ªî£ì‚è‚ è™m g¬ôj™ èeî‹...

bjhl¡f¡ fšé MÁça® g£la¥ go¥ò

(D.El.Ed)

ªî£ì‚è‚ è™M G¬ôJ™èEî‹ èŸHˆî™

ð£ìŠHK¾ - 504

ªî£°F - 1

ªî£ì‚è‚ è™M G¬ôJ™èEî‹ èŸøL¡ º‚Aòˆ¶õ‹

«îCòˆ Fø‰îªõO ðœO Þò‚èè‹A-24/25, GÁõùŠ ð°F, ªî£°F -62 ªï£Œì£ªè÷î‹ ¹ˆî˜ ïè˜, àˆFóŠHó«îê‹ -201309

õ¬ôî÷‹: www.nios.ac.in

î¬ôõ˜ à¬ó....Ü¼¬ñ ñ£íõ˜è«÷ !

«îCòˆ Fø‰îªõO ðœO Þò‚èè‹ (NIOS) Þ‰Fò ÜóC™, ñQî õ÷«ñ‹ð£†´ Ü¬ñ„êèˆF¡ (MHRD) W› Þòƒ°‹ å¼ î¡ù£†C Ü¬ñŠð£°‹.Þç¶ àôè÷M™ IèŠ ªðKò Fø‰îG¬ô ðœOèO¡ Ü¬ñŠð£°‹. ÞF™ ²ñ£˜2.02 I™Lò¡ ñ£íõ˜èœ Þ¬ìG¬ô ñŸÁ‹ àò˜G¬ô‚è™M ðJ½A¡øù˜.NIOS «îCò ñŸÁ‹ ê˜õ«îê Ü÷M™ å¼ƒè¬ñŠH¬ù‚ ªè£‡ì¶ - ÞF™ 15õ†ì£ó ¬ñòƒèœ, 2 ¶¬í ¬ñòƒèœ, 5000 è™M ¬ñòƒèœ àœï£†®½‹ªõOï£†®½‹ è™Mê£˜ ñŸÁ‹ ªî£N™ê£˜ è™MJ¬ù õöƒ°A¡ø¶. Þ¶èŸðõ˜ ¬ñò îóñ£ù è™M ñŸÁ‹ Fø¡ «ñ‹ð£†´ ðJŸCJ¬ùˆ Fø‰îñŸÁ‹ ªî£¬ôÉó‚ è™MJ¡ õNò£è õöƒ°A¡ø¶. Þ‚è™Mò£ù¶Ü„CìŠð†ì ð£ìŠªð£¼œ, «ïK¬ìò£ù õ°Š¹èœ (îQŠð†ì ªî£ì˜õ°Š¹èœ), îèõ™ ªî£ì˜¹ ªî£N™¸†ð àîM»ì¡ (åL,åO ï£ì£ õ†´‚èœ,õ£ªù£L ñŸÁ‹ ªî£¬ô‚è£†C åLðóŠ¹èœ) ïìˆîŠð´A¡ø¶. NIOSªî£ì‚èG¬ôJ™ º¬øò£ù ðJŸCªðø£î ÝCKò˜è¬÷Š ðJŸÁMŠðîŸè£ùÜFè£óˆ¬îŠ ªðŸÁœ÷¶. D.El.Ed. ªî£ì‚è‚ è™M ð†ìòŠ ðJŸC‚è£ùè†ìèˆ¬î NIOS Þˆ¶¬øJ™ ðEò£ŸÁ‹ Hø GÁõùƒèO¡ åˆ¶¬öŠ¹ì¡à¼õ£‚A»œ÷¶. Þ‰GÁõù‹ ¹¶¬ñò£ù ñŸÁ‹ êõ£™èœ G¬ø‰î Þó‡´õ¼ìˆ ªî£ì‚èG¬ô ÝCKò˜ ð†ìòŠ ðJŸC¬ò RTE 2009 õN‚è£†´îL¡ð®ðEò£ŸÁ‹ º¬øò£ù ðJŸCªðø£î ÝCKò˜èÀ‚è£è à¼õ£‚A»œ÷¶.

àƒè¬÷, Þ‰î «îCò Fø‰îªõO ðœO Þò‚èè‹ õöƒ°‹ ªî£ì‚èG¬ôÝCKò˜èÀ‚è£ù ð†ìòŠðJŸC‚° õó«õŸðF™ ñA›„C Ü¬ìA«ø¡. àƒèœñ£GôˆF¡ ðƒèOŠHŸè£è õ£›ˆ¶A«ø¡. ÝCKò˜ RTE ê†ì‹ 2009õNè£†´îL¡ð® ªî£N™ê£˜‰î ðJŸCªðÁî™ è†ì£òñ£°‹. àƒèO¡ÜÂðõ‹ Fø¡õ£Œ‰î ÝCKò¼‚°ˆ «î¬õò£ù Ü®Šð¬ìˆ Fø¡è¬÷õöƒAJ¼‚°‹ â¡ð¬î ¹K‰¶œ«÷£‹. Ýù£½‹ ê†ìˆF¡ Ü®Šð¬ìJ™ÞŠðJŸC¬ò G¬ø¾ ªêŒî™ è†ì£òñ£Aø¶. cƒèœ ãŸèù«õ ªðŸø ÜP¾‹ÜÂðõº‹ ÞŠðJŸCJ™ G„êòñ£è àƒèÀ‚° àî¾‹.

Fø‰îG¬ôJ™ ªðÁ‹ ªî£ì‚è‚ è™M ð†ìòŠ ðJŸC àƒèO¡ Ü¡ø£ìÝCKò˜ ðE¬ò âšMîˆF½‹ ð£F‚è£¶, ªî£N™ê£˜‰î Fø¬ù õ÷˜ˆ¶‚ªè£œõîŸè£ù õ£ŒŠð£è«õ Ü¬ñ»‹. Þƒ° à¼õ£‚èŠð†´œ÷ î£«ù èŸø™ð£ìŠªð£¼O™ ¹Kî¬ô ãŸð´ˆF, cƒèœ ðEò£ŸÁ‹ ÝCKò˜ «õ¬ô‚°î°FŠð´ˆ¶õ¶ì¡, Cø‰î ÝCKòó£è à¼õ£‚è¾‹ õNõ¬è ªêŒAø¶.

àƒèO¡ ÞŠªðKò ºòŸC‚° âù¶ õ£›ˆ¶‚èœ.

î¬ôõ˜ (NIOS)

njhFjp

myF

ghlj;jiyg;G

fw;gpj;jy; Neuk;

nray;Kiw

fs;

ghlg; nghUs;

nray;ghL

njhlf;f epiyg;gs;spfspy; fzpjk; fw;wypd; Kf;fpaj;Jtk;.

1 Foe;ijfs; fzpjk; fw;gJ vg;gb?

3

2

fzpjk; gw;wpa gaj;ij Nghf;f fUj;juq;F elj;Jjy;.

2 fzpjKk; fzpjf; fy;tpAk;.

4

2

3 fzpjf; fy;tpapd; Fwpf;Nfhs;fSk; njhiyNehf;Fk;;.

4

2

tFg;giw #oypy; fzpjk; fw;wypd; NghJ Vw;gLk; FiwghLfis milahsk; fz;L jPu;f;f toptif fhzy;.

4 njhlf;ff;fy;tpapy; fw;wy; kw;Wk; fw;g;gpj;jy; ikaKiwfs

5

3

fzpj kd;wk; elj;Jjy;

TOTAL

16

9

Grand Total

16

9

25 hrs

ªð£¼÷ì‚è‹

õ.â‡ Üô° ð‚è â‡.

1. Foe;ijfs; fzpjk; fw;gJ vg;gb? 001

2. fzpjKk; fzpjf; fy;tpAk 041

3. fzpjf; fy;tpapd; Fwpf;Nfhs;fSk njhiyNehf;Fk;; 067

4. njhlf;ff;fy;tpapy; fw;wy; kw;Wk;

fw;g;gpj;jy; ikaKiwfs 092

1

ªî£ì‚è è™M G¬ôJ™ èEî‹ èŸHˆî™

MáÇa® fšÉ g£la¥ go¥ò (D.El.Ed)

1.0. ÜPºè‹

1.1 èŸ°‹ «ï£‚èƒèœ

1.2 °ö‰¬îèO¡ C‰î¬ùõNG¬ôèœ

1.2.1. ÜPFø¡ õ÷˜„CJ¡ ð®G¬ôèœ

1.2.2. èEî‚è¼ˆ¶‚èO¡ õ÷˜„C G¬ôèœ

1.3. °ö‰¬îŠð¼õˆF™ èEî‹ èŸø™

1.3.1. èEî‹ èŸø™ º¬øèœ

1.3.2. èEîˆ¬îŠ ðŸPò Ü„ê‹

1.3.3. èEî‹ èŸø¬ô ÞQ¬ñò£ù Åöô£è à¼õ£‚°î™

1.4. ªî£°ˆ¶ Ãø™

1.5. î¡ùP¾„ «ê£î¬ù ñ£FK Mù£ˆî£œ

1.6. «ñŸ«è£œ Ë™èœ

1.7. Üô°ˆ «î˜¾

1.1 ÜPºè‹ : ( Introduction)

Ü¬ùˆ¶Š ðœOŠð£ìƒèO½‹ èŸøL™ èEîˆFŸ°Iè¾‹

º‚Aòˆ¶õ‹ ÜO‚èŠð´Aø¶. àƒèœ ñ£íõ ð¼õˆF½‹ ñŸÁ‹ å¼

ÝCKòó£è Þ¼‚°‹«ð£¶‹ ñŸøð£ìƒè¬÷ åŠH´‹ «ð£¶

ÜFèŠð®ò£ù Ü¿ˆî‹ èEî ªêò™ð£´èœ ªêŒõîŸ°

«î¬õŠð´Aø¶. ªðŸ«ø£¼‹ ñŸø ð£ìƒè¬÷ Üõ˜è÷¶ è™Mˆ

î°F¬òŠ ªð£¼†ð´ˆî£¶ Üõ˜è÷¶ °ö‰¬îè¬÷ Iè‚è®ùñ£è

ñŸø ð£ìƒè¬÷‚ è£†®½‹ èEîŠ ð£ìˆFŸ° ÜFèñ£ù «ïó‹

å¶‚°Aø£˜èœ. ªð¼‹ð£½‹ â™ô£ G¬ôèO½‹ èEî‹ èŸøL™

Üô° - 1

°ö‰¬îèœ èEî‹ èŸð¶ âŠð®?(How Children Learn Mathematics)

2


ªî£°F 1 : ªî£ì‚è è™M G¬ôJ™ èEî‹ èŸøL¡ º‚Aòˆ¶õ‹

èŸ«ð£˜ G¬ô ÜKî£è«õ «ï£‚èŠð´Aø¶. å¼ ªð£¶õ£ù àí˜¾,

°ö‰¬î â¡ð¶ å¼ CPò Ü÷¾ ºF˜‰î Üõ˜èœ èEîˆF™ â‡

ê£˜‰î Fø¬ñè¬÷ º¬ùŠ«ð£´ ªðÁî™ â¡ð¶ å¡Á ¹Kî™Fø¬ù

õ÷˜ˆî™ Ü™ô¶ ñùŠð£ì‹ ªêŒî™ Ý°‹. Þšõ£ø£ù ï‹H‚¬è«ò

ªðŸ«ø£˜ ,ÝCKò˜èOì‹ GôM õ¼A¡ø¶. Þî¡ M¬÷¾

â¡ùªõ¡ø£™ ªð¼‹ð£ô£ù °ö‰¬îèœ Ü®Šð¬ìè¬÷ˆ

ªîK‰¶ªè£œ÷£ñ™ ñùŠð£ì‹ ªêŒõî£™ èEî‹ ðŸPò Ü„ê‹

ðœO º¡ð¼õˆF™ «î£¡P Ü¬õ «ñ½‹ õ÷˜‰¶ õ£›ï£œ

º¿õ¶‹ c®‚Aø¶. èEîˆF¡ è¼ˆ¶‚èœ ñŸÁ‹ ªêò™ð£´è¬÷

ðô G¬ôèO™ cƒèÀ‹ ÜP‰¶ àœk˜èœ. CøŠð£è àƒèœ ºî¡¬ñ

G¬ôèO™ âOòõŸP™ Þ¼‰¶ C‚èô£ùõŸ¬ø èŸÁ‚ ªè£œÀ‹ «ð£¶

“cƒèœ âŠ«ð£î£õ¶ G¬ùˆî¶ à‡ì£? Üšõ£Á ãŸð£´ ªêŒî£™

èŸ«ð£˜ õ÷˜„C º¡«ùŸø‹ ñŸÁ‹ C‰î¬ùFø¡ «ñ‹ð´õF™

ªî£ì˜¹¬ìòî£?”

âù Ýó£Œ„Cèœ G¼H‚èŠð†´œ÷ù. C‰î¬ù õ÷˜„C ñŸÁ‹

º¡«ùŸøˆF™ èEîˆF¡ ªêò™ð£´èÀ‚° ªï¼ƒAò ªî£ì˜¹

àœ÷¶. å¼ ÝCKòó£è àƒèO¡ åšªõ£¼ ñ£íõK¡ ðô‹ ñŸÁ‹

ðôiùˆ¬îŠ ¹K‰¶ªè£‡´ êKò£ù èŸø™ º¬øè¬÷ «î˜‰ªî´ˆ¶

èŸHˆî™ «õ‡´‹.

âŠªð£¿¶‹ èŸ«ð£K¡ «î¬õ ñŸÁ‹ Ý˜õˆ¬î êKò£è

¹K‰¶ªè£‡´ ÜõŸ¬ø âOî£‚°î™ Ü™ô£ñ™ Üõ˜è¬÷ èEî‹

èŸÁ‚ªè£œ÷ è†ì£òŠð´ˆ¶‹ «ð£¶ ªð¼‹ð£ô£ù ñ£íõ˜èOì‹

èEî‹ èŸø™ ²¬ñò£Aø¶. å¼ ÝCKòó£è °ö‰¬îèÀ‚° èEî‹

èŸø™, èŸHˆîL™ ªîOõ£ù è‡«í£†ìˆ¶ì¡ âO¬ñò£ù

õNº¬øèO™ ñA›„C»ì¡ èŸÁ‚ ªè£œ÷ ï£‹ Üõ˜èO¡

ªî£ì‚èG¬ôJ™ ðJŸC ªè£´‚è«õ‡´‹.

Þ‰î Üô° Üî£õ¶ èEî‚ èŸø™ ðŸPò ð£ìˆF†ìˆF¡ ºî™

Üô° °ö‰¬îèO¡ ¹ôµí˜¾ õ÷˜„CJ¡ ð® èEî‚ è¼ˆ¶è¬÷

ð®Šðî¡ Íô‹ èEî‹ èŸÁ‚ ªè£œÀ‹ õNè¬÷Š ðŸP Mõ£F‚è

ºòŸCˆ¶œ«÷£‹. èEîŠ ðJŸCJ¡ õ÷˜„C‚° õNõ°‚°‹

3



¸†ðƒè¬÷»‹ èŸÁ‚ªè£œõîŸè£è õNè¬÷»‹ «ñ½‹ ÜõŸP™

°ö‰¬îèO¡ ªð£¶õ£ù Hó„C¬ù¬ò è‡ìP»‹ õNè¬÷Š ðŸP

è£‡«ð£‹.

1 . 1 . èŸø™ «ï£‚èƒèœ : (Learning Objectives)

Þ‰îÜôA¬ù ð®Šðî¡ Íô‹ èŸø™ «ï£‚èƒè¬÷

ÜP‰¶ªè£œ÷º®»‹.

º¡Hœ¬÷/Ýó‹ðè£ô °ö‰¬î ð¼õˆF™ èEîˆ¬î ÜP‰¶

ªè£œ÷º®»‹. Ýó‹ðè£ô °ö‰¬îŠð¼õˆF™ èEî‹ èŸø™ ñŸÁ‹

«ñ‹ð£´ «ð£‚°è¬÷ Ü¬ìò£÷‹ è£í¾‹. ð™«õÁ G¬ôèO™

èEî‹ èŸÁ‚ ªè£œõF™ âO¬ñò£ù õNè¬÷ Ýó£Œî™. èŸøL™

Ýó‹ð‚ è†ìˆF™ °ö‰¬îèœ âF˜ªè£œÀ‹ Cóñƒè¬÷‚ è‡ìP‰¶

¶¬í‚ è¼Mè¬÷ ¬èò£‡´ èŸø¬ô ñA›„Cò£ùî£è ñ£ŸP

Ü¬ñ‚è ÜP‰¶ ªè£œÀî™.

1.2 å¼°ö‰¬îJ¡ C‰î¬ù õN G¬ôèœ:(A Ways a Child Thinks )

ÜFè â‡E‚¬èJô£ù °ö‰¬îè¬÷ cƒèœ åšªõ£¼ï£À‹

àŸÁ«ï£‚A Þ¼Šd˜èœ. àƒèœ °´‹ðˆF™ , àƒèœ ðœOJô, ;

è¬ìˆªî¼M™ Ü™ô¶ ê£¬ôJ¡ æóñ£è àƒè¬÷ ²ŸP½‹

°ö‰¬îèÀì¡ åšªõ£¼ï£À‹ ðôº¬ø «ðC»‹ ðöA»‹ Þ¼Šd˜èœ.

.Üšõ£Á Ü‚°ö‰¬îèÀì¡ «ð²‹ «ð£¶ °ö‰¬îè¬÷Š ðŸP cƒèœ

â¡ù G¬ùˆF¼Šd˜èœ. Üõ¡-Üõœ G¬ù‚°‹ Mîñ£è¾‹

Üõ˜è÷¶ C‰î¬ù ñŸÁ‹ èŸÁ‚ªè£œÀ‹ º¬øJ™ ºF˜‰îõ˜è÷£è

«î£¡P»œ÷ùó£? cƒèœ G¬ùˆî¶ à‡ì£? Üõ˜èœ C‰F‚è èŸÁ‚

ªè£œÀõ¶ ðœOJô£? â.è£ °ö‰¬îJ¡ C‰î¬ù G¬ôJ™

C‰F‚°‹ Cô õNèœ.

“å¼°ö‰¬îJ¡ ñù‹ â¡ð¶ å¼ â¿îŠðì£î è£Aî‹ «ð£ô”

“å¼°ö‰¬îJ¡ ñù‹ â¡ð¶ ºŸP½‹ Þ¼†ì£ù¶, Üî¬ù

ÜPM¡ Íô‹ Hóè£C‚è ¬õ‚è«õ‡´‹”.

4



“å¼°ö‰¬î èOñ‡ «ð£¡ø¶. ï£‹ M¼‹Hò â‰î õ®õˆ¬î»‹

ªè£´‚èº®»‹”.

“ å¼ °ö‰¬î ð²‹ îO¼‚° «ð£¡ø¶. îO˜‚° c˜ áŸP

õ÷˜‚è«õ‡´‹”.

“å¼ °ö‰¬îJ¡ ñù‹ ªõŸÁŠ ð£¬ù «ð£¡ø¶. Üî¬ù

ÜPõ£™ GóŠðº®»‹”.

°ö‰¬îJ¡ ñù¬î êKò£ù º¬øJ™ MõK‚è Þ‰î

õ£‚AòƒèO¡ Íô‹ ï‹¹Al˜è÷£? â‰îªõ£¼ îQ ïðK¡ ñù

â‡íƒè¬÷ ªîK‰¶ªè£œî™ â¡ð¶ Iè¾‹ è®ù‹ .Üõ˜èO¡

õòF¬ùŠ ªð£¼†ð´ˆî£ñ™ ï£‹ èŸø™ èŸHŠðF™ Ý˜õñ£è

àœ«÷£‹. èŸø™ ðœOJ™ Þ¼‰¶ å¼ °ö‰¬îJ¡ ñùF™ â¡ù

Þ¼‚A¡ø¶. °ö‰¬îèœ âŠð® C‰F‚Aø£˜èœ. C‰FˆîõŸ¬ø âšõ£Á

ðò¡ð´ˆ¶A¡øù˜ â¡ð¶ ªîKòM™¬ô.

C‰î¬ù‚° Ü®Šð¬ì‚ è¼ˆ¶ ñŸÁ‹ ÅöL™ ªð£¼†èÀì¡

ªî£ì˜¹ð´ˆ¶î™. Üšõ£Á ªî£ì˜¹ð´ˆ¶õî£™ ÜÂðõ‹

A¬ì‚A¡ø¶. “Þ÷‹ °ö‰¬îJ¡ C‰î¬ù â¡ð¶ Å›G¬ôèO™

îƒè¬÷ ªð£¼ˆF‚ ªè£œõF™ A¬ì‚°‹ ÜÂðõƒè¬÷

Ü®Šð¬ìò£è‚ªè£‡ì¶.”

°ö‰¬îèÀ‚° îƒèœ Þ¼‚°‹ ÅöL™ A¬ì‚è‚Ã®ò

¹ô¡è£†Cè÷£ù ð£˜ˆî™ , ªî£´î™ ,«è†ì™ ,ñŸÁ‹ ²¬õˆî™

,Íô‹ C‰î¬ù «î£¡ÁA¡ø¶. (ÜP¬õŠ ðŸPò ÜP¬õ ªðŸø¶

â¡Á ÜP‰îõ˜ ¹è›ªðŸø ²Mv à÷Mòô£÷£.; Hò£«ü ñŸÁ‹

¹¼í˜.

Hò£«ü ñŸÁ‹ ¹¼íK¡ è¼ˆ¶ð® ¹ô¡ àí˜¾ â¡ð¶ å¼

G¬ôò£ù ªð£¼†è¬÷ ðŸPò C‰î¬ù¬ò ªêò™ð´ˆî™;. ¹ô¡

àí˜¾ (ªð£¼†ê£˜‰î ÜP¾ «ïó®ò£ù ªî£ì˜H¡ Íô‹) «ñ½‹

HóFGFˆ¶õ‹ (àíóŠð†ì ªð£¼O¡ ñù‚ èŸð¬ù) G„êòñ£è

HóFGFˆ¶õˆFŸ° õ®õ‹ ªè£´‚°‹ õ¬èJ™ ªñ£N º‚AòŠ

ðƒ¬è õA‚A¡ø¶.

5



å¼ ÝCKòó£è ï‹º¬ìò ñùF™ ¹ô¡è£†CèO¡ MFè¬÷

ñùF™ ¬õˆ¶‚ªè£‡´ °ö‰¬îèO¡ ¹ô¡è£†CJ™ ãŸð´‹

î¬ìè¬÷ c‚è«õ‡´‹. Üì™«ð˜† Ü‹v ü˜ð˜ 1938 Þ™ õL»ÁˆFò

Cô ªè£œ¬èèœ ñùF™ ªè£œ÷«õ‡´‹.

ïñ¶ ¹ô¡è£†Cèœ ï‹¬ñ ²ŸP»œ÷ ªð£¼†èO™ Þ¼‰¶

ªðøº®ò£¶. ï‹º¬ìò àí˜¾èœ ï‹IìI¼‰¶ «î£¡ÁA¡øù. Þ¶

ªð£¼†èO¡ º‚Aòˆ¶õˆ¬î °¬øˆ¶ ñFŠHìM™¬ô. Ýù£™

åšªõ£¼ ð£˜¬õ»‹ Ü‰îŠªð£¼¬÷ àí˜‰¶ªè£œÀ‹ õN¬ò

º¡Q¬ôŠð´ˆî «õ‡´‹. ªðKòõ˜èœ ñŸÁ‹ °ö‰¬îèœ ñŸÁ‹

ªð¼‹ð£ô£ù ñ‚èÀ‚° Þ¶ Iè¾‹ ÜK¶. îñ¶ º‰¬îò ÜÂðõƒèœ

îƒè÷¶ º‰¬îò ÜÂñ£ùƒèœ â¡ð¶ ï‹º¬ìò «ï£‚èƒèœ ñŸÁ‹

«î¬õèO¡ Ü®Šð¬ìJ™ Ü¬ñ‰î å¼ ªêò™ð£ì£è àíÃ¡«ø£‹.

ï£‹ ï‹º¬ìò è¼ˆ¶‚è¬÷ ñ£ŸÁ‹ õ¬ó‚°‹ è¼ˆ¶è¬÷

ñ£ŸÁõîŸ° ê£ˆFò‹ A¬ìò£¶. b¬ñ Ü®Šð¬ìJô£ù å¡¬ø

ªêŒî£™ ï‹º¬ìò ºòŸCèœ Mó‚Fò¬ìA¡øù. å¼ °ö‰¬î î¡

î£Œ å¼õó£™ î£¡ Þó‡´ à¼‡¬ìè¬÷‚ ªè£‡´ å¼ ªó£†®¬ò

îò£˜ ªêŒõî£è G¬ùˆî£™ 1+1=2 â¡Á Ã´îô£è Þ¼‚è«õ‡´‹.

Üõ÷¶ àí˜¬õ ñ£ŸÁõîŸ° Üõœ å¼ Fìñ£ùªð£¼¬÷ (å¼

ðOƒ° «ð£¡ø) ñŸªø£¡Áì¡ «ê˜‚è«õ‡´‹. ï‹ àí˜¾èœ ï‹

èì‰îè£ô ÜÂðõƒèOL¼‰¶ õ‰F¼Šðî£™ åšªõ£¼õ¼‹

îQˆ¶õñ£ù õ¬èJ™ å«óªð£¼¬÷ àí˜‰¶ªè£œõ¶ ªîOõ£è

àœ÷¶. ªî£ì˜¹ð´ˆ¶î™ â¡ð¶ «ï£‚èƒèœ áèƒèœ â¡ø

Ü®Šð¬ìJ™ ñ†´«ñ ê£ˆFò‹.

å¼õ˜ ªêò™ð´‹ «ð£¶ ¹ô¡è£†C‚è£ù Ü˜ˆî‹ ¹ôŠð´‹.

ñ¬ö õ¼‹ ªð£¿¶ ñ‚èœ Cô Þ¼ŠHìˆ¶‚° æÁ¡øù˜. Ýù£™

Cô˜ ñ¬ö¬ò óCˆ¶ ïìù‹ ÝÁ¡øù˜. Üõ˜èO¡ ¹ô¡

è£†CèO¡ Íô‹ à‡ì£°‹ ¹ô¡ àí˜¾èœ Üõ˜èO¡

â‡íƒèO¡ HóFGFˆ¶õˆF¡ Ü®Šð¬ìJ™ àí˜¾èœ /

ªêò™èœ ªõOŠðÁ¡øù.

6



Üõ˜èO¡ â‡íƒèO¡ HóFðLŠð¬õ â¡ð¶ ªð£¼œèO¡

à¼õƒè¬÷ à¼õ£‚°‹ ªêò™º¬øò£°‹. Þ¶ «ïó®ò£ù

è‡è£EŠ¹‚° ªð£¼‰î£î¶ Ü™ô. Ü‰î G¬ôJ™ °ö‰¬î îù¶

ñùF™ àœ÷ å¼ ªð£¼œ ªñ£NJ™ Ü¶ «õÁ õ®õˆF™

ÜŠªð£¼¬÷ °P‚°‹ â¡ð¬î MõK‚è «õ‡´‹. âù«õ ªñ£N

â¡ð¶ C‰î¬ùJ¡ õ£èù‹ Ý°‹.

