working with the cartesian plane and algebraic formulas...

Working with the Cartesian plane and algebraic formulas seems to be easy for

students with Down syndrome.

Elisabetta Monari Martinez, University of Padua, Italy.

Katia Neodo, University of Padua, Italy.

Strengths and difficulties in learning (1)

• Children and adolescents with Down syndrome (DS) have some difficulties in language, memory, and cognition, but relying on their strengths (e.g. using visual prompts), their social abilities, and their motivation to stay with their peers, they can learn with pleasure (Chapman and Hesketh, 2001; Couzen & Cuskelly, 2014).

• About mathematics, they have difficulties in the language-dependent numerical functions, as counting, times tables, and word problems, but not everywhere, and rulers, visual devices, and calculators are used to help them in those basic abilities that are not yet mastered and to improve their self-confidence (Faragher & Clarke, 2014).

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Strengths and difficulties in learning (2)

• Bypassing their basic difficulties in numeracy with the use of devices (rulersand calculators), it was relatively easy for them to learn algebra and even to use it in the solution of simple word problems (Monari Martinez, 1998; Monari Martinez, 2002; Monari Martinez & Pellegrini, 2010; Monari Martinez & Benedetti, 2011; Faragher & Clarke, 2014).

• As they can learn a procedure and follow it in the right order (Chapman and Hesketh, 2001), they were able to use the algebraic rules to find the solution of an equation, with the help of a calculator for the numerical computation, as well as to replace the letters with the correct numbers in an algebraic formula.

• Relying on their well-preserved implicit memory (Vicari, Bellucci, Carlesimo, 2000), they learn easily to use even scientific calculators (Monari Martinez & Benedetti, 2011) and new technologies, as my Facebook friends with DS.

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Why to study analytic geometry?

• Since 1992, Italian public schools have been under a duty to include all disable students (aged 3–19 years) within their mainstream classes, in accord with their chronological age and regardless of their level of disability or impairment, and to provide them with the help of a special education teacher (Cottini & Nota, 2007).

• In this context, it seemed natural to try to teach them the same topics as their typically developing schoolmates and then to teach them also analytic geometry, as Nives Benedetti did with two students (Monari Martinez & Benedetti, 2011). The surprise was that the student with severe difficulties in counting and in measuring a segment, learned it using the algebraic formula of the distance between to points on the Cartesian plane.

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Schools: Elementary (E), Middle (M), Second. (S) E M S

The student does not know fractions, 88% 87% 76%

The student is not able to find a point in the

Cartesian plane, - 82% 78%

The student is not able to measure with a given

unit of length, 85% 68% 58%

The student is able to recognize basic geometrical

shapes and even solid shapes. 80% 79% 87%

Previous study: Results of Gherardini and Nocera (2000).

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This educational study.• This explorative study was carried out in the North-East of Italy by the

second author, with the supervision of the first, with 6 adolescents with Down syndrome.

• They were taught, on the same programme and exercises, individually at home in the afternoon once a week for nine months by the second author that was both teacher and experimenter.

• At the end of this course, all of them were asked to do the same test.

• This study was designed to verify if the previous single case positive experience on algebra and analytic geometry (Monari Martinez & Benedetti, 2011) could be extended to more students with Down syndrome.

• The small size of this sample allows just a descriptive analysis of the data.

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The students

Students St.1 St.2 St.3 St.4 St.5 St.6Gender

(Male/Female) F M M M M MChronological Age

(years) 14 15,5 16 14 15,5 14,5IQ (Raven's Matrices) 70 69 74 71 65 66

Mental Age (years) 9,8 10,7 11,8 9,9 10,1 9,6

School Inclusion Good Good Good Poor Moderate PoorAssoc. Course

attendance No Yes Yes No Yes Yes

Verbal ability Good Good Good Poor Good GoodReading and writing

ability Good Good Good Very poor Good Good

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The programme (1)

• Fractions: to use fractions as operators, the visual representation, the simplification, and the four operations;

• Positive and negative integer numbers: the visualization on the number line, the sum, and the multiplication;

• Algebraic equations, with rational coefficients, of the first degree (ax= b);

• The Cartesian plane: given the coordinates of a point, to find the point on the plane and vice versa. To draw a straight line through 2 points. Given the coordinates of a third point, to check if it belongs to the line. To check if two lines are either parallel or intersecting.

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The programme (2)

• To measure the distance between two points on the Cartesian plane with a ruler.

• The equation of a straight line as y=ax+b, or y=c, or x=d: to find the coordinates of some points belonging to the line and to draw the line through them.

• Given the equation of a straight line, to check if a point, indicated by its coordinates, belongs to the line: geometrical check and algebraic check.

• The distance between 2 points, using the Pythagorean Theorem andto check the measurements with the ruler.

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Scoring method.

0 Not able to reply to the question

1 Able to reply to the question only with a strong help (more than 50%)

2 Able to reply to the question only with a littlehelp (less than 50%)

3 Able to reply to the question without anyhelp.

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ESERCIZIO 14 : Usa i principi di equivalenza per risolvere queste equazioni.

a) 2 x = 10 b) 3 x = 21 c) 9 x = 9 d) 5 x = 15

e) 10 x = 40 f) 6 x = 36 g) 7 x = 35 h) 11 x = 88

i) 20 x = 40 l) 12 x = 36

Soggetto 1 Soggetto 2 Soggetto 3 Soggetto 4 Soggetto 5 Soggetto 6

a 1 1 1 1 1 0

b 1 2 2 1 1 1

c 2 2 3 1 1 1

d 2 2 3 1 2 1

e 2 3 3 2 3 1

f 3 3 3 3 3 1

g 3 3 3 2 3 2

h 3 3 3 2 3 2

i 3 3 2 3 3 2

l 3 3 3 3 3 2

Percentuale

di riuscita77% 83% 87% 63% 77% 43%

Punteggio grezzo ottenuto dai soggetti nell'esercizio n°14

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Results in the Course (C) and in the Final Test (FT) per each student and per each topic

Students St.1 St.2 St.3 St.4 St.5 St.6

Verbal ability Good Good Good Poor Good Good

Reading/ writing abil. Good Good Good Very poor Good GoodPercent mean accuracy in each topic

Topics of the exercises C FT C FT C FT C FT C FT C FT

Fractions 90,9 83,4 88,0 86,6 84,2 87,0 80,2 88,0 88,5 93,4 63,8 57,6Positive/Negative numbers 97 87,5 81,5 79 75 87,5 62,5 71 78 96 57 87,5

Equations 73,5 79 78 83,5 77 62,5 48 54 67 66,5 38 33

Cartesian plane 65 100 86 96 63,5 83,5 81,5 75 67,5 83,5 26 62,5

Measure of segments 78 89 78 78 100 78 78 89 67 78 67 78Parallel/Intersecting lines 81,8 100,0 94,5 97,3 95,3 91,7 74,5 77,7 72,8 89,0 75,0 86,0Equation of a straight line 92 82 85,5 92 84,25 91 83 91 88,75 95 75,5 81

Geometric check: point on a line? 94 93,75 91,5 91,75 77,5 93,75 72 93,75 78 95,75 87 81,25

Algebraic check: point on a line? 44 33 44 33 33 33 33 33 33 33 33 33

Pythagorean theorem 71 78 75 89 58 56 50 44 71 100 38 6727/07/2018 E. Monari Martinez 13

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Results.• All the students learned the most of the programme and learned to work on it without

any help, except for the algebraic verification that a point belongs to a line that probably needs more exercises to be mastered.

• Working with a fixed and graduated reference system on the Cartesian plane and with algebraic formulas seemed to be relatively easy for these students, because the algebraic approach and the geometrical one seemed to strengthen each other.

• Visual prompts helped them in understanding, but are false the prejudices that they can learn only what can be represented or that the rules with a geometrical representation are learned better than the others. In fact, in this study, the multiplication of integers (not representable) was easier than the sum of integers that had a visual explanation.

• Also, they had some difficulties in the exercise on the distance between two points, even if was given a geometrical representation with the theorem of Pythagoras. On the contrary, the student described in (Monari Martinez & Benedetti, 2011) performed it easily, using the algebraic formula.

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Conclusions• As any prejudice on the learning is often arbitrary, the students with Down syndrome should share the

same educative opportunities with their peers and the inclusion in the mainstream schools seems to be the most natural way to get that sharing.

• In fact, also in this study, a student, who had severe difficulties in speaking, reading, and writing, was able to learn this programme and to enjoy it. Without the mainstream inclusion, he and the teacher might not have the motivation to do it.

• Even so, a poor inclusion can lower the self-esteem and the motivation for learning, as described here for another student that was reluctant in the beginning, but later learned and enjoyed the programme.

• According to the teacher, all the students that participated in this programme enhanced their self-esteem, also for the social approval in learning something that was difficult for many people, and had a general improvement in other fields too.

• More studies are needed to confirm these results with more participants and also an extension of the programme of analytic geometry should be possible: e.g. to study curves as the parabola, the hyperbole, and the circle.

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Autism,

Intellectual

Disabilities, and

beyond

References• Chapman R. S, and Hesketh L. J. (2001). Language, cognition, and short-term memory in individuals with Down

syndrome. Down Syndrome Research and Practice, 7(1), 1-7.

• Cottini L. & Nota L. (2007) School Inclusion: The Italian model. In J.A. Rondal & A. Rasore-Quartino (Eds.). Therapies and rehabilitation in Down Sindrome (pp. 144- 162). Chichester, England: Wiley.

• Couzen D. & Cuskelly M. (2014) Cognitive strengths and weaknesses for informing educational practice. In Faragher R. & Clarke B. (eds) Educating Learners with Down Syndrome: Research, theory, and practice with children and adolescents. Routledge, New York.

• Faragher R. & Clarke B. (2014) Mathematics profile of the learner with Down syndrome. In Faragher R. & Clarke B. (eds) Educating Learners with Down Syndrome: Research, theory, and practice with children and adolescents. Routledge, New York.

• Gherardini, P. & Nocera, S. (2000) L’integrazione scolastica delle persone Down: una ricerca sugli indicatori di qualità in Italia [The inclusion in mainstream schools of persons with Down syndrome: A study on quality indicators in Italy]. Trento, Italy: Centro Studi Erichson.

• Monari Martinez E. (1998) Teenagers with Down Syndrome Study Algebra in High School. Down Syndrome: Research and Practice. 5 (1), 34-38.

• Monari Martinez E. (2002), Learning mathematics at school and.... later on. Down syndrome news and update, 2 (1), 19-23, ISSN: 1463-6212.

• Monari Martinez E. & Benedetti N. (2011) Learning mathematics in mainstream secondary schools: experiences of students with Down syndrome. European Journal of Special Needs Education, 26(4), 531-540.

• Monari Martinez E. & Pellegrini K. (2010) Algebra and problem solving in Down syndrome: a study with 15 teenagers. European Journal of Special Needs Education, 25(1), 13–29.

• Vicari S., Bellucci S., Carlesimo G.A. (2000) Implicit and Explicit memory: a functional dissociation in persons with Down Syndrome. Neuropsychologia, 38, 240-251.

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