BED II- CPS 9
(I)
Pedagogy of Mathematics (Part I)
,
,
ISBN: 13-978-93-85740-76-3
BED II- CPS 9 (BAR CODE)
BED II- CPS 9
गिणत का िश%णशा' (भाग I)
Pedagogy of Mathematics (Part I)
िश#क िश#ा िवभाग, िश#ाशा% िव'ाशा खा
उ"राख&ड म
ु
* िव-िव.ालय, ह"#ानी
अ"ययन बोड(
िवशेष& सिमित
!ोफे सर एच० पी० श
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उ"राख&ड म
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* िव-िव.ालय
!ोफे सर म
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व$ अिध(ाता, िश#ा स
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काय,
जािमया िमि&लया इ)लािमया व प
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लपित, मौलाना आजाद रा*+ीय उद
ू
0
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!ोफे सर एन० एन० पा#डेय (बा# िवशेष)- सद#य), िवभागा&य(, िश#ा िवभाग,
एम० जे० पी० !हेलख&ड िव*िव+ालय, बरेली
!ोफे सर के ० बी० ब
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धोरी (बा# िवशेष)- सद#य), प
ू
व( अिध,ाता, िश0ा स
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काय,
एच० एन० बी० गढ़वाल िव'िव(ालय, *ीनगर, उ/राख1ड
!ोफे सर जे० के ० जोशी (िवशेष आम
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उ"राख&ड म
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!ोफे सर र'भा जोशी (िवशेष आम
ं
)ी- सद#य), िश'ाशा) िव+ाशाखा, उ.राख0ड
म
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# िव&िव'ालय
डॉ० िदनेश क
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मार (सद#य), सहायक (ोफे सर, िश/ाशा0 िव2ाशाखा, उ5राख!ड
म
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# िव&िव'ालय
डॉ० भावना पलिड़या (सद#य), सहायक (ोफे सर, िश/ाशा0 िव2ाशाखा,
उ"राख&ड म
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स
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#ी ममता क
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मारी (सद#य), सहायक (ोफे सर, िश/ाशा0 िव2ाशाखा एव
ं
सह-
सम#वयक बी० एड० काय$%म, उ(राख+ड म
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डॉ० !वीण क
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सम+वयक बी० एड० काय$%म, उ(राख+ड म
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!ोफे सर एच० पी० श
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िव#ाशाखा, उ(राख*ड म
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म
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िव#िव$ालय, भोपाल
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िव#ाशाखा, उ(राख*ड म
ु
. िव/िव#ालय
डॉ० िदनेश क
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मार (सद#य), सहायक (ोफे सर, िश/ाशा0
िव#ाशाखा, उ%राख'ड म
ु
+ िव.िव/ालय
डॉ० भावना पलिड़या (सद#य), सहायक (ोफे सर, िश/ाशा0
िव#ाशाखा, उ(राख*ड म
ु
. िव/िव#ालय
स
ु
#ी ममता क
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मारी (सद#य), सहायक (ोफे सर, िश/ाशा0
िव#ाशाखा एव
ं
सह-सम#वयक बी० एड० काय$%म, उ(राख+ड म
ु
.
िव#िव$ालय
डॉ० !वीण क
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मार ितवारी (सद#य एव
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स
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योजक), सहायक -ोफे सर,
िश#ाशा% िव'ाशाखा एव
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सम-वयक बी० एड० काय$%म, उ(राख+ड
म
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िदशाबोध: (ोफे सर जे० के ० जोशी, प
ू
व$ िनदेशक, िश+ाशा- िव.ाशाखा, उ1राख3ड म
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7 िव!िव$ालय, ह)*ानी
काय$%म सम(वयक:
डॉ० !वीण क
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मार ितवारी
सम#वयक, िश)क िश)ा िवभाग,
िश#ाशा% िव'ाशाखा,
उ"राख&ड म
ु
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ह"#ानी, नैनीताल, उ+राख.ड
काय$%म सह-सम#वयक:
स
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#ी ममता क
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मारी
सह-सम#वयक, िश)क िश)ा िवभाग,
िश#ाशा% िव'ाशाखा, उ*राख,ड
म
ु
# िव&िव'ालय, ह,-ानी, नैनीताल,
उ"राख&ड
पाठ
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य&म सम)वयक:
डॉ० अिखलेश क
ु
मार
सहायक &ोफे सर, िश#ा िव&ापीठ,
वध#मान महावीर ख
ु
ला िव.िव/ालय,
कोटा, राज'थान
पाठ
्
य&म सह सम*वयक:
स
ु
#ी ममता क
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मारी
सह-सम#वयक, िश)क िश)ा िवभाग,
िश#ाशा% िव'ाशाखा,
उ"राख&ड म
ु
* िव-िव.ालय, ह23ानी,
नैनीताल, उ(राख+ड
!धान स&पादक
डॉ० !वीण क
ु
मार ितवारी
सम#वयक, िश)क िश)ा िवभाग, िश)ाशा- िव.ाशाखा,
उ"राख&ड म
ु
* िव-िव.ालय, ह"#ानी, नैनीताल, उ+राख.ड
उप स$पादक
डॉ० अिखलेश क
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मार
सहायक &ोफे सर, िश#ा िव&ापीठ,
वध#मान महावीर ख
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ला िव.िव/ालय, कोटा, राज5थान
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स)पादक
भाषा स%पादक
!ा#प स&पादक
!