ªêò™ð£´ :1

å¼ ªðò¬ó‚ ªè£´ƒèœ (ªð¡C™ â¡ø ªê£™) ªð£¼†èO¡

ªðò¬ó‚ «è†ì¾ì¡ àìù®ò£è ñùF™ «î£¡Á‹ Mûòƒè¬÷„

ªê£™½‹ ð® ñ£íõ˜èÀ‚°‚ ÃÁƒèœ. ñ£íõ˜èO¡ ðF™è¬÷

W«ö ðF¾ ªêŒò¾‹. â™ô£ Å›G¬ôèO™ îQïð˜èOìI¼‰¶

ñ£Áð£ì£ù è¼ˆ¶‚èœ Üõ˜èœ ÜP¾ ñŸÁ‹ ÜPõ£Ÿø™ è†ì¬ñŠ¹

ªêò™º¬øJ™ «î£¡ÁAø¶ â¡Á Hò£«ü è¼¶A¡ø£˜; . Üõ˜

åšªõ£¼ °ö‰¬î»‹ å¼ ªêò™º¬ø ªêŒ»‹ «ð£¶‹ C‰î¬ùJ™

Ü®Šð¬ìJ™ Gè›¾è¬÷ ªêò™ð´ˆ¶Aø¶ (¹ô¡è£†CJ™

A¬ìˆîõŸ¬ø ªð£¼œè¬÷ Gè›¾è÷£è ñ£ŸP Ü¬ñ‚°‹ å¼

Ü¬ñŠ¹ )

å¼ °ö‰¬î Þó‡´ ªêò™ð£´èO™ ( àí¼‹ ªð£¼†è¬÷

M÷‚°î™ â¡ð¶ C‰î¬ùJ™ ãŸèù«õ àœ÷ ñù Ü¬ñŠ«ð£´

Gè›¬õ ãŸð´ˆ¶A¡ø¶ )

°ö‰¬îèO¡ å¼ƒA¬íŠ¹ â¡ð¶ êñ„YóŸø G¬ô Þ¶

Hò£«üM¡ õ÷˜„C «è£†ð£†®™ º‚Aòñ£ù å¼ G¬ô. åšªõ£¼

ïðK¡ C‰î¬ù ªêò™º¬øò£ù¶ Þ¬íˆî™ ñŸÁ‹ å¼ƒè¬ñˆî™

â¡ðî¡ Ü®Šð¬ìJ™ êñ¡ð£´è¬÷ ñ£ŸP Ü¬ñˆî™;

˜¡ðŸøŠð´Aø¶. â™«ô£¼‹ Üõ¼¬ìò C‰î¬ù õNèO™

îQˆ¶õñ£è àœ÷ù˜.

ÞŠ«ð£¶ àôè÷£MògFJ™ ãŸÁ‚ªè£œ÷Šð†ì Hò£«üM¡

¹ôµ˜¾ õ÷˜„C G¬ôè¬÷ ï£‹ å¼ ²¼‚èñ£ù è‡«í£†ìˆ¶ì¡

7



ð£˜‚èô£‹. Üõ¡ ç Üõœ C‰î¬ùèœ õ÷˜‰î H¡ ïñ‚° ¹K»‹.

èEî‹ èŸHŠðîŸè£ù °PŠH†ì °í ïô¡è¬÷ ªè£‡®¼Šð£˜.

Hò£«ü èEî‹ èŸø™ ÝŒ¾è¬÷ MKõ£è Ýó£Œ‰¶œ÷£˜.

E1 : C‰î¬ù õ÷˜„CJ;¡ Þó‡´ Ü®Šð¬ì ªêò™º¬øè¬÷

àî£óíˆ¶ì¡ M÷‚°è.

E2 : C‰î¬ù¬ò ªêò™ð´ˆ¶õF™ âšõ£Á êñG¬ô¬ò‚

¬èò£‡´ C‰î¬ù õ÷˜„C¬ò «ñ‹ð´ˆ¶î™.

ÜPî™ Fø¡ õ÷˜„C ð® G¬ôèœ (Stages of Cognitive Development)

ÝCKò ñ£íõ˜è«÷ cƒèœ Þ¡Á õ÷˜‰îõ÷£è Þ¼‚A¡l˜èœ

õ°ŠH™ ÝCKò˜ ªê£™õ¬î àƒè÷£™ ¹K‰¶ªè£œ÷ º®A¡ø¶;

Ü¬îŠðŸP C‰F‚è º®A¡ø¶; â¶ êK â¶ îõÁ âù º®¾ ªêŒò

º®A¡ø¶; Ü¬î ñùF™ Þ¼ˆFˆ «î˜M™ MKõ£‚A Ü™ô¶ ²¼‚A

M¬ìè¬÷ â¿î º®A¡ø¶. Þˆî¬èò ñùˆFø¡èª÷™ô£‹

àƒèOìˆF™ âšõ£Á õ÷˜„Cò¬ì‰î¶ â¡Á àƒèÀ‚°ˆ ªîK»ñ£?

àƒèOì‹ è™M ðJô õ¼‹ °ö‰¬îèOì‹ àƒèOì‹ àœ÷ Þˆî¬èò

ñùˆFø¡èœ àœ÷ùõ£? °ö‰¬îèOìˆF™ Þˆî¬èò ñùˆFø¡èœ

âšõ£Á õ÷˜„Cò¬ìA¡øù â¡Á ÞŠð£ìŠð°FJ™ è£‡«ð£‹.

î¡¬ù»‹ ²ŸÁŠ¹ø Å›G¬ô»‹ ¹K‰¶ ªè£œÀ‹ ñùˆFø¬ù

ÜPî™ Fø¡ (Cognitive Ability) â¡A«ø£‹. ÜPî™ FøQ™ C‰Fˆî™,

G¬ù¾Ã˜î™ , ªñ£N¬òŠ ðò¡ð´ˆ¶î™ , ð¬ìŠð£Ÿø™ ,

¸‡íP¾, ¹K‰¶ ªè£œÀîô,; Hó„ê¬ù¬ò b˜ˆî™ º®¾ ªêŒî™

«ð£¡ø ðô Fø¡èœ ÜìƒA»œ÷ù. ÞˆFø¡èO¡ õ÷˜„C¬ò«ò

ÜPî™ Fø¡ õ÷˜„C â¡A«ø£‹. ÜPî™ Fø¡ õ÷˜„C C²Š

ð¼õˆFL¼‰¶ å¼ F†ìI†ì õK¬ê‚ Aóññ£ù ð® G¬ôèO™

ï¬ìªðÁA¡ø¶. Þ‰îŠð® G¬ôèœ º¡H¡ù£è ñ£ø£ñ™ Ü™ô¶

ñ¬øò£ñ™ â™ô£‚ °ö‰¬îèÀ‚°‹ å¼ õK¬ê‚AóñŠð®«ò

ï¬ìªðÁA¡øù. åšªõ£¼ õ÷˜„C G¬ôJ½‹ ÜîŸ° º¡ àœ÷

ð® G¬ô¬ò Mì îóˆF™ º¡«ùŸøñ£ù õ÷˜„C «î£¡ÁAø¶.

8



ÜP;î™ Fø¡ õ÷˜„C H¡õ¼‹ ï£¡° ð® G¬ôèO™

ï¬ìªðÁA¡ø¶ â¡Á Hò£«ü â¡Â‹ à÷MòôPë˜ è¼¶A¡ø£˜.

1. ¹ô¡ Þò‚èŠð¼õ‹ (Sensory Motor Stage:0-2 Ý‡´èœ)

2. ñù„ ªêò™ð£†´‚° º‰¬îò ð¼õ‹ (Preoperational Stage :2-7

Ý‡´èœ.

3. è‡Ãì£è ð£˜Šð¬î ªè£‡´ C‰F‚°‹ ð¼õ‹ (Concrete Operation

stage :7-12 Ý‡´èœ.)

4. º¬øò£ù ñù„ªêò™ð£†´ ð¼õ‹ (Formal Operational Stage

:12Ý‡´èÀ‚° «ñ™.)

1.¹ô¡ Þò‚èŠð¼õ‹ (Sensory Motor Stage :0-2 Ý‡´èœ)

ÜPî™ Fø¡ õ÷˜„CJ™ °ö‰¬î Hø‰î¶ ºî™ Þó‡´

Ý‡´èœ õ¬ó»œ÷ ð¼õ‹, ¹ô¡ Þò‚è ð¼õ‹ âùŠð´‹. Ýó‹ð

ï£†èO™ °ö‰¬îJ¡ ªêò™èœ ð£˜ˆî™, «è†ì™ â¡ø ¹ô¡

àí˜¾èÀì¡ ð£™°®ˆî™ , àøƒ°î™ , Ü¿î™, ªð£¼†è¬÷

H´ƒ°î™ «ð£¡ø Þò‚èƒèÀì¡ G¡Á M´A¡ø¶. Þ¬õò£¾‹

ñÁM¬ù ªêò™è÷£è àœ÷ù. ï£÷¬ìM™ ïó‹¹ ñ‡;ìô‹ õ÷ó

õ÷ó °ö‰¬î ¹ô¡ àí˜¾è¬÷»‹ , àì™ Þò‚èƒè¬÷»‹ ,å¼ƒA¬íŠð¶ Íô‹ àôèˆ¬î ðŸP ¹K‰¶ ªè£œA¡ø¶. Þîù£™

ÞŠð¼õˆF¬ù ¹ô¡ Þò‚èŠð¼õ‹ â¡ø£˜ Hò£«ü.

ÞŠð¼õˆF¡ ªî£ì‚èˆF™ Üî£õ¶ Hø‰î ï£¡° ñ£îƒèœ

õ¬ó °ö‰¬îèœ ªð£¼O¡ G¬ôŠ¹ˆî¡¬ñ¬ò (Object Permanence)àí˜õF™¬ô. å¼ ªð£¼œ è‡º¡ Þ™¬ô â¡ø£½‹ ÜŠªð£¼œ

âƒ«è£ å˜ ÞìˆF™ Þ¼‚A¡ø¶ â¡Á àí˜‰¶ ªè£œÀî™

ªð£¼O¡ G¬ôŠ¹ˆî¡¬ñ¬ò àí˜î™ Ý°‹. Ýù£™ ï£¡° ñ£î‚

°ö‰¬îèOìˆF™ å¼ ªð£‹¬ñ¬ò‚ è£†® H¡ù˜ Üî¬ù å¼

î¬ôò¬íJ¡ W› ñ¬øˆ¶M†ì£™ °ö‰¬î Ü‰îŠ ªð£‹¬ñ¬òˆ

«îì ºòŸCŠðF™¬ô. è£óí‹ ªð£‹¬ñ î¬ôò¬íJ¡ W› àœ÷¶

â¡Á °ö‰¬î àí˜‰¶ ªè£œ÷ º®õF™¬ô.

9



ï£÷¬ìM™ ð£Fò÷¾ ñ¬ø‚èŠð†ì ªð£‹¬ñ¬ò Ü™ô¶

ªð£¼œè¬÷‚ °ö‰¬î ð£˜‚°‹ «ð£¶ ªð£¼œèœ å¼ êñò‹ è‡

º¡ «î£¡P H¡ù˜ ñ¬ø‰¶, e‡´‹ «î£¡Á‹ ÜÂðõˆ¬îŠ

ªðÁ‹ «ð£¶ ÜŠªð£¼œ ÞŠ«ð£¶ è‡ º¡ Þ™¬ô â¡ø£½‹

âƒ«è£ Þ¼‚A¡ø¶. â¡Á °ö‰¬î è¼¶Aø¶. ÜŠ«ð£¶ Ü‰î

ªð£¼O¡ à¼õ‹ °ö‰¬îJ¡ ñùF™ «î£¡ÁAø¶. Þ¶«õ

°ö‰¬îJ¡ ÜPî™ FøQ¡ ºî™ õ÷˜„C Ý°‹. ÞîŸè£ù ÜP°P 4

- 6 ñ£îƒèO™ ºî¡ ºî™ «î£¡ÁAø¶. Þšõ÷˜„C å«ó ÞóM™

ïì‰¶ M´õF™¬ô Iè¾‹ ªñ¶õ£è ï£À‚° ï£œ ðô

ðK«ê£î¬ùèO¡ Ü®Šð¬ìJ™ ï¬ìªðÁA¡ø¶. ãø‚°¬øò 18

ñ£îƒèœ Ýù H¡¹ °ö‰¬î ªð£¼O¡ G¬ôŠ¹ˆ î¡¬ñ¬ò

ºŸP½‹ àí˜‰¶ªè£œAø¶. Þšõ£Á ªð£¼O¡ G¬ôŠ¹ˆ

î¡¬ñ¬ò àí˜î½‹ Þ ªð£¼O¡ à¼õ‹ ñùF™ «î£¡Áî½‹ Þ

¹ô¡ Þò‚èŠ ð¼õˆF¡ º‚Aò õ÷˜„C Ý°‹.

2. ñù„ ªêò™ð£†´‚° º‰¬îò ð¼õ‹:

(Preoperational stage 2-7:Ý‡´èœ) :

ÞŠð¼õˆF™ °ö‰¬îèœ ñù à¼õƒè¬÷ ðò¡ð´ˆ¶õF™

ð®Šð®ò£è º¡«ùÁA¡øù˜. Ü«î êñòˆF™ °ö‰¬îJ¡ ªñ£N»‹

õ÷˜„C Ü¬ìAø¶. Þîù£™ °ö‰¬î Þ‰î àô¬è õ£˜ˆ¬îèœ Íô‹

õ¼;E‚è ªî£ìƒ°Aø¶. Þ‰î õ£˜ˆ¬îè¬÷»‹, ñù à¼õƒè¬÷»‹,

ðò¡ð´ˆ¶‹ Fø¡ õ÷˜„C Ü¬ì‰î °lf†´ C‰î¬ù¬ò

HóFðLŠð«î£´ ¹ô¡ àí˜„CèÀ‚°‹ ªêò™èÀ‚°‹ ÜŠð£Ÿð†ì

ªî£ì˜¹è¬÷»‹ HóFðL‚A¡ø¶.

ñù à¼õƒè¬÷‚ ªè£‡´ C‰î¬ùè¬÷ˆ ªî£ì˜‰î£½‹

ÞŠð¼õˆF™ °ö‰¬îJ¡ ÜPî™ Fø¡ º¿õ÷˜„CÜ¬ìõF™¬ô.

ÞŠð¼õˆF™ °ö‰¬îèO¡ C‰î¬ùèO¡ Cô °¬øð£´èœ

è£íŠð´A¡øù. â´ˆ¶‚è£†ì£è °ö‰¬îèÀ‚° ªð£¼œèO¡

ñ£ø£ˆî¡¬ñ (Convervation) ¹KõF™¬ô. ªð£¼œèO¡ ñ£ø£ˆî¡¬ñ

¹Kò£î G¬ô â¡ð¶ ªð£¼œèO¡ õ®õº‹ «î£Ÿøº‹

«õÁð†ì£½‹ Ü÷M™ ñ£ÁõF™¬ô .â¡ð¬î ÜP‰¶ ªè£œ÷

10



Þòô£î G¬ô .ÞîŸ° è£óí‹ °ö‰¬îèO¡ C‰î¬ùJ™ è£íŠð´‹

å«ó å¼ ÃP™ ñ†´‹ èõù‹ ªê½ˆ¶‹ î¡¬ñ (Concentration ), ïì‰¶

º®‰î æ˜ Gè›„C¬ò ñùˆî÷M™ e‡´‹ ð¬öò G¬ô‚° ªè£‡´

õó Þòô£î î¡¬ñ (Irreversibility), î¡¬ù ¬ñòñ£è‚ ªè£‡´ C‰F‚°‹

î¡¬ñ (Egocentrism), àJóŸø ªð£¼œè¬÷»‹ àJ¼œ÷¬õè÷£è

ð£M‚°‹ î¡¬ñ (Animism) Ý°‹.

å«ó å¼ ÃP™ ñ†´‹ èõù‹ ªê½ˆ¶‹ î¡¬ñ (Centration)

Þâ¡ð¶ °ö‰¬îèœ å¼ Hó„ê¬ùJ¡ å«ó å¼ ÃP™ ñ†´‹ èõù‹

ªê½ˆF ñŸø º‚Aò ÃÁè¬÷Š ¹ø‚èE‚°‹ ð‡ð£°‹. å¼

ðK«ê£î¬ùJ™ Hò£«ü åˆî è‡í£® °õ¬÷èœ Þó‡®¬ù

â´ˆî‚ªè£‡´ ÜõŸP™ êñ Ü÷¾ c¬ó GóŠHù˜. °ö‰¬îèœ

ÜõŸP™ àœ÷ c˜ Þó‡´‹ êñ Ü÷¾ à¬ìò¶. â¡Á

ãŸÁ‚ªè£‡ìù˜. H¡ù˜ å¼ °õ¬÷J™ Þ¼‰î c¬ó °ÁAò Þ

Ýù£™ àòóñ£ù °õ¬÷J™ áŸPù£˜. ÞŠ«ð£¶ àœ÷ Þó‡´

°õ¬÷èO½‹ êññ£ù c˜ àœ÷ùõ£? â¡Á «è†ì£˜. Þ‰î

Å›G¬ôJ™ ñù„ªêò™ð£†´‚° º‰¬îò ð¼õˆF™ (Preoperational stage)

àœ÷ °ö‰¬îèœ ªð¼‹ð£ô£ùõ˜èœ Þ™¬ô â¡Á ðFôOˆîù˜.

°ÁAò - àòóñ£ù °õ¬÷J™ àœ÷ c˜ ÜFèñ£ù¶ â¡øù˜

.Ü‚°ö‰¬îèœ °ÁAò àòóñ£ù °õ¬÷J¡ °Á‚°ªõ†´Š ðóŠ¬ð

¹ø‚èEˆ¶ àòóˆF™ ñ†´‹ èõù‹ ªê½ˆF Ü‚°õ¬÷J™ î£¡

ÜFè c˜ àœ÷¶ â¡Á õL»ÁˆFù˜. ÞFL¼‰¶ °ö‰¬îèœ å¼

êñòˆF™ àòó‹Þ Üèô‹ Þ â‡E‚¬è Þõ‡í‹ «ð£¡ø

ã«îÂ‹ å«ó å¼ î¡¬ñ¬ò ñ†´‹ èõQ‚A¡øù˜ â¡Á ªîKAø¶.

ªð£¼O¡ â‰îˆî¡¬ñ¬ò °ö‰¬î Hóî£ùñ£èŠ

ð£˜‚A¡ø«î£ Ü¶ °ö‰¬îJ¡ èõù‹ ¬ñòñ£è ñKM´A¡ø¶

.ÞŠðK«ê£î¬ùJ™ Þ‚°ö‰¬îèœ °õ¬÷J¡ àòóˆF™ ñ†´‹

èõù‹ ªê½ˆFù£˜èœ. Ýù£™ Ü‚°õ¬÷J¡ °Á‚°ªõ†´

ðóŠ¬ðŠ ¹ø‚èEˆî£˜èœ.Þšõ£Á ÞŠð¼õˆF™ àœ÷

°ö‰¬îèÀ‚° å¼ Hó„ê¬ùJ¡ å¡Á‚° «ñŸð†ì ÃÁèO™ è¼ˆ¬î

ªê½ˆF C‰F‚è ÞòôM™¬ô.

11



«ñ½‹ Þõ˜èœ àôè‹ î¡¬ù ¬ñòñ£è‚ªè£‡´ Þòƒ°A¡ø¶

.â¡Á G¬ù‚Aø£˜èœ. Üîù£™ Þš¾ô¬è îù‚«è àKò º¬øJ™

ð£˜‚A¡øù˜. â´ˆî‚è£†ì£è î¡Â¬ìò Ü‹ñ£ ñŸø Ü¬ùõ¼‚°‹

Ü‹ñ£î£¡ âù G¬ù‚A¡øù˜. ÞŠð¼õˆF™ àœ÷ °ö‰¬îèœ å¼

Gè›„C¬ò «õÁ å¼õK¡ ð£˜¬õJL¼‰¶ è£µ‹

Fø¬ñòŸøõ˜è÷£Œ î¡ ð£˜¬õJ™ Þ¼‰¶ ñ†´«ñ ð£˜‚è

Ã®òõ˜è÷£Œ (Egocentrism) àœ÷ù˜.

ÞŠð¼õˆF™ ÜPî™ FøQ™ æó÷¾ ñù à¼õƒè¬÷Š

ðò¡ð´ˆ¶‹ Fø¡ õ÷˜‰F¼‰¶‹ Ü¶ º¿¬ñò£è õ÷˜„C

Ü¬ìò£ñ™ Ýù£™ õ÷˜„C «ï£‚A„ªê™õî£™ Hò£«ü Þî¬ù ñù

ªêò™ð£†´‚° º‰¬îò ð¼õ‹ (Preoperational Stage) âù ªðòK†ì£˜.

3. è‡Ãì£è ð£˜Šð¬î ªè£‡´ C‰F‚°‹ ð¼õ‹

(Concrete operational Stage :7-12 Ý‡´èœ.)

º¬øò£ù ñù„ªêò™ð£´ (Formal Operational Thinking) àœ÷£˜‰î

ñ£Ÿøƒèœ ñù à¼õƒè¬÷ ¬èò£Àî™ ñù Ü¬ñŠ¹è¬÷ ñ£ŸP

Ü¬ñˆî™ ÝAò «ñ‹ð†ì ñù„ªêò™ð£´èÀ‹ Ü´ˆ¶ õ¼‹

ð¼õˆF™ «î£¡ÁA¡øù.

ÝCKò˜èœ M¬÷ò£†´ ªêŒ¶ èŸø™ Þ ï®ŠH¬êŠ ð£†´Þ

è¬î «ð£¡ø Å›G¬ôè¬÷ à¼õ£‚A‚ èŸHˆî£™ èŸøô£ù¶

°ö‰¬îèÀ‚° âO¬ñò£è¾‹ Þ ñA›„Cò£è¾‹ Þ¼‚°‹. «ñ½‹ Ü¶

°ö‰¬îJ¡ ÜPî™ Fø¡ õ÷˜„C‚° àî¾‹. èEî‹ ð£ìˆF™

I°Fò£è °Pf´èÀ‹, è¼ˆFò™èÀ‹ ðò¡ð´ˆîŠð´õî£½‹

ÞÝó‹ðŠ ðœOJ™ ðJ½‹ °ö‰¬îèœ è¼ˆF™ C‰î¬ùJ™ õ÷˜„C

Ü¬ìò£ñ™ Þ¼‰î£½‹ ÝCKò˜ èEî ð£ìˆF¬ù ªð£¼œè¬÷‚

ªè£‡´ èŸH‚è «õ‡´‹. â´ˆ¶‚è£†ì£è 2+3=5 â¡Á °Pf´èO™

è¼‹ðô¬èJ™ â¿¶õ¬î MìÞ Þó‡´ ¹ˆîèƒèÀì¡ Í¡Á

¹ˆîèƒè¬÷ «ê˜ˆî£™ ªñ£ˆî‹ ä‰¶ ¹ˆîèƒèœ â¡Á‹ ¹ˆîèƒèœ

ñŸÁ‹, ñEèœ «ð£¡ø ðô ªð£¼†è¬÷ ¬õˆî 2+3=5 â¡ø

°Pf´è÷£™ â¿Fù£™ °ö‰¬îèœ Cóñ‹ Þ™ô£ñ™ ¹K‰¶

12



ªè£œõ£˜èœ. Þ‹ñQî¬ù ï£èKè õ÷˜„CJ¡ à„C‚° ªè£‡´

ªê™ô «ñ‹ð†ì C‰î¬ù‚° ªðK¶‹ ¶¬í¹Kõ¶ ÞŠð¼õˆF™

°ö‰¬îèOì‹ ãŸð´õ¶‹ ªñ£Nõ÷˜„C Ý°‹ .

4. º¬øò£ù ñù„ªêò™ð£†´ ð¼õ‹ :

(Formal Operational Stage :12 Ý‡´èÀ‚° «ñ™.)

ÞŠð¼õˆF™ àœ÷ °ö‰¬îèœ ªð£¼œèO¡ ªî£ì˜;¹è¬÷

ÜP‰¶ Ü¬õèO¡ Ü¬ñŠ¹è¬÷ «ò£Cˆ¶ îƒèœ ñùˆF«ô«ò

ªð£¼œèO¡ G¬ôè¬÷ ñ£ŸP ¬õˆ¶ ð£˜‚èˆ ªî£ìƒ°Aø£˜èœ.

ê‰«îèƒèœ ãŸð´‹ «ð£¶ ªð£¼œè¬÷ Ü™ô¶ ÜõŸP¡

Ü¬ñŠ¹è¬÷ ñ£ŸP ¬õˆ¶ è‡ Ãì£è ð£˜ˆ¶ ê‰«îèƒè¬÷

«ð£‚A‚ªè£œA¡øù˜. .ªð£¼œèO¡ ñ£ø£ˆî¡¬ñ¬ò ¹K‰¶

ªè£œA¡øù˜. 6+3=9 â¡ø£™ 3+6=9 Ýèˆî£¡ Þ¼‚è «õ‡´‹ â¡Á

èŸèˆ ªî£ìƒ°Aø£˜èœ.

ÞŠð¼õˆFù˜ î£º‹ ¹K‰¶ ªè£‡´ ñŸøõ˜èÀ‚°‹ M÷‚è

ºŸð´A¡øù˜. «ïó® èŸHˆî™ º¬øŠð´ˆîŠðì£î ÜÂðõƒèœ

ñŸÁ‹ ºF˜„CJ¡ è£óíñ£èˆ îƒèÀ¬ìò è¼ˆ¶‚èO™

ñ£Ÿøƒè¬÷‚ ªè£‡´ õó ºò™A¡øù˜. ; .èŸH‚èŠð´‹

è¼ˆ¶‚èO™ àœ÷ à‡¬ñè¬÷ ¹K‰¶ ªè£œ÷ ÜîŸè£ù è£óí

è£Kòƒè¬÷ åŠH†´Š ð£˜‚èˆ ªî£ìƒ°A¡øù˜.

Hò£«üM¡ è¼ˆ¶‚è¬÷ ÜPî™ Fø¡ õ÷˜„CJ™

ðò¡ð´ˆ¶î™:

ªî£ì‚èŠðœO‚ °ö‰¬îèœ Hò£«ü ÃÁõ¬îMì ÜFèŠ

ð®ò£ù ÜP¾ˆFø¡è¬÷ ªõOŠð´ˆ¶õî£è«õ Ýó£Œ„Cò£÷˜èœ

è¼ˆ¶ ªîKM‚A¡øù˜. îŸ«ð£¶œ÷ è™MˆF†ìˆF™ Hò£«üM¡

è¼ˆ¶‚è¬÷ âšMîˆF™ ðò¡ð´ˆîô£‹. â¡ð¬î è£‡«ð£‹.

