TouchMath

Colorado Springs, CO

SupplementalInformation

This is how most of us were taught mathematics.This is how traditional teachers continue to teach inspite of wholesale failure in elementary school mathe-matics and the distressing number of remedial mathcourses in American high schools, colleges and uni-versities. Different methods of teaching mathematicsmust be employed.

Failure in mathematics begins with the numberfacts. We expect students to memorize all 390 ofthem — 100 for addition, 100 for subtraction, 100 formultiplication and 90 for division — to the point ofinstant recall and to be able to do so FOREVER. Thisstandard ensures failure for all but a handful of stu-dents. It foolishly denies the use-it-or-lose-it maximthat governs remembering and forgetting. What weuse, we remember. What we don’t, we forget.

Fact memorization favors auditory and visuallearners. What about tactile learners? Most at-riskand special education students are tactile learners.So are many CEOs for America’s Fortune 500 com-panies. TouchMath’s method of touching and countingon numerals to generate number facts is effectivewith all three major learning types: tactile, as well asauditory and visual.

What do students taught traditionally do whenthey forget a number fact? Most guess, and oftenincorrectly. Some invent their own TouchMath methodfor figuring it out. They count as they nod their headsback and forth, tap their fingers on their desks or clicktheir tongues against their teeth. Some count onsticks they draw on their papers but erase afterwardsso no one will know they used a crutch.

The number facts are often referred to as thebasic facts, and for good reason. They are exactlythat to success in mathematics: BASIC.Why do somany students fail algebra? They’re shaky on theirnumber facts.They can’t work (4X+7)(5X+8) unlessthey know 4x5, 4x8, 7x5 and 7x8, and many 8th and9th graders don’t, as their teachers verify. Not know-ing even ten of the 390 number facts can result inenough wrong answers to erode self-confidence andlower self-esteem to the point a student becomes amath dropout.

With TouchMath, there is no shaky on the numberfacts. If students remember a fact, that is, it pops intotheir head, they write it down. If they don’t rememberit, they figure it out. THEY NEVER GUESS!TouchMath is a dictionary for the number facts. It’s away of looking up a number fact — a way of knowingyou know, not hoping you know. When will a studentquit looking up a number fact? When they know it, thesame as when they’ll quit looking up the meaning ofa word or how to spell it in a regular dictionary.

“TouchMath is a dictionary for the number facts … a way of knowing you know, not hoping you know.”

Flash cards, Speed Drills,Worksheets, Memorization…

The following interview reveals howMary Todd Allen has been helping youngeducators help their students overcomecommon math learning obstacles.ILC: Why did you adopt TouchMath in yourclassroom teaching?

Todd Allen: The concept made sense for somany of my special high school students. Theywere unable to remember basic facts, so theycounted on their fingers, trying to add largenumbers. With TouchMath, they could use aproven touching/counting method, and it was somuch easier and more effective. They made fewererrors and quickly became proficient.

ILC: Why do you now teach TouchMath to new teachers?

Todd Allen: There are two reasons. At VirginiaTech, where I coordinated a school-to-worktransition program, I kept running into a majorbarrier that these teachers found with theirstudents; the kids were unable to handle basicmath facts so they could not work successfully atjobs that required them to do quick addition andsubtraction. TouchMath was an answer.

Furthermore, in 1997, I had an opportunity atWinston-Salem State University to teach a classcalled Math for Students with LearningDisabilities. I realized that student teachers neededstrategies they could use right away in theclassroom. They needed a way to build confidenceas instructors and help children at the same time,something easy to learn. TouchMath is aphenomenal discovery for tactile or kinestheticlearners. It’s also a great confidence-builder fornew teachers.

ILC: What specific features do you likeabout the program?

Todd Allen: TouchMath is based on researchand on what we know about how children learn ina variety of ways. It’s also stood the test of time.

ILC: What advice do you have for aspiringteachers?

Todd Allen: I have a strong belief that allchildren can learn … but not in the same way. Iencourage student teachers to find out how eachof their children learn and then use the toolsthat work.

