move on” crowd.

I will teach these students to create a partial multiplication chart quickly using skip counting: 2, 3, 5, 9 plus the special patterned products in the upper times table.  From these brief answers they can reason others. This allows the students to perform better and it encourages them to keep trying.  It allows them not to feel like abject failures. 

Ultimately, I feel very strongly that we must shift to a concept based approach to math for in class work which allows students to use friendly number facts for group work.  I advocate having students create their own near point references to use during class and for teachers to offer sufficient practice with a limited set of facts at a time to develop automaticity.  We must continue to work on fluency development for all students. For independent work, homework and tests appropriate  challenges for more rigor or accommodations could be used. 

The point is, that we must take the time to build fluency and reasoning capacity in our students so that the mind can be free to do the math. Students who exclusively use calculators often do not understand the underlying concepts.  They may be pushing numbers around according to that unit’s procedures. The learning doesn’t last and they lose the ability to apply.

In other words, do not think that by skip counting with older students that you are teaching multiplication facts. You are not reviewing them or practicing using them.  You are, according to the standards, laying the foundation only. 

Post # 27 There is No Time Table for Times Tables

I often speak about fluency and math facts. Recent articles address the conflicts between memorization, the need for fluency and math drills, especially timed ones.

In this course we stress the load on memory that retrieval of verbal information causes and the fact that some students will not benefit from timed drills.  Their unique learning differences make them unsuitable candidates for the pressures of automatic retrieval in timed settings.  These timed drills can create stress and anxiety, feelings of defeat and loss of self-worth. That does not mean we should give up on gradual and incremental growth toward fluency and fact retrieval.  Rather it would suggest that by using the same facts repeatedly in problem solving activities and through games, children can eventually achieve the fluency we desire.

Remember that "chunking" is a viable study skills strategy.  We chunk those facts or a specific number of facts that a child can master with relative ease.  We review mastered facts periodically to preserve fluency.  We do not drill without meaning attached.

Woodin Math has a book aimed at teaching multiplication tables to students with disabilities and there are a variety of vehicles available other than "drill and kill" and timed Mad Minute exercises. 

Here is a link to one of the two articles on timed drills and fluency. It includes research links which you may follow if you wish.  At this time, the article is a free pdf download.

  

http://youcubed.stanford.edu/fluency-without-fear/

You can also find some very interesting research on numeracy in the UK which more than likely would be replicated here in the USA.  The simple graphics make the lack of numeracy awareness in the general population to be quite shocking.  The research from the UK is an outgrowth of Brian Butterworth's work on numeracy. 

Post # 26 The Benefits of Multisensory Math for ELL's

As this course evolves to address the needs of all learners, I am continually reminded of the many students who are impacted by language deficits not only those with disabilities.  I have worked on occasion with teachers of a unique group of ELL students, those who have gaps in school attendance in their home countries.  Some of these students have had severe circumstances which have affected their ability to even attend school and thus, through no fault of their own, have huge gaps in content learning.  Many have suffered trauma.  They are in need of a safe learning environment and one which can nurture them to feel whole again.  They need a place where learning builds confidence and where they can feel safe to learn even basic skills. 

The multisensory approach is particularly appropriate for these teachers and students because of its emphasis on instructional language.  As I work with these teachers I am continually impressed with their willingness to embrace even difficult concepts.  In my most recent professional development session, we spent the entire day on pre-algebra and algebra, instruction intended for students performing at an elementary level of math.  Why?  Many of these students are adolescents, young adults who will be entering the work force sooner rather than later.  These teachers have a heavy load to bear.  They understand the importance of the vertical approach to content mentioned in the last post. They must navigate the curriculum with students who are often conceptually at the level of place value and regrouping and move them through multiplication/division concepts to fractions and decimal fractions in a very short amount of time. 

Again I am very impressed with the dedication of these teachers who see even older students in need of basic skills.  Even with this knowledge, we have students struggling to learn a new language, as they find themselves in a new culture with high educational demands.  How then do we address the need for filling those gaps and bridging the great divide between what they know and what they need to know. 

In our work, the teachers and I are exploring ways to use manipulatives to model and then ways to pair the instructional language of integers, linear functions, graphs and grids to those manipulatives.  We are finding that unifix cubes work well for many of these concepts.  We build linear functions with the cubes - which at this level I call "alge-blocks."  We use the state high school standards exam language of "constant rate of change" and "starting value."  The students explore real life applications of the algebra before we even introduce numbers. 

Many of the dedicated teachers who work with these students are not math educators themselves.  They are English as a second language teachers who are teaching content. They must find a way to teach the language while teaching content that is at more than a basic level.  We know that even many elementary school general educators tend to teach math the way they were taught.  So it may also be for the ESOL instructor.  The district has invested in professional development to support these teachers.  As I work with them, I see them embrace new methods to meet the demands of serving these challenging students.  There is immediacy to their work and though it is frightening to consider what some of these students face, the commitment of their instructors is heartening.

Post # 25 Thinking Vertically - Working with Older Students

For intervention 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. 

Start by using manipulatives to model whole number operations with a large place value mat.  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 including regrouping from the whole to the part.

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. 

I have also used pattern blocks to model this concept and procedure.  The new place value mat using only the 1’s place and Fractions of 1 is a great place value practice sheet.  You might even create a mat with 10’s, 1’s, and “Fractions of 1.”  Using this mat, you can ask students to model regrouping from 10 to 1’s and then from 1’s to Fractions of 1.   This enables you to link to prior learning very efficiently and move on to regrouping with fractions and decimal fractions.  

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 fundamental math concepts which form the foundations of higher level skills.

I also want to emphasize the need for fluency and familiarity with multiplication facts.  Too many older students were given calculators as an accommodation at which point teachers stopped having them devote time to developing fluency.  For students with language based disabilities, multiplication fact memorization has been demonstrated to be largely a language retrieval problem not a math disability.  These students may take a very long time to develop a degree of fluency which can support reasoning. This is not to say they cannot develop it, simply that it can take a very long time.

By offering targeted practice, especially in what I call “high value products” students can continue to work toward the fluency they need.  By high value products, I mean those which have several factors.  They are often the ones students must factor in order to simplify fractions or expressions at many levels of math. Think twelve, twenty-four, thirty-six, forty-eight, even seventy two. Students will see these numbers again and again throughout fraction studies and algebra. Repeated, targeted practice to automaticity is one ticket to independence.

Post # 24 Fluency and Measurement

For intervention 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. 

Start by using manipulatives to model whole number operations with a large place value mat.  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 including regrouping from the whole to the part.

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. 

I have also used pattern blocks to model this concept and procedure.  The new place value mat using only the 1’s place and Fractions of 1 is a great place value practice sheet.  You might even create a mat with 10’s, 1’s, and “Fractions of 1.”  Using this mat, you can ask students to model regrouping from 10 to 1’s and then from 1’s to Fractions of 1.   This enables you to link to prior learning very efficiently and move on to regrouping with fractions and decimal fractions.  

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 fundamental math concepts which form the foundations of higher level skills.

I also want to emphasize the need for fluency and familiarity with multiplication facts.  Too many older students were given calculators as an accommodation at which point teachers stopped having them devote time to developing fluency.  For students with language based disabilities, multiplication fact memorization has been demonstrated to be largely a language retrieval problem not a math disability.  These students may take a very long time to develop a degree of fluency which can support reasoning. This is not to say they cannot develop it, simply that it can take a very long time.

By offering targeted practice, especially in what I call “high value products” students can continue to work toward the fluency they need.  By high value products, I mean those which have several factors.  They are often the ones students must factor in order to simplify fractions or expressions at many levels of math. Think twelve, twenty-four, thirty-six, forty-eight, even seventy two. Students will see these numbers again and again throughout fraction studies and algebra. Repeated, targeted practice to automaticity is one ticket to independence.

Post # 23 The Importance of Imagery and Simultaneous Processing

Students inspire.  They often teach us as much as we teach them.  My recent work at a school inspired me to find ways of teaching older students including some with moderate to severe disabilities.  I was tasked with coming up with a way to make the link between fractions and decimals real and the conversions between them more meaningful…not just a set of procedures based on words.  I came up with an idea based on a previous practice you will see modeled in the class, an extension so to speak.  It worked and I believe the students gained a new understanding of the link between the concrete representations. 

In the MSM I class, you will see clay used to demonstrate decimal fractions on a place value chart.  Before having the students create the decimal fractions though, I used a new and simple chart available in the supplemental materials section of your folder.  It simply has “ones” and “fractions of one” as its sections.  I asked students to use clay to model halves and fourths using our “handy dandy fraction creator tool.”  They had no difficulty in doing so.  Then we changed colors of clay and created the decimal fractions emphasizing the visual link to the based ten blocks.  We created tenths, hundredths and thousandths.  They realized that unlike the traditional fractions, they had to create fractions consistently by cutting each previous shape into tenths.  This stressed the essential differences between fractions and decimals.  We ended with applications involving whole numbers using fractions and decimals interchangeably.  At some point, I will film this lesson and post it on the website as a demonstration, but that is for another time.  We can talk about this in our video conferences. 

The second component of my demonstrations lessons involved simultaneous processing and the importance of “near point” references.  We were making the link between money and place value using the organizer in your manuals.  These were high school students in a consumer math class learning to make change.  Each student was given a hundreds chart from which we practiced counting by ten from any number:  sub-skill.  Then as we constructed quantities with real change on our place value mat, we used the hundreds chart to keep our place as we counted by tens to near 100 and the by ones to make the final change from a dollar.  Using the simultaneous processing and both hands helped the students keep their places and successfully make change with coins.  I believe it was a successful lesson and as closure at the end, one of the students remarked that it was really “neat” the link between money and place value which she finally understood.  “It really helped to see that,” she said. 

Of note:  the final piece from that lesson was later applied to a second grade class- the importance of having a counting chart within near point copy range.  The number line high on the wall does not provide the intimate access that many students need for counting.  Those who must follow with their fingers, need desk copy access to number lines and counting charts. We would call this a “near point copy.”  For many students it is an essential tool for self-monitoring and metacognition. 

Post # 22 The Language of Math: When Reading & Math Meet

This past year at the IDA International Conference in Connecticut.  We had some wonderful presentations but again, not as many math as I had hoped.  I presented some new material which was recorded for viewing on the IDA website.  I  refined my presentation on word problems.  

The What Works Clearinghouse Practice Guide on problem solving was updated just this month.  I am attempting to get a lock on the convergence between the reading comprehension work and the evidence from math research. John Woodward's group has come to some interesting conclusions.  

As a reading specialist as well as a math specialist, I have some unique insights.  The WWC Practice guide suggests that students who can draw representations have a better grip on solving them.  It also suggests that teachers help students reflect on the problem solving process.  

I am making connections to the continuous reflection of math word problems to Eileen Marzola's strategy on reading comprehension:  Think Before (activate prior knowledge), Think Along (reflect, summarize, predict) and Think After. In math I am altering the sequence to Think Before (activate prior knowledge), Do-Along (organize, draw, note and calculate), Think After (State the solution in a complete sentence).  

The student with language processing deficits may struggle to hold all the meaning and operations in working memory.  We should begin to look at this as a teachable moment for teachers.  Catherine demonstrated this in our recent video conference and Jennifer will also talk about it in her session.  I will try to film the session at some point so that you can all access the video. 

Post # 21 Math Made Real When They Hold It In Their Hands

As I prepared a new presentation for educators working with English Language Learners, students who have missed class time in their home countries, I thought about the dual difficulties they encounter: content and language.  In this program we have students who are disadvantaged because they – through no fault of their own- have missed instructional time in their native countries AND now must attempt grade level content in a new language.  How does an educator even begin to think about ways to reach these students?

Many of us may encounter older students who will all too soon be in the work force and for all our talk about “career and college readiness” are at an extreme position on that scale.  These youngsters will confront an impossible time table to gain basic skills which will form a foundation for district level content mandates.  Thankfully these students are in a unique program that does attempt to address their needs.  I will also soon be working with educators in an adult education program whose students missed gaining fluency in math for other reasons.  It inspires me to think about those students who are too often pushed too quickly or are simply left behind.  

One of the features of this multisensory approach is a focus on instructional language.  It must be concept based, memorable and retrievable.  Another feature of this approach is what I call vertical thinking.  With a core set of fluent facts, complex concepts can be taught. 

There is no getting around the meaning behind the math.  A student must understand what operations mean.  Calculators are no substitute for conceptual understanding.  Technology is an essential element of today’s math, a tool for expanding competence beyond fact recall; but it is no substitute for reasoning. 

So as I prepare for my ELL presentation, I am focused on an algebra mandate.  These students who may just be learning to be competent at basic operations may be thrust into a more advanced curriculum.  Their conceptual understanding of pre-algebra must be rock solid.  We must decide which concepts form the conceptual underpinnings  and focus intently on student comprehension. 

We will begin with language and constructions.  Simple language paired with concrete manipulatives forms a bridge for many gaps.  As a student constructs solutions to applications involving integers, linear functions, ratios, square numbers and roots, meaning can be extracted while visual and tactile memories take root.  In constructing solutions the language input is enhanced.  The student sees the meaning behind the math take shape as he holds it in his hands. 

Simultaneous processing is a major key here.  The student struggling with language is the language made real, concrete and observable.  Then we can move to the representational:  number lines, graphs and the like.  For the student struggling with language for any reason, the concrete level of experience is essential, communal and explicit.  It is an elegant blend of math made real and meaningful as we read the math concepts with our hands.  

Here a student models the creation of an "improper" number in order to subtract. Composing or decomposing (formerly known as regrouping)  involves either the simplification of an improper number OR the creation of one in order to subtract. Newer math programs use only the terms composing and decomposing. Parents and those of us who came of age in the past are admonished to banish the words "borrow and carry." Teachers need to thoroughly understand the linkages between older terminology and newer descriptions that may be in use.  Our language matters.  The words we choose must be mathematically accurate and make sense to a child who is reasoning through a problem. We may need to conference with parents to illustrate connections between these terms and the older "borrow and carry" terms parents know and which remain largely procedural in nature. 

Post # 20 The Importance of Sub-Skills

I sometimes work with schools for at risk students.  These students come in all categories imaginable but with one common denominator.  Many of them are below grade level in skills not only math, but reading, writing and critical thinking.  They are still required to pass state mandated tests and exit exams.  This includes algebra I.  I have students who are not allowed to use calculators on these tests and are still counting on fingers and ignoring all word problems.  They guess at random or simply put their heads down in defeat.

I see teachers trying to teach content for which the students are wholly unprepared.  I see them feeling frustrated because they feel they do not have time to teach content deeply and must take short cuts.  They throw their hands up and teach procedures hoping they will stick for the week or so until the testing is done. They practice random sample test prep questions, public release items which cover a year's worth of content in random order, one problem isolated from the next as if it all makes perfect sense.

In this type of school there is frequently high turnover in both students and teachers.  There are often students who enter the week before the tests.  Their skills are a patchwork of misunderstood procedures and a lack of numeracy is almost a given.  The students push numbers around with no meaning.

Throughout the year I refer to the What Works Clearinghouse suggestions and Universal Design for Learning.  I strongly suggest numeracy activities at all levels of math.  I mandate teaching some facts to mastery and using those facts for all new introductions.  We use linkages and emphasize practice in sub-skills before embedding those subs-kills in larger a context.  

I will give one example I used recently.  The object of the lesson was simplifying radicals for an upcoming algebra 1 exit exam.  I began with multiplication.  We constructed perfect squares using simple cubes to demonstrate square numbers.  We placed them on a multiplication chart to illustrate the products growing diagonally down the page.  I only constructed the first few, to make the point of perfect squares and then transitioned to the representational level.  We inserted the meaning of the square root and began choral recitation of the facts.  Two squared is four and the square root of four is two.  Three squared is nine and the square root of nine is three.  We counted by perfect squares to one hundred sixty nine and back by squares to one.  We alternated using the meaning of "squared" occasionally asking "what number times itself is forty-nine?" They did not use calculators.

We discussed why other radicals are irrational and how the most accurate way to write them was with the radical.  Finally I introduced the concept that the square root could be simplified by factoring out a perfect square and "pulling out its root."  Within the hour these remedial students were factoring radicals and simplifying them.

The process:  Begin with something they know, link it to something they can relate to and learn quickly, use visual representations and manipulatives conceptually,  teach incrementally in logical steps using repetitive procedures that make sense and relate to the final solution, teach the sub-skills and integrate them into the final skill set, solidify learning with practice and problem solving.   Time:  one class 50 minutes to one hour.

And at the end, the students were leaving for the day.  One rather tall young man hung back as they exited and said,"I just wanted to thank you lady, for teaching me that today.  I got it.  I got it really fast."    Priceless.

Post #19 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 primary grade teachers, we need to understand how what we do at basic levels forms the foundation of what is to come.  As secondary teachers, we need to understand the basic concepts 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.  Think about the skills that form the basis of these operations.  A student's knowledge of fractions for example has ramification throughout high school and college math; yet, it is often the least developed of math skills as student’s progress