1. Í¡Á ºî™ ã¿ õò¶œ÷ °ö‰¬îèœ ¹ôQò‚èˆFø¡Þ

ªñ£NˆFø¡ ªðŸøõ˜è÷£è àœ÷ù˜. Þõ˜èÀ‚° æMò‹

õ¬óî™Þ õ‡í‹ b†´î™ èOñ‡í£™ ªð£‹¬ñèœ ªêŒî™

13



«ð£¡ø ¬è«õ¬ôèœ Þ ïìù‹ Þï£ìè‹ Þ Þ¬êŠðJŸCÞ «ð„²Þ

«ð£¡ø ªêò™èO™ ðJŸCòO‚è «õ‡´‹;. ªð£¼†èO¡

î¡¬ñ õ¬èŠð´ˆ¶î™ Þ;õK¬êŠð´ˆ¶î™ Þ è£ôñÞ;

Þ¬ìªõOÞ ªî£¬ô¾Þ â‡èœ Þ Ã†ì™ Þ èNˆî™ Þ ªð¼‚è™

Þ õ°ˆî™ Þ «ð£¡ø èEî„ ªêò™èœ ºîLòùõŸPŸ°

º‚Aòˆ¶õ‹ ªè£´ˆ¶ ðJŸCèœ ðô ÜO‚è «õ‡´‹. â†´

ºî™ ðQªó‡´ õò¶ õ¬ó»œ÷ °ö‰¬îèÀ‚°ˆ îˆ¶õ

KFò£èŠ ¹K‰¶ ªè÷;÷ º®ò£î è¼ˆ¶‚è¬÷ Üõ˜èÀ¬ìò

Þ÷ñùF™ FEŠð¬î îM˜‚è «õ‡´‹.

2. èŸø™ - èŸHˆîL™ àðèóíƒèÀ‹, è¼MèÀ‹ °ö‰¬îèO¡

ÜP¾ˆFø¡ õ÷˜„C‚«èŸð ÜP‰¶ õöƒè «õ‡´‹ .¹Fò

è¼ˆ¶‚è¬÷ ÜPºèŠð´ˆ¶‹ «ð£¶ °ö‰¬îèO¡ Ý˜õˆ¬î

É‡´‹ õ¬èJ™ èŸH‚°‹ ð£ìƒèÀ‹ è¼MèÀ‹ ªîK¾

ªêŒî™ «õ‡´‹. ÜŠ«ð£¶î£¡ èŸH‚èŠð´‹ è¼ˆ¶‚èœ

ñùF™ àÁFŠð´‹ ªêŒFè¬÷ «ïK¬ìò£è õ£Œªñ£N õNò£è

ñ†´‹ Ãø£ñ™ ªêò™õN‚ èŸHŠð¶ ï¡¬ñ ðò‚°‹. ñùgFò£ù

ñ£Ÿøƒè¬÷ ãŸð´ˆ¶‹.

3. ¹Fò è¼ˆ¶‚è¬÷ ÜPºèŠð´ˆ¶‹ «ð£¶ ð¬öò

ÜÂðõƒèÀì¡ åŠH†´‚ ÃP Ü¬õ Þó‡®Ÿ°‹ àœ÷

ªî£ì˜H¬ù â´ˆ¶‚è£†® °ö‰¬îèÀ‚° ¹Kò ¬õ‚è

«õ‡´‹. ªêŒFè¬÷ ÜŠð®«ò ñùŠð£ì‹ ªêŒò á‚°M‚è

Ãì£¶.

1.2.2 : èEî‚ «è£†ð£´èO¡ õ÷˜„C :

Í¡Á Ü®Šð¬ì‚ èEî‚ °¿‚è÷£è èEî‚ «è£†ð£´èœ

HK‚èŠð†´ ÜõŸP¡ Ü®Šð¬ìJ™ ªî£ì‚èŠ ðœOèO™ èEî

ªêò™ð£´èœ ð£ìˆF™ «ê˜‚èŠð†´œ÷ù. ð£ìˆF†ìˆF™

«ê˜‚èŠð†´œ÷ Ü¬ùˆ¶ MûòƒèO½‹ ÜõCòñ£ù¶ â‡èœÞ

ñŸÁ‹ â‡èœ ê£˜‰î C‰î¬ù ñŸÁ‹ Ü÷i´èœ Ý°‹. Þ‰îŠ

ð°Fèœ ñŸÁ‹ ªî£ì‚è G¬ôJ™ àœ÷ èEîˆF¡ è¼ˆ¶‚èœ

14



HŸð°FèO™ àœ÷ ªî£ì˜¹¬ìò è¼ˆ¶‚èœ Þ‰îŠ ð£ìˆF†ìˆF™

ªî£°F -2 Þ;™ MKõ£è èô‰¶¬óò£ìŠð´‹. Þ‰î HKM™ Þ÷‹

°ö‰¬îèÀ‚° èEî‚ è¼ˆ¶‚è¬÷ èŸHˆî™ º¬øè¬÷ âšõ£Á

F†ìI´î™ «õ‡´‹ â¡ðîŸè£ù õNè¬÷»‹ èŸHŠðîŸè£ù

Í¡Á Ü‹êƒè¬÷»‹ ²†®‚è£†´Aø¶.

â‡ è¼ˆ¶‚èO¡ õ÷˜„C :

õö‚èñ£è â‡µ‹ Þ â‡ è¼ˆ¶‚è¬÷ ÜPºèŠð´ˆîL™

ºî™ ð®ò£è è¼îŠð´Aø¶. «ñ½‹ å¡ø£‹ õ°ŠH™ ªð¼‹ð£½‹

â‡èO¡ ªðò˜è¬÷ ñùŠð£ì‹ ªêŒ»‹ º¬øèœ

è‡Í®ˆîùñ£ù ñùŠð£ì„ ªêò™è÷£è«õ Ü¬ñA¡ø¶. Ýù£™

Fø¡ õ£Œ‰î èŸø™ º¬øJ™ â‡èO™ è¼ˆ¶‚è¬÷ ªðø ªêŒ»‹

º¡ ÜõŸP¡ º‰¬îò â‡ è¼ˆ¶‚è¬÷ èŸÁˆîó «õ‡´‹.

º‰¬îò â‡ è¼ˆ¶‚èœ :

Þ¬õ °ö‰¬îèO¡ º‰¬îò ðœO Ý‡´èO™ â‡

è¼ˆ¶‚è¬÷ õ÷˜„C Ü¬ìò ªêŒò «õ‡´‹. Üî£õ¶ 7 õòFŸ°

º¡ù˜ (º¡ ªêò™ð£†´ G¬ô‚° º¡¹).

ªð£¼ˆ¶î™ :

Þ‰î ªð£¼ˆ¶î™ è¼ˆ¶‚è¬÷ ¹K‰¶ ªè£œ÷¾‹ å¡Á

ñŸªø£¡«ø£´ ªî£ì˜¹¬ìò¶; â¡ð¬î»‹ èŸÁˆî¼A¡ø¶. å¼

°ö‰¬î ê£‚«ô† ç èì‰¶ ªê™½‹ «ð£¶ Þ åšªõ£¼ Ü¬øJL¼‰¶‹

å¼ °ö‰¬î‚° å¼ ê£‚«ô† A¬ì‚Aø¶. êKò£ù Ü÷¾ ñ†´«ñ

à÷÷¶. Ü™ô¶ Ã´î™ ê£‚«ô†èœ Þ¼‚°‹.

°ö‰¬îèœ â‡º¬øJ¡ Ü®Šð¬ìJ™ ªð£¼‰¶Aø¶. å¼

°ö‰¬î “Ü¶«õ” à¼õ£‚°‹ «ð£¶ Þó‡®¬ù ªð£¼ˆîô£‹. Þ‰î

ªêòô£ù¶ è®ùñ£ù ðEèÀ‚° å¼ º¡ «î¬õò£ù Fø¡

ÝA¡ø¶. °ö‰¬î åšªõ£¼ ªð£‹¬ñ‚°‹ å¼ ê£‚«ô† ¬õ‚°‹«ð£¶

Üõ¡ ç Üõœ åšªõ£¼ ªð£‹¬ñ‚°‹ å¼ ê£‚«ô† àœ÷¶ â¡Á

15



ªõO;Šð¬ìò£è ªõOŠð´ˆî º®»‹ Ü™ô¶ ªõŸPèóñ£è

Þó‡®¬ù»‹ ªð£¼ˆî¾‹ º®»‹.

õK¬êò£‚è‹:

°ö‰¬îèœ ð™«õÁ CøŠHò™¹è¬÷ ð£˜‚è¾‹ Ü«î«ð£™

Þ¼‚°‹ ð‡¹è¬÷‚ è‡ìPò¾‹ ªîK‰¶ ªè£œ÷«õ‡´‹.

ÜŠ«ð£¶ °ö‰¬îèœ Ü¬õ îMó Hø õ¬èò£ù ð‡¹è¬÷ èŸÁ‚

ªè£œA¡øù.

æŠH´¬èJ™ :

°ö‰¬îèœ ªð£¼†è¬÷ åŠd´ ªêŒ¶ ÜõŸP¡ î¡¬ñè¬÷

¹K‰¶ ªè£œA¡øù˜. ªð£¼†è¬÷ èõQˆ¶ ªðKò / CPò ÞÅì£ù /

°O˜‰î ªñ¡¬ñò£ù / è®ùñ£ù. àòóñ£ù / °œ÷ñ£ù, èùñ£ù

/ å™Lò£ù / î®ñù£ù Mûòƒè¬÷ åŠH´A¡øù˜. °ö‰¬îèœ

Þ¼õ¼‚°‹ Þ¬ì«ò àœ÷ ªõOf†¬ìŠ ð£˜‚°‹ «ð£¶ Þ¶ «ð£¡ø

åŠd´ MFº¬øè¬÷Š ðò¡ð´ˆ¶õ¶ Iè º‚Aòñ£ù¬î ÜFèñ£ù

/ °¬øõ£ù/ Ü«î åŠd†¬ì b˜ñ£Qˆ¶ °ö‰¬îèO¡ ªî£ì‚è

G¬ôò£ù è¼ˆ¶‚è¬÷ èŸè «õ‡´‹. «ñ½‹ ÜõŸ¬ø åŠd´ ªêŒò«õ‡´‹. CPò °ö‰¬îèœ ð£˜¬õ åŠd´èœ Íô‹ ÜFè ç °¬øõ£ùåŠd´ ªêŒò «õ‡´‹.

õK¬êŠð´ˆî™ (Ordering) :

õK¬êð´ˆ¶î™ â¡ð¶ â‡Eô‚èˆF¡ Ü®Šð¬ì.°ö‰¬îèœ â‡µ¼‚è¬÷ â‡í¾‹ ÜõŸ¬ø êKò£ù õK¬êJ™Ü¬ñ‚è¾‹ èŸÁ‚ ªè£´‚Aø¶. â‡µ¼‚è¬÷ º¡ Gð‰î¬ùJ¡PõK¬êŠð´ˆî¾‹ õK¬êò£è Ü÷¾, c÷‹ ñŸÁ‹ àòóˆF¡Ü®Šð¬ìJ™ ªð£¼†è¬÷ ¬èò£ÀA¡ø¶. °ö‰¬îèÀ‚° F¬êè¬÷ÃÁ‹«ð£¶ õ£˜ˆ¬îè÷£™ ºî™, Ü´ˆ¶, è¬ìC â¡Á ÃÁî™.

Subitizing àì«ù/àìù®ò£è :

W›‚è£µ‹ â‡Eì£î ç â‡íŸø õ¬èèO¡ àìù®

ÜƒWè£ó‹ â¡ð¶ ªîK‰¶ ªè£œ÷¾‹ CÁªî£°Š¬ð å¼ Üô°

16



â¡Á ð£˜‚è àî¾Aø¶. Þ¶ å¼ Ýó‹ð G¬ôJ¡ ªî£ì‚è‹

¹ôµí˜¾ Ü®Šð¬ìJô£ù â‡¬í õöƒ°A¡ø¶. Ýù£™

Þ¶¾‹; “â‡ ê£˜‰î ÜP¾ Þ™¬ô”.

â‡ è¼ˆ¶èœ :

â‡è¬÷ êKò£è ¹K‰¶ ªè£‡´ â‡è¬÷Š ðò¡ð´ˆ¶î™

â‡èœ ñŸÁ‹ ªêò™ð£´è¬÷ â‡EŠ ð£˜ˆî™ â¡ð¶ â‡íŸø

è¼ˆî£‚èƒèO¡ º‚Aò ¬ñ™è÷£è è¼îŠð´A¡ø¶. â‡èO¡

ªð£¶õ£ù ðò¡ð£ì£ù¶ èí‚AìŠð´Aø¶. â‡ ªêò™º¬øèœ

Þó‡´ ð®G¬ôèœ ªè£‡ì¶ . å¼ °PŠH†ì ªð£¼À‚° å¼

â‡¬í å¶‚°õ¶ ºî™ ªð£¼O¡ õK¬êJ™

õK¬êŠð´ˆ¶õ¶.å¼ «êèKŠH™ àœ÷ ªð£¼œèO¡ â‡ õK¬ê

º¬ø Ü‹êƒè¬÷ ÜP‰¶ ªè£œõ¶.

Þšõ£Á ºF˜„Cò¬ìî™. 3 ºî™ 5 àœ÷ G¬ôJ™

õ÷˜„Cò¬ìA¡ø¶ .ðô â‡ ªðò˜èœ ÜPòŠð†ì¾ì¡ °ö‰¬î

å¡Á Þó‡´ â¡ø ªî£°Š¹èO™ ªð£¼†è¬÷ ªð£¼ˆ¶õîŸ°

ºò½Aø¶. 1 ºî™ 9 õ¬óò£ù â‡èO¡ ªðò˜èœ ðŸPò ÜP¾ 2

ºî™ 3 õò¶ õ¬óò£ù â‡èO¡ Fø¬ùŠ ªð£¼ˆîŠðì£ñ™

ªñ£NJ¡ õ÷˜„C «ð£™ õ÷˜„C ªðÁA¡ø¶.

õ£˜ˆ¬îèœ å¡Á (Þó‡´Þ Í¡Á Þï£¡°Þ ..... å¡ð¶) â¡ø

ªð£¼†èO¡ â‡Eò™ Fø¡èO¡ õ÷˜„C G¬ôJ¡ Ýó‹ð

è†ìñ£°‹;.ªð£¼†èœ ñŸÁ‹ â‡ ªðò˜èÀ‚° Þ¬ì«òò£ù å¡Á‚°

«ñŸð†ì ªî£ì˜¹è¬÷ å¿ƒ°Šð´ˆ¶‹ ªêò™ º¬øèœ Ý°‹.

Þó‡´ è£óíƒèÀ‚è£è ªð£¼†èO¡ «êèKŠ¹ Ü÷¾è¬÷

õK¬êŠð´ˆ¶õ¶ àÁF ªêŒòŠðìM™¬ô.

I. °ö‰¬î 2 ºî™ 4 Ý‡´èœ õ¬ó â‡ ªðò˜èO¡ ªî£ì˜¹¬ìò

Ü÷M¬ù ªîK‰¶ ªè£œA¡ø¶.

II.. Üõ˜èœ Þ¡Â‹ â‡ ð£¶è£Š¹ G¬ôèœ à¼õ£‚èŠðì Þ™¬ô

â¡ð¶‹ ªîK»‹.

17



â´ˆ¶‚è£†ì£è 5 õòFŸ° à†ð†ì °ö‰¬îò£™ ºî™ õK¬ê¬ò

Mì Þó‡ì£õ¶ õK¬êJ™ ÜFèñ£ù ªð£¼†;èœ àœ÷ù â¡Á

â´ˆ¶‚ªè£œÀ‹. Þƒ«è °ö‰¬î Ü˜ˆîñŸø ªð£¼†èœ õK¬êJ™

Üî¬ù Ü¬ìò£÷‹ è£í¾‹ õK¬ê ñ£ø£ñ™ Üî¡ â‡è¬÷

°PŠHì¾‹ º®Aø¶.

Ü¬î «ð£¡Á °ö‰¬î, c÷‹, ðóŠð÷¾, Üì˜ˆF, ñŸÁ‹ Üî¡

ªî£°Fè¬÷ ð¼Šªð£¼œ G¬ôJ¡ H¡ ï£†èO«ô«ò ÜPò º®»‹.

(7 õò¶ ºî™ 11 Ü™ô¶ 12 õòFù˜) â‡è¬÷ å¼ º¬ø ¬èò£÷

èŸÁ‚ ªè£œA¡øù˜. 4 ºî™ 5 õò¶ °ö‰¬îèœ ªð£¼†èO¡

Ü®Šð¬ìèO™ °¿õ£è HKˆî™ ñŸÁ‹ Üî¡ ÜôA¬ù‚ °PŠHì

º®»‹.

â‡è¬÷ ¬èò£÷™ : ( Use of Numerals)

â‡èœ °Pf´è÷£è 1,2,3..,, âùŠ ðò¡ð´ˆîŠð´A¡øù.

æ¡Á, Þó‡´, Í¡Á â¡Á õ£˜ˆ¬îè÷£™ Ãø¾‹ Þò½‹.

â‡èO¡ èŸø™ â¡ð¶ ªð£¼†èO¡ ªî£ì˜¹è¬÷ ñŸªø£¡«ø£´

Þ¬íˆ¶ ªð£¼†è¬÷‚ ªè£‡´‹Þ â‡èO¡ Ü¬ñŠH¬ù‚

ªè£‡´‹ ð™«õÁ Mîñ£ù â‡è¬÷ ÜPºèŠð´ˆî™ «õ‡´‹.

îêñ º¬øèO™ â‡è¬÷‚ ¬èò£÷ å¼ °ö‰¬î ºîL™ Ü‰î

â‡è¬÷Š ðöè «õ‡´‹. åŸ¬øŠð¬ì â‡èœ 0-9 å¼ â‡èOL¼‰¶

ñŸªø£¡¬ø à¼õ£‚è¾‹ èŸÁ‚ªè£´‚Aø¶. °ö‰¬î Üî¬ùŠ ¹K‰¶

ªè£œ÷ˆ îò£ó£è¾‹ «ñ½‹ Üî¬ù ðò¡ð´ˆî¾‹,

èŸÁ‚ªè£œAø¶. Ýù£™ ÜFè Ü÷Mô£ù Þì ñFŠ¹è¬÷‚ ªè£‡ì

â‡è¬÷‚ èŸ°‹ Fø¬ù 11 õòF™ ªðø º®»‹. 10 â¡ø â‡¬í

â¿î ñŸÁ‹ Üî¬ù Mì ÜFèñ£ù â‡è¬÷ â¿î â¿¶‹ ÜP¾

ÜõŸP¡ ÞìñFŠ¹èO¡ º‚Aòˆ¶õˆ¬î ê£˜‰¶ Ü¬ñ»‹

18



.ÞìñFŠ¹èœ ê£˜‰î â‡èO¡ ÜP¾ õ÷˜‰î¾ì¡. °ö‰¬î

Üšªõ‡è¬÷ æŠd´ ªêŒò èŸÁ‚ ªè£œAø¶.

â‡ ê£˜‰î ªêò™èœ : (Operation of Numbers )

Ã†ì™ ñŸÁ‹ Üî¡ î¬ôW› èNˆî™ ªêŒ»‹ ªð£¿¶

°ö‰¬îè¬÷ àŸÁ «ï£‚è™ â¡ð¶ Üõ˜èO¡ 6 õòFŸ°œ èŸÁˆîó

«õ‡´‹ . Ã†ì™ â¡ð¬î ¹ômì£ù ªð£¼†èœ ñŸÁ‹ Ü¬î ê£˜‰î

àŸÁ«ï£‚°î™ â¡ð¶ ðœO‚° ªê™ô£î °ö‰¬îè÷£™ èŸè Þòô£¶.

Ýù£™Þ à‡¬ñò£è ¹K‰¶ ªè£œ÷™ â¡ð¶ ÜõŸP¡ Ü¬ñŠ¹èœ

ñŸÁ‹ Üî¡ ªêò™ð£´è¬÷ 9 ñŸÁ‹ 11 õò¶ õ¬ó ïì‚°‹.

õ÷˜„C ð£¬îJ™ °ö‰¬îè÷£™ ªð¼‚è™ ñŸÁ‹ Ã†ì¬ô å«ó

êñòˆF™ ðŸÁ‚ªè£œ÷ º®»‹. Ýù£™ ðœOèO™ ªð¼‚è¬ô

õ°ˆî«ô£´ «ê˜‰¶ 3 ‹ G¬ôJ™ èŸÁ‚ªè£œA¡øù˜. Üõ˜èÀ¬ìò

9 ‹ õòF™ «ñ½‹ Üõ˜èœ ªð¼‚è™ õ®õ¬ñŠ¹è¬÷»‹ ñŸÁ‹

õ°ˆîL¡ Ü¬ñŠ¹è¬÷»‹ Þò™¹ â‡èÀì¡ ð¼Š ªð£¼œ

G¬ôJ™ 11 õòF™ H¡ù˜ ÜPºèŠð´ˆ¶î™.

Ü÷i†´‚è¼ˆ¶èO¡ õ÷˜„C :

( Development of Measurement Concept)

Hò£«üM;Â¬ìò ÜPFø¡ õ÷˜„C‚ «è£†ð£´ Ü÷i´èO¡

è¼ˆ¶è¬÷ °ö‰¬î ¹K‰¶ ªè£œõF™ ªðÁ‹ ðƒ° õ°‚A¡ø¶. Üõ˜

2 õNèO™ Ü¬ìò£÷‹ î‰¶œ÷£˜. Üî£õ¶ Ü÷i´èO¡ è¼ˆ¶‚èœ

°¬øò£¶ ÜõŸ¬ø îQˆ¶õñ£è ð£¶è£ˆî™ «õ‡´‹. ï£‹ Þ¬î

ðŸPò â‡íƒè¬÷ º¡¹œ÷ ð°FèO™ Mõ£Fˆ¶œ«÷£‹.

ñ£Áð£´èœ â¡ø è¼ˆ¶ â.è£ Þì‹ ªðø™ «õ‡´‹. å¼

°ö‰¬îJì‹ ðœOˆ «î£†ìˆ¬î G¬ù×†´‹ «ð£¶ Üî«ù£´

ªî£ì˜¹¬ìò Ü«î Ü÷Mô£ù ð‚èƒè¬÷ à¬ìò ªêšõè

«î£†ìˆ¬î â´ˆ¶‚è£†ì™ «õ‡´‹.

å¼ «õ¬ô Üõ˜èÀ‚° ªè£´‚è‚ Ã®ò GôˆF¡ c÷‹ A â¡ø£™

ÜîÂ¬ìò Þ¡ªù£¼ ð‚è c÷ˆ¬î °ö‰¬î B â¡Á Ü÷¾ «è£ô£™

19



Ü÷ˆî™ «õ‡´‹. H¡¹ °ö‰¬î Ü«î Ü÷Mô£ù c÷ˆ¬î à¬ìò

Gôˆ¬î ªê¶‚°î™, C å¼«õ¬ô Ü‰î ªêò¬ô ªêŒ»‹ «ð£¶ êKò£ù

c÷ˆF™ ªêŒî™. Üõ¡ Ü¬î âƒA¼‰¶ ÜP‰¶ ªè£‡ì£¡ â¡ð¬î

A = B ñŸÁ‹ B = C H¡ù˜ A = C Üî£õ¶ Þ‰î GôˆF¡ êñÜ÷¾ c÷‹

ñK ñK ªè£´‚èŠð†´ Üî¬ù åŠH†´ ð£˜‚è º®Aø¶. Þ‰î

Å›G¬ôJ™ Ü÷i´ ªêŒî™ .«ñ½‹ Üšõ£Á è¼M¬ò ¬èò£Àî™

ñŸÁ‹ ñ£ŸP Ü¬ñˆî™ «ð£¡øõŸ¬ø ¹K‰¶ ªè£œ÷ º®»‹.

ªð¼‹ð£½‹ Ýó£Œ„Cèœ Ü÷i†´ è¼ˆ¶‚èO¡ õ÷˜„CJ¡ ªõŸP

â¡ð¶ Þì ÝŸø¬ô Ü÷ˆî™ «ð£¡ø¶.

ºîL™ å¼ CPò °ö‰¬î º¡ð¼õ ðœOJ™ Üî£õ¶ 6

õ¼ìƒèÀ‚°œ c÷ ñ£Ÿøˆ¬î‚ è£†´õ¶ Þ™¬ô. Üõ˜èO¡ º®¾

ºîL™ CPò / åŸ¬øŠ ¹ô¡è£†CJ¡ Ü®Šð¬ìJ™ Ü¬ñ»‹. Þ‰î

õòF™ °ö‰¬î Þó‡´ «è£´è¬÷ (ðì‹ 1.2) Þó‡´‹ êññŸø

¹œOèÀ¬ìò «è£´èœ â¡ð¬î ªîK‰¶ªè£œAø¶.

_________________________________________________________

_________________________________________________________

ðì‹ 1.

ð°F ñŸÁ‹ ªî£°FèO¡ G˜íòƒèœ Iè c‡ì «ï˜;;«è£†´

ðKñ£íƒèœ (Þ¶ ªðKò¶ ãªù¡ø£™ Þ¶ c÷ñ£ù¶ â¡ð¬î

°ö‰¬î àí˜Aø¶. 6 to 7 õò¶œ÷ °ö‰¬î Ü÷M™ô£î G¬ôò£ù

Üô°è¬÷ ¬èè÷£™ îƒè÷¶ àòóˆ¬î Ü÷ˆî™ Ý°‹.

å¼ °ö‰¬î ªî£°FèO™ ñ£Áð£´è¬÷ ¹K‰¶ ªè£œ÷ 7 to 8

õò¶œ÷ °ö‰¬î âŠ«ð£¶ FóõˆF¡ Ü÷M¬ù Üèô‹ ñŸÁ‹

Üî¡ àòóˆ¬î ªè£‡´ èí‚A´‹ ªð£¿¶ ªîK‰¶ ªè£œA¡øù.

²ñ£˜ 8 to 10 õò¶œ÷ °ö‰¬îò£™ Ü÷i†´ ªî£°Fè¬÷ Ãø¾‹

Üî¬ù CPò Ü÷¾è÷£è HK‚è¾‹ Þò½‹. Þ‰î G¬ô õ¬ó Ü÷i´

â¡ð¶ ºòŸC(ñ)H¬öè÷£™ è‡´H®‚A¡øù. ÞŠªð£¿¶

20



°ö‰¬îè÷£™ ¶™Lòñ£è èí‚W´ ªêŒò Þò½‹. ðóŠ¹ ñŸÁ‹

ªî£°FèO¡ Ü÷¾œ÷ ªð£¼†èO¡ ÞìñFŠ¬ð H¡Â‚°

îœÀA¡øù˜.

å¼ °ö‰¬î ÞÁF G¬ôJ™ ðóŠð÷¾ (ñ) ªî£°Fè¬÷ èí‚Aì

«ï˜;«è£†´ ðKñ£íƒè¬÷ (c÷‹, Üèô‹, àòó‹ (ñ) Üî¡ Üì˜ˆF )

10 õö 11 õòF¡ H¡ G¬ôèO™ Þò™A¡øù.

â‡Eò™ C‰î¬ùèO¡ õ÷˜„C (Development of the Spatial Thinking)

ºîL™ °ö‰¬îò£ù¶ àí˜¬õ î£¡ õ£¿‹ àô¬è å¿ƒèŸø

å¡ø£è è¼¶Aø¶. â‰î ðìˆ¬î Üõ¡ / Üõù£™ «õÁð´ˆî «ñ½‹

Ü¬î G¬ô»Áˆî c‡ì «ïó‹ º®A¡ø«î£ Üšõ£Á °ö‰¬î ¹ô¡

ßì£ù G¬ô¬òî£‡® Í¡Á Ü™ô¶ õòF™ Í®ò (ñ) Fø‰î

ðìƒè¬÷ MˆFò£êŠð´ˆî º®»‹.Ýù£™ Ü¬ùˆ¶ CPò Í®ò

ðìƒè÷£™ ªêšõè‹Þ ê¶ó‹ , õ†ì‹ (Ü) Üô°, º‚«è£í‹ Þ¬õ

Ü¬ùˆ¶‹ å¡Á â¡Á Üî¬ù Ü«î «ð£™ õ¬óò¾‹ ðö°A¡ø£¡.