Abraham Maslow reportedly observed that if everyproblem looks like a nail, the only tool you’llhave is a hammer. I encourage new teachers to fill their toolboxes, to add at least one newtool every year they’re in this profession.TouchMath is a tool they can incorporateinto their teaching repertoire easily andinexpensively. They will see immediate successwith students struggling to master basic facts.

Mary Todd Allen is an instructor inSpecial Education at Winston-Salem StateUniversity. She has been an adjunctprofessor at Wake Forest University andNorth Carolina Agricultural and TechnicalState University. Mary has also workedwith the Virginia and US Departmentsof Education.

This past summer she worked forWinston-Salem State where sheinstructed teaching candidateswho have baccalaureate degreesbut need additional courses forcertification.

At Wake Forest, she served asan adjunct instructor in theeducation department. And atA&T, her primaryresponsibilities have beento develop online coursesfor distance learning; shehas also taughtelementary educationclasses for studentteachers.

A graduate of Radford Universitywith emphasis in special education, sheholds a master’s degree in education fromVirginia Polytechnic Institute and StateUniversity. Mary has been employed in thefield of education since 1982 and was a highschool teacher for over ten years.

Teaching aspiring educators today means exposing themto tools designed to get fast, confidence-building results.

Dr. Beth McCulloch Vinson’smany academic and professionalachievements suggest someonewho in childhood excelled atmath quizzes. She must havebeen the student who alwaysthrust her hand in the air first.On the contrary, Dr. Vinsononce struggled with math.Perhaps facing and overcomingthis common dread instilleduncommon passion in aneducator on a mission.

In 1989 and 1992, she received herbachelor’s and master’s degrees inelementary education from theUniversity of North Alabama. In 1993,she received an EdS and, in 1994, aPhD from the University of Alabama,with majors in curriculum andinstruction. Her refined Spanishspeaking and writing skills becamecritical in Lima, Peru, and Bogotá,

Colombia, where she taught graduate-level courses.

She has taught third grade, as well as K-6 enrichment classes in art, music, andSpanish. At Athens State University,where she has taught teacher education

courses since 1992, she became anassistant professor in 1996 and anassociate professor in 2002.

Dr. Vinson is also an editorial boardmember of the Alabama Journal ofMathematics, an editorial panel reviewerfor the NCTM Journal of Mathematics,and vice president of the NorthAlabama Council of Teachers ofMathematics.

She is the author of Mathematics in thePreschool and Primary Grades, which hasbeen slated for release by Merrill-Prentice Hall in August 2004.

The following interview reveals howDr. Vinson has been helping neweducators overcome math-learningobstacles, both personally andprofessionally.

A leading Alabama authorand educator has a mission:to help teachers and theirstudents overcome math-learning obstacles.

“Love your subject.Your students will dothe same.

A well-founded positiveattitude toward thesubject you teach – theprocess of learning andteaching – and a genuinelove of and confidence inyour students areimperative.”

– Dr. Beth McCulloch Vinson

TouchMath has been based onsound research. Jean Piagetand Jerome Bruner bothsuggested that learning ofconcepts should and do followa predicted set of stages.

– Dr. Beth McCulloch Vinson CONTINUED

FOLLOWS APREDICTED

SET OF STAGES.

Learning of Concepts

Jean Piaget

ILC: Why didyou choosemathematicseducation?

Vinson: Mathematicsprovides the most joyand challenge to me fortwo reasons. First, Ilearned math on the symbolic level, so Istruggled with it. I found ways tounderstand and simplify the concepts formyself. I want to do the same for newteachers. Second, many of myundergraduate students come to me withmath anxiety. The challenge is to helpthem overcome that while showing themhow to teach mathematics. The jewel of mycourse is when they realize that they havelearned math while learning to teach it.

ILC: Why do you thinkteachers should includeTouchMath in their curriculum?

Vinson: TouchMath has been based onsound research. Jean Piaget and JeromeBruner both suggested that learning ofconcepts should and do follow a predictedset of stages.