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फ़ स
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शोधक
स
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#ी ममता क
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मारी
सहायक &ोफे सर, िश-ाशा.
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,
िव#िव$ालय
स
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#ी ममता क
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मारी
सहायक &ोफे सर, िश-ाशा.
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स
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स
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#ी ममता क
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सहायक &ोफे सर, िश%ाशा'
िव#ाशाखा, उ(राख*ड म
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साम$ी िनमा(ण
!ोफे सर एच० पी० श
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#ल
िनदेशक, िश'ाशा) िव+ाशाखा, उ.राख0ड म
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4 िव5िव+ालय
!ोफे सर आर० सी० िम#
िनदेशक, एम० पी० डी० डी०, उ"राख&ड म
ु
* िव-िव.ालय
© उ"राख"ड म
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# िव&िव'ालय, 2017
ISBN-13 -978-93-85740-76-3
!थम स
ं
&करण: 2017 (पाठ
्
य&म का नाम: गिणत का िश%णशा' (भाग I), पाठ
्
य&म कोड- BED II- CPS 9)
सवा$िधकार स
ु
रि%त। इस प
ु
%तक के िकसी भी अ
ं
श को !ान के ि कसी भी मा+य म म- .योग करने से प
ू
व5 उ"राख&ड म
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* िव-ि व.ालय से िलिखत अन
ु
मित लेन ा आव7यक है। इकाई
लेखन से स
ं
ब
ं
िधत िकसी भी िववाद के िलए प
ू
ण56पेण लेखक िज8मेदार होगा। िकस ी भ ी िव व ाद का िनपटारा उ"राख&ड उ(च *य ायालय, नैनीताल म2 होगा।
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* िव-िव.ालय।
काय$ %म का नाम: बी० एड०, काय$%म कोड: BED- 17
पाठ
्
य$म का नाम: गिणत का िश%णशा' (भाग I), पाठ
्
य&म कोड- BED II- CPS 9
इकाई लेखक
ख"ड
स
ं
#या
इकाई
स
ं
#या
डॉ० िवनोद क
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सहायक &ोफे सर, िश#ा स
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1
1
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1
2
2
5
डॉ ० आ"ाशि& राय
सह #ोफे स र , िविश+ िश,ा स
ं
काय, शक
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नवा<स िव=िव>ालय, लखनऊ
1
3
2
4
डॉ० ए० के ० जैन
सह !ोफे सर एव
ं
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1
4
2
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ु
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1
5
2
3
डॉ० अ"बरीश क
ु
मार ितवार ी
सोनार टोली, सासाराम, रोहतास, िबहार
2
2
BED II- CPS 9
गिणत का िश!णशा% (भाग I)
Pedagogy of Mathematics (Part I)
ख"ड 1
इकाई स
ं
०
इकाई का नाम
प
ृ
# स
ं
०
1
गिणत िश&ण: इितहास, +क
ृ
ित एव
ं
&े2
2-17
2
गिणत और समाज
18-34
3
िव#ालयी !तर पर गिणत िश)ण का उ-े/य
35-52
4
गिणत िश&ण म( अिधगम उ,े.य
53-68
5
िव#ाथ& को समझ न ा: रचनावाद एव
ं
िवधेयवाद का प/र0े1य
69-84
ख"ड 2
इकाई स
ं
०
इकाई का नाम
प
ृ
# स
ं
०
1
िव#ालयी गिण त पाठ
्
यचया/
86-100
2
रा#$ीय शैि*क शोध एव
ं
1िश*ण प4रष द 7ारा िवकिसत गिणत पाठ
्
य=म क? 1व
ृ
ितयाँ
101-116
3
गिणत िश&ण क" िविभ&न िविधयाँ तथा कौशल
117-137
4
गिणत िवषय म& िश(णश ा+ ीय ब द ल ाव
138-155
5
गिणत िश&ा म) नवाचार और िश&क क0 भ
ू
िमका
156-173
CPS-9
1
1
Block 1
CPS-9
2
1- ,
Mathematics Teaching: History, Nature and Scope
1.1
1.2
1.3
1.3.1 500.
.
1.3.2
500.
.400.
1.3.3 400.1200.
1.3.4 1200.1800.
1.3.5 1800.
1.4
1.4.1
1.4.2
1.4.3
1.4.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.1
CPS-9
3
, , ,
,
-
,
,
,
-,
-
1.2
-
1.
2.
- , , ,
, ,
,
3. ,
4.
, ,
,
5.
6.
7.
,
8.
9.
-
10.
CPS-9
4
1.3
-
,
-
,
,
-
, ,
‘’
‘’
-
1. 500
a. 1000
b. 1000
500
-
i.
ii.
2.
500.
.400.
3. 400.1200.
4. 1200.1800.
5. 1800.
1.3.5 500
,
CPS-9
5
(1000.
. -
-
‘‘
’’
,
,
‘’
,
19
‘
’ 0, 1, 2, 3, 4, 5,
6, 7, 8, 9)
Indo-Arabic Numerals
(1000.