7 to 8 õòFô£ù °ö‰¬îè÷£™ ê¶ó‹, ªêšõè‹ ñŸø‹ ï£Ÿèó‹

«ð£¡øõŸP¡ «õÁð£´è¬÷ ÜPò¾‹, °ö‰¬îJ¡ 10 õò¶ õ¬ó

Þ‰î ðìˆ¬î êKò£è Ü¬ìò£÷‹ è‡´ «õÁð´ˆF è£†ì º®»‹.

«ñ½‹ Cô C‚èô£ù G¬ôèO¡ C‰î¬ù õ÷˜„C¬ò H¡ õ¼‹

ï£†èO™ ð¼Šªð£¼œ G¬ôJ™ ªêò™ðÁø¶.

E3 : º¡ â‡èO¡ è¼ˆ¶Š ªð£¼†è¬÷‚ ªè£‡´ «õÁð´ˆF

ÜPè. â‰î G¬ôJ™ º¡ â‡ èŸø¬ô «õÁð´ˆF è£í

Þò½‹.

E4 : â‰î ÜPFø¡ G¬ôJ™ ªð¼‹ð£ô£ù èí‚W†´

è¼ˆ¶‚èœ õ÷Ã¡øù.?

E5 : Þ‰î ÜPFø¡ G¬ôJ™ ªð¼‹ð£ôù èí‚W†´ è¼ˆ¶‚èœ

õ÷Ã¡øù..

E6 : Ü÷i´è¬÷ è£ˆî™ â¡ø£™ â¡ù?

21



1.3 °ö‰¬îŠ ð¼õˆF™ èEî‹ èŸø™:(Mathematics Learning during Early Childhood)

«ñ«ô è‡ì Mõ£îƒèO™ Þ¼‰¶, èEî è¼ˆ¶õ÷˜„CJù£™

â‡íŸø C‰î¬ùè¬÷ ðŸP C‰F‚èô£‹. Üç¶ ò£ªîQ™ âOî£ù

õNJ™ èí‚°è¬÷ °ö‰¬îèœ âšõ£Á ¬èò£œõ£˜èœ?

â¡ð¬îŠðŸP»‹ õòFŸ«èŸð ðœOŠð¼õˆF™ èEî‹ èŸø™

õNè¬÷Š ðŸP è£‡«ð£‹.

1.3.1 èEî‹ èŸÁ‚ªè£œõîŸè£ù âOò õNº¬øèœ :

(Ways of Learnign Mahematics)

Ýó‹ðè£ô èŸøL™Ãì èEî‹ èŸÁ‚ ªè£œõîŸè£ù â‰î å¼

îQˆ¶õñ£ù ñŸÁ‹ àÁFò£ù õN Þ™¬ô. º‰¬îò

Mõ£îˆFL¼‰¶ ªî£ì‚è ï£†èOL¼‰¶ èEî‹ èŸøL¡

î¡¬ñèÀ‚° Cô è¼ˆ¶‚èœ à¼õ£AJ¼‚èô£‹ «ñ½‹ Ã´îô£è

èEî‹ èŸHˆîL¡; ð‡¹èœ ðŸPò Cô °PŠ¹èœ Þƒ«è àœ÷ù

âù ªõJ†Hó† (ÜƒAªôK 1995 ) â¡ðõ˜ ÃP»œ÷£˜.

ðœO‚°„ ªê™½‹ ð¼õˆFŸ° º¡ù«ó Ü‚°ö‰¬îèO¡

i†®L¼‰«î èEî‹ èŸø™ - èŸHˆî™ ªî£ìƒ°Aø¶. èEî‹ ¹Kî™

â¡ð¶ æ˜ Ü®Šð¬ì„ ªêòô£°ñ.; èí‚A´õîŸ° °ö‰¬îèO¡

ªê£‰î è¼ˆ¶‚è¬÷ õL»Áˆî «õ‡´‹. Üõ˜èO¡ C‚è™è¬÷ˆ

b˜‚è â¿îŠð†ì G¬ôò£ù ªð£¼œe¶ ÜFè º‚Aòˆ¶õ‹ îó

«õ‡´‹. Þî¡ eî£ù º‰¬îò ï¬ìº¬øè¬÷»‹ Gó£èK‚Aø¶.

èEî‹ å¼ ê‚Fõ£Œ‰î è¼Mò£è¾‹Þ àôèˆ¬îŠ ¹K‰¶

ªè£œõîŸ°‹Þ à‡¬ñò£ù ÜÂðõˆF™ «õÏ¡P Þ¼‚è¾‹

ðò¡ð´Aø¶.

Ü¡ø£ì õ£›‚¬è„ ÅöL™ èEî‹ èŸø™ °ö‰¬îèO¡

â‡íƒèOL¼‰¶ ªõOõ¼A¡øù. à‡¬ñò£ù Cô

è£óíƒèOù£™ èEî‹ Cô ªêò™ð£´èO¡ Íô‹ «õÏ¡P

àœ÷¶. ªêŒ¶ èŸøLù£™ èEî ªêò™ð£´èœ °ö‰¬îèœ Íô‹

ªõOŠð´A¡øù.

22



è£óí è£Kòƒèœ Íô‹ èEî‹ èŸø™ Iè¾‹ °¬øõ£è«õ

õL»ÁˆîŠðÁø¶. °ö‰¬îèO¡ ñù‹ à¼õƒè¬÷ õ÷˜„C ªðø

ªêŒò¾‹ ðí‹ ñŸÁ‹ è£AîˆF™ â‡íƒè¬÷ HóFðL‚è„

ªêŒAø¶.

°ö‰¬îèÀ‚°‹ ÝCKò˜èÀ‚°‹ Þ¬ì«ò õ½õ£ù àø¬õ

õ÷˜‚°‹ º‚Aò è¼Mò£è èEî‹ àœ÷¶. â¿¶«è£™ (ªð¡C™)

ñŸÁ‹ è£Aî ðJŸCJ¡ àîMò£™ °ö‰¬îèœ Iè¾‹ âOî£è¾‹ Þ

ÞòŸ¬è»ì¡ Ã®ò Å›G¬ôJ½‹ îƒèœ G¬ùŠð¬î á‚°MŠðî¡

Íô‹ èEî‹ «ñ‹ðÁ¡ø¶.

èEî‹ èŸðF™ ãŸð´‹ H¬öèœ º‚Aòñ£ù å¡ø£è

ãŸÁ‚ªè£œ÷ŠðÁø¶. ÜŠH¬öèœ Þ™ô£ õ‡í‹ F†ðñ£è

èí‚°èœ ªêŒò °ö‰¬îèœ èŸÁ‚ªè£œ÷™ «õ‡´‹. °ö‰¬îèœ

î¡Â¬ìò Føù£Œ¾ e¶ ê‰«îè‹ ãŸðì£õ‡í‹ M¬óõ£è

b˜¾è£µî™ «õ‡´‹;.

«ñŸè‡ì Mõ£îˆFL¼‰¶ ªî£ì‚è‚è™MJ™ èEî‹ âšõ£Á

èŸèô£‹ âù Cô Ü®Šð¬ì õNèO™ èõù‹ ªê½ˆî ºòŸC ªêŒî™

«õ‡´‹. Þ¬õ ò£¾‹ õ°Šð¬ø Å›G¬ôJ™ èEî è™M èŸøL¡

õNº¬øò£°‹.

èEî «ï£‚èƒè¬÷ ¬èò£Àî™ : (Manipulation of Objects)

èEî «ï£‚èƒè¬÷ ¬èò£Àî™ â¡ð¶ ªî£ì‚è‚è™Mñ£íõ˜èœ èEî ÜP¬õŠ ªðÁõ¬îŠðŸP‚ ÃÁAø¶.ªî£ì‚è‚è™M ñ£íõ˜èœ èEî ÜP¬õŠ ªðÁõî¡ Íô‹åŠH´î™, õ¬èŠð´ˆ¶î™ â‡µî™ «ð£¡ø Ü®Šð¬ì„ªêò™ð£´è¬÷ ÞòŸ¬è»ì¡ ¬èò£÷ «õ‡´‹ .

ªî£ì‚èŠ ðœO ñ£íõ˜èœ èEî‚ è¼‚¶‚è¬÷ âOF™ðJ½‹ Mîñ£è ñ£íõ˜èO¡ ðö‚èõö‚èƒèÀ‹, ð£ìŠ¹ˆîèÜô°èÀ‹, ð™«õÁ ªð£¼†è¬÷»‹, è¼ˆ¶‚è¬÷»‹ ñ£íõ˜èœ

M¼‹H èŸ°‹ Mîñ£è ð£ì‚ è¼ˆ¶‚è¬÷ èEîˆF™ Ü¬ñ‚è

«õ‡´‹.

23



ðEèœ ªêŒõîŸè£ù ÞQ¬ñò£ù Åö™è¬÷ à¼õ£‚°î™; :

(Placing tasks in Meaning Contents)

èEî‹ èŸø™ à‡¬ñò£ù ÅöL™, ðE;è¬÷„ ªêŒò

àî¾A¡ø¶. ñ£íõ˜èœ âOî£è¾‹, «õèñ£è¾‹ î¡ ªê£‰î

ºòŸCJ™ î¡Â¬ìò «î¬õè¬÷ º¬øê£ó£ ñŸÁ‹ º¬ø ê£˜‰î

º¬øèœ Íôº‹ «î¬õ¬ò Gõ˜ˆF ªêŒA¡ø¶. Cô ñ£íõ˜èœ

ðœOJ™ ¸¬ö‰î¾ì¡ °¬øð£´èÀì¡ ªêò™ðÁ¡øù˜.

Üî£õ¶ º¬øò£ù ªð¡C™ ñŸÁ‹ è£Aî â¿¶ º¬øò£ù¶

ªîOõ£ù «ï£‚èƒèÀì¡ èí‚° ªêŒò ã¶õ£Aø¶. ñ£íõ˜èœ

èŸÁ‚ªè£œÀ‹ õNè¬÷Š ðŸP Ýó£Œ„C ê£¡Áèœ â¡ù

ªîKM‚Aø¶ â¡ø£™ ï£‹ ªêŒò «õ‡®ò ªêò¬ô à‡¬ñò£ù

Hó„ê¬ù»ì¡ Ýó‹H‚è «õ‡´‹ H¡ù˜ ÜõŸPL¼‰¶

HóFGF¶õˆFŸ° ²¼‚èñ£è «õ¬ô ªêŒò «õ‡´‹ â¡ðî£°‹;.

°ö‰¬îèœ î‹ Ü¡ø£ì õ£›M™ èEî ÜP¬õŠ

ðò¡ð´ˆ¶õîŸ° ãó£÷ñ£ù õ£ŒŠ¹èœ àœ÷ù. M¬÷ò£´‹

«ð£¶‹ F‡ð‡ìƒè¬÷Š HKˆ¶ à‡µ‹ «ð£¶‹, ªõš«õÁ

ªêò™è¬÷ °¿õ£èŠ HKˆ¶ ªêò™ð´ˆ¶‹ «ð£¶‹ , M´º¬ø

ï£†è¬÷ â‡µî™ «ð£¡ø ªêò™ð£´èO½‹ èEî‹ ðò¡ðÁø¶.

°PŠð£è Þ÷‹ °ö‰¬îèOìˆF™ Hó„ê¬ùèœ

èŸð¬ùèOL¼‰¶ Hø‚è «ïKÁø¶. î£«ñ èŸð¬ùJL¼‰¶

à¼õ£‚Aò è¬îèÀ‹ M¬÷ò£†´ º¬øèÀ‹ Üõóõ˜ ãŸð´ˆFò

ê†ìƒèÀ‚° W› à‡ì£‚A Hóè£êñ£Œ Þ¼‚è ºòŸC ªêŒõ˜. Þ¬õ

èEîˆF¡ ðò¡ð£†®Ÿè£ù Cô â´ˆ¶‚è£†´èœ Ý°‹.

Þ÷‹ °ö‰¬îèœ î‹º¬ìò èEî Fø¬ñ¬ò î£«ñ º¡«ùŸø‹

Ü¬ìò ªêŒA¡øù. î£«ñ à¼õ£‚Aò à‡¬ñò£ù C‚è™è¬÷

âF˜ªè£œÀ‹ õ¬èJ™ ¹K‰¶ ªè£œAø£˜èœ. Þ¶«ð£¡ø Åö™èO™

ÝCKò˜èœ ñ£íõ˜èÀ‚° õNè£†´î™ ñ†´‹ ªè£´ˆî£™

«ð£¶ñ£ù¶

24



\Üî£õ¶ ªêò™º¬ø»‹ è¼ˆ¶‚èÀ‹ ð™«õÁ Åö™èO™

Üõ˜èÀ‚° õöƒèŠðì «õ‡´‹. Þ‰î º¬øJù£™ É‡ìô£ù¶

ÞòŸ¬èJ¡ ªêò™º¬ø»ì¡ ªî£ì˜¹ Þ™ô£îFL¼‰¶

ªî£ì˜¹¬ìòõŸ¬ø b˜ˆ¶ ªè£œ÷ º®»‹. ÞÁFJ™ îƒèO¡

ªêò™ð£†®¡ Ü®Šð¬ì‚ ÃÁð£´èÀ‚è£è ²¼‚èô£‹ â™ô£

êñòƒèO½‹ ï£‹ ñùF™ ªè£œ÷ «õ‡®ò¶ ò£ªîQ™ èEî‹

õL¬ñ¬ò ªðø °ö‰¬îèœ à‡¬ñG¬ôJ¡ ï‹H‚¬è»ìÂ‹

ÜõŸ¬ø ªêŒ¶ èŸÁ ¹K‰¶ ªè£œÀî™ «õ‡´‹.

HóFGFˆ¶õ‹ ªðÁõîŸè£ù ð™«õÁ õNG¬ôèœ:

(Representation in Multiple Ways)

²¼‚èñ£ù õNJ™ èEîˆ¬î ñ£íõKì‹ FEˆî™ â¡ð¶

å¼ º¬øò£°‹. Þ‹º¬ø ñ£íõ˜èœ HóFGFˆ¶õˆF¡

º‚Aòˆ¶õˆ¬î à¼õ£‚°Aø¶. ñùF¡ HóFGFˆ¶õ‹ â¡ð¶

èŸð¬ù õ£Jô£è Gè›¾èœ Ü™ô¶ ªêò™º¬øè¬÷ èŸÁ àí˜î™

Ý°‹. Þ‹º¬ø °ö‰¬îèÀ‚° î£«ñ èEîˆF¡ b˜¾ ªêŒõ¶ å¼

ï™ô õ£ŒŠð£è Ü¬ñAø¶.

ÞîŸè£ù ªêò™º¬ø»‹Þ îò£KŠ¹‹; ê£î£óí º¬øJ™

îò£K‚èŠð´Aø¶. Þî¡Íô‹ °ö‰¬îèœ èEîˆF¬ù

âO¬ñò£è¾‹ à‡¬ñJ¡ õ£Jô£è¾‹ ñŸø ð£ìƒèÀì¡ ªî£ì˜¹

ð´ˆF èŸÁ‚ ªè£œA¡øù˜. îŸ«ð£¶ Ü¬ùˆ¶ èEî

ÝŒõ£÷˜èÀ‹ °ö‰¬îèœ èEî ÜPM¬ù ªðø Ü™ô¶ C‰F‚è

â¿F ðö°õîŸ° º¡¹‹ , °Pf´è¬÷ ðò¡ð´ˆ¶õîŸ° º¡¹‹ «ðC

èŸA¡øù˜ â¡ð¬î àÁFªêŒ¶œ÷ù˜. «ü‹v 1985 â¡ðõK¡

ÝŒM¡ð® do, Talk and Record ªêŒ»ƒèœ , «ð²ƒèœ , ðF¾ ªêŒ»ƒèœ

ú âù‚ÃP»œ÷£˜ . Þî¡Íô‹ èŸÁ‚ ªè£œ÷ 5 ð®G¬ôè¬÷

à¼õ£‚A»œ÷ù˜.

èŸðõ˜ î‹ñ¬ìò èŸð¬ùˆ Fø¬ù HøKì‹ Ãø™.

î£‹ èŸð¬ù ªêŒî¬î â¿¶«è£™ ( ªð¡C™ ) Íôñ£è«õ M÷‚è

; õ¬è ªêŒA¡øù˜

25



î£‹ èŸÁ ªè£‡ì¬î õ‡í â¿¶ «è£L¡ Íô‹ ªêŒ¶

è£†Á¡øù˜

Üõ˜èœ ðò¡ð´ˆFò ªêò™ º¬øJ¡ ªî£ì˜„Cò£ù

è¼ˆ¶‚è¬÷ ²¼‚èñ£è ªðÁA¡øù˜.

ÞÁFJ™ î‹ñ£™ ãŸÁ‚ªè£œ÷Šð†ì îóG¬ô ÜPMŠ¬ðŠ

ªðÁA¡øù˜.

å¼ õ°ŠHL¼‰¶ ñŸªø£¼ õ°Šð¬ø‚° ªê™½‹ Üšõ°Š¹

ñ£íõ˜èO¡ ªêò™ð£´èÀ‹ C‰î¬ùèÀ‹ àòKò ÞìˆFŸ°„

ªê™A¡øù.

º¡«ùŸøˆFŸè£ù ñ£ŸÁ õNº¬øèœ:

(Developing Alternative Strategies)

ñ£íõ˜èO¡ îQˆFø¬ù èí‚W´ ªêŒõîŸ°

ð£ìŠ¹ˆîèˆFL¼‚°‹ õNJ¬ùˆ îM˜ˆ¶ «õÁõNJ½‹

èí‚Aìô£‹. ðœO‚° ªê™ô£îõ˜èœ Ãì è®ùñ£ù èí‚¬è

âOòº¬øJ™ ªêŒ¶ M´õ˜. ðœOJ™ ð®ˆî å¼õ˜ º¬øò£è å¼

èí‚AŸ° b˜¾è£í CóñŠð´‹ «ð£¶ ðœO‚°„ ªê™ô£ñ™

º¬øò£è ð®‚è£î æ¼õ˜ Ü‚èí‚¬è âOF™ b˜Šð£˜. Üîù£™

ðœOJ™ ð®ˆî èí‚° õ£›‚¬èJ™ ï¬ìº¬ø ªêò™èÀ‚°

ðò¡ð´õF™¬ô â¡ø å¼õ¬è ñùŠ«ð£‚° ãŸðÁø¶

¹Fò õNº¬øèœ ò£¾‹ ñ£íõ˜èÀ‚° àî¾õF™¬ô Ýù£™

å¼õ˜ ñŸªø£¼õ¼ì¡ Þ¬í»‹ «ð£¶ î¡Â¬ìò Fø¬ñ¬ò

ªõOŠð´ˆ¶A¡øù˜. Þ¼ ïð˜èÀ‚° Þ¬ì«ò è¼ˆ¶Š ðKñ£Ÿø‹

ï¬ìªðÁ‹ «ð£¶ å¼õ¼¬ìò è¼ˆ¶ ñŸªø£¼õ¼‚°

ðKñ£ŸøŠðÁø¶. Þšõ£Á ðKñ£ŸøŠð´‹ «ð£¶ C‚è½‚è£ù b˜¾

è‡´H®‚èŠð†´ ï¬ìº¬øŠð´ˆîŠðÁø¶. Üîù£™

ñ£íõ˜èO¬ì«ò èEî‹ èŸøL™ Ý˜õ‹ ãŸðÁ¡øù. èEî‹

èŸH‚°‹ ÝCKò˜ ñ£íõ˜èOì‹ è´¬ñò£è¾‹ âOF™ ñ£íõ˜èœ

Üõ¬ó Üµèº®ò£îõó£Œ Þ¼‰î£½‹ å¼ ñ£íõ¬ùŠ ðŸPò

26



î¡Â¬ìò è¼ˆ¶‚è¬÷ ñ£ŸP‚ªè£œ÷£îõó£è¾‹ Þ¼‚°‹ «ð£¶

ñ£íõÂ‚° èEî ð£ìˆF™ Ý˜õ‹ ãøðì£¶ â‰î ð£ìˆF¡ ÝCKò˜

ñ£íõ¼‚° H®ˆîõó£è àœ÷£«ó£ Ü‰î ð£ì‹ ñ£íõ¼‚° H®ˆ¶

M¼‹H èŸð˜ . ñ£íõQ¡ õò¶ ÜÂðõˆFŸ° ãŸø èEî

C‚è™è¬÷»‹ ï¬ìº¬ø õ£›‚¬è»ì¡ ªî£ì˜¹¬ìò èí‚°è¬÷»‹

âOò èŸHˆî™ º¬øèÀì¡ ¬èò£‡´ ñ£íõ¬ù á‚èŠð´ˆF

èŸøL™ Ý˜õˆ¬îˆ É‡® èŸH‚°‹«ð£¶ èEî‹ å¼ ÞQ¬ñò£ù

ð£ìñ£è Ü¬ñ»‹.

Hó„ê¬ùè¬÷ b˜ˆî½‹ Hó„ê¬ù‚è£ù Mù£ â¿Š¹î½ñ:;;

( Problem Solving and Problem Posing)

èEîˆF™ b˜¾ è£µ‹«ð£¶ õ£›‚è¬èJ™ âšõ¬èJ™ b˜¾

è£íº®»‹ âù ÜP‰¶ªè£œ÷ô£‹. Þî¡ ªêò™ º¬øèœ

ñ£Áð†ì£½‹ ÜõŸP™ Cô åŠ¹¬ñèÀ‹ ¹Kî½‹ ªõš«õÁ

º¬øèO™ b˜¾ è£íŠ ;ðò¡ðÁø¶ Þ‰î b˜¾ è£µ‹ º¬øò£ù¶

ñ£íõ¬ù îQò£è¾‹, °¿õ£è¾‹ ªêò™ðì ñ£íõ¬ùˆ

É‡Áø¶. èEî C‚è¬ô ¹K‰¶ ªè£‡´ Üî¡ªêò™ º¬øè¬÷Š

H¡ðŸP Ü¶ ªî£ì˜ð£ù Mù£M¬ù â¿ŠH ñ£íõQ¡ ÜP¬õ

«ê£Fˆî™. Þ‰î ºòŸC¬ò ªî£ì˜ ªêòô£è H¡ðŸø¾‹. Þî¬ù ðŸP

îèõ™èœ ð£ì‹ 4 ™ MKõ£è ÃøŠð†´œ÷¶.

E7 Mù£M¡ Íô‹ èEî ÜP¬õ õ÷˜‚è º®»ñ£? àù¶

è¼ˆF¬ù â´ˆ¶‚è£†´ì¡ M÷‚°è.

E8 ªð£¼œè¬÷ ¬èò£ÀõFL¼‰¶ â‡Eò™ ÜP¬õ õ÷˜‚è

º®»ñ£? àî£óí‹ î¼è.

1.3.2 èEî‹ ðŸPò ðò‹ : (Mathematics Phopia )

Cô ªêò™èœ; ñ£íõ˜èœ èEîˆF™ â‰î Ü÷¾ ß´ð£´ì¡àœ÷ù˜ â¡ð¬î ÃÁA¡øù˜. “èEîˆF™ b˜¾ è£µ‹; «ð£¶ ñù‹ñŸÁ‹ ÜP¾ ªõÁ¬ñò£è àœ÷î£è àíÃ«ø¡, èEî ÜP¾

ÜŸøõù£è â‡µA«ø¡, Þ‰î‚ è£óíˆFù£™ å¼ CPò èí‚AŸ°

Ãì b˜¾ è£í º®òM™¬ô

27



èí‚AŸ° b˜¾ è£í ºò½‹ «ð£¶ ñù‹ ñŸÁ‹ ÜP¾

ªõÁ¬ñò£è àœ÷î£è àí˜A«ø¡.èEî ÜPM™ô£îõù£è

â‡µA«ø¡.Þ‚è£óíˆFù£™ å¼ CPò èí‚AŸ° Ãì b˜¾ è£í

º®òM™¬ô.

“èí‚AŸ° b˜¾ è£í Üî¡ M¬ì å¡Á ñ†´«ñ ÜîŸ°

«ñŸð†ì G¬ôJ™¬ô. Ü‰î M¬ì¬ò ï£¡ êKõó è‡´H®‚èM™¬ô

â¡ø£™ «î£™M Ü¬ì‰îõ¡ â¡Á å¶‚èŠð´«õ¡. Þîù£™ èEî‹

èŸè Þòô£¶ â¡ø â‡í‹ à¼õ£Aø¶.”

“èEîˆ«î˜¾ â¡¬ù ðòŠð´ˆ¶Aø¶ â¡ø ðòˆî£™

àœ÷ƒ¬è Mò˜‚Aø¶. «õèñ£è ²õ£C‚A«ø¡. â¡ù£™ â¡ «î˜¾ˆ;

î£¬÷ êKò£èŠ ð£˜‚èº®òM™¬ô. Ü¼A™ àœ÷õ˜è¬÷ ð£˜‚°‹

«ð£¶ Üõ˜èœ Ü¬ùõ¼‹ CøŠð£è ªêŒõ¶ «ð£™ â‡í‹

«î£¡ÁAø¶ .

“ ï£¡ èEî õ°ŠH™ å¡P™ Ãì êKò£ù M¬ì ÃPòF™¬ô

â¡Á ñ£íõ˜ àí˜Aø£¡. ÝCKò˜ â¡ù ÃÁAø£˜; âù ¹K‰¶

ªè£œ÷ º®òM™¬ô. Þîù£™ èEî‹ «î¬õJ™¬ô â¡ø º®¾‚°

õ¼A¡«ø¡.”

“å¡ð¶ õò¶ ºî«ô èEîˆ¬î ªõÁ‚è Ýó‹Hˆ¶ M†«ì¡.

î‰¬î ªð¼‚è™ õ£ŒŠð£´ ªîKòM™¬ô â¡Á î‡ì¬ùèœ

õöƒAù£˜. â¡ î‰¬îò£˜ å¼ èEî ÝCKòó£è Þ¼‰î «ð£¶ èEî‹

«ð£ì õŸ¹ÁˆFù£˜. ªêŒò º®ò£î «ð£¶ âù‚° î‡ì¬ù»‹

ªè£´ˆî£˜.

“èEîˆF™ ÅˆFóƒèœ, MFèœ, «îŸøƒèœ ÜFèñ£è ñùù‹

ªêŒ¶ ë£ðèˆFŸ° ªè£‡´ õó «õ‡®»œ÷¶. èEî‹

õ£›‚¬è»ì¡ ªî£ì˜¹¬ìòî£è Þ™¬ô. èEî‹ å¼ êLŠÌ†´‹

ð£ì‹”

«ñŸè£µ‹ õ£˜ˆ¬îèO™ ã«îÂ‹ å¡¬øò£õ¶ ñ£íõ˜èœ

ºî™ ªî£ì‚è‚è™M ÝCKò˜èœ õ¬ó èEîˆFŸ° b˜¾ è£µ‹

«ð£¶ ÃÁõ¬î «è†´œ«÷£‹.

28



ðœOŠ ð£ìƒèœ Ü¬ùˆFL¼‰¶‹ èEî ð£ì‹ ñ†´‹

«õÁð†´œ÷¶.

ñ£íõ˜èœ; õ°Šð¬øJ™ èEîˆ¬î âO¬ñò£è èŸè 4

õNº¬øèœ îóŠð†´œ÷ù.