ILC: What other features doyou like about TouchMath?

Vinson: Especially in mathematics,effective instruction deals with appropriateportrayal of the concept, modeling, guidedpractice, and plenty of independentpractice. It all begins at the right pointand progresses toward more complex skillsand concepts. TouchMath’s sequentiallearning strategy follows a research basisby providing the sense of numeration andthen progressing to addition andsubtraction.

From the student’s perspective, one of thegood things about TouchMath is thatactivity pages do not have extraneousimages and symbols. Many children haveproblems with focus. TouchMath helpsthem focus on the numerals by providingthem with the pictorial representation ofhow many (number) as well as the symbol(numeral).

ILC: How can TouchMath helpovercome the limitations oftraditional math curricula?

Vinson: In order for most of us to do wellin mathematics, we have to understand it.We cannot simply continue to memorizeroutines, rules, and procedures, withoutbecoming bogged down in the process. It isa natural process to be curious and towonder, “Why?” Many people believe thatmath is only for the select few. In reality,mathematics can and should be one of themost understood subjects.

Teachers can present mathematics in a moreconcrete, hands-on way. Students cannot seemany abstract concepts, such as freedom(social studies) and ionization (science).They can see mathematics on their desktopswhen using manipulatives and symbolicstrategies such as TouchMath.

ILC: What final thoughts wouldyou like to share with othereducators about TouchMath?

Vinson: TouchMath is the best methodavailable for making mathematicalsymbols more computable. It bridges thegap between the hands-on and pictoriallevels to the symbolic.

To take full advantage of this program, Ibelieve that teachers need more support inthe areas of classroom instructional money(to purchase well-founded programs such asTouchMath) and time away from theclassroom to receive practical, hands-onprofessional development.

TouchMath combines allthree learning stages,especially the pictorial andsymbolic. This providesstudents with a supportsystem like no other.

– Dr. Beth McCulloch Vinson

Early years mathematics: Have we got it right?Ian Thompson (British Author of Enhancing Primary Mathematics Teaching)

For the following article in its entirety, please refer to:http://www.standards.dfes.gov.uk/numeracy/prof_dev/self_study/effectiveteaching/14699/?leaf=3

This paper shows that there is now a respectable body of research evidence which suggests thatan approach to early number that concentrates on the development of rational counting skillsshould be allocated a much more important place in the mathematical development of pre-schooland school children than is currently the case... It is now time to re-evaluate the structure andcontent of the number strand of the early years’ mathematics curriculum in the light of theplethora of recent research findings into the way that young children use counting at all stagesin their growing development of an understanding of number.

Counting and early number acquisition An increasing awareness has developed in recent years of the fact that children generally bring withthem to school a much richer experience of number than had previously been assumed. Thisknowledge is seen as being different in kind, however, from the culturally developed body ofknowledge that the school system attempts to impart. There is a substantial corpus of researchevidence concerning the important part played by counting strategies in the development ofchildren’s number understanding. Young children hear the number words being used in a variety ofdifferent situations in their daily life, and these words vary in meaning according to the context inwhich they are used. However, their experience of actually saying the number words is more likelyto have been gained in a counting context. Children learn to count out of practical necessity but alsofor reasons of intrinsic motivation. Counting, like the recitation of nursery rhymes, is a traditionalshared activity enjoyed by most children and their parents in all cultures.

Fuson et al. (1982) have shown that children learn the difference between counting words and non-counting words at a very early age. They asked a group of children between the ages of two andfive to count collections of objects. All of the three-, four-, and five-year-olds and most of the two-year-olds used counting words on every occasion. There were just two children, both two-year-olds,who used non-counting words instead, and in both cases they chose to recite letters of the alphabetrather than the number words. In another small-scale study Gelman and Meck (1983) asked youngchildren to comment on the counting ability of a puppet. Nearly all of the three- and four-year-oldsdetected the puppet’s double counting or skipping errors, even though the numbers involved wereoften larger than their own counting ceiling. The authors suggest that the children knew in principlewhat it was to count large numbers even though they might well have difficulty putting theseprinciples into practice themselves.

Hebbeler (1977) has researched the extent to which counting strategies are used by pre-schoolchildren to solve simple problems, and reports that the normal course of development forKindergarten and first grade children is to progress spontaneously from counting to the use ofnumber facts as a problem solving strategy. Similarly, Houlihan and Ginsburg (1981) havedemonstrated that first and second graders can apply non-taught counting algorithms in the solutionof a number of problem solving types, and have suggested that young children select their countingstrategies according to the size and familiarity of the numbers involved.

Aubrey (1993) details the level of understanding of a group of children (average age 4 years and 6months) on ten separate aspects of number knowledge. The sixteen children studied were in thereception class of an urban infant school in the north-east of England with a catchment area thatreflected a wide social and ethnic mix. She found that twelve of the sixteen children weresuccessful in counting two separate arrays of objects containing three and seven items. Two of theothers were able to count the group of three items only. These results are consistent with those ofSchaeffer et al. (1974) who had earlier found that a group of children of average age 3 years and 5months successfully counted 71% of arrays containing between five and seven poker chips.

It was also the case that three quarters of the children in Aubrey’s (1993) study were successful onan activity which suggested an appreciation of the fact that counting an array in a variety ofdifferent orders should yield the same result. She also found that half of the group could give thecorrect answer 8O% of the time to the question ‘How many sweets would there be altogether?’when asked in the context of two teddy bears and separate piles of sweets. These results suggestthat many of these children, who had been in school for just two weeks, already possessed anunexpectedly good intuitive understanding of some of the number concepts that the school plannedto teach them over the next few years.

The young children in the research studies reported above are able to count various arrayssuccessfully, and in so doing are demonstrating their command of some or all of the followingsubskills: • reciting the standard counting word sequence up to the required number and in the correct order; • matching these number names in one-to-one correspondence with the objects to be counted; • ensuring that the same number name is not matched to two different objects; • ensuring that each object to be counted has one, and only one, number name assigned to it; • co-ordinating the recitation of the number names with the action of moving or pointing at the

objects.

In addition to this, the children are showing some understanding of the cardinal aspect of number,where the number name assigned to the final object in the count is used to describe the numerosityor size of the collection of objects.

The level of performance manifested by these young children would appear to raise questionsconcerning the content of the early years mathematics curriculum. If these children are revealingthese competences at the start of their school careers it seems appropriate to ask why they need tobe spending their time on such prenumber activities as sorting, classifying, and matching: activitiesdesigned to prepare them for an introduction to number.

Counting and the arithmetical operations Piaget (1952) argued that those children who had not reached the concrete operational stage ofthinking - normally attained at about the age of seven - could not possibly understand the operationsof addition and subtraction. However, Hughes (1986) and Starkie et al. (1982), with their respective‘Box’ and ‘Pennies’ tasks, have shown that children between the ages of three and five can performaddition and subtraction tasks with some understanding, provided that the numbers involved aresmall, and the operations are carried out on real items in such a way that the objects and actions arevisible to the children but the final total is not.

Other researchers have shown the important role of counting in the acquisition of number facts andin the execution of the basic mathematical operations. During a long-term study Carpenter andMoser (1983) suggested the following developmental stages that children between the ages of sixand nine pass through - albeit at different rates - when finding the sum of two numbers:

• counting all - where children solving a simple addition problem such as 2 + 3 first count out twoblocks followed by three other blocks, and then find the total by counting the number of blocksaltogether;

• counting on from the first number - where children, finding 2 + 3, begin the count by repeatingthe first number and then continue the count by starting from that number. For example, a childmight say: ‘Two ... three, four, five. There are five’;

• counting on from the larger number - where children proceed as in the previous example, butbegin the count from three, because they realise that starting from the larger number means thatless counting will be involved;

• using known number facts - where children give immediate responses to those number bondswhich they know by heart - usually the simpler number bonds such as the smaller doubles like 2 +2 and 3 + 3;

• using derived number facts - where children use a number bond that they know by heart tocalculate one that they do not know. In the initial stages there is a tendency to use the doubles, sothat 6 + 5 might be found by saying: ‘Five and five is ten and one more makes eleven’, or ‘Sixand six is twelve, but it’s one less so it must be eleven’.

Groen and Resnick (1977) demonstrated that pre-school children can invent for themselvescalculation procedures that they have not previously been taught. A group of five-year-olds wastaught the ‘count-all’ strategy, and after several practice sessions, many children spontaneouslyprogressed from ‘counting-all’ to ‘counting-on’, and some even adapted the latter strategy to‘counting-on-from-larger’: a strategy which implies an intuitive understanding of commutativity. Inaddition to this, ‘derived fact strategies’ have been shown to be used by more than just a smallpercentage of children (Carpenter and Moser, 1983; Steinberg, 1985).

Finally, my own research (Thompson, 1989, 1990, 1991, 1992, 1993), which involved 103 childrenfrom Years 2 and 3, suggests that children employ a wider range of these strategies than haspreviously been documented in the literature. The following examples illustrate the extent to whichyoung children make use of counting at different stages in the development of their understandingof mathematical operations, and show how they combine counting techniques with other calculationskills and number bonds to derive idiosyncratic strategies possessing ‘local’ originality. Three of theexamples deal with addition and three with subtraction.

John (6y. 4m.)The earliest number bonds that children learn are the doubles (Thompson, 1990). John, finding 5 +7, said: ‘10 ... 11, 12. I counted in my head’.

He actually counted aloud, but when pressed it transpired that he meant that he had counted twoobjects in his head. This could be described as a ‘doubles-with-counting-on’ strategy

Paul (6y. 9m.)Paul’s explanation of his answer to 5 + 8 was: ‘I made the eight into ten and went 11, 12, 13’.

In order to use this ‘complements-in-ten’ strategy effectively children need to be able to do thefollowing: ascertain what is needed to build one of the numbers up to ten; partition the othernumber into two appropriate parts, and then add these two parts separately by counting-on or bymaking use of their knowledge of the effect of adding a single-digit number onto ten (Thompson,1989). Other children used this strategy quite frequently.

Ben (6y. 3m.)Ben’s answer to the calculation 4 + 5 was: ‘4. . .6. . .8. . .9’

It was difficult to ascertain exactly why Ben had tackled the problem in this way, but it related tohis visualising the five in standard ‘domino’ formation, then adding the two pairs of dots bycounting on in twos from four, and finally adding on the remaining dot.

Patrick (7y. 1m.)Patrick, working out 7 - 3, said: ‘Four ... I just knock down skittles in my head’.Further into the interview having just given the correct answer to 13 - 6 he explained: ‘I just had a game of skittles’.

Patrick used a mental representation of the situation, whereas other children use a method thatinvolves modelling - usually on the fingers - the number to be operated upon (the minuend). Therequired number of fingers is set up and then the number to be taken away (the subtrahend) isremoved either by counting the fingers down, or by using prior knowledge of finger totals. Theremainder is then dealt with in a similar way.

This ‘counting-out’ strategy can be very successful when dealing with numbers up to ten, but -because of a finger shortage problem - is less useful with larger numbers. Many children perceivethe need for a more sophisticated technique when dealing with numbers greater than ten. Others,however, do not, although several children showed surprising ingenuity in modifying a strategy thathad brought them success.

Anna (6y. 7m.)Anna had been correctly using ‘counting-out’ to answer some simple subtractions, and was asked tocalculate 11 - 6 with a view to making her aware of the inadequacies of this technique withnumbers larger than ten. Fingers were set up and the correct answer was given. When asked howshe had worked out the answer even though she only had ten fingers, she replied: ‘I counted thenewspaper’.

Anna had used a newspaper, which had been lying on the table, as the eleventh object in hercollection. She proceeded to ‘remove’ that object first before returning to the familiar territory ofher ten fingers to deftly remove the remaining five objects. She later calculated 15 - 9 by imaginingfive extra objects.

Joanne (6y. 6m.)Joanne, however, was the expert in this particular technique. The following explanations for threedifferent subtractions reveal her creative talents and her apparent obsession with her bodily parts: ‘Iused my two legs’ (12 - 4) ‘I used both strips of my tracksuit as well as my legs. You could use your arms and your legs

instead’ (14 - 6)‘I took that one away (points to one arm). . . then that (points to other arm) ... and then my head’(13 - 7).

Conclusions There is no doubt about the extensive use that pre-school and Early Years children make ofcounting skills when solving problems involving mental arithmetic calculations, particularly insituations that make ‘human sense’ to them. This helps them become gradually aware of the variousprinciples that underlie rational counting. As they progress through school they continue to usecounting as an important part of their problem solving armoury, and develop creative adaptations ofthis basic skill by learning to combine counting with other newly acquired mathematical skills, factsand knowledge. Number concepts and meaningful counting appear to develop in tandem, and it isthrough the application of increasingly more efficient counting procedures that young childrengradually discover or construct for themselves many of the basic number concepts.

Clements’ (1983) study suggests that a ‘rational counting skills’ approach to early number can bemore successful overall than the more traditional ‘logical operations’ approach. The young childrenin Aubrey’s (1993) study had no formal experience of sorting, classifying, ordering, matching ormapping activities, having been in the Reception class for only two weeks, and yet, in general, theyshowed an unexpectedly high level of understanding of early number. These findings suggest thatthere is more likelihood of young children developing an implicit understanding of a concept suchas one-to-one correspondence by actually indulging in the counting process itself, rather than by,say, joining the members of a set of four cups to the members of a set of four saucers - a pre-number activity common to many commercial mathematics schemes. The ability to use countingcompetently and successfully would appear to be more crucial to a child’s development of anunderstanding of basic number concepts than the ability to sort, match and order sets of objects.

This paper has shown that there is now a respectable body of research evidence which suggests thatan approach to early number that concentrates on the development of rational counting skills shouldbe allocated a much more important place in the mathematical development of pre-school andschool children than is currently the case - even to the exclusion of work on set theory and thelogical foundations of numbers. It is now time to re-evaluate the structure and content of thenumber strand of the Early Years’ mathematics curriculum in the light of the plethora of recentresearch findings into the way that young children use counting at all stages in their growingdevelopment of an understanding of number.

From Curriculum 15 (1), 42-48.

Quotes from Referenced Sources Utilizing and ResearchingTouchMath

“TouchMath was introduced as a replacement for the ineffective strategiesthat the children were using prior to intervention and proved to be successfulsignificantly improving the results from pretest to posttest . . . Students’scores on a multiple probe measure rose from an average of 16.2 percentcorrect prior to intervention to an average of 95.6 percent correct followingintervention.” – Strafford and Gunn (20)

“Math tends to be more challenging than reading for most of our children.It’s important to plan instruction so that a child is well grounded in aconcrete understanding of quantity and then moves on to an understanding ofsymbols before progressing to abstract problem-solving . . . TouchMathfeatures multisensory math materials for all grade levels, with accents onlearning by touching, seeing and hearing, enlivened by engaging,entertaining graphics.” – Scheiber (16)

“The TouchMath program is an increasingly popular, commercially-produced, multisensory math remediation program that is being used byconventional educators and private educational clinicians across the country.We believe this program would prove to be both cost effective and successful for parents in home schools who have children that demonstratelearning deficits in the skill areas of addition, subtraction, multiplication,division, number concepts, sequence counting, place value, and shapes andsizes.” – Sutton and Sutton (21)

“Many first and second-grade teachers use this type of method, in conjunction with existing math programs, to help students improve theiraccuracy in addition and subtraction and to aid with memorization of basicfacts. Children are encouraged to eliminate touching the points as theybecome more proficient.” – Goldstein and Mather (6)

“To compensate for the student’s inability to memorize math facts, teachhim/her to use TouchMath and a calculator.” – Mather and Jaffe (11)

“Children with mild disabilities must understand all critical mathematicsconcepts and be able to apply them successfully. This means that the specialeducation teacher must minimize the amount of memorization required whenlearning basic facts and maximize the child’s motivation to developautomaticity with basic skills.” – Ruth Meese (12)

“If math is hard for you, try the great new method called TouchMath, whichwoks even better than using your fingers (which is okay). It is easy to learnand helps many students learn to add, subtract, multiply and divide.” –Joanne East (4)

“Dyslexics who cannot do mental arithmetic without whispering and‘counting on their fingers’ are crippled in typical math assignments if adultsdemand that they work silently with no finger involvement. A remarkablyeffective touch/see/say math program has been developed by a creativegroup of teachers in Colorado. The TouchMath program is excellent forestablishing basic math skills with dyslexic children.” – Dale Jordan (9)

“Use designs and color for helping students visualize the pattern andsequence of multiples. The program “TouchMath” has posters and designsfor each of the multiples . . . The ‘spatial’ approach seems to be the answerfor some of these children. It is amazing to watch them close their eyes andpicture where the number is placed spatially.” – Sandra Reif (15)

“Kinesthetic learners who are pushed too early into abstract math conceptsand symbols, made to memorize by rote and apply concepts they don;tunderstand often develop “math phobia.” Although they may have a lot ofpotential to be excellent math students, they are being taught backwards.Math should be taught from the concrete to the abstract.” – Cheri Fuller (5)

“TouchMath is a program that has helped many tactile-kinesthetic learnersunderstand basic math operations. It provides hands-on learning withoutactual manipulatives.” – Susan Winebrenner (24)

TOUCHMATH®

ADDITIONAL RESOURCES

Innovative Learning Concepts, Inc. ~ 6760 Corporate Drive ~ Colorado Springs, CO 80919-1999

Telephone Number: (800) 888-9191

1) Bender, W. N. (1995). Learning disabilities: characteristics, identification, and teaching

strategies (p.240-241). Needham Heights, MA: Allyn & Bacon.

2) Bozzone, M. A. (1994, January). 16 Resources to help you do your job better. Instructor, 67. New York: Scholastic, Inc.

3) DeRuvo, S. & Latno, J. (2001, Fall). Need curriculum help? (6) California Association of Resource specialists and Special Education Teachers 21(5).

4) East, J. (1993) Yes you can! (24). Chicago: The Easter Seal Society.

5) Fuller, C. (1994). Unlocking your child’s learning potential (191). New Press, Colorado

Springs, Colorado.

6) Goldstein, S. & Mather, N. (1998). Overcoming underachieving (245-247). New York:

John Wiley & Sons, Inc.

7) Hanrahan, J., & Newman, T. (1996). Teaching addition to children. In B. Stratfford &

Gunn (eds), New approaches to downs syndrome, (300-308). London: Cassell.

8) Hanrahan, J., Pupo, M., & Newman, T., (July, 2000). An analysis of an imprinted-dot

approach to teaching arithmetic to intellectually disabled children and its potential for

facilitating inclusion. Published Papers of the Fifth International Special Education

Congress, Manchester, England.

9) Jordon, D. R. (1989). Overcoming dyslexia in children, adolescents and adults (48, 78).

Austin, TX: Pro-Ed, Inc.

10) Mather, N. & Goldstein, S. (2001). Learning disabilities and challenging behaviors: A

guide to intervention and classroom management. Baltimore, MD: Paul H Brookes.

11) Mather, N. & Jaffe, L. E. (1992). Woodcock-Johnson psycho-educational battery -

Revised, recommendations & reports (367). Brandon, VT: Clinical Psychology

Publishing Company, Inc.

12) Meese, R. L. (1993). Teaching learners with mild disabilities (286-287). Pacific Grove,

CA: Brooks/Cole Publishing Company, Inc.

13) Newman, T., (1994). The effect of a multisensory approach for teaching addition to

children with Downs Syndrome. Unpublished masters thesis, McGill University, Canada.

14) Pupo, M., and Hanrahan, J. (2000). Teaching intellectually disabled students addition

through a dot-notation approach. International Journal of Practical Approaches to

Disability, 24(3). 50-55.

TOUCHMATH®

ADDITIONAL RESOURCES

Innovative Learning Concepts, Inc. ~ 6760 Corporate Drive ~ Colorado Springs, CO 80919-1999

Telephone Number: (800) 888-9191

15) Rief, S. F. (1993). How to reach and teach ADD/ADHD children (89). New York: The

center for applied research in education.

16) Scheiber, B. (2002) Fulfilling Dreams (131). Williams Syndrome Association, Royal Oak, Michigan.

17) Scott, K. S. (1993). Multisensory mathematics for children with mild disabilities. Exceptionality, 4(2), 97-111.

18) Scott, K. S. (1993). Reflections on multisensory mathematics for children with mild disabilities. Exceptionality, 4(2), 125-129.

19) Spiridigliozzi, G. A. & Lachiewicz, A. M. (1994). Educating boys with Fragile X Syndrome

(11). Durham, NC: Office of Publications, Duke University Medical Center.

20) Strafford, B. & Gunn, P. (1996) New Approaches to Down’s

Syndrome (300-307). Cassell. London.

21) Sutton, C. J. & Joe, P. (1995). Strategies for struggling learners (144-145). Simpsonville,

SC: Exceptional Diagnostics.

22) Shimota, K. (1988, Spring). Touch Math: Teaching math effectively. The Elementary

Mathematician, 3-4.

23) Williamson, V.C. (1982). The touchmath method of teaching basic addition verses

traditional teaching methods. Unpublished masters thesis, College of Notre Dame,

Belmont, California.

24) Winebrenner, S. (1996). Teaching kids with learning difficulties in the regular classroom

(152-153). Minneapolis, MN: Free Spirit Publishing, Inc.

TOUCHMATH®

COLLEGE AND UNIVERSITY RESOURCES

(Partial listing of colleges and universities incorporating TouchMath in teacher training classes)

Athens State University – Athens, AlabamaBowling Green State University – Bowling Green, OhioCalifornia State University – Carson, CaliforniaCalifornia State University – Los Angeles, CaliforniaCapital University – Columbus, OhioCollege of Mt. St. Joseph – Cincinnati, OhioFort Hays State University – Hays, KansasHannibal La Grange College – Hannibal, MissouriIndiana Wesleyan University – Marion, IndianaKean College – Middleton, New JerseyKentucky Christian College – Grayson, KentuckyLees College – Jackson, KentuckyLock Haven University – Lock Haven, PennsylvaniaMcGill University – Montreal, QuebecMonmouth College – Monmouth, IllinoisNorthwestern State University – Natchitoches, LouisianaPeabody College at Vanderbilt University – Nashville, TennesseeRiverside Community College – Riverside, CaliforniaSaint Francis College – Fort Wayne, IndianaSaint Joseph College – West Hartford, ConnecticutSan Francisco State University – San Francisco, CaliforniaSlippery Rock University – Slippery Rock, PennsylvaniaSoutheastern Oklahoma State University – Durant, OklahomaSoutheastern Louisiana University – Hammond, LouisianaTroy State University – Troy, AlabamaUniversity of California – Los Angeles, CaliforniaUniversity of Colorado – Boulder, ColoradoUniversity of Houston - Victoria – Victoria, TexasUniversity of Indianapolis – Indianapolis, IndianaUniversity of Nebraska - Omaha – Omaha, NebraskaUniversity of Northern Iowa – Cedar Falls, IowaUniversity of Pittsburgh – Pittsburgh, PennsylvaniaUniversity of Portland – Portland, OregonUniversity of Saint Thomas – Houston, TexasUniversity of Washington – Seattle, WashingtonUniversity of Wisconsin - Oshkosh – Oshkosh, WisconsinWesleyan College – Macon, GeorgiaWinston Salem State University – Winston Salem, North Carolina

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