.500.
.
i.
-
,
-,
, ,
-
1000.
.)
,
-
‘‘
’’
,
,
-
CPS-9
6
1000
)
, ,
-
11)
,
ii.
-
-
500
) ,
‘
’ , ‘
’
350
)
300
)
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(1) 2)
, , ,
-
,
,
1550-1617)
-
,
1.3.2
500
400
,
,
,
135
)
176 -
CPS-9
7
1.3.3 400.1200.
,
, , ,
,
1.3.4 1200.1800.
‘’
1500.
‘
’
‘’ 1356.), 1587.),
1608.) 1731.)
1.3.51800.
1831.),
1884-1960.)
19,
Abstract
1777-1855 ) ,
Convergence
19,
-
,
, Hyperbolic
Elliptical,
,
,
1.4
--
CPS-9
8
1. Language-,
Mathematical concept,
Formulae, Principle
Signs
2.
Numbers, Place, Direction-
Measurement
3. , ,
Test
4. Senses
5. Real, Systematic, Logical
More clear,
6.
(Abstract concepts
Concrete concepts
,
7.
Goods
Association
8.
9. , ,
Doubt
10. Inductive, Deductive
Generalisation
11. - Self-confidence -
Self-dependence
12.
,
13.
14.
15. (Application
-,
,
CPS-9
9
16. ,
,
17. Scientific View
,
,
1.4.1Definitions
,
’’
’’ ’’
,
,
,
1.4.2
’’ ’’ ,
, ,
,
’’ ,
’’
’369’ ,
’348’
,
’’9, 36?’’ ,
,
1.4.3
,
’’?’’
,
CPS-9
10
,
, ’’
,
,
,
-
1.
2.
3.
4.
1.4.4
,
.5
,
,
i.
4
ii.
,
iii.
,
iv.
4
’’ ,
’’
,
, ,
,
CPS-9
11
, 1)
3) , 4) ’
’’ ,
, ’’
’’
’’
,’’ , (1) ,
4
,
4) ,
,
,
1.6
,
, ,
’, ’
,
i. -
, ,
, ,
,
,
, ,
ii. -,
, ,
, ,
,
,
iii. -
, ,
,
,
CPS-9
12
, ’’
’’
,
,
,
,
1.7
,
,
,
,
,
’’
,
, ’
’’
’
,
’’
,
,
,
1.8
,
, ,
CPS-9
13
,
, -,
-
i.
-,
’’
’’
ii.
-
,
,
’
’
,
iii.
-
-
,
,
iv.
-, ,
CPS-9
14
v.
- -,
,
’’
,
,
’ ’
’’
? , ,
1.9
, -
,
, ,
,
,
,
,
,
, ,
CPS-9
15
,
,
, ,
,
,
,
, -
,
,
,
,
,
1.9.1
,
’ - ’ ’’
,
,
,
, , ,
,
,
,
CPS-9
16
,
, ,
1.10
,
,
, ,
-
, , ,
, ,
,
1.11
1. Bhatnagar, A.B. (2009). Teaching of Mathematics, R. Lal Book Depo,
Meeruth.
2. IGNOU (2006). Teaching of Mathematics, New Delhi.
3. Jain, S.L. (2007). Teaching of Mathematics, Rajasthan Hindi Granth Acadamy,
Jaipur.
4. Kulshreshtha, A.K. (2005). Teaching of Mathematics, R. Lal Book Depo,
Meeruth.
5. NCERT (2006). Focus Group Report on Teaching of Mathematics, New Delhi.
6. NCERT (2012). Pedagogy of Mathematics, New Delhi.
7. NCTE (2009). National Curriculum Framework for Teacher Education, New
Delhi.
8. www.en.wikipedia.org/wiki/history_of_mathematics
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CPS-9
17
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CPS-9
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31
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CPS-9
32
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CPS-9
33
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1. Fatima R. ‘Role of Mathematics in the Development of Society’. Accessble
- http://www.ncert.nic.in/pdf_files/Final-ArticleRole%20of%20Mathematics
%20in% 20the% 20Development %20ofSociety-NCER-.pdf
2. James, Anice (2010). “Teaching of mathematics”, Neel Kamal Publication
Pvt. Ltd. Hyderabad
3. Kulshreshtha, A.K. (2010). “Teaching of Mathematics”, R. Lall, Book
Depot, Meerut
4. Lal, D. (2015); Geometry and Geometrical Patterns in Nature, The Signage,
Jamshedpur, Vol. 3 No.-1, JanJun, 2015, Pp 107-116
5. Laure Gouba. (2008). ‘The importance of Mathematics in everyday life’
African Institute for Mathematical Sciences 6 Melrose Road, Muizenberg
7945, South Africa, Accesseble-
http://users.aims.ac.za/~laure/math_life.pdf
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CPS-9
35
-
Aims of Teaching Mathematics at School Level
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CPS-9
36
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CPS-9
39
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CPS-9
40
3.4
(Aims of Goals and specific objective)
-
(Aims or Goals)
,
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CPS-9
41
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CPS-9
42
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46
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49
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CPS-9
50
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CPS-9
51
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2005,
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4.
,
, , 2006
5. Chambers, P. (2010).Teaching Mathematics, Sage Publication, New Delhi.
6. Chapman, L.R. (1970). The Process of Learning Mathematics, New York:
Pregamon Press.
7. David, A.H., Maggie, M.K., & Louann, H.L. (2007). Teaching
Mathematics Meaningfully: Solutions for Reaching Struggling Learners,
Canada: Amazon Books.
8. Gupta, H. N., & Shankaran, V. (Ed.), (1984). Content-Cum-Methodology
of Teaching Mathematics. NCERT, New Delhi.
9. James, A. (2005). Teaching of Mathematics, New Delhi: Neelkamal
Publication.
CPS-9
52
10. Kumar, S. (2009). Teaching of Mathematics, New Delhi: Anmol
Publications.
11. Mangal, S.K. (1993). Teaching of Mathematics, New Delhi: Arya Book
Depot.
12. Siddhu, K.S. (1990). Teaching of Mathematics, New Delhi: Sterling
Publishers.
13. Suggested Readings
14. Keeley, P. K., & Cheryl, T. R. (2011). Mathematics Formative
Assessment, Canada: Sage Publications.
15. National Curriculum Framework for Teacher Education. (2009). NCTE,
New Delhi.
3.9
1.
2.
3.
4.
5. ,
CPS-9
53
-
Learning Objectives in Mathematics Teaching
4.1
4.2
4.3
4.3.1
4.3.2
4.4
4.5
4.6
4.6.1
4.6.2
4.6.3
4.7
4.8
4.9
,
,
,
-
--
CPS-9
54
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Mathematics Teaching)
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CPS-9
55
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CPS-9
56
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-
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defined learning objectives
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CPS-9
57
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-
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iii.
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CPS-9
58
B.B. Masia)
-
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-
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i. Knowing) ,
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CPS-9
59
ii.
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different content areas in mathematics)
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of Learners
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CPS-9
60
labeling)
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CPS-9
61
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CPS-9
62
whole)
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CPS-9
63
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CPS-9
64
conflict)
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CPS-9
65
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CPS-9
66
Psychomotor Objectives/Domain)
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-
-
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-
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may be defined as visible activity. Behavioral Objectives State how a person
(student) is to act, think or feel.
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-
CPS-9
67
i. -
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-
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-
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-
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Review)
-
,
,
,
4.8
1. Kapoor J.N (; Some Aspects of school Mathematics, Arya Book Depot
Nai wala, Karolbagh N. Delhi.
2. Kumar Sudhir (; Teaching of Mathematics Anmol Publications, Pvt
Ltd. N. Delhi-
3. Sidhu, Kulbir Singh (; The Teaching of Mathematics, sterling
Publishers (P) Ltd.Morigate, Delhi-
4. N.C.E.R.T. New Delhi (“National Curriculum Framework for School
Education.
CPS-9
68
5. N.C.E.R.T. New Delhi (“National Curriculum Framework for School
Education.
6. N.C.E.R.T. New Delhi ( Report of Position paper of Mathematics
focus group “NCF
9
1. ?
2.
3.
4.
5.
6.
7.
CPS-9
69
- :
Understanding Learners : Behaviourist vs Constructivist vs
Enactivist Perspective
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
.1
-
1915
CPS-9
70
.2
-
1.
2.
3.
4.
5.
6.
7.
.3 (Constructivism)
(Jean Piaget)
CPS-9
71
Cognitive)
‘ ’
Lev Vygotsky)
,
,
(Cognitive Constructivism)
(Jean Piajet) ,
(Jerome Bruner), (Gagne)
(Socio-Culturist Perspective)
Lev Vygotsky)
Tabula Rusa)
,
,
,
, ,
CPS-9
72
,
,
CPS-9
73
,
:
,
,
,
(
)
. (Cognitive Constructivism)
,
()
CPS-9
74
Assimilation) Accommodation)
Scheme)
Assimilation)
Adaptation)
Inquiry-based learning),
Active learning),
Experiential learning,
Knowledge Building)
(Stages of Cognitive development
proposed by Piaget)
/ (Stages)
(Main characteristics)
1. 2
(Form birth to 2
years)
(Sensory – motor stage)
(reflex
action) (Use of symbols)
,
(Object
permanence)
2. 2 7
(2 to 7 years)
(Pre-operational stage)
(Symbols and
images)
(ego centric)
5. 7 12
(From 7 to 12
(Concrete-operational
CPS-9
75
years )
stage)
(Categorization)
(Conservation)
4. 12
(above 12 years
)
(Formal operational stage)
.
(Object Permanence): -
(Object Permanence)
A-B
A-B
(Mental Representation):
- -
- (Analogical Problem
Solving)
-
CPS-9
76
Understanding law of conservation)
, -
,
.
Lev Vyogotsky)
-
Socio-Culturist Approach)
Lev Vygotsky)
–
Socio-Cultural Theory)
,
CPS-9
77
Egocentrism)
-
-
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,
,
, , ,
,
,
,
CPS-9
78
,,
,
‘
’ ‘
’
Zone of Proximal Development or ZPD)
Inter-subjectivity) (
Scaffolding)
Scaffolding)
,
,
Scaffolding)
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A B
,
,
A
B
AB
CPS-9
79
?
‘’
,
.
Scaffolding)
Scaffolding)
,
,
Scaffolding)
Scaffolding)
Scaffolding)
Zone of Proximal Development or ZPD)
CPS-9
80
. Enactivism)
(Enactivism)-
(Enactivism
describes living as systems that produce themselves endlesslyReid .
(Producer of Information) ,
CPS-9
81
Living systems are not simply observation objects or
interacting systems, but rather autonomous, self-contained, self- referencing and
self-constructing closed systems).Maturana & Varela )
Autopoietic)
(Autopoiesis is the property of a living system such as a
bacterial cell or a multicellular organism) that allows it to maintain and
renew itself by regulating its composition and conserving its boundaries)
.9
(Enactivism & Teaching and Learning of
Mathematics)
(Enactivism)
CPS-9
82
Enactivism= Constructivism + Embodied Cognition
5.10
Jean Piaget)
Cognitive Constructivism)
Jean Piaget) ,
Jerome Bruner), Gagne)
Socio-Culturist Perspective)
Lev
Vygotsky)
,
, - ,
(Producer of Information)
CPS-9
83
5.11
1. Mangal S.K. (2009) Advanced Educational Psychology, PHI, India
2. Mangal S.K. (2014) Educational Technology, PHI, India
3. NCERT (2005) Constructivist Approach of Teaching,
4. Singh A.K. (2007) General Psychology
5. https://froese.files.wordpress.com/2014/04/froese-14-radicalizing-
enactivism-basic-minds-without-content.pdf
6. http://www.embodiment.org.uk/topics/enactivism.htm
7. http://www.acadiau.ca/~dreid/enactivism/EnactivismMathEd.html
8. http://www.eucognition.org/index.php?page=embodied-dynamicism-and-
enactivism
9. http://etec.ctlt.ubc.ca/510wiki/Enactivist_Theory
5.12
1.
2.
3.
4. -
5.
6.
7.
CPS-9
84
2
Block 2
CPS-9
85
1-
School Mathematics Curriculum
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
.
,
,
-
-
CPS-9
86
1.2
1.
2.
3.
4.
5.
6.
1. 3 (Concept of Mathematics
Curriculum)
(Curriculum)''''
(Cunningham)
''
( )
((
(
''
CPS-9
87
1.4
0
(Samuel) ''
(learning
Experience
(Aspect
I.
i. (i, (Economical
(ii (iii
(iv(v
,,
ii.
,
,
Thorndike
“Teach nothing because of its
disciplinary value,but everything so as to get what disciplinary value it does
have”)
iii.
(
CPS-9
88
iv.
II.
-
-
1.
2.
3.
4.
5.
CPS-9
89
6.
7.
15 (Principles of Desigining
Curriculum n Mathematics)
,
-
i.
(Mathematics Curriculum
Should be Child Centered) -
CPS-9
90
ii.
(Curriculum Should be action based)
-
-
iii. -
(Subject Matter
should be arranged according to Psychological way) -
- ?
iv.
(Consideration of personal
thoughts-
(
v. (Consideration of difficulty level-
CPS-9
91
45910
vi.
(Consideration of Teacher’s
Consent in Curriculum Construction-
vii.
(Flexible Objectives according
to Need-
,,
viii. (Expression of Correlation among Subject-
ix. (Maximum use of Mathematics in
Science-
x.
(Flexibility of Curriculum-
CPS-9
92
xi.
(Interest and Aptitude -
xii.
(Absorption of
riddles and other Entertainment Activities in Mathematics Curriculum
-
,
100 100 -
1100
?
xiii.
(Principle of Knowledge for
the Conservation of Culture and Civilization)-
,
,-‘’
‘’
xiv.
(Mathematical
Curriculum should be life related -
CPS-9
93
,
i. (
15678
ii.
iii.
-
16 (Vision of School
Mathematics)
400
(Plato)
(Napoleon)
, “
(Bacon
''
''
CPS-9
94
(Bernard Shaw
(Herbert), (Froebel))
(Pestalozzi), (Maria Montessori), (T.P.
Nun)
“
,
“
1(Skilled
Subject) (1986)
17
(Curriculum Choices at Different Stages of School
Mathematics Education
(Caribbean
CPS-9
95
(CBSE
18 Care Dreads of
Concerns in School Mathematics
i.
ii.
iii.
(attitude
//
CPS-9
96
i. -
(express
ii.
- (accurate, reasoning,
thinking and judgment
iii. -
(Personal
error
(difference of opinion
iv. -
(Intelligent Reasoning
CPS-9
97
v. -
-
vi.
(economical living
(technology
(“playback–planner”)
CPS-9
98
19
(Pedagogical Analysis of Various Topics in
Mathematics of Various Levels of School
-
(operations
(notation and numeration
-
: (revitalize)
i.
ii.
iii.
a.
(identities (solutions of triangles
(height and distance
b.
“”
“”
i.
CPS-9
99
ii.
(topical(arrangement
110
i.
ii.
iii.
(
CPS-9
100
Child Centered)
(Community Centered)
(Creativity Based)
(Activity Based) (Integrated)
(Re-
source based)
4CAIR
111
1. Kapoor J.N (1967); Some Aspects of school Mathematics, Arya Book Depot
Naiwala, Karolbagh N. Delhi.
2. Kumar Sudhir (1997); Teaching of Mathematics Anmol Publications, Pvt
Ltd. N. Delhi-2.
3. Sidhu, Kulbir Singh (1967); The Teaching of Mathematics, sterling
Publishers (P) Ltd.Morigate, Delhi-6.
4. N.C.E.R.T. New Delhi (2000) “National Curriculum Framework for School
Education.
5. N.C.E.R.T. New Delhi (2005) “National Curriculum Framework for School
Education.
6. N.C.E.R.T. New Delhi (2005) Report of Position paper of Mathematics
focus group “NCF 2005
112
1.
2.
3.
4.
5.
6.
CPS-9
101
-
Trends of NCERT Syllabi of Mathematics
2.1
2.2
2.3
-
2.3.1
-
2.3.2
-
2.4
2.5
2.6
2.7
2.8
2.9
- -
-
, , -
, -
-
CPS-9
102
-
,
-
1.
-
;
2.
;
3.
-
--
-
, -,
-
Foley & McPhee, 2008)
CPS-9
103
,
,
--
-
-
-
1.
2.
3.
4.
2.3.2
-
-
-
, ,
CPS-9
104
-
--
-
,
1.
-
2.
-
?
,
,
,
,
,
CPS-9
105
,
,
, -
,
3.
CPS-9
106
90
1
-
-
-
-
-
-
-
-
9
i.
ii.
-
iii.
iv.
‘’
CPS-9
107
v.
, ,
,
,
vi.
vii.
viii.
ix.
x.
xi.
xii.
xiii.
,
xiv.
,
-
-
-
-
-
-
-
CPS-9
108
-
-
-
-
-
-
-
-
-
-
10
CPS-9
109
i.
ii.
-
iii.
iv.
‘’
v.
vi.
vii.
viii.
ix.
x.
xi.
xii.
xiii.
xiv.
,
xv.
,
CPS-9
110
-
-
-
-
-
-
-
-
-
-
-
-
-
-
CPS-9
111
-
-
-
-
11
i.
ii.
-
iii.
iv.
‘
’
v.
vi.
vii.
viii.
ix.
x.
xi.
xii.
CPS-9
112
-
-
-
-
-
-
-
-
-
-
-
-
12
i.
ii.
-
CPS-9
113
iii.
iv.
‘
’
v.
vi. ‘
’
vii.
viii.
ix.
x.
xi.
xii.
xiii.
,
-
,
4.
?
CPS-9
114
5.
?
6.
?
7.
?
8.
9.
10.
?
11.
12.
-
,
,
,
,
CPS-9
115
7
1.
2.
3.
4.
5.
6.
7.
8. , 6
9.
10.
11.
12.
8
1. Carl, A. (2009). Teacher empowerment through curriculum development
theory into practice. Juta&Company Ltd.
2. DeRose, J. (2009). Back to the future with textbooks: Using textbook
passages from the past to help teach historiography. The History Teacher,
42(2). Long Beach, CA: The Society for History Education.
3. Falknor, J. (2010). Instruction of reading comprehension strategies in
literature textbooks. Retrieved from ProQuest's website:
http://udini.proquest.com/view/instruction-of-reading-goid:840627294/
4. Foley, B., & McPhee, C. (2008). Students' attitudes towards science in
classes using hands-on or textbook based curriculum. Washington, DC:
American Educational Research Association.
5. Johnson, J. A. (2001, August 28). Curriculum revision that works. In
principles of effective change. Retrieved March 10, 2014 .
6. Stambaugh, J., & Trank, C. (2010). Not so simple: Integrating new research
into textbooks. Academy of Management Learning and Education, 9(4) 663–
681. Briarcliff Manor, NY: Academy of Management.
CPS-9
116
7. Teten, R. (2010). When in Rome, do as Jon Stewart does: Using America:
The book as a textbook for introductory-level classes in American politics.
Journal of Political Science Education, 6(2) 163–187. London: Routledge.
9
1. ‘
- ’
2.
?
3.
-
?
4.
CPS-9
117
3 -
Methods and Strategies of Teaching Mathematics
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
.1
-
CPS-9
118
.2
-
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
.
(Lecture Method)
(Inductive-Deductive Method)
(Analytic-Synthetic Method)
(Problem Solving Method)
(Project Method)
CPS-9
119
i. Lecture Method)-
(Chalk and talk)
(Abstract)
review)
,
(Cost Effective)
CPS-9
120
-
/
,
,
,
,
, ,
(Specially AbledChildren)
(Unipolar)
ii.
Inductive-Deduvtive Method) -
,
concrete)
abstract), general)
,
concrete facts)
CPS-9
121
,
,
-
(,
, )
, ,
,
180
''
180 ''
1. (a+b) x (a –b)
a
2
– b
2
,
(b+c) (b- c)
b
2
–c
(x+y) (x-y)=x
2
y
2
,
,
,
,
CPS-9
122
,
,
(Deductive Method)-
(General) Specific) ,
(Abstract)
(Concrete)
-
CPS-9
123
(abstract formula)
,
(concrete)
(crammingbased)
,
-
-
1
2
3
4
5
CPS-9
124
iii. -
Analytic–Synthetic Method)
(Analytic Method)-
: tan
2
sec
2
θ
-
tan θsec θ ,
1
secθ
cosθ
sinθ
tanθ
cosθ
2 2 2 2 2 2
22
sin sin θ sin sin θ cos cos θ
1
sin sin θ cos 2θ
sin
2
θ+cos
2
θ
2
2
1
sec θ
cos θ
22
1 tan sec
tan
2
sec
2
θ
-
(Merits of Analytic Method)
1.
2.
3.
4.
5.
6.
CPS-9
125
(Demerits of Analystic Method)
1. ,
2.
3.
4.
(Synthetic Method)
,
“-
”
,
(Merits of Synthetic Method)
(Limitations of Synthetic Method)
,
CPS-9
126
iv. (Project Method)-
(Pragmatic)
(Kilpatrick, 1921)
(Stevenson, 1922)
,
“
” (Learning by Doing)
(Basic Principles of Project Method)
CPS-9
127
(Steps involved in Project Method)
1.
2.
3.
4.
5.
6.
(Merits of Project Method)
,
,
, , ,
CPS-9
128
- (Demerits and Limitations of Project
Strategy)
,
v. -Problem-Solving Method) -
Psychological and Scientific)
'' '' (Learning by doing)
,
(Steps of Problem Solving Method)
1. (Identification of problem):
2. (Defining the
problem)
CPS-9
129
3.
(Data Collection):
4.
(Analysis of Data)
(Possible solutions)
5.
(Accurate Solution)
6. Verification of Solution):
-
-
(Data Collection):
(Analysis of Data)
–
............. (i)
(90
0
) .................. (ii)
CPS-9
130
-
-
-
-
-
5.
(Accurate Solution
6. Verification):
Merits of Problem Solving Method)
(information processing)
(Limitations of Problem Solving Method)
vi. Drill work in Mathematics Teaching) -
Drill work in Mathematics Teaching) -
CPS-9
131
(Importance and Uses) -
,
,
,
(Limitations of Drill and Practice in
Mathematics)
CPS-9
132
,
3.4
Oral and
Written Work in Maathematics)
(Oral Work in Mathematics) :
CPS-9
133
(Written Workin Mathematics) :
,
3.5
Brain Stormingin Teaching of
Mathematics)
(Brain Storming)
(Meaning) –
(Alex Osburn)
(Creative Problem Solving
Method) (Applied Imagination)
1953
CPS-9
134
(Higher cognitive
abilities)
(Merits of Brain Storming)
(Limitations of Brain Storming)
CPS-9
135
,
3.6
Computer Based Learning)
(Mathetics)
(Mathetics)
i.
(Principle ofsmallsteps)
ii.
(Principle of active response)
iii.
(Principleofimmediate confirmation)
iv. -
(Principleof selfpacing)
v.
(Principleofstudenttesting)
-
CPS-9
136
(Merits of Computer Based Learning)
-
Limitations of Computer Based Learning):
(Techno-Friendly)
3.7
(Lecture
Method),
(Inductive-Deductive Method),
(Analytic-Synthetic Method), (Problem Solving Method)
(Project Method)
(Higher
cognitive abilities)
CPS-9
137
3.8
1. Chambers P. (2008) Teaching of Mathematics, Sage South Asia
2. Kulshreshtha S. (2013) Teachingof Mathematics
3. Mangal SK, (2016) Teaching of Mathematics, Tandon Publishers,
Ludhiyana
4. Mangal SK.2013) Education Technology, Prentice Hall of India
3.9
1.
2.
3.
4.
5.
6.
7.
8.
CPS-9
138
4-
Pedagogical Shift in Mathematics)
4.1
4.2
4.3 ,
4.4
,
4.5 Es (
,, ,
,
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
,
,
Es (
,, ,
, , ;
;
CPS-9
139
-
1. ,
2.
3. Es (
, ,
;
4.
5.
6.
7.
8.
,
,
-
CPS-9
140
- -
-
- -
,
-,
,
CPS-9
141
(1915 – 2016)
,
,
i.
,
ii.
,
iii.
,
iv.
,
-
1. _________
2. _________
3. _________
4. _________
5. _________
6.
4.4 (Difference
between teaching of mathematics and teaching of science)
,
CPS-9
142
,
, , ,
,
,
,
180
-
,
,
,
,
, ,
,
,
-
,
,
,
,
-
reason Empirical
logical Positivism
logical
CPS-9
143
7. _________
8. _________-
9. _________
Es ( ,, ,
,
E
-
E,
E
-
Engage, Explore, Explain, Elaborate, Evaluate E
,
5 Es
i.
Engage
,
ii. Explore
,
iii. Explain-
, ,
, ,
iv. Elaborate
CPS-9
144
v.
Evaluate
-
,
-
10. Es _________
11. _________
12. _________
4.6 (Collaborative Learning Approach)
i.
,
,
ii.
iii.
,
,
iv.
–
CPS-9
145
,
,
-
-
i. Positive Interdependence
ii. (Promotive Interaction)
-
iii.
Individual and group accountability
iv.
,
-
v.
Group processing:
13. _________
CPS-9
146
14. _________
4.7(Problem Solving Approach)
,
CPS-9
147
--
i. -
ii. -?
iii.
-
iv. -
,
v. -
,
CPS-9
148
15.
4.8 (Concept Mapping)
,
,
,
,
,
,
(2006)
CPS-9
149
IGNOU (2016) Learning and Teaching (BES 123), New Delhi
i.
ii.
iii.
CPS-9
150
16. ____________
17.
_________
(Experiential learning)
,
,
,
, ,
Concrete experience )
Reflective
observation)
Abstract
conceptualization)
Active experimentation)
CPS-9
151
,
18.
___________
19.
4.8
,
,
,
,
,
-
-
CPS-9
152
,
,
-
,-
-
Es
-
,
E,
E
-
Engage, Explore, Explain, Elaborate, Evaluate E
,
–
,
--, ,
,
,
4 .9
1.
CPS-9
153
2.
3.
4.
5.
6.
i.
,
ii.
,
iii.
,
iv.
,
7.
8.
9.
10.
11.
12.
13.
14.
15. -
i. -
ii. -
?
iii.
-
iv. -
,
v. -
16.
17.
18.
19.
Concrete experience)
CPS-9
154
Reflective observation)
Abstract conceptualization)
Active experimentation)
4 .10
1. Bybee, R. W., Taylor, J. A., Gardner, A., Van Scotter, P., Powell, J. C.,
Westbrook, A., and N. Landes. 2006. The BSCS 5E instructional model:
origins, effectiveness, and applications. Colorado Springs: BSCS.
http://www.bscs.org/curriculumdevelopment/features/bscs5es.html
2. Birbili, M (Mapping Knowledge: Concept Maps in Early Childhood
Education," in Early Childhood Research & Practice ,
3. Encyclopedia of psychology.(4 vol.). (1994). 2nd ed. New York: Wiley.
The most comprehensive encyclopedia of general psychology available.
4. Graham, G. (2005). Behaviorism. In E. N. Zalta (Ed.), The Stanford
encyclopedia of philosophy (Fall 2007 ed.). Retrieved from
http://plato.standford.edu/entries/behaviorism/ VandenBos, G. R.
(Ed.).(2007). APA dictionary of psychology. Washington, DC: American
Psychological Association.
5. IGNOU (2016) Learning and Teaching (BES 123), New Delhi
6. National Council of Educational Research and Training ( National
Curriculum Framework (NCF). New Delhi: NCERT.
7. National Council of Educational Research and Training ( National
Curriculum Framework for Teacher Education (NCFTE). New Delhi:
NCERT.
8. Siltala, R. (Innovativity and cooperative learning in business life and
teaching.University of Turku.
4 .11
1. ,
2.
CPS-9
155
3. 5 Es
4.
5.
6.
7.
8.
CPS-9
156
-
Recent Trends in Mathematics Education and the Role of
Teacher
5.1
5.2
5.3
5.4
5.5 :
5.6
5.7
5.1
,
,
-,
-
-
, |
:
-
-
CPS-9
157
,
,
-
-
,
,
,
,
,
- ,
-
,
-,
(...) -
-
-
,
-
5.2.
1. ...
2.
3.
4.
5.
-
6. ..
7. :
CPS-9
158
8. :
, , ,
9. ,,,
10. /
5.3. ...
, ...
,
-
, , , , , ,
, ,
,
, -
,
-
-
,
-
-,
-
- -
-
CPS-9
159
-
-
-
- -
-
,
,
,
, ,
...
, ,
,
, ,
,
, ,
CPS-9
160
...
........:- http://www.mathisfun.com/; http://www.educationworld.com/;
http://www.nctm.org/
...
...
...,
-
...
,
-
-
-
-
CPS-9
161
-
,
,
,
-
,
, , ,
,
5.4.
,
-
-
i. -
/
CPS-9
162
ii. -
,
iii.
-
, ,
-
-
CPS-9
163
-
-
,
,
,
iv. -
,
,
,
, ,
,
, -
, ,
v. -
,
,
,
CPS-9
164
vi. ?
,
/
, ,
,
-
, ,
,
, ,
,
,
,
:
: ,
:
:-,
,
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:
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-
CPS-9
165
,
-
,
:
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:
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:
CPS-9
166
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-
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3.
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4.
5.
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, :
CPS-9
167
7.
,
,
,
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9.
,
,
10.
-
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, ,
12.
13. :
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CPS-9
168
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CPS-9
169
- ; ;
; ;
,
; ;
, ,
,
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,
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-
CPS-9
170
,
,
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,
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,
i. - ,
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,
CPS-9
171
iii. ... - ,
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iv.
-
,
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vi.
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172
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173
2.
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5.
-
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7. :
8. :
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9. , , ,
10. /
Sources
1. http://www.techlearning.com/blogentry/8716
2. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1705977/
3. http://www.techlearning.com/blogentry/8716
4. http://www.education.gov.gy/web/index.php/teachers/tips-for-
teaching/item/1570-importance-of-communicating-in-the-classroom