ºîô£õî£è IèŠªðKò èEî «ñ¬îèœ Ãì îõÁèÀ‚° H¡«ð

êKò£ù MFè¬÷«ò£ «îŸøƒè¬÷«ò£ è‡ìP‰¶œ÷ù˜.

èEî‹ ªêŒ»‹ «ð£¶ ñ£íõ˜èÀ‚° ãŸð´‹ îõÁè¬÷

êKªêŒ¶ èí‚¬è êKò£è ªêŒò á‚èŠð´ˆîô£‹. ðœOJ™

èEîˆF™ ²¼‚èñ£è‚ Ãø °Pf´è¬÷ ðò¡ð´ˆ¶A«ø£‹ . Þ¶

ñ£íõ˜èÀ‚° è®ùˆî¡¬ñ¬ò à‡ì£‚°Aø¶.

Þó‡ì£õî£è ñ£íõ˜èœ î£œ, â¿¶«è£™ ¬õˆ¶ ñ†´«ñ

èEî‹ ªêŒ¶ M¬ì ÃÁA¡øù˜. ñùF«ô èí‚A†´ M¬ì

ÃÁ‹ Fø¬ñ ÜŸøõ˜è÷£è àœ÷ù˜ ÜîŸè£ù õNº¬øè¬÷

èŸÁîó «õ‡®»œ÷¶ .

Í¡ø£õî£è ðœO èEî‹ Ýù¶ å¼ °PŠH†ì º¬øJ™

ñ†´«ñ ªêŒò«õ‡®»œ÷¶. °ö‰¬îèœ ñŸøõ˜èœ

àîMJ™ô£ñ™ â‡è¬÷ ¬õˆ¶ ªè£‡´ ñ†´«ñ èí‚°‚°

M¬ì è‡´H®‚A¡øù˜. Þ‰î M¬ì¶™Lòñ£è Þ¼ŠH¡

Ü¬ùˆ¶ ñ£íõÂ‹ Þ‰î º¬ø¬ò ¬èò£ÀA¡øù˜. Þîù£™

èEî‹ êŸÁ è®ùñ£è àœ÷¶

ï£¡è£õî£è à÷Mò™ º¬øJ™ Ýó£Œ‰î£™ àôA™ àœ÷

°ö‰¬îèœ Ü¬ùõ¼‹ èEî‹ èŸø½‚è£ù õNº¬øè¬÷

ªðÁA¡øù.

É‡ì™ Íô‹ èŸø™ (Learning Induction)

É‡ì™ Íô‹ èŸø™ Iè¾‹ ê£ˆFòñ£è àœ÷¶ (å¼

õó‹HL¼‰¶ ªð£¶ MFèœ Ü™ô¶ õ®õƒè¬÷

á´¼MŠð£˜Šð¶) Ýù£™ ¶™Lòñ£ù M÷‚èˆFŸ° åŠd†´

29



Ü÷M™ °¬øõ£è àœ÷¶ (ªð£¶MF º¬øJL¼‰¶ °PŠH†ì

õö‚èˆ¬î î£‡®ò ªêò™º¬ø) ªêò™º¬øò£è Þ¼‰î£½‹

«õÁõ¬èò£è Þ¼ŠHÂ‹ É‡ì™ Íô‹ èŸ°‹ è™M

ñ£íõ˜èO¬ì«ò âOF™ ªê¡ø¬ìAø¶ . Þ‰î º¬øJ™ èŸø™

âOF™ ñùF™ ðFò ¬õ‚è¾‹ ë£ðèˆFŸ° ªè£‡´õó¾‹

É‡ì™ è™M º¬ø ðò¡ðÁø¶.

õ¬óòÁ‚èŠð†ì ðE G¬ù¾ Fø¡

(Limited “Working Memory” Capacity )

õòFŸ° ãŸø ë£ðèˆFø¡ ñ†´«ñ ñ£íõ˜èOì‹ Þ¼‚°‹

(â.è£) Miller ÃŸÁð® 7 õ¬èò£ù ªêò™º¬øè¬÷ ïñ¶ °ÁAò è£ô

G¬ùM™ ªêò™º¬ø G¬ù¾è¬÷ ¬õˆ¶ªè£œ÷ º®»‹. Þ¶

°ö‰¬îèO¡ 17å9 â¡ø â‡¬í âOF™ ªð¼‚A M¬ì ÃÁõ˜. Þ¬î

184 å 596 â‡è¬÷ ªð¼‚A M¬ì ÃÁ‹ ªêò™º¬øèœ ªêŒõF™

è®ù‹. å¼ ð°F¬ò «î£ŸÁMˆî£½‹ º‰¬îò èí‚W†®¡ M¬÷¾

ñø‚èŠðì‚Ã´‹. Iè¾‹ CPò â‡E‚¬èJô£ù °ö‰¬îèÀ‚°

Ü¬ùˆ¶ «ïóƒèO½‹ °¬ø‰î C‚èô£ù b˜¾è¬÷ b˜‚è Þò½‹ .

3.ÜPî¬ô ÜP‰¶í˜î™ ðŸPò MNŠ¹í˜¾ ñŸÁ‹ è†´Šð£´

(Development of ‘Meta – Cognitive’ Awareness and Control )

ñQî ªêòô£‚èˆF¡;; Í¡ø£õ¶ ªð£¶õ£ù Ü‹ê‹ Þ‰î º¬ø

ÜPî¬ô ÜP‰¶í˜‰¶ èŸÁ‚ªè£œÀî™ Ý°‹. C‰î¬ù ñŸÁ‹

èŸøô£ù¶ Üõó¶ õNèO™ Þ¼‰¶ ªðøŠðÁ¡øù. ÜªñK‚è

à÷Mòô£÷˜ è¼ˆ¶Šð® ÜPî¬ô ÜP‰¶í˜îô£ù¶ Üõó¶

ïìõ®‚¬è ñŸÁ‹ è†´Šð£´èO™ ÜFè Fø¡ ªðÁA¡øù .

ñ£íõ˜èœ èEîŠ Hó„ê¬ùè¬÷ b˜‚°‹ õNè¬÷ ÜP‰¶

b˜‚A¡øù˜ . ð™«õÁ Hó„ê¬ùè¬÷ ÜP‰¶ b˜Šðî¡ Íô‹

ñ£íõ˜èœ á‚èŠð´ˆîŠðÁ¡øù˜. C‚è¬ô b˜ŠðîŸè£ù

M÷‚èˆ¬î ªðÁA¡øù ñŸÁ‹ C‚èL¡ b˜‚°‹ õNº¬øèO™ Cô

«ïóƒèO™ êLŠð£ùî£è¾‹Þ ð£óñ£è¾‹Þ Þ¼‚A¡øù . ÜPî¬ô

ÜP‰¶í˜î™ â¡ð¶ Åö™ ñŸÁ‹ Gð‰î¬ù‚°†ð†ì ÜP¾

30



î¡¬ùˆî£«ù ÜPî™ ÝAòõŸ¬ø àœ÷ì‚Aò C‚è¬ôˆ b˜‚è‚

Ã®ò ÜP¾ ê£˜‰î ªêò¬ô °P‚A¡ø¶.

ÜPî¬ô ÜP‰¶í˜îô£ù¶ â¡ð¶ àí˜¾ Ì˜õñ£ù¶ .

«ñ½‹Þ Þ¶ ð¬ìŠð£Ÿø™ è™M º¬ø‚° Ü®Šð¬ìò£è

Ü¬ñ‰¶œ÷¶ Üî£õ¶ èŸÁ‚ªè£œõîŸ° H¡ðŸÁ‹ õNº¬øè¬÷

à‡ì£‚A»œ÷¶ . Þî¬ù ð£ìñ£è ªõOŠð´ˆ¶õî¡ Íô‹ b˜M¡

¸†ðƒèœ èŸ«ð£K¡ Åö™ Þ Gð‰î¬ù‚°†ð†ì ÜP¾ˆFø¡

ÝAòõŸ¬ø ªîK‰¶ ªè£œ÷ º®Aø¶. èŸ«ð£˜‚° èŸø™ ð£¬îèO™

ªê™õîŸ°‹ Þ õNè£†´î™ ªðÁõîŸ°‹ ã¶õ£Aø¶. è¬î ïñ¶

Ü¡ø£ì õ£›‚¬èJ¡ ªî£ì˜¹¬ìò¶ . ñ£íõ˜èÀ‚A¬ì«ò

îèõ™è¬÷ ðKñK‚ ªè£œõF™ ðôMî Þì˜ð£´èœ ãŸðÁ¡øù .

èEîˆF™ èŸÁ‚ªè£œõF™ Cô Þì˜ð£´èœ Þ¼Šðî£™ ñ£íõ˜èœ

Þ¬ì«ò å¼Mî ðò‹ à¼õ£A¡;øù. èEî‹ èŸÁ‚ªè£œõF™ ãŸð´‹

ªð£¶õ£ù ðò‹ H¡õ¼ñ£Á.

èEîˆ¶ì¡ âF˜ñ¬øò£ù ÜÂðõ‹:

(Prior negative Experiences with Methematics )

ê£îèñŸø ðœO Å›G¬ô : (Unfavourable School Climate)

ê£îèñŸø ðœO;„ Å›G¬ôJ™ Iè¾‹ è´¬ñò£ù å¿‚è ªïP

º¬øèœ è¬ìH®‚èŠðÁ¡øù. ñ£íõ˜èO¡ C‰î¬ù‚° º¿

²î‰Fó‹ ªè£´‚è£ñ™ è®¬ñò£ù ð£ìˆF†ì‹

è¬ìH®‚èŠðÁ¡øù.èEî C‚è¬ô b˜‚è ñ£ŸÁ õNè¬÷

ñ£íõ˜èÀ‚° «î˜‰ªî´ˆ¶ ªè£´‚è «õ‡´‹.

ªðŸ«ø£˜ ñŸÁ‹ ÝCKòK¡ á‚èI¡¬ñ :-

(Lack of encouragement from parents and or teachers)

èEî è¼ˆ¶‚è¬÷ èŸÁ‚ªè£œõF™ ñ£íõ˜èÀ‚° å¼ Mî

ðò‹ ãŸðÁø¶.ÞšMî ðòˆ¬î «ð£‚°õîŸ° ÝCKò˜ ñŸÁ‹

ªðŸ«ø£˜èO¡ á‚èI¡¬ñ«ò º¿¬ñò£ù è£óí‹ Ý°‹.

31



èEî‹ èŸøLù£™ ãŸð´‹ ðò‹ è£óíñ£è ñùÜ¿ˆî‹

ãŸðÁ¡ø¶. èEî ÝCKò˜èœ ñŸÁ‹ ªðŸ«ø£˜èœ ñ£íõ˜èœ

èEîˆF™ «ñ½‹ Fø¡ªðø «õ‡´‹.â¡Á ªî£ó†„Cò£ù Ü¿ˆî‹

ªè£´Šðî£™ êó£êK ñŸÁ‹ eˆFø¡ I‚è ñ£íõ˜èœ Ãì èEî‹

èŸðF™ Ü„ê‹ ªè£œA¡øù˜.

«ï˜¬ñò£ù º¡ ñ£FKèO¡ ðŸø£‚°¬ø:

(Lack of Positive role models)

èEîˆ Fø¡èœ èŸøL™ °¬ø‰î îóˆF™ àœ÷

°ö‰¬îè¬÷»‹ á‚èŠð´ˆ¶A¡øù. º¡ ñ£FK¬òŠ ðŸPò

«ð£¶ñ£ù ²¼‚èŠªð£¼œ Þ™ô£îî£™ Þîù£™ °ö‰¬îèÀ‚°

«ð£¶ñ£ù àŸê£è G¬ô Þ™ô£¬ñ ãŸðÁ¡ø¶. Ýîô£™ èEî

èŸøL™ ðò‹ ãŸðÁ¡ø¶.

Þù ð£Lù åŸÁ¬ñ: (Ethnic and or gender stereotypes)

ªð‡èÀ‹ °ö‰¬îèÀ‹ êÍèˆF™ H¡îƒA Þ¼‰îî£™

Üõ˜èÀ‚°; «ð£Fò º¡«ùŸø‹ Þ™ô£ñ™ Þ¼‰î¶.èEî

õ°Šð¬øJ™ Cô ñ£íõ˜èO¡ Fø¬ù ð£˜‚°‹ ªð£¿¶ Iè¾‹

H¡îƒAò G¬ôJ™ Þ¼‚A¡ø£˜èœ.

C‚è™ b˜ˆî G¬ôJ™ Cô ðœOèO™ î‡ì¬ù õöƒ°î™ :

(Mathematics problems being used as punishment in School)

èEîˆF™ àœ÷ C‚è™è¬÷ˆb˜‚°‹ «ð£¶ ãŸð´‹ îõÁèÀ‚°

Cô ÝCKò˜èœ î¼‹ î‡ì¬ù ñ£íõ˜èÀ‚° Ü„;êÍ†´õî£è¾‹

ªõÁŠÌ†´õî£è¾‹ Ü¬ñAø¶.

«ê£î¬ù «î˜¾ ªêŒ»‹ º¬ø: (The pressure of taking timed tests)

ðœOJ™ «î˜M¡ º‚Aòˆ¶õˆ¬î àí˜‰¶ èŸÁ‚ªè£œõî¡

Íô‹ «î˜¾èO™ «î˜„C ªðÁA¡ø£˜èœ.°´‹ðˆF™ ñ£íõ˜èÀ‚°

ÜFè Ü¿ˆî‹ ªè£´Šðîù£™ MˆFò£êñ£ù ê‰î˜ŠðƒèO™

º¡«ùŸø‹ â¡ð¶ î¬ìðÁø¶.«î˜¾èœ â¡ð¶ ñ£íõ˜èÀ‚°

32



õ£óˆFŸ° å¼º¬ø»‹ Þó‡´ õ£óƒèÀ‚° å¼º¬ø»‹ ñ£îˆFŸ°

å¼ º¬ø; Üî¡ ªî£ì˜„Cò£è è£ô£‡´ «î˜¾, Ü¬óò£‡´ «î˜¾,

º¿ Ý‡´ «î˜¾ âù ªî£ì˜‰¶ «î˜¾èœ ï¬ìªðÁA¡øù.Þîù£™

ñ£íõ˜èÀ‚° ÜFè Ü¿ˆî‹ ãŸð´Aø¶.ÞF™; ñFŠd´èœ ªêŒ»‹

«ð£¶ ñ£íõ˜èO¡ èŸÁ‚ªè£‡ì¶ â‰î Ü÷MŸ° àœ;÷ù âù

ªîO;õ£èŠ ¹K‰¶ ªè£œ÷ º®Aø¶ .ÞF™ «î˜„C MAî‹ °¬ø‰¶

è£íŠð†ì£™, ñ£íõ˜èœ èEîˆF¡ «ñ™ ªè£‡´œ÷

Ý˜õI¡¬ñ«ò è£óí‹ â¡ð¶ ªîOõ£A¡ø¶.

èEî‹ èŸðîŸ° º¡Â‹ H¡Â‹ ðòˆ¬î «ð£‚°õîŸè£ù

õNº¬øèœ : (The fear of looking or feeling ‘stupid’ in front of others)

èEîˆ¬î ÜFèñ£ù ñ£íõ˜èœ ï¡ø£è ¹K‰¶

ªè£œõF™¬ô. Þîù£™ ãŸð´‹ ê‰«îèƒè¬÷ «è†´ ÜîŸè£ù

b˜¾è¬÷ ªðÁî™ «õ‡´‹. èEî ÜP¾ °¬øð£´ à¬ìò

°ö‰¬îèœ Ü™ô¶ ñ£íõ˜èœ èEî‹ â¡ø£™ ðò‹ â¡ø ñù

G¬ô‚° õ¼A¡øù˜;. Ýè«õ ÜŠðòˆ¬î Gõ˜ˆF ªêŒ¶ ªè£œÀî™

ïô‹ ðò‚°‹.

èEî‚°¬øè¬÷ «ð£‚°‹ õNèœ: (Lack of Preparedness) :

ñ£íõ˜èœ îƒèO¡ «î˜M™ à‡ì£ù ê‰«îèƒè¬÷ «è†´

ÜîŸè£ù M÷‚èƒè¬÷ ï¡ø£è ¹K‰¶ ªè£œõîù£½‹

«î¬õ‚«èŸ;ð ÜîŸè£ù ÅˆFóƒèO¡ Íô‹ Ü¬î ªîO¾ð´ˆF,

èõ¬ô cƒA èEîˆF¡ «ñ™ Ý˜õ‹ ãŸð†´ ï¡ø£è èŸÁ‚ªè£œ÷

º®Aø¶.

1.3.3 èEîˆ¬î ÞQ¬ñò£è èŸÁ‚ªè£œÀî™

(Making Mathematics Learning pleasurable):

èEî‹ â¡ð¶ è®ùñ£ùî£è àíóŠð´A¡ø¶.Ü¬î

ÝCKò˜èœ Íô‹ ªîO¾ªðŸÁ èŸ°‹ «ð£¶ Ü¬î âO¬ñò£è ¹K‰¶

ñ£íõ˜èœ èŸÁ‚ªè£œ÷ º®Aø¶.Ýù£™ ñ£íõ˜èœ èŸÁ‚

ªè£œÀî™ â¡ð¶ ªî£ì‚è G¬ôJ«ô«ò àœ÷ù. ðœOJ™ èEî‹

33



Iè¾‹ âO¬ñò£ù¶. ÞQ¬ñò£ù¶ â¡ð¬î àíó„ªêŒîõ¡ Íô‹

ñ£íõ˜èÀ‚° èEîˆF¡ «ñ™ ÜFè ß´ð£´ ãŸðì„ªêŒòô£‹.

ÞF™«îK„C ªðø£î ñ£íõ˜èÀ‚° âO¬ñò£è èŸÁ‚ªè£´ˆ¶

ðœOJ¡ «î˜„C MAîˆ¬î ÜFèK‚è ªêŒòô£‹.ñ£íõ˜èO¡

èŸø™ â¡ð¶ ªêò™õN èŸø™ Íôñ£èˆî£¡ ñ£íõ˜èÀ‚°

èŸÁ‚ªè£´ˆî™ «õ‡´‹. åšªõ£¼ °ö‰¬îèÀ‚°‹ M¬÷ò£†

´ì¡ Ã®ò èŸø™ â¡ð¶ ðò¡ ÜO‚°‹. Þ¶ ñ£íõ˜èÀ‚°

Ý˜õˆ¬î «î£ŸÁM‚A¡ø¶. â‰î å¼ è®ùñ£ù ð£ìˆ¬î»‹

âO¬ñò£è èŸÁ‚ªè£œ÷ º®Aø¶.

â‡ M¬÷ò£†´ : (Number Race):

å¼ õ°ŠH™ àœ÷ ñ£íõ˜è¬÷ ðô °¿õ£èŠ HKˆ¶ Üî¡

î¬ôõ˜è¬÷ «î˜‰ªî´ˆî™ «õ‡´‹. å¼ °¿¬õ è¼‹ðô¬è¬ò

«ï£‚A GŸè ¬õ‚è¾‹. Üõ˜èÀ‚° 2e ªî£¬ôM™ àœ÷

Ãö£ƒèŸè¬÷ ªè£´‚è «õ‡´‹. å¼ Ü†¬ìJ™ 5 â¡ø â‡¬í

â¿F‚ è£†ì «õ‡´‹. àì«ù °¿M™ àœ÷ å¼õ˜ æ®„ªê¡Á

5èŸè¬÷ â´ˆ¶‚ ªè£‡´ õ‰¶ î¡; °¿ GŸ°‹ ÞìˆF™ ¬õ‚è

«õ‡´‹. ò£˜ î¡ °¿M™ ºîL™ Ãö£ƒè™¬ô ¬õ‚A¡ø£«ó£

Üõ˜ î£¡ ªõŸP ªðŸøõó£è ¬è¬ò àò˜ˆî «õ‡´‹ . Þ‰î

ªêò™ð£´èœ e‡´‹ ªî£ìÃø¶ . Þ‰î M¬÷ò£†¬ì ªî£ì˜‰¶

ªêŒ¶ ò£˜ ÜFè â‡E‚¬èJ™ èŸè¬÷ «ê˜‚A¡ø£˜è«÷£Þ

Üõ˜è«÷ ªõŸPò£÷˜è÷£è‚; è¼îŠðÁ¡ø£˜èœ .

Þì ñFŠ¹ (place value )

Þ‰î M¬÷ò£†®¬ù Þ¼õ«ó£ Ü™ô¶ Þó‡´ °¿‚è«÷£

M¬÷ò£ìô£‹. Þ¼ °¿‚èÀ‚°‹ õ¬óŠðìˆî£œèœ îóŠð´‹ .

Þó‡´ æóˆF½‹ ßKô‚è ñŸÁ‹ å˜ Þô‚è â‡èœ â¿îŠðÁ¡øù.

â‡ Ü†¬ìèO™ 0 ºî™ 9 àœ÷ â‡è¬÷ â¿F Ü‰î Ü†¬ìè¬÷

°½‚AŠ «ð£ìŠðÁø¶. ºî™ ñ£íõ¡ ãî£õ¶ å¼ Ü†¬ì¬ò

â´ˆ¶ Ü‰î Ü†¬ì¬ò æKô‚è â‡èœ àœ÷ ÞìˆFô£ Ü™ô¶

ßKô‚è â‡èœ àœ÷ ÞìˆFô£ â¡ð¬î º®¾ ªêŒ¶ Ü‰î

34



Ü†¬ì¬ò «î˜‰ªî´ˆ¶œ÷ ÞìˆF™ ¬õ‚A¡ø£¡. Ü´ˆî °¿M™

å¼ ñ£íõ¡ õ‰¶ æ˜ Ü†¬ì¬ò â´ˆ¶ ºî™ ñ£íõ¡ ªêŒî

ðEJ¬ù«ò ÜõÂ‹ ªêŒAø£¡.

\e‡´‹ ºî™ °¿ ñ£íõ¡ õ‰¶ å¼ Ü†¬ì¬ò ªõÁñ«ù

àœ÷ ªð†®J™ «ð£Áø£¡. Ü´ˆî õ¼‹ ñ£íõÂ‹ Üšõ£«ø

ªêŒA¡ø£¡. ÞÁFJ™ ñ£íõ˜èœ «ê˜ˆî â‡ Ü†¬ìèO¡

Ã´î¬ô ÃÁA¡øù˜. ÜFè â‡ ªðŸøõ«ó ªõŸPò£÷£˜. Þšõ£Á

ªõŸPò£÷˜è¬÷ «î˜¾ ªêŒA¡øù˜ . W«ö ðô ðìƒèœ

îóŠð†´œ÷ù .

ªè£´‚èŠð†´œ÷ ðìƒè¬÷ å¿ƒ°ð´ˆF æšªõ£¼

ñ£íõ¼‹ ñŸø ñ£íõ˜ M¼‹¹‹ âˆî¬ù ðìƒè¬÷

õ¬óA¡øù˜.å¶‚èŠð†´œ÷ «ïóˆF™ âˆî¬ù ðìƒè¬÷

õ¬ó‰îù˜.ÜF™ ÜFè ðìƒè¬÷ ò£˜ õ¬ó‰î£«ó£ Üõ«ó

ªõŸPò£÷˜ âù„ê£¡øO‚èŠðÁø¶.

Ã†ì™ M¬÷ò£†´ (Addition Game)

ÞšM¬÷ò£†®¬ù Þ¼ °¿ ñ£íõ˜èÀ‹ M¬÷ò£ìô£‹.

ÞšM¬÷ò£†®Ÿ° â‡ Ü†¬ìèœ ðò¡ð´ˆîŠðÁø¶.

ñ£íõ˜è¬÷ Ü¬óõ†ì õ®M™ Üñó„ªêŒ¶ â‡ Ü†¬ìè¬÷

°ö™ «ð£™ ²¼†® Ü‹ñ£íõ˜èœ º¡ «ð£ìŠð´‹.åšªõ£¼õ¼‹

æ˜ Ü†¬ìJ™ àœ÷ â‡ °Pˆ¶‚ªè£œ÷Šð´‹. ÜšM¬÷ò£†®¬ù

ªî£ìó„ ªêŒ¶ ÜFè ñFŠªð‡ ªðŸøõ˜ ò£˜? â¡Á è‡´H®ˆ¶

ªõŸPò£÷£˜è¬÷ «î˜¾ ªêŒî™ «õ‡´‹.

35



C‰Fˆ¶ M¬ì î¼‹ M¬÷ò£†´ (Guessing Game)

Þ‰î M¬÷ò£†ì£ù¶ «ñ™G¬ô ñ£íõ˜èœ ñ†´«ñ M¬÷ò£ì

ã¶õ£ù¶. ñ£íõ˜è¬÷ Þ¼ °¿‚è÷£è HKˆ¶ A, B âùŠ ªðòKì

«õ‡´‹. ºî™ °¿M™ àœ÷ å¼ ñ£íõ˜ å¼ î£O™ 1 ºî™ 100

º®ò àœ÷ â‡èO™ ãî£õ¶ æ˜ â‡¬í â¿F ÝCKòKì‹

î¼õ£˜èœ.ñŸªø£¼ °¿M™ àœ÷ å¼ ñ£íõ˜ Ü¶ â¡ù â‡ â¡Á

áAˆî Ãø™ «õ‡´‹ êKò£è‚ ÜFè ð†êñ£è ãî£õ¶ 10 «èœMè¬÷

°¿ B J™ ñ£íõ¡ «è†èô£‹. «è†èŠð´‹ 10 Mù£‚èO™ ºî™

Mù£MŸ° °¿ B è‡´H®ˆ¶M†ì£™ 10 ñFŠªð‡µ‹ 2

Mù£‚èÀ‚° M¬ì¬ò è‡´H®ˆî£™ 9 ñFŠªð‡ âù¾‹

ñFŠªð‡èœ °Pˆ¶‚ªè£œ÷Šð´‹. Þ¶ «ð£™ 2 °öMù¼‹ ñK ñK

M¬÷ò£ì «õ‡´‹. º®M™ °PŠH†ì Gè›M™ ò£˜ ÜFè

ñFŠªð‡ ªðŸø£«ø£ Ü‚°¿Mù«ó ªõŸP ªðŸøõó£è

è¼îŠð´õ˜.

ªêò™º¬ø :2

ªî£ì‚èè™MJ™ ðJ¡ø c†ì™ Ü÷¬õè¬÷Š ðò¡ð´ˆF

º¬øŠð´ˆ¶‹ ªêò™º¬øè¬÷ âOî£è èŸø™.

Þ¶«ð£¡ø â‡Eò™ ªî£ì˜ð£ù M¬÷ò£†´‚èœ

ñ£íõK¬ì«ò èEî ÝŸø¬ô ÜFèŠð´ˆ¶‹ ê¶ó õ®õñ£ù

õ¬óðìˆî£O™ õ‡íñ£ù Ì‚è¬÷ (óƒ«è£L) õ‡íŠ

Ì„²èÀì¡ õ¬ó‰¶ ñ£íõ˜èÀ‚° è£†´î™.Ü‰î õ¬óðìˆ

î£O¬ù ñ®ˆ¶ ¹Fò õ¬óŠðìˆ¬î Þ¼ðKñ£í‹ (2D) ñŸÁ‹

ºŠðKñ£í‹ (3D) Ü¬ñŠH™ ñ£íõ˜èÀ‚° è£†´î™.Þ¶ «ð£¡ø

M¬÷ò£†´è¬÷»‹ M¬÷ò£ì„ªêŒòô£‹.

M¬÷ò£†´ º¬øJ™ èEî‹ èŸø™ â¡ð¶ ñ£íõ˜èO¬ì«ò

âOî£è¾‹ Þ M¼Šðñ£ùî£è¾‹ Ü¬ñ»‹. Þ¶ «ð£¡ø

M¬÷ò£†´‚èO¡ Íô‹ â‰î M¬÷ò£†®™ ñ£íõ˜èœ ÜFè

ß´ð£†´ì¡ èŸA¡øù˜ â¡Á ð£˜ˆ¶ Ü‹º¬øJ¡ Íô‹

36



âO¬ñò£è ñ£íõ˜èOì‹ èEîˆ¬î ªè£‡´ ªê™ô™ ï¡¬ñ

ðò‚°‹.

îŸ«ð£¶ “Hˆ¶” Üî£õ¶ ãö£ƒè™ â¡ø M¬÷ò£†´

º¬ø¬òŠðŸP‚è£‡«ð£‹. Ü¬ùˆ¶ ï£´èO½‹ ªõš«õÁ

ªðò˜è¬÷ ªè£‡´ Þ‰î M¬÷ò£†ì£ù¶ ñ£íõ˜èO¬ì«ò

M¬÷ò£ìŠðÁ¡ø¶.9 ºî™ 10 õ¬óJô£ù èŸè¬÷«ò£ Ü™ô¶

«õÁ ã«îÂ‹ ªð£¼†è¬÷«ò£ å¼ õ†ìˆF¡ ï´M™

¬õ‚è«õ‡´‹. H¡ù˜ å¼ ð‰F¬ù ªè£‡´ Ü‚èŸè«÷£ Ü™ô¶

ªð£¼†è«÷£ Ü®‚èŠðÁ¡øù.

CîPò èŸè¬÷ å¿ƒ°ð´ˆî å¼ °¿¾‹ Þšªõ£¿ƒ° ð´ˆ¶‹

°¿M¬ù î´‚è ð‰¬î‚ªè£‡´ âPò å¼ °¿¾‹ M¬÷ò£†®™

ß´ðÁø¶.H¡ù˜ Ü´ˆî °¿ ñK ñK M¬÷ò£ìŠðÁø¶.ÜFè

ñFŠªð‡ ªðŸø °¿ ªõŸPò£÷£˜ ÝAø¶.

Þ‰î õ¬èò£ù M¬÷ò£†´‚èO¡ Íô‹ â‡Eò½‹ Þ

Ã†ì½‹ ñ£íõ˜èO¬ì«ò âOî£è ªè£‡´ ªê™ôŠðÁø¶.

Þšõ¬è M¬÷ò£†´‚èO¡ Íô‹ èEî‚ ªè£œ¬èè¬÷

âOF™ èŸÁ‚ªè£œ÷ô£‹.

ñŸªø£¼ èEî M¬÷ò£†´ º¬øò£ù¶ ñ£íõ˜èO¬ì«ò

èEî‹ èŸ°‹ Ý˜õˆ¬î á†Á¡ø¶.ÜšM¬÷ò£†ì£ù¶

¶Šð£‚A ²´õ¶ «ð£¡ø â‡íŸø õ†ìƒè¬÷ õ¬ó‰¶ ªè£‡´

õ†ìˆFŸ° ªõO«ò G¡Á ñ£íõ˜èœ õ†ìˆFŸ°œ è™¬ô âPî™

«õ‡´‹. åšªõ£¼ õ†ìˆFŸ°‹ å¼ ñFŠªð‡.ÞF™ â‰î ñ£íõ˜

ÜFè ñFŠªð‡ ªðÁA¡ø£«ó£ Üõ«ó ªõŸPò£÷˜.

ñŸÁ‹ å¼ M¬÷ò£†´ º¬ø

Í¡Á «õÁð†ì õ‡íƒèO™ ªð¡C™è¬÷ â´ˆ¶‚ªè£œ÷

«õ‡´‹. ÜõŸ¬ø ï£¡° ¶‡´è÷£è ªõ†®‚ªè£œ÷ «õ‡´‹.

(ªõ÷;¬÷ - 1 Þ cô‹ - 2 Þ CèŠ¹ - 3;) âù ñFŠªð‡ îó ðì «õ‡´‹.

37



ªð¡C™ ¶‡´è¬÷ °¿‚A ñ£íõ˜è¬÷ Ü¬î â´‚è„ ªêŒò

«õ‡´‹. â´‚°‹ ªð¡C™èœ õ‡íˆFŸ° î°‰î£Ÿ«ð£™

ñFŠªð‡ õöƒè «õ‡´‹. Þ‰î õ¬èò£ù M¬÷ò£†´

ñ£íõ˜èO¡ ªð¼‚è™ ñŸø‹ Ã†ì™ èí‚°è¬÷ èí‚W´ ªêŒò

àî¾A¡ø¶.åšªõ£¼ õ‡íˆFŸ°‹ 3 Ü™ô¶ 4 ¶‡´÷£è

â´ˆ¶‚ªè£œ÷ «õ‡´‹.

Þ‹º¬øJ™ â‡èœ ðò¡ð´õF™¬ô.Üîù£™ ñ£íõ˜èœ

ß´ð£†´ì¡ ªêò™ð´A¡øù˜.Þ¶ «ð£¡ø â‡Eò¬ô»‹

ðò¡ð´ˆî£ñ™ M¬÷ò£ì£ñ™ ; ªêŒòô£‹.

ªêò™º¬ø :3

à¡ i†¬ì„ ²ŸP Cô ñ£íõ˜èœ Cô M¬÷ò£†´‚è¬÷

M¬÷ò£´õ¬î ð£˜ˆF¼Šð£Œ .ÜšM¬÷ò£†´‚èO™ âšõ£Á

èEî ðò¡ð£´ àœ÷¶ â¡Á àŸÁ «ï£‚A â¿¶è.

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

E9 : õ°Šð¬ø„ ÅöL™ èEî‹ èŸ°‹ «ð£¶ âšMîñ£ù ðò‹

«î£¡ÁAø¶.ÜîŸè£ù è£óíˆ¬î‚ ÃÁ.

E10 : õ°Šð¬ø„ ÅöL™ èEî ðòˆF¬ùŠ «ð£‚è ï£¡°

õNº¬øè¬÷‚ ÃÁè.

èEîˆF¬ù Ý˜õˆ¶ì¡ èŸè ÞÞòŸ¬è»ì¡ Ã®ò

M¬÷ò£†´ º¬øJ™ èEî‹ èŸè «õ‡´‹;. Þ‹º¬ø Íô‹ èEî

ÃŸÁ‚è¬÷ âOF™ M÷ƒA‚ªè£œ÷ô£‹.ÝCKò¼‚°‹ Þñ£íõ˜

èÀ‚°‹ å«ó Mîñ£ù ñA›„C îó‚Ã®ò H¬íŠ¹ Þ¼ˆî™ ÜõCò‹.

ÞšMîñ£ù H¬íŠ¹ Þ¼ŠH¡Þ; ñ£íõK¬ì«ò ðòˆ¬î «ð£‚A

ï™ô å¼ èŸø™ Åö™ CøŠð£ùî£è Ü¬ñ»‹.

38



1.4 ªî£°ˆî™

èEîˆF™ Þó‡´ º¬øèœ ðò¡ð´ˆîŠðÁø¶.è‡´í˜î™

(Perception) è‡èOù£™ è‡´ èEîˆ¬î b˜ˆî™.

èŸð¬ù (Representation) :

è‡è÷£™ è£í º®ò£î¬î èŸð¬ùJ™ ¬õˆ¶ b˜ˆî™.

çHò£«üJ¡ è¼ˆ¶Šð® Þªêò™º¬øèO;¡ Íôº‹

Ü¬ñŠ¹èO¡ Íôº‹ î¿¾î™ â¡ø ¹Fò ÅöL™ èŸ°‹ «ð£¶

Ü¬ùˆ¶ ñ£íõ˜èÀ‚°‹ ªð£¶õ£ùªî£¼ è™M CøŠð£ùî£è

Ü¬ñAø¶.

°ö‰¬îèOìˆF™ èŸð¬ù Íô‹ èŸøL™ ï£¡° Mîñ£ù

ªêò™ð£´èœ àœ÷ù. â¡Á H«ò£«üM¡ è¼ˆ¶ ÃÁAø¶.

1. ¹ô¡ Þò‚èG¬ô ( 0 ºî™ 2 õò¶ õ¬ó)

2. ªêò½‚° ºŸð†ì G¬ô ( 2 ºî™ 7 õò¶ õ¬ó)

3. ¹ômì£ù ªêò™G¬ô Ü™ô¶ ð¼Šªð£¼œ G¬ô ( 7 ºî™ 11

õò¶ õ¬ó )

4. è¼ˆFò™ G¬ô( 14 -15 õò¶ õ¬ó)

èEî‚ è¼ˆ¶‚è¬÷ ñ£íõ˜èO¬ì«ò CøŠð£ùî£è‚ ªè£‡´

ªê™ô â‡íŸø õNº¬øèœ îóŠðÁ¡øù. â‡è¬÷Š ðŸPò

è¼ˆ¶, ²¼‚°î™, ªð£¼ˆ¶î™, åŠH´î™ «ð£¡ø¬õ ðŸPò

M÷‚èƒèœ ªî£ì‚èG¬ôJ™ ( 6 õò¶ õ¬ó ) èŸH‚èŠðÁø¶.

â‡Eò¬ô»‹ c†ì™ Ü÷¬õ è¼ˆ¶‚è¬÷»‹ å¼ ªð£¼¬÷‚

è‡´ ªîO¾ ð´ˆF‚ ªè£œ÷ô£‹. ÞõŸ¬ø 11 õò¶ õ¬ó

ñ£íõK¬ì«ò èŸH‚èô£‹.

â‡èO¡ ¶¬íò£™ å¼ ªð£¼O¡ c÷‹, Üèô‹ , àòó‹, G¬ø

«ð£¡ø¬õè¬÷ è‡ìPò èŸH‚èŠðÁø¶.Þî¬ù‚ èŸH‚è è£ô‹

ÜFèŠð´ˆîŠðÁø¶.

39



ªî£ì‚è G¬ôJ™ èEî ð£ìƒè¬÷ âOF™ ð®‚è ÜFè õN

º¬øèœ àœ÷ù. «ñ½‹ ªð£¼œè¬÷ ¬èò£ÀõF½‹, Ü¡ø£ì

ªêò™ð£´èO¡ è¼ˆ¶‚è¬÷ ¹K‰¶ ªè£œ÷¾‹ Þ ªî£°ˆ

¶¬ó‚è¾‹ ñ£ŸÁ õNº¬øèœ àœ÷ù. èí‚¬è b˜¾ è£í¾‹

Mù£M¬ù â¿Š¹õîŸ°‹ ðò¡ð´Aø¶.

èEî‹ èŸðîù£™ ãŸð´‹ ðî†ì‹ Þ ðò‹ ãŸðì ðôîóŠð†ì

è£óEèœ àœ÷ù.ðœO õ£›‚¬èJ½‹ ñQî õ£›‚¬èJ½‹

ãŸð´‹ ð™«õÁ ªêò™º¬øèœ ñŸÁ‹ õ°Šð¬ø i†´„ÅöL™

«ð£¡øù.

èEîˆ¬î ñA›¾ìÂ‹ å¡P»‹ M¼ŠðºìÂ‹ èŸè åŠð£˜

°¿õ£è¾‹ M¬÷ò£†ì£è¾‹ èŸHˆîL™ èŸø™ àðèóíƒè¬÷

ðò¡ð´ˆF»‹ Mù£® Mù£ ÞèEî è‡è£†C, ªð£¼†è£†C, ¹F˜,

M¬÷ò£†´, èEî ñ¡ø‹ «ð£¡ø ªêò™ð£´è¬÷ ï¬ìº¬øð´ˆF

Ý˜õÍ†ìô£‹.

1.5 î¡ùP¾ «î˜¾ -M¬ìèœ :

E : 2 : å¼ƒè¬ñˆî™ ñŸÁ‹ ªð£¼ˆ¶î™

E : 3 : ªð£¼ˆ¶î½‹ Þ õ¬èŠð´î¶;î½‹

E : 4 :ð¼Šªð£¼œ G¬ô (7 to 11 years)

E : 5 : è¼ˆFò™ G¬ô (above 11 years)

E : 6 : ªð£¼†èœ à¼õ‹ Þì‹ ñKù£½‹ ÜõŸP¡ Ü÷¾ °¬øò£¶

â¡Â‹ è¼ˆ¶ (Conservation of lengths)àíóŠð´Aø¶.

E : 7 : Ý‹. GÏðí‹ ªêŒ.

40



1.6 «ñŸ«è£œ Ë™èœ :

Anghileri , julial (ed)(1995).Children’s mathematical thinking in primary

years.perspectives on children’s learning London :Casse 11.

Copeland , Richard W.(1979).How Children learn methematics : Teaching

implications of piaget’sresearch(3rd Edn)New York.Macmillan publishing .co.

Dickson ,Linda,Brown,Margaret,&Gibson, Olwen (1984).Children Learn

mathematics.New York : Holt , Rinehart &Winston.

1.7 Üô°„«ê£î¬ù (Unit - End Exercises)

1. MõK‚è :- º¡ ðœOŠ ð¼õˆF™ â‡èOù£™ âšõ£Á è¼ˆ¶

ñŸÁ‹ HóFGFˆ¶õ‹ ß´ð´Aø¶.

2. ÜPFø¡ õ÷˜„CJ¡ Íô‹ èEî‚ è¼ˆF¬ù âšõ£Á

º¡«ùŸÁõ£Œ; ?

3. èEîˆF¡ Ü®Šð¬ì‚ è¼ˆ¬î âšõ£Á «ð£FŠð£Œ?

ñ£íõ˜è¬÷ âšõ£Á ñA›¾ì¡ èŸè„ªêŒõ£Œ?.

41



è†ì¬ñŠ¹ :

2.0 ÜPºè‹

2.1 èŸø™ °P‚«è£œèœ

2.2 èEîˆF¡ î¡¬ñ

2.3 èEî‚ è™MJ¡ º‚Aòˆ¶õ‹

2.3.1 Ü¡ø£ì õ£›‚¬èJ™ èEîˆF¡ ðò¡ð£´.

2.3.2 ñŸø ð£ìƒèO™ èEîˆF¡ ðò¡ð£´. èEîˆ«î£´ ñŸø

ð£ìƒèO¡ ªî£ì˜¹‹ ðò¡ð£´‹.

2.3.3 èEîˆF™ b˜õ£Œ¾ º¬ø èEîˆF¡ Íô‹ b˜¾ ªêŒ»‹

º¬ø.

2.3.4 èEîˆF™ ÜPî™ º¬ø

2.4 èŸø¬õ

2.5 ñ£FK Mù£ˆî£œ.

2.6 «ñŸ«è£œ Ë™èœ

2.7 Üô°ˆ«î˜¾

2.0 . ÜPºè‹

î£Œªñ£N‚° Ü´ˆîð®ò£è õ£›‚¬èJ¡ åšªõ£¼ ªêòL½‹

èEî‹ ªî£ì˜¹ ªè£‡´œ÷¶. ð®ŠðP¾ Þ™ô£î åšªõ£¼õ¼‚°‹

CPò èEî ÜP¾ ÜFèñ£ù ðô¬ù‚ ªè£´‚°‹ .Fù‚ÃL ß†´‹

ªî£Nô£O ºî™ Mõê£JÞ è¬ôë˜è÷Þ; ÝCKò˜èœÞ ÜPMò™

M…ë£Qèœ õ¬ó»œ÷ Ü¬ùõ¼‹ Üõóõ˜ ªî£N½‚° ãŸð

Üô° - 2

èEîº‹ èEî è™M»‹

42



èEî ÜP¾ ðò¡ð´Aø¶. ðœO‚è¬ôˆF†ìˆF™ èEîˆFŸ° ºî™

G¬ô»‹Þ îQ„CøŠ¹‹ ÜO‚èŠð†´œ÷¶.;

«îCò‚è¬ôˆF†ì õ®õñ£ù¶ (National Curricululm Frame Work -

NCF) èEîˆF¡ Ü¬ñŠH¬ù “ïñ¶ ð£˜¬õJ™ èEî‚

è™Mò£ù¶ Þ¼ î˜‚èKFò£ù õ£êèƒè¬÷‚ ªè£‡´œ÷¶. Ü¬ùˆ¶

ñ£íõ˜èÀ‹ âOF™ èEîˆ¬î‚ èŸÁ‚ ªè£œ÷ Þò½‹. Ü¬ùˆ¶

ñ£íõ˜èÀ‚°‹ èEî èŸø™ Iè¾‹ ÜõCò‹.”

“All Students should Learn Mathematics and Solve the Problems. All Students

should Learn Mathematics and that all students need to learn Mathematics”

ï£‹ H¡õ¼‹ C‚è™è¬÷Š ðŸP Mõ£F‚èô£‹.

1. ðœO‚è™MJ™ ; èEî‹ èŸHŠðî¡; «ï£‚è‹ â¡ù?

2. ÝCKò˜ ñ£íõ˜èÀ‚° âšõ£Á èEî ÝŸø¬ô»‹

Ý˜õˆ¬î»‹ õ÷˜‚èô£‹ ?

3. âŠð®Šð†ì èEî ÜP¾‹ Fø¬ñ»‹ ñ£íõ˜èO¬ì«ò

ãŸð´Aø¶?

4. èEî‹ èŸøL¡ î¡¬ñèœ ò£¬õ?

«ñŸÃP»œ÷ b˜¾èO™ Þ¼‰¶ âOF™ M¬ìè£í º®»‹.

ñ£íõ˜èœ âOF™ ¹K‰¶ ªè£œÀ‹ð® Ü®Šð¬ì‚ èEîˆF¡

Íô‹ ñ£íõ˜èÀ‚° èŸH‚°‹ º¬ø¬ò õ®õ¬ñ‚è «õ‡´‹.

H¡ù˜ èEîˆF¡ º‚Aòˆ¶õˆF¬ù ªî£ì‚èG¬ô ñ£íõ˜èÀ‚°

M÷‚;A‚ Ãøô£‹.

ð£ì«õ¬÷ - 8 ñE «ïó‹;

43



2.1 èŸø™ °P‚«è£œèœ

Þ‰îŠ ð°F¬ò èŸ°‹ «ð£¶ «î£¡ÁA¡ø «ï£‚èƒèœ :-

èEîˆF¡ î¡¬ñè¬÷ ðœOèO½‹ ï¬ìº¬ø õ£›‚¬èJ¡

Íôº‹ ªîK‰¶ªè£œ÷ô£‹.

Ü¡ø£ì õ£›‚¬èJ™ èEîˆF¡ ðò¡ð£†¬ì àíóô£‹.

âšMîñ£è‚ èEî ÜP¾ ðò¡ð´Aø¶ â¡ð¬î ÜPòô£‹.

èEîˆFŸ°‹ ñŸø ð£ìÜPMŸ°‹ àœ÷ ªî£ì˜¬ð

ÜPò„ªêŒî™.

2.2 èEîˆF¡ î¡¬ñèœ :

æ˜ ÝCKòó£è Þ÷‹ °ö‰¬îèÀ‚° èEî‹ èŸHŠðîŸ°

ÜÂðõ‹ «î¬õ. Cô «ïóƒèO™ èEî‹ èŸHˆî™ Fø¬ù ñŸø

ð£ìƒèÀì¡ ªî£ì˜¹ ð´ˆF èŸHˆî™; «ï£‚èƒè¬÷ Þ ñA›„C»ì¡

èŸÁ‚ ªè£œ÷ô£‹.

Ý‹. â¡ø£™ â¡ªù¡ù è£óEèœ «î£¡ÁA¡øù?

«ñŸè‡ì Mù£M¡ Íô‹ M¬ìJ¬ù b˜¾ è£µ‹ ªð£¿¶

èEîˆF¡ ð‡¹è¬÷Š ¹K‰¶ ªè£‡´ èEîñ£ù¶ ñŸø

ð£ìƒèOL¼‰¶ «õÁð†´œ÷¶ â¡ð¬î ÜPòô£‹. èEîˆF¡

î¡¬ñò£ù¶ èEî‹ èŸHŠðF½‹, ¹K‰¶ ªè£œÀîL½‹ CøŠð£ù

ðƒèO‚A¡ø¶. Üîù£™ ªî£ì‚è‚è™M ÝCKò˜ èEîˆF¡

Fø¡è¬÷ ÜP‰¶ ªè£œÀî™ «õ‡´‹. èEîˆFø¡ ñŸø

ð£ìŠð°FJ™ Þ¼‰¶ «õÁð†´œ÷¶.

ÞQ èEîˆF¡ Fø¡ Þ º‚Aòˆ¶õˆF¬ùŠ ðŸP Mõ£FŠ«ð£‹

èEîˆF™ î˜‚èº¬ø

èEîˆF¡ ñŸªø£¼ Cø‰î î¡¬ñ ÜîÂì¡ Hø‰î

î˜‚èº¬øò£°‹. å¼ èí‚A¬ùˆ b˜‚è ºò½‹ «ð£¶ ªè£´‚èŠð†ì

Mõóƒè¬÷ î°‰î G¬ôJ™ å¡Á‚ªè£¡Á ªî£ì˜¹¬ìòî£è,

44



õK¬êŠð´ˆF è£óí è£Kòˆ«î£´ ªî£ì˜¹ð´ˆFÞ Ü¬ñ‚°‹

ªð£¿¶, è£íŠð´‹ b˜M™ îõÁ ã¶‹ ãŸðì£¶. á.¹. «ý‹ªð™

è¼ˆF¡ ð® “å¼ °PŠH†ì G¬ùõ£ŸøL™ èEî‹ b˜‚èŠð´Aø¶.”

â¡Aø£˜.

èEîˆF¡ è¼ˆFò™ ªè£œ¬èJ¡ ð® Üî£õ¶ Ã†´ˆªî£ì˜Þ

ÞòŸèEî‹ ñŸÁ‹ â‡Eò™ ÝŒ¾ Þ¬õò£¾‹ î˜‚èº¬øŠð®

G¬ø«õŸøŠð´Aø¶.

èEîˆF¡ Ü¬ùˆ¶ ÅˆFóƒèÀ‹ ñŸÁ‹ M÷‚èƒèÀ‹ å¼

°PŠH†ì î˜‚è õ®õ¬ñŠH™ î£¡ àœ÷¶. Üî£õ¶ èEîˆF¡

à‡¬ñ‚ ÃÁèœ ò£¾‹ î˜‚è º¬øŠð® Gõ˜ˆF ªêŒòŠð†´œ÷¶.

å¼ Cô MFèO¡ ð® èEîˆF¡ ªêò™ð£´èœ å¼ î˜‚èˆF¡ ð®

ñŸÁ‹ õ¬óò¬øJ¡ ð®»‹ ï¬ìªðÁAø¶.

Þšõ£Á ªð£¼ˆîñ£ù õK¬êŠð® ãŸÁ‚ªè£œ÷ˆ î°‰î

è£óí è£Kòƒè«÷£´ â™ô£ ð®G¬ôè¬÷»‹ Ü¬ñˆ¶ b˜¾ è£µ‹

º¬ø‚° “î˜‚èº¬ø” â¡ø¬ö‚A«ø£‹.

SI : Þó‡´ Þó†¬ìŠð¬ì â‡è¬÷‚ Ã†®ù£™; A¬ì‚è‹

M¬ìò£ù¶ å¼ Þó†¬ìŠð¬ì â‡í£è«õ A¬ì‚°‹.

Þ‰î õ¬èò£ù èí‚A¬ù âšõ£Á àŸÁ«ï£‚Aù£½‹ b˜¾

è£‡ð¶ âOî™ô. ÞîŸ° å¼ Cô ðJŸC èí‚°è¬÷ b˜¾ ªêŒ¶

ðJŸC ªðŸø Hø° SI â¡ø ÃŸø£ù¶. êK âù àÁF Ãø º®»‹.

Þó†¬ìŠð¬ì â‡ â¡ø£™ â¡ù? â¡ð¬î êKõó ªîK‰¶ ªè£‡ì

Hø° î£¡ Þ‰î õ¬è‚ èí‚A¬ù êñ¡ ªêŒò º®»‹. â‰î å¼

Þó†¬ìŠð¬ì â‡¬í»‹ “2n” â¡Á â¿îô£‹. ÜF™ n â¡ð¶

ãî£õ¶ å¼ Þò™ â‡ Ý°‹. Þó†¬ì ð¬ì â‡è¬÷ 2n, &2n2 âù‚

°P‚èô£‹. (ÞF™ n & n2 â¡ðù Þòªô‡èœ Þó‡´ â‡èO¡

Ã´î™ 2n, +2n2 = 2 (n1+n2 , 2m m = n1+n2 (Þòªô‡ Ý°‹)

Þƒ° 2m â¡ð¶ 2 - Ý™ õ°‚è º®»‹ è£óE âù«õ Þ¶

Þó†¬ìŠð¬ì â‡í£°‹ .

45



âù«õ Þó‡´ Þó†¬ìŠð¬ì â‡èO¡ Ã´î™ Þó†¬ìŠð¬ì

â‡í£è«õ Þ¼‚°‹ . Þšõ¬èò£ù ñ£FK Mù£‚èœ º®¾è¬÷ˆ

b˜‚è„ ªêŒò¾‹ à‡¬ñ‚ ÃŸ¬ø Ü¬ìò¾‹ ðò¡ð´õî£™ Þ‰î

º¬ø “MFM÷‚è º¬ø ;” âùŠð´Aø¶.

W›‚è£µ‹; Mù£M¬ùˆ b˜¾ ªêŒ¶ «ñŸÃP»œ÷ ÃŸP¬ù

êK ð£˜‚è :

E1 : MFM÷‚è î˜‚è º¬ø¬òŠ ðò¡ð´ˆF Þó‡´ åŸ¬øŠð¬ì

â‡èO¡ Ã´î™ Þó†¬ìŠð¬ì â‡ âùˆ b˜‚è.

ªêò™ : 1

ªî£ì‚è‚è™M èEî ¹ˆîèˆF™ Þ¼‰¶ ãî£õ¶ ä‰¶ MF

M÷‚è î˜‚è º¬ø ðò¡ð£†¬ì‚ ÃÁ.

èŸHˆî™ º¬øJ™ “MFõ¼º¬ø î˜‚è‹” â¡ø ñŸªø£¼ º¬ø

ðò¡ð´Aø¶. Üî¡ ªî£ì˜ð£ù Cô âOò èí‚°è¬÷‚ è£‡«ð£‹.

2,4,6,8,10,16,36,54,68&102 Þ¬õò£¾‹ Þó†¬ìŠð¬ì â‡èœ.

ÞõŸP™ ãî£õ¶ Þó‡´ Þó†¬ìŠð¬ì â‡è¬÷‚ Ã†®ù£™ õ¼‹

M¬ìò£ù¶ Þó†¬ìŠð¬ì â‡í£? Ü™ô¶ åŸ¬øŠð¬ì â‡í£?

âù‚ è‡´H®.

b˜‚è :

2+4 = 6 - Þƒ° 6 â¡ð¶ Þó†¬ìŠð¬ì âí.;

6+4 = 10 - ÞF™ 10; â¡ð¶ Þó†¬ìŠð¬ì â‡.

10+8 = 18 - ÞF™ 18; â¡ð¶ Þó†¬ìŠð¬ì â‡.

54+22 = 76 - Þƒ° 76 â¡ð¶ Þó†¬ìŠð¬ì â‡.

46



ñ£íõ˜è¬÷ àŸÁ«ï£‚è„ ªêŒ¶ ; ñŸÁ‹ Ü‹ñ£FKò£ù

Þó‡´ Þó†¬ìŠð¬ì â‡è¬÷ Ã†®ù£™ õ¼‹ M¬ì Þó†¬ìŠ

ð¬ì â‡ â¡ð¬î êñ¡ ªêŒè.

Þ‹ñ£FKò£ù ðôîóŠð†ì èí‚°è¬÷ˆ b˜¾ è£µ‹ «ð£¶

ï£‹ ÜP‰¶ ªè£œõ¶ Þó‡´ Þó†¬ìŠð¬ì â‡èO¡ Ã´î™ å¼

Þó†¬ìŠð¬ì â‡ âù ÜPòô£‹.

MFM÷‚è î˜‚èº¬ø Íô‹ ðô èEîˆ b˜¾èO¡ Íô‹

à‡¬ñ¬ò GÏH‚èô£‹. ÞQ õ®Mò™ Íô‹ MFM÷‚è

î˜‚èº¬ø¬ò Mõ£FŠ«ð£‹.

å¼ º‚«è£íˆF™ ºî™ «è£í‹ 80º âù¾‹ Þó‡ì£‹

«è£í‹ 60º âù¾‹ â´ˆî‚ªè£‡ì£™ Í¡ø£õ¶ «è£íˆF¡

ñFŠ¹ â¡ù?

å¼ º‚«è£íˆF¬ù õ¬ó‰¶ ªè£´‚èŠð†´œ÷ «è£í

Ü÷¾è¬÷‚ °P‚è¾‹. å¼ º‚«è£íˆF¡ Í¡Á «è£íˆF¡ Ã´î™

180º Ý°‹.

âù«õ Í¡ø£õ¶ «è£íñ£ù¶ 180º - (80º -60º ); ; 180º - 140º =

40º Ý°‹. Í¡ø£õ¶ «è£íˆF¡ ñFŠ¹ 40 Ý°‹. Þ‹ñ£FKò£ù

ÃŸÁ‚° º®¾ è£µ‹ «ð£¶ èEîˆF¡ º‚Aòˆ¶õ‹ ñŸÁ‹

MFè¬÷ Þù‹ è‡´ “MF M÷‚è î˜‚èº¬ø¬ò” õ¬óò¬ø

ªêŒòô£‹ .

47



å¼ ÃŸÁ à‡¬ñ âQ™ n (â‡íŸø)Þ ñŸÁ‹ (n+1) â¡ø

ÃŸÁ‹ à‡¬ñò£°‹.

b˜¾ è£µî™ :

E2 : â™ô£ ðè£ â‡èÀ‚°‹ Þó‡´ è£óEèœ àœ÷ù.

âšMîñ£ù î˜‚è º¬ø Þî¬ù àÁF ªêŒòŠ ðò¡ð´Aø¶

âù‚ è£‡«ð£‹.

«ñŸè‡ì Mõ£îˆF¡ Íô‹ ï£‹ àí˜‰¶ ªè£œõ¶

â¡ùªõ¡ø£™ èEî‹ â¡ð¶ à‡¬ñò£ù î˜‚è º¬øJ™ Þ¼‰¶

õ‰¶œ÷¶.èEîˆF™ ðò¡ð´‹ “MF õ¼º¬ø” èEîˆF¡ Fø¬ù‚

ÃÁAø¶. Fø¡ àœ÷ å¼ î˜‚èº¬øò£ù¶ ñŸªø£¼ î˜‚èº¬øJ™

Þ¼‰¶ HKˆî™ ÃŸÁ ðôîóŠð†ì î˜‚èº¬øJ™ Þ¼‰«î

HK‚èŠð†ì¬õ “Î‚O†” â‡ èEî «ñ¬îJ¡ õ®Mò™ ð°F å¼

Cø‰î â´ˆ¶‚è£†ì£°‹. «ñ½‹ Þõó¶ º¬øèœ Þƒ° îóŠð†´œ÷

ðô î¬ìè¬÷ à¬ìˆî¶. î˜‚èº¬ø âšMîñ£ù ¹¶ñ£Ÿøƒè¬÷»‹

ªè£‡´ õóM™¬ô.

Þ¼ ÃŸÁ‚è¬÷ â´ˆ¶‚ ªè£œ«õ£‹.

ÃŸÁ 1 : Þ¼Ë¬ø ðˆî£™ ªð¼‚Aù£™ A¬ìŠð¶ Þó‡ì£Jó‹

ÃŸÁ 2 : ãî£õ¶ Þó‡´ Þòªô‡èœ òÞ® J¡ Ã´î¬ô

õ˜‚èŠð´ˆFù£™ b ¡ õ˜‚è‹ ñŸÁ‹ b ¡ õ˜‚è‹ ÞõŸP¡

Ã´î«ô£´ ò® â¡ø Þ¼ â‡èO¡ Þ¼ñìƒ°è¬÷‚ Ã†´î™.

Þ‚ÃŸÁè¬÷ ï£‹ èEî‚ °Pf´èœ Íô‹ Þšõ£Á â¿îô£‹

1. 200 x 10 = 2000

2. (a+b)2 = a2+b2+2ab

Þ‹º¬øèO™ èEîˆF¡ Ü®Šð¬ì„ªêò™èœ (+,-, ñŸÁ‹

)ñŸÁ‹ õ¬óðìƒèO¡ Íô‹ «è£´, «è£í‹, º‚«è£í‹,Þ ï£Ÿèó‹

ñŸÁ‹ õ†ì‹ «ð£¡ø¬õ å¡Á‚ ªè£¡Á ªî£ì˜¹ ªè£‡ì¬õèœ

48



âù ÜPòô£‹. Þ¬õò£¾‹ ïñ¶ Ü¡ø£ì õ£›‚¬èJ™ ÜFèñ£è

ðò¡ðì‚ Ã®ò¬õ.

b˜‚èº®ò£î ðô õ¬èò£ù èEî‚ ÃŸÁè¬÷ MKõ£è è£µ‹

«ð£¶ MFõ¼ º¬ø¬òŠ ðò¡ð´ˆF âO¬ñò£è ñ£íõ˜‚° èŸH‚è

Þò½‹.

èEîˆF™ ¶™Lò‹ (Precision)

èEîˆF™ å¼ èí‚AŸ°ˆ b˜¾ è£µ‹ ªð£¿¶ Üˆb˜¾ êK

â¡ø£™ êK, Ü™ô¶ îõÁ â¡ø£™ îõÁ â¡P¼‚è «õ‡´«ñ îMó

êKò£è Þ¼‚èô£‹Þ Ü™ô¶ îõø£è Þ¼‚èô£ñ, â¡ø

ÜÂñ£ùƒèÀ‚° ÞìI™¬ô. èí‚AŸè£ù M¬ì Iè„êKò£è

Þ¼‰î£™ ñ†´«ñÞ èEî‹ ãŸÁ‚ ªè£œ÷Šð´‹. âù«õ âˆî¬èò

C‚è™èÀ‹ ¶™Lòñ£è Þ¼‚è «õ‡´‹.

â´ˆ¶‚è£†´ :

Ã‹H¡ è¼ˆ¶‚èœ ªîOõ£è Þ¼ŠH¡ Ã‹H¡ õ¬óò¬ø¬ò‚

è£íô£‹.

Ã‹¹ â¡ð¶ ºŠðKñ£í õ¬óðìñ£°‹. Þ¶ W› ð‚è‹

î†¬ìò£è¾‹ õ†ìñ£è¾‹ «ñŸ¹ø‹ õ¬÷‰¶‹ è£íŠð´‹.«ñŸð°F

à„C âù Ü¬ö‚èŠð´‹. à„C

î†¬ìò£ù ð°F

49



ÞFL¼‰¶ Ã‹H¬ùŠ ðŸP ªîOõ£ù è¼ˆ¶‚è¬÷ ¹K‰¶

ªè£‡«ì£‹. â¡ø£™ ãî£õ¶ å¼ ªð£¼¬÷‚ è£µ‹ ªð£¿¶

ÜŠªð£¼œ Ã‹¹ õ®õñ£ ? Þ™¬ôò£ ? âù âOF™ Ãø Þò½‹.

ªêò™ - 1

å¼ ªè†® Ã‹H¬ù â´ˆ¶‚ ªè£‡ì£™ ÜF™ âˆî¬ù êñî÷

ðóŠ¹‚èœ àœ÷ù. âˆî¬ù à„Cèœ àœ÷ù âù¾‹ ÜP‰¶

ªè£œ÷ô£‹.

Ã‹H¬ù õ¬ó»‹ º¬ø :

W›‚è‡ì ªð£¼†è¬÷ «êèK:

ðè¬ì, ªêƒè™ ñ†¬ìð‰¶, ävAg‹ «è£¡, bŠªð†® «êèKˆî

ªð£¼†èO™ Þ¼‰¶ Ã‹¹ õ®õñ£ù ªð£¼†è¬÷ «î˜‰ªî´.

C.J «èê˜ â¡ðõK¡ ÃŸP¡ð® “îóñ£ù èEî‹ â¡ð¶

ªî£ì˜„Cèœ ªè£‡ì¶. à‡¬ñ‚°Š ¹ø‹ð£ù¶ Ü™ô. ÃŸP¡

º®¾‚° êKò£ùî£è â´ˆ¶„ªê™ô âOF™ º®MŸ° õó

ðò¡ð´Aø¶. Ü‰î º¬ø èEî ¶™Lòˆ¶ì¡ ªî£ì˜¹¬ìò¶.”

èEî ÝCKòó£è ci˜?; ÞŠð®Šð†ì CøŠ¹ õ£Œ‰î èEîˆ¬î

ñ£íõ˜èÀ‚° ªî£¬ô«ï£‚° ð£˜¬õ»ì¡ èŸHˆî™ «õ‡´‹.

¶™Lò‹ â¡ð¶ èEîˆF¡ º‚Aòñ£ù î¡¬ñ Ý°‹.èEî‹

â¡ð¶ Iè„êKò£è Ü™ô¶ Ièˆ ¶™Lòñ£è èŸHˆî™ º¬ø Ý°‹.

ÜF™ “Iè„êKò£ù”â¡ø õ£˜ˆ¬îò£ù¶ â™ô£º¬øJ½‹

èEîˆ¬î b˜‚è‹ ªêŒî™ Ý°‹.

°ö‰¬îèœ à‡¬ñˆ¶õˆ¬î ¹K‰¶ ªè£œ÷¾‹, ºòŸC

ªêŒò¾‹, º®MŸ° õó¾‹ èEî‹ ðò¡ð´Aø¶. ñŸø ð£ìŠð°F¬ò

èEîˆ¶ì¡ ªî£ì˜¹ð´ˆF ð£˜‚°‹ ªð£¿¶ ñŸø ð£ìŠð°Fèœ

M¬ìJ¬ù â¿î¾‹Þ MFJ¬ù â¿î¾‹; ªêŒº¬ø ªêŒ¶ M÷‚è‚

Ã®òî£è«õ Þ¼‚°‹. Þ¶ Ü‰î‰î ð£ìŠð°FJ¡ G¬ô«ò Ý°‹.

Ýù£™ èEîŠ ð£ìˆF™ ÜîŸ° ÞìI™¬ô. ªê£‰î‚ è¼ˆ¶‚°‹

50



ÞìI™¬ô. èEîˆ¬î ð®‚°‹ ñ£íõ˜èœ à‡¬ñˆ¶õˆ¬î ¹K‰¶

ªè£‡´ î‹º¬ìò õ£›‚¬è º¬øJ™ ðò¡ð´ˆ¶Aø£˜èœ.

ñ£íõ˜èÀ‚° ãŸð´‹ Hó„ê¬ùè¬÷ âOF™ b˜ŠðîŸ°‹ èEî‹

ðò¡ð´Aø¶.

èEî‹ è†ì¬ñŠ¹ ðŸPò è™M :

è†ì¬ñŠ¹ â¡ð¶ õK¬êò£è å¿ƒ°ð´ˆî™Þ «ê˜ˆî™Þ

àœ÷¬ñ¾èœÞ; ð®õƒèœÞå¿ƒ°ð´ˆîŠð†ì º¬øèœ Ý°‹. èEî

è¼ˆ¶‚èœ ò£¾‹ Yó£ù º¬øJ™ õK¬êŠð´ˆîŠ ð†´œ÷ùõ£?

ãî£õ¶ å¼ èEî àœ÷¬ñ¾è¬÷ cƒèœ àŸÁ«ï£‚A Þ¼‚Al˜è÷£?

èEî‚ è¼ˆ¶‚èÀ‚° Þ¬ì«ò ãî£õ¶ ªî£ì˜¹ àœ÷î£?. èEîˆF¡

î¡¬ñè¬÷ ci˜ Ýó£Œ‰¶ ð£˜‚°‹ ªð£¿¶ èEî‹ â¡ð¶ Yó£ù

º¬øJ™ àœ÷ è†ì¬ñŠ¹ â¡ð¶ ªîK»‹. ªî£ì‚èG¬ô

èEîŠð£ìˆF™ °ö‰¬îèœÞ Þò™ â‡èœÞ º¿ â‡èœÞ

º¿‚èœÞH¡ùâ‡èœÞMAîºÁ â‡èœ ñŸÁ‹ ªñŒªò‡èœ

â¡ø õK¬êJ™ Þ ²¼œ ¬ñò º¬øJ™ ð£ì‹ èŸH‚èŠð´A¡ø¶.

ªî£ì‚è‚è™M ñ£íõ˜èœ Þò™ â‡èœ Þº¿â‡èœÞ º¿‚èœÞ

MAîºÁ â‡èœÞ H¡ù â‡èœ ñŸÁ‹ ªñŒªò‡èœ ðŸP

èŸA¡øù˜.

ªêò™ - 2

èEî ¹ˆîèƒèœ õN«ò «ñŸÃPò â‡èO¡ õ¬óò¬øè¬÷

â¿F‚ªè£œè. å¼ â‡µ‚°‹ ñŸªø£¼ â‡µ‚°‹ ªî£ì˜¹

àœ÷î£? âù‚ è£‡è. Üˆªî£ì˜H¬ù ðì‹ õ¬ó‰¶ M÷‚°è.

Q Z

W

N

51



ªêò™ð£´ :

E2 : “èEî‹ â¡ð¶ ¶™Lòñ£ù ñŸÁ‹ M‰¬îò£ù Ü¬ñŠHŸ°

ªî£ì˜¹¬ìò¶. Þ‰î ÃŸÁ à‡¬ñªòù G¬ù‚A¡l˜è÷£?

Ý‹ âQ™ M÷‚°è.

èEîˆF¡ è¼ˆFò™ ªè£œ¬èJ¡ «ï£‚è‹ :

Ý™M¡ â¡ø õ°Š¹ ÝCKò˜ ºî™ îó õ°Šð¬øJ™ ªêò™

º¬ø «î˜¬õ ªî£ìƒ°Aø£˜. Þó‡´ °¿‚è÷£è HK‚Aø£˜.

Ý‡èœ°¿Þ ªð‡èœ°¿ â¡Á HK‚Aø£˜. eF àœ÷ °ö‰¬îè¬÷

Üõ˜èœ M¼‹¹‹ ÞìˆF™ ªê¡Á Üñó„ ªê£™Aø£˜. °ö‰¬îèœ

ªêò™ º¬øJ¬ù ªî£ì˜A¡øù˜. °ö‰¬îèOì‹ ÝCKò˜ ã¡ Ü‰î

Þìˆ¬î «î˜‰ªî´ˆî£Œ â¡Á Mù¾Aø£˜? Þ„ªêò™º¬ø¬ò à¡

õ°Šð¬øJ™ Gè›ˆî¾‹. W›‚;è‡ì Mù£M¬ù «è†´ M¬ìòO‚è

¬õ‚è¾‹.

1. Üõ˜è«÷ Þìˆ¬î «î˜‰ªî´ˆ¶ Üñ˜‰î£˜è÷£?

2. Üõ˜è«÷ Ü‰îŠ ðE¬ò„ ªêŒî£˜è÷£?

èEîˆF¡ è¼ˆFò™ î¡¬ñ¬ò ðŸP MõKŠ«ð£‹ :

î‰¬îJ¡ õò¶ ñèQ¡ õò¬î «ð£™ Þ¼ ñìƒ°. .30 õ¼ìƒèÀ‚°

º¡¹ î‰¬îJ¡ õò¶ ñèQ¡ õò¬îŠ «ð£™ ï£¡° ñìƒ° ÜFè‹.

âQ™ Þ î‰¬îJ¡ õò¬î‚ è£‡è.

î‰¬îJ¡ õò¶ = x â¡è.

ñèQ¡ õò¶ = 2x

â¡è.

30 õ¼ìƒèÀ‚° º¡

(x-30) = 4 ( 2x

- 30 )

x = 90.

52



î‰¬îJ¡ õò¶ = 90 ñèQ¡ õò¶ = 45

L. Bers “èEîˆF™ è¼ˆFò™ â¡ð¶ â‡èœ Ü¬ùˆ¶«ñ å¼

è¼ˆ¬î °PŠðî£°‹. Ü¬õ à¼õ Ü¬ñŠHŸ° ÜŠð£Ÿð†ì¶”.

èEîˆF™ è¼ˆFò™ G¬ô ñŸªø£¼ CøŠ¹ˆ î¡¬ñò£°‹. èEîˆF™

ÞòŸèEî‹ â¡ø å¼ ¹Fò ð°F «î£¡ÁõîŸ° Þ‰î è¼ˆFò™ G¬ô«ò

è£óí‹. ÞòŸèEî‹ èEîˆF¡ å¼ ð°F.[ªîKò£î â‡µ‚°

ðFô£è â¿ˆ¶‚è¬÷«ò£ Ü™ô¶ °Pf´è¬÷«ò£ Þ¬íˆ¶

ðò¡ð´ˆîŠð´‹ å¼ èEî HK¾ - ÞòŸèEî‹.] è¼ˆFò™ G¬ô

â¡ð¶ Iè¾‹ ðó‰¶ MK‰¶œ÷ èEîˆF™ º‚Aò ðƒ° à¬ìòî£°‹.

ªêò™ð£´

E3 : º‚«è£íˆF¬ùŠ ðŸPò Í¡ø£‹ G¬ô ñ£íõ˜èO¡

è¼ˆFò™ C‰î¬ù â¡ù?

2.3 è™MJ™ èEîˆF¡ º‚Aòˆ¶õ‹ :

èEî è™Mò£ù¶ ïñ¶ êÍè‹ , ªð£¼÷£î£ó‹, ð‡ð£´,

èô£„ê£ó‹ ê£˜‰î ; èEî è¬ôˆF†ìˆF¬ù»‹ ð£ìŠªð£¼¬÷»‹ ,

èŸHˆî™ º¬øè¬÷»‹ ªè£‡´œ÷¶. Þî¡ º‚Aòˆ¶õ‹ ªêò™

ðJŸC«ò Ý°‹. Þ¶ ðô ¶¬øèO½‹ CøŠð£ù ðƒèO‚Aø¶.

èEî‹ âšõ£Á ï‹ Ü¡ø£ì õ£›M™ ðò¡ð´Aø¶ â¡Á

ð£˜Š«ð£‹. «îCò è¬ôˆF†ì‹ 2005 (NCF-2005) ¡ ð® ªî÷;põ£ù

C‰î¬ùÞ èŸð¬ù Fø¡Þ î˜‚;èKbò£ù C‰î¬ù «ð£¡ø¬õ èEî

Íôñ£è«õ ãŸð´Aø¶. Þšõ¬èò£ù C‰î¬ù èEî‹ èŸ«ð£¼‚°

Þ¼‰î£™ C‚è™èÀ‚° âOF™ b˜¾ è£íô£‹. Þ‰î ð°FJ™ èEî‹

èŸðî¡ ÜõCòˆ¬îŠ ðŸP ÜP‰¶ªè£œ÷Š«ð£A«ø£‹;.

2:3:1 Ü¡ø£ì õ£›M™ èEî‹ :

c˜ ðœO º®‰î¾ì¡ °ö‰¬îèœ ð™«õÁ Mîñ£ù

M¬÷ò£†´èœ M¬÷ò£´õ¬î èõQˆF¼Šd˜.

è£™ð‰¶ M¬÷ò£†®™ î¡ ÜEJù¬ó «èŠì¡ 5+3+2 Ü™ô¶

4+3 +3 â¡ø õK¬êJ™ GŸè ¬õŠð£˜. Ü«î «ð£¡Á AK‚ªè†

53



M¬÷ò£†®½‹ «èŠì¡ d™ì˜è¬÷ êKò£ù ÞìˆF™ GŸè ¬õŠð£˜.

Þîù£™ âšMîñ£ù Gè›¾ ãŸðÁ¡ø¶? M¬÷ò£†®¡ º®¾

CøŠð£ùî£è Ü¬ñ»‹. ÞF™ Þ¼‰¶ ió˜è¬÷ â‰ªî‰î ÞìˆF™

GŸè¬õ‚è «õ‡´‹ â¡ø MNŠ¹í˜¾ «èŠìÂ‚° à¼õ£Aø¶.

ªêò™ð£´ :

E4 : c˜ ðœOJ™ M¬÷ò£®ò ªð£¿¶ M¬÷ò£†´èO™ èEî

è¼ˆ¶‚è¬÷ ðò¡ð´ˆFò¶ à‡ì£?

å¼ Mõê£J ðJ˜ ªî£N™ ªêŒò F†ìI´‹; ªð£¿¶Þ

âšõ÷¾ ðóŠH™ ðJ˜ ªêŒò «õ‡´‹ â¡ð¬î»‹, ÜîŸ° M¬î,

àó‹, Ì„C‚ªè£™L ñ¼‰¶,ªî£Nô£÷˜èœ Þðí‹ âšõ÷¾ «î¬õ

âù ÜP‰¶ ªè£œ÷ èEî ÜP¾ «î¬õŠ ðÁø¶.

ªêò™ : 3

à¡ ð°FJ™ àœ÷ ä‰¶ ïð˜è¬÷ «î˜¾ ªêŒè. Üõ˜èœ ªêŒ»‹

ðE¬ò èõQ. Üõ˜èœ ªêŒ»‹ ðE‚°‹ èEîˆFŸ°‹ ªî£ì˜¹

àœ÷î£? ÞõŸ¬øŠ ðŸP à¡ ï‡ð˜èÀì¡ Mõ£F.

èEî ðò¡ð£†´ ÜP¾ õ£›‚¬èJ¡ Ü¬ùˆ¶ ð°FJ½‹

ðóM»œ÷¶. åšªõ£¼ ñ£íõ˜èÀ‹ ðôîóŠð†ì êÍèˆFL¼‰¶

õ¼A¡øù˜. ÜõŸP™ å¼ ªêòL¬ù ðŸP W«ö îóŠð†´œ÷¶.

Þ‰î ªêò™º¬ø¬ò ªêŒõîŸ° å¼ Cô õ£óƒèœ Ý°‹.

õ°Šð¬øJ™ ÅKò åO‚èF˜èœ ê¡ùL¡ õN«ò õ¼‹. æO‚èF˜èœ

õ¼‹ õN«ò 5 ªê.e c÷ˆFŸ° ð¬ê ï£ì£¬õ å†Á¡«ø£‹.

ðôîóŠð†ì Mù£M¬ù «î£ŸÁM‚A¡ø¶.

1. âšõ£Á ñ£íš˜èœ «ïóˆ¬î «î˜‰ªî´‚A¡øù˜.?

2. M¿‹ Göô£ù¶ åšªõ£¼ ï£À‹ å«óñ£FK G¬ôJ™ àœ÷î£?

3. Cô ï£†èÀ‚° Hø° ñ£íõ˜èœ ÜîŸ° õ¬óðì‹ âšõ£Á

õ¬ó‰¶œ÷ù˜?

54



«ñŸè‡ì Mù£MŸ° M¬ìòO‚è èEî ÜP¾ «î¬õ. ºî™

«èœMJL¼‰¶

«ïóˆ¬î‚ èí‚Aì ªîK‰¶ªè£œA¡«ø£‹. Í¡ø£õ¶

«èœMJL¼‰¶ ï‹º¬ìò ÜÂðõ ÜPM™ ðFôO‚°ñ£Á àœ÷¶.

ï£‹ èõùñ£è «ïóˆF¬ù êKò£è èEˆî™ «õ‡´‹. ÞõŸPL¼‰¶

«ïóˆ¬î º¡ùî£è«õ èí‚Aì º®»‹. Þšõ£ø£ù º¬ø¬ò

ðò¡ð´ˆ¶è.

2.3.2 èEîˆFŸ°‹ Hø ð£ìƒèÀ‚°‹ àœ÷ ªî£ì˜¹ :

“èEî‹ Þ™ô£î â‰î å¼ ÜPMò™ è™M»‹ Üî¡Ü®ˆî÷ˆF«ô«ò °¬øð£´ àœ÷î£è Þ¼‚°‹” â¡Aø£˜ è£‹«ìâ¡ø ÜPë˜. èEî‹ â¡ðî¡ ªñ£Nò£‚è‹ ÞñQî õ£›MŸ°‹ÞñQî ï£ègè õ÷˜„C‚°‹ ðò¡ð´Aø¶.â‡èœ. èEî õ¬óðìƒèœÞÅˆFóƒèœÞ õ£ŒŠð£´èœ ºîLò¬õ Íô‹ Ièˆ¶™Lòñ£ù ÜP¾õ÷˜„CJ¬ùŠ ªðøº®Aø¶.

èEîº‹ Þô‚Aòº‹ :

ñŸø ªñ£NèÀ‹ Þô‚AòƒèÀ‹ ÞòŸ¬è‚° ñ£ø£è«õ àœ÷ù.ïñ¶ â‡íƒè¬÷ ªõOŠð´ˆ¶õ¶ ªñ£N, ªî£ì‚è G¬ôJ™ªñ£NŠ ð£ìƒè¬÷ èŸ°‹ ªð£¿¶, °ö‰¬îèÀ‚° õ£˜ˆ¬îèœîõø£è Þ¼‰î£½‹ «ð²õîŸ° ²î‰Fó‹ ÜO‚èŠðì«õ‡´‹.ªî£ì‚è G¬ô è™MJ™ ªñ£N ðJŸCèœ ªêŒò «õ‡´‹. ªî£ì‚èG¬ô è™M¬ò °ö‰¬îèœ º®‚°‹ ªð£¿¶ 5000 õ£˜ˆ¬îè÷£õ¶èŸÁˆ«î˜„C ªðø«õ‡´‹.

ªî£ì‚è G¬ô è™M‚°Š H¡ ñ£íõ˜èœ õ£˜ˆ¬îèO¡Ü˜ˆîƒè¬÷Š ¹K‰¶ Ü¬î ªõOŠð´ˆî «õ‡´‹. ¶™Lòñ£ùÜ˜ˆîƒè¬÷ ªîK‰¶ ªè£œ÷ ñ£íõ˜è¬÷ á‚°M‚è«õ‡´‹. å¼ÝƒAôŠ ð£ìˆF™ àœ÷ õKè¬÷»‹,åˆî æ¬ê à¬ìò õ£˜ˆ¬îèœò£¬õ»‹ ¹K‰¶ ªè£œ÷™ «õ‡´‹. «ñŸÃPò ðJŸCè¬÷

ñ£íõ˜èœ ªêŒ»‹ ªð£¿¶ ð£ì‹ è†ìõ¬ó¾ ñŸÁ‹ ªñ£N

ÜPFø¡ «ð£¡øõŸP™ «ñ‹ð´‹.

55



èEîº‹ ÜPMò½‹ :

ÞòŸHòL™ èEîˆF¡ ðò¡ð£´ Iè ÜFèñ£è Þ¼Šðî£™

ÞòŸHò™ ñŸø ÜPMò™è¬÷ Mì èEîˆFŸ° ªï¼‚èñ£è Þ¼Šð¬î‚

è£íô£‹. ÞòŸHò¬ô ï‹H‚¬è«ò£´ ðJô «õ‡´ªñQ™ èEî

ÜP¾ Þ¡Pò¬ñò£î¶ ÝAø¶. ÞòŸHòL™ àœ÷ åšªõ£¼ MF»‹Þ

ªè£œ¬è»‹ èEî Ü¬ñŠH™ Þ¼Šð¬î è£‡A«ø£‹ .ÞòŸHò™

MFèO¡ ÞÁF õ®õ‹ èEîˆF¡ Ü¬ñŠH™ î£¡ Ü¬ñAø¶.

â´ˆ¶‚è£†ì£è cK¡ ªè£FG¬ô 100º â¡ð¶ ÜPMò™ ªð£¶

à‡¬ñ .Þ‰î à‡¬ñò£ù¶ ÜPMò™ ðK«ê£î¬ùJ¡ Íô«ñ

GÏH‚èŠð´Aø¶. Ü¶«õ è£ŸP¡ Ü¿ˆî‹ ÜFèK‚°‹ «ð£¶ cK¡

ªè£FG¬ô»‹ ÜFèK‚°‹ â¡ð¶ «ð£¡ø à‡¬ñè¬÷ èEîˆF¡

Íô‹ ªîK‰¶ªè£œ÷ô£‹.

æšªõ£¼ ¶¬øèO½‹ ÞòŸHò™, «õFJò™, Þò‰FóMò™, åL,

åO, «õF M¬ùèœ, èEî M¬÷ò£†´èœ «ð£¡ø¬õ èEî º¬ø

Íô«ñ MõK‚èŠð´‹.

Ü¬ùˆ¶‹ «õFJò™ èô¬õèÀ‹, ÜõŸP¡ êñ¡ð£´èÀ‹

°PŠH†ì Cô èEî MFè¬÷ Ü®Šð¬ìò£è‚ ªè£‡«ì

G˜íJ‚èŠð´A¡øù. «õFJò™ «ê˜ñƒèO¡ à¼õ£‚è‹

èEîˆFù MAî ; èí‚W´è¬÷‚ ªè£‡«ì Ü¬ñAø¶.

àJKò™ ðJ™«õ£¼‚°‹, ðJ¡«ø£¼‚°‹ èEîñ£ù¶

Wö‚;è‡ì Þó‡´ è£óíƒèÀ‚è£è Þ¡Pò¬ñò£îî£Aø¶. ºî™

è£óí‹, ð«ò£ - ÞòŸHò™ ñŸÁ‹ ð«ò£ - «õFJò™ ÝAò

ð£ìƒè¬÷Š ðJ™õîŸ°‹, ¹K‰¶ ªè£œõîŸ°‹ èEî‹ ªîK‰F¼‚è

«õ‡´‹

Þó‡ì£õ¶ è£óí‹ Þæ˜ àJKò™ õ™½ï˜ î‹ ÝŒ¬õ

«ñŸªè£œ÷ æ¼ Gè›¬õ «ïó®ò£è àŸÁ«ï£‚°õî¡ Íô‹

ªî£ìƒ°A¡ø£˜. H¡ Ü‰Gè›¬õ M÷‚°î™, õ¬èŠð´ˆî™; ñŸÁ‹

åŠHì™ «ð£¡øõŸP¡ Íô‹ å¼ MF¬ò ¹¶¬ñŠð´ˆ¶A¡ø£˜.

56



ÜõŸ¬ø ð°ˆî£Œ¾ ªêŒAø£˜. ÞõŸÁ‚ªè™ô£‹ è‡pî ÜP¾

ÜõCòñ£Aø¶. Üõ˜ î‹ è‡´H®Š¹è¬÷ ªõOJ´‹ «ð£¶

°lf´è÷£è¾‹; ²¼‚èñ£è¾‹ èEî«ñ ðò¡ð´Aø¶.

ªêò™ : 4

ãî£õ¶ Þ¼ î¬ôŠ¹è¬÷ â´ˆ¶‚ªè£œè.(å¡Á ÞòŸHò™

«õFJò™ Þ¼‰¶‹ ñŸªø£¡Á àJKòL™ Þ¼‰¶‹) î¬ôŠ¹èœ

àòó ªî£ì‚è‚è™M ð£ìˆF™ èEî ÜP¾ì¡ ¹K‰¶ªè£œ÷™

è¼ˆ¶ì¡ Þ¼ˆî™ «õ‡´‹.

__________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________

__________________________________________________________________________________________________________________________

èEîº‹ ²ŸÁ„Åö™ è™M»‹ :

ªî£ì‚è G¬ô‚ è™MJ™ àœ÷ è¬ôˆF†ìˆF™ EVS â¡ø

ð£ìñ£ù¶ ðô î¬ôŠ¹è¬÷‚ ªè£‡´œ÷¶. àî£óíñ£è, ðœOˆ

«î£†ìˆF¡ c÷‹, Üèô‹, ðóŠ¹ ºîLò¬õè¬÷ ðœO õ÷£èˆF™

«î˜‰ªî´ˆ¶ Ü¬ñ‚è «î¬õò£ù èí‚W´è¬÷ ñ£íõ˜èœ

¹Kî½ì¡ ªîK‰¶ Þ¼‚è «õ‡´‹ .

êKMAî àí¾ ñŸÁ‹ êˆ¶í¾ «ð£¡øõŸP™ èô‰¶œ÷

ªð£¼†èœ â‰î MAîˆF™ èô‰F¼‚è«õ‡´‹ â¡ð¬î èí‚W´èœ

Íô‹ ªêŒò ñ£íõ˜èœ

ÜP‰F¼‚è «õ‡´‹ .ï£‹ à‡µ‹ àíM™ àœ÷

¬õ†ìI¡èœ ñŸÁ‹ î£¶Š ªð£¼†èœ â™ô£‹ â‰î MAîˆF™ Þ¼‚è

«õ‡´‹ â¡ð¬î èí‚W´ Íô‹ ªîK‰¶ªè£œA¡øù˜.

àíM™ àœ÷ î£¶Šªð£¼†èœ ñŸÁ‹ ¬õ†ìI¡èœ Ü÷¾

°¬øõ£è àœ÷¶.Üšõ£Á °¬øõ£è Þ¼‰î£™ ãŸð´‹ «ï£ŒèO¡

57



õ¬èè¬÷»‹ è‡ìPòô£‹. «ñ½‹ àíM™ àœ÷

êˆ¶Šªð£¼†èO¡ Íô‹ ãŸð´‹ õ÷˜„CJ¬ù õ¬óðì‹ Þêîiî‹

Íô‹ èí‚Aìô£‹.

èEîº‹ ¹MJò½‹ :

ï£‹ õ£¿‹ ¹M ñŸÁ‹ Þ Üî¡ Hóð…êˆ¬îŠ ðŸP»‹ èEî‹

ñŸÁ‹ ÜPMò™ Ì˜õñ£è M÷‚èñO‚°‹ ÜPMò™ î£¡ ¹MJò™

Ý°‹. ¹MJòL™ èEîˆF¡ ðò¡ð£´ Iè ÜFèñ£è«õ àœ÷¶.

¹MJ¡ ðKñ£í‹ ñŸÁ‹ Ü÷¾ Üî¡ Åö™ ñŸÁ‹ Hóð…êˆF™

¹MJ¡ G¬ô, Þó¾, ðè™ ãŸð´î™ ê‰Fó ñŸÁ‹ ÅKò Aóèíƒèœ,

Ü†ê«ó¬è ñŸÁ‹ b˜‚è«ó¬è èì™ ñ†ìˆ¬îŠ ªð£¼ˆ¶ àòó‹,

ñ¬öŠªð£N¾, ªõŠðG¬ô, ãŸøˆî£›¾, Ü¿ˆî‹ «ð£¡øõŸ¬ø ðŸP

¹MJòL™; ÜP‰¶ ªè£œ÷ ðò¡ð£†´‚ èEî‹ ðò¡ðÁø¶.

åšªõ£¼ õ¬óðì‹; õ¬óò Ü÷¾«è£™ «î¬õ. Gô Ü÷¬õ‚

è¼MèO™ èEî‹ ðò¡ðÁø¶. õ£Qò™ î£‹ Þ¼‚°‹ ÞìˆF¡

F†ì«ïó‹ ; ñŸÁ‹ ð¡ù£†´ «ïó‹ ÝAòõŸ¬ø ªîK‰¶ ªè£œ÷

èEî‹ àî¾Aø¶. º‚«è£íMò™ ñ¬ôJ¡ àòóˆ¬î‚ è‡´H®‚è

àî¾A¡ø¶. æ˜ ÞìˆF¡ ñ¬öŠªð£N¬õ‚ èí‚AìŠ

ðò¡ðÁ¡ø¶.

èEîº‹ õóô£Á‹ :

å¼ °PŠH†ì è£ôè†ìˆF™ Gè›‰î Gè›„Cèœ ñŸÁ‹ ªêò™èœ

ðŸPò ªêŒFè¬÷Š ð®Šð¶ õóô£Á Ý°‹. ðô ËŸø£‡´èÀ‚°

º¡¹ ïì‰î õóô£ŸÁ„ ê‹ðõƒè¬÷ ÜPõîŸ°‹ èEî‹ ÜõCò‹

«î¬õŠðÁø¶. .â‰î è£ôˆF™ â‰î Üóê˜ Ý†C ¹K‰î£˜. Üõ˜

Ý†CJ™ â¡ªù¡ù Gè›„Cèœ ï¬ìªðŸøù. î°‰î ¹œO

Mõóƒè«÷£´ ÜP‰¶ ªè£œ÷ èEî‹ àî¾Aø¶. ðöƒè£ô

ï£èKèˆ¬î ÜPò àî¾‹ è†ìì è¬ôÞ CŸð‚è¬ôÞ æMòƒèœ

Ü¬ùˆFŸ°‹ èEî«ñ Ü®Šð¬ìò£è Ü¬ñ‰F¼Šð¬î‚ è£íô£‹.

ªî£™ªð£¼œ ÞòL™ è‡´H®‚èŠð´‹ ªð£¼†èœ â‰î è£ôˆFŸ°

àKò¬õ â¡ð¬î ÜPõîŸ° èEî‹ «î¬õŠðÁø¶. Üšõ£ø£è

58



ðöƒè£ô ñ‚èœ õ£›‰î è£ô‹ àî£óíñ£è ð¬öò èŸè£ô‹, ¹Fò

èŸè£ôñ, Þ¼‹¹‚ è£ô‹ ªï¼ŠH¡ è‡´H®Š¹, ê‚èóˆF¡

è‡´H®Š¹ «ð£¡øõŸ¬øˆ ªîK‰¶ ªè£œ÷ èEî‹ ÜõCòñ£Aø¶.

Ü¶ñì´ñ™ô£¶ ä«ó£ŠHò õóô£Á, Þ‰Fò õóô£Á Þ ïiù

Þ‰Fò£M¡ õóô£Á ÝAòõŸ¬ø ï£‹ ÜP‰¶ ªè£œ÷ èEî«ñ

àî¾A¡ø¶.

èEîº‹ ¸‡è¬ô»‹ :

Þ¬ê‚è¼MJ™ ã¿ ²óƒèÀ‹ Þ¬êŠðîŸ° êƒWî ªñ£NJ™

èEî«ñ ðò¡ð´Aø¶ .Þ¬êJ™ W›â™¬ô à„êvî£J

«ð£¡øõŸPŸ«èŸø£Ÿ «ð£™ èEîˆF¡ ðò¡ð£´ àœ÷¶. æMò‹

õ¬óò¾‹Þ CŸðƒè¬÷ ªê¶‚è¾‹ ÞÜôƒè£óŠªð£¼†è¬÷

Ü´‚è¾‹ Þ²õŸP™ å†ì¾‹ Þè¬ô ïòˆ«î£´ è†®ìƒèœ è†ì¾‹Þ

ï£†®òˆF™ Ü¬ê¾èÀ‹, ð£ì™èÀ‹, î°‰î ãŸø Þø‚èˆ«î£´

ð£ìè˜ ð£ì¾‹, ï£ìè‹ ñŸÁ‹ °ó™ ñ£ŸP «ð²õF™ è£ô Üõè£ê‹

(Timing) «î¬õŠð´Aø¶. ð™«õÁ ¸‡è¬ô Ü‹êƒèO½‹ èEî‹

Þó‡ìø‚ èô‰F¼Šð¬î ï£‹ è£íô£‹. àî£óíñ£è å¼ ñQî¡

Ü™ô¶ MôƒA¡ ðìˆF¬ù õ¬ó»‹ «ð£¶ î¬ô, àì™, ¬èèœ,

è£™è÷, «ð£¡øõŸPŸ° å¼ MAî£ê£ó ªî£ì˜¹ Þ¼‰î£™ ñ†´«ñ

ÜŠðì‹ à‡¬ñˆî¡¬ñ»ì¡ M÷ƒ°‹.

èEîº‹ àìŸè™M»‹ :

â‰îªõ£¼ õ®õº‹ Ýó‹H‚°‹ «ð£¶ â‡èO¡ õK¬ê Iè¾‹

ÜõCò‹ õK¬ê â‡ G¬ôJ«ô«ò â‰î å¼ Ü†ìõ¬í»‹ ¶õƒ°‹.

àìŸðJŸCèœ îQïð˜ «ò£è ðJŸCèÀ‚° â‡èO¡ àîM «î¬õ.

â‰î å¼ àìŸðJŸC M¬÷ò£†´‚èœ îìè÷Š«ð£†®èœ

«ð£¡øõŸP™ ðJŸC â´Šðõ˜ ñŸÁ‹ ðJŸÁMŠ«ð£˜

«ð£¡øõ˜èÀ‚° èEî‹ Iè¾‹ ÜõCòñ£Aø¶.

2.3.3 èEîˆF™ b˜õ£Œ¾ º¬ø:

èí‚A™ ªè£´‚èŠð†´œ÷ è¼ˆ¶‚è¬÷‚ ªè£‡´ Hø

«î¬õò£ù ¶¬í‚ è¼ˆ¶‚è¬÷‚ è‡´H®ˆ¶ ªè£´‚èŠð†ì

59



èí‚AŸ°ˆ b˜¾ è£‡ðîŸ° «ñŸªè£œÀ‹ èŸHˆî™ º¬ø

b˜õ£Œ¾ º¬ø âùŠð´‹.

èEîŠ ð£ìñ£ù¶ èí‚°èœ Gó‹Hò¶. Üî¬ù‚

èŸHˆî½‚°‹ èŸø½‚°‹ â‡íŸø ªîì˜¹èœ; àœ÷ù. Fø‹ðì

èí‚°è¬÷ˆ b˜‚°‹ Fø¬ñ«ò èEîŠ ð£ì‹ èŸøL¡ ªõŸP‚°

Ü®«è£½‹.

õ£›‚¬è‚ èí‚°è¬÷ õ£‚Aò‚ èí‚°èœ âù ï£‹ Þƒ°

°PŠH´A¡«ø£‹. â‡µ‹ â¿ˆ¶‹ Þ¬í‰î ªê£Ÿªø£ì˜è¬÷

õ£‚Aò èí‚°èO™ ðò¡ð´ˆ¶A«ø£‹. ªñ£NˆFø¡, ªê£™ô£†Cˆ

Fø¡, ªð£¼÷P Fø¡ ºîLòù ; õ£‚Aò‚ èí‚°è¬÷Š ¹K‰¶

ªè£œõîŸ°Kò Ã†´ˆ Fø¡è÷£è‚ è¼îŠð´Aø¶.

àî£óíñ£è :

«ñ£è¡ 8 ªð†®èœ ÞQŠ¹è¬÷»‹ ªè÷K 3 ªð†®èœ

ÞQŠ¬ð»‹ MŸA¡øù˜ .ªè÷K «ñ£è¡ MŸø Ü÷MŸ° ÞQŠ¹Š

ªð†®è¬÷ MŸð¬ù ªêŒõîŸ° Þ¡Â‹ âˆî¬ù ªð†®è¬÷

MŸð¬ù ªêŒò «õ‡´‹?.

Þ‚èí‚A™ èí‚A´‹ Fø¬ù õì ªñ£NˆFø¡ ðò¡ð£´

ÜFèñ£è àœ÷î™ôõ£? âù«õ õ£‚Aò èí‚°è¬÷ ¹K‰¶

ªè£œõîŸ° ªñ£NˆFø¡ Iè¾‹ ÜõCò‹.

âù«õ õ£‚Aò‚ èí‚°è÷£™ èEîŠ Hó„ê¬ùè¬÷

âO¬ñò£ù õNJ™ b˜¾ è£í H¡õ¼‹ àˆF¬ò‚ è‡ìP‰¶

¬èò£‡ì£˜.

1. ªè£´‚èŠðì´œ÷ õ£‚Aò‚èí‚°èO¡ è¼ˆ¶è¬÷ Ýó£Œ‰¶

èí‚A¡ î¡¬ñ¬ò Ýó£Œî™.

2. õ£‚Aò‚ èí‚A¬ùŠ ¹K‰î ÜîŸ°ˆ b˜¾ è£í ÜîŸè£ù

ð®G¬ôè÷£èŠ HKˆî™.

60



3. õ£‚Aò‚; èí‚AŸè£ù ðôîóŠð†ì b˜¾è¬÷ Ýó£Œ‰¶ êKò£ù

b˜M¬ù‚ è‡ìPî™.

4. èí‚°è¬÷ ð°ˆîP‰¶ H¡ù˜ ªî£°ˆîP‰¶ õ£›‚¬è‚

èí‚°èœ ªî£ì˜ð£è ªè£´‚èŠð†ì õ£‚Aò‚èí‚°è¬÷

b˜ˆî™.

àˆFèO¡ ð®G¬ôèœ :

1. àˆFè¬÷Š ðò¡ð´ˆ¶õîŸè£ù è¼ˆ¶ «êèKŠ¹

õNº¬øè¬÷‚ è‡ìPî™

2. ªð£¼ˆîñ£ù è¼ˆ¶‚è¬÷Š ðò¡ð´ˆF b˜¾ è£‡ðîŸ«èŸð

ªð£¼÷£‚è‹ ªêŒî™

3. õNè£†´î½ì¡ ðJŸC ÜOˆî™

4. îQˆîQò£è ðJŸC ÜOˆî™

5. eœð£˜¬õ ñŸø‹ F¼ˆî‹ ªêòî™

6. ªð£¶MFè¬÷ ÜPî™

7. ªî£ì˜ ðE «ñŸªè£œÀî™.

2.3.4 èEîˆ ªî£ì˜ð£ù C‰F‚°‹ Fø¡

ð£ìˆFø¡ ñŸÁ‹ ªñ£öˆFø¡ Þ¼‰î£™ ñ†´«ñ å¼

èí‚A¬ùŠ ¹K‰¶ ªêŒò èŸðõó£™ º®»‹. àî£óíñ£è ãî£õ¶

å¼ Þò™ â‡èO¡ e.ªð£.õ (HCF) ñŸÁ‹ e.C.ñ. (LCM) - ‚°‹

Þ¬ì«òò£ù ªî£ì˜H¬ù è‡ìPî™.

e.ªð£.õ (HCF) ‚°‹ ñŸÁ‹ e.C.ñ (LCM) ‚°‹ Þ¬ì«òò£ù

ªî£ì˜H¬ù âšõ£Á ªðÁõ£Œ?

61



Þò™ â‡èœ e.ªð£.õ e.C.ñ °PŠ¹èœ

«è£® (HCF) (LCM)

(4,6) 2 12

(3,8) 1 24

(6,6) 6 6

(3,7) 1 21

«ñŸè£‡ Ü†ìõ¬í Íô‹ â¡ù ªîK‰¶ ªè£œAø£Œ? ªð£¶

MF â¡ù? e.C.ñ âŠªð£¿¶‹ e.ªð£.õ MìŠ ªðKòî£è«õ Þ¼‚°ñ£?

ãî£õ¶ Þ¼ â‡èO¡ e.ªð£.õ °¬øõ£è Ü™ô¶ êññ£è Þ¼‚°ñ£?

ÞšMîñ£ù «ñŸè£‡ Ü†ìõ¬íJ™ Þ¼‰¶ Cô ªð£¶

MFè¬÷ ï£‹ ªðÁA¡«ø£‹.

Þ¼ â‡èO¡ e.ªð£.õ - Ýù¶ Þ¼ â‡è¬÷ Mì CPòî£è«õ£

Þ êññ£è«õ£ Þ¼‚è£¶.

Þ¼ ðè£ â‡èO¡ e.ªð£.õ âŠªð£¿¶‹ 1 Ýè Þ¼‚°‹. Ýù£™

ðè£ â‡èO¡ e.C.ñ â¡ð¶ Þ¼ â‡èO¡ ªð¼‚è™ ðôù£è«õ

Þ¼‚°‹.

Þ¼ Þò™ â‡èO¡ ªð¼‚èŸðôù£ù¶ ÜšM¼ â‡èœp¡

e.ªð£.õ ñŸÁ‹ e.C.ñ ÞõŸP¡ ªð¼‚èŸ ðôÂ‹ êññ£è Þ¼‚è‹.

«ñŸè£‡ Ü†ìõ¬íJ™ Þ¼‰¶ «õÁ ã«îÂ‹ ªð£¶

MFè¬÷Š ªðøº®»ñ£?

«ñŸè‡ì 3 ªð£¶MFèÀ‹ Í¡Á «õÁð†ì â‡èÀ‚°

ªð£¼‰¶ñ£ ? «ñŸè£‡ MFèœ 10000 ‚°‹ «ñŸð†ì â‡èÀ‚°Š

ªð£¼‰¶ñ£ ? ÞšMFèO™ cƒèœ Hó„ê¬ùèÀ‚° b˜¾

è£‡ðîŸè£ù ºùÂî£óíñ£è Fè›i˜èœ. å¼º¬ø

Þšõ¬èŠð†ì èí‚°è¬÷ ¹K‰¶ èŸø™ ªêòL™ ß´ð†´ M†ì£™

62



Þ Hø° ñŸø èí‚°èO¡ b˜¾è¬÷ ÜÂñ£Qˆ¶ GÏðí‹ ªêŒ¶

Mìô£‹. Þšõ¬è MFè¬÷Š ¹K‰¶ èŸø™ Þòô£î G¬ôJ™ e‡´‹

ñ£ŸÁ‚ èí‚°èO¡ Íô‹ ð®G¬ôè¬÷ ÜÂñ£Qˆ¶ ¹¶¬ñŠ

ð´ˆF e‡´‹ e‡´‹ eœ ÝŒ¾ ªêŒî H¡ Ü‚èí‚°è¬÷ ªêŒò

«õ‡´‹.

âù«õ å¼ èí‚AŸ° º¡õ®õ‹ ªè£´ˆ¶ H¡ èí‚A¬ù

b˜¾ ªêŒî½‚è£ù ð®G¬ôèœ.

îQŠð‡¹ªð£¶õ£ùÜÂñ£ù‹

GÏðí‹

GÏðí‹Þ™¬ô

ñÁ‚‚Šð†ì¶

èEî «ñ¬î H. ¬õ™ â¡ðõK¡ õ¬óò¬øŠð® “èEîMò™

C‰î¬ùò£ù¶ ªõO»ôèˆF™ àœ÷ ñŸø ÜPMò™ C‰î¬ù

õNò£èˆ î¡ Ü¡ø£ì õ£›‚¬èJ™ â‡èO™ HóFðL‚A¡ø¶”

â¡Aø£˜.

èEîñ£ù¶ ¶™Lòñ£è C‰F‚è ªî÷;põ£è ªõOŠð´ˆî

î˜‚ègFò£è C‰F‚èÞ F†ìI†ìÞ ªð£¶¬ñŠð´ˆîÞ ÝAò ªêò™

º¬øè¬÷„ ªêŒò èEî‹ ðò¡ðÁø¶.

èEî‹ èŸø™ ê£óêK ñŸÁ‹ ÞÁFJ™ Þ¼ ðKñ£ŸøˆFŸ°‹

ðò¡ðÁø¶. ÜŠð®ªòQ™ î˜‚è C‰î¬ù õ÷˜Þ °ö‰¬îèœ

î‹¬ñî£«ñ ÞòŸ¬èò£è«õ Fø¬ñè¬÷ õ÷ó„ ªêŒòŠ

ðò¡ðÁ¡ø¶. ñ£íõ˜èÀ‚° èŸð¬ùˆ Fø¬ù ñA›„C»ì¡ âF˜

63



ªè£œ÷ ðò¡ð´Aø¶. Þ‹ ñ£íõ˜èœ â‰îªõ£¼ ÞòŸ;¬è

Gè›¾è¬÷»‹ èEîˆF¡ õ£Jô£è«õ ªîK‰¶ ªè£œAøù˜.

èEî‹ èŸø ñ£íõ˜èOì‹ à†è£†C õN èŸø™ ÜFè‹

è£íŠð´‹.; à†è£†C â¡ø£™ «ê£î¬ù Íô‹ ÜPî™, àŸÁ«ï£‚A;

ÜPî™, «ñ½‹ HóFðLŠ¹ â¡ð¶ è£óíƒè¬÷ «ê£î¬ùJ¡P

ÜPõ¶. Ü‰î HóFðLŠH™ â‡íŸø ªêò™ð£´èœ

ï¬ìªðÁA¡øù. .HóFðLˆî™ Þè‡´H®ˆî™, èŸð¬ù ªêŒî™,

M¬÷ò£†´ì¡ ãŸð´‹ C‰î¬ù «è£†ð£´ Þ è†ì¬ñŠ¹Þ

ªð£¶¬ñŠð´ˆî™ ÝAòù.

î¡ùP «ê£î¬ù :

E :5 àœÀí˜¾ ñŸÁ‹ ñŸÁ‹ ªõO»í˜¾ C‰î¬ù Cô

â´ˆ¶‚è£†´èœ î¼è? Þ¬õ âšõ£Á èEî C‰î¬ù‚°

ðò¡ð´A¡øù.?

2.4 ªî£°Š¹¬ó :

èEî ÃŸÁèO¡ GÏðíƒèœ î˜‚è Mõ£îƒèO¡ ªî£ì˜„C

ñŸÁ‹ èEî MFè¬÷ ðò¡ð´ˆ¶î™ «ð£¡øõŸ¬ø ê£˜‰¶œ÷¶.

F†ð‹ Ü™ô¶ ¶™Lò‹; : (Precision)

èEîˆFŸ° â¡«ø Cô î¡pˆî¡¬ñèœ àœ÷ù. èEîˆ

b˜M¬ù M´MŠðFù£™ A¬ì‚°‹ M¬ì êK Ü™ô¶ îõÁ â¡«ø

Ü¬ñ»‹. ÞšMó‡´‹ Þ™ô£î Þ¬ìG¬ô‚ è¼ˆ¶‚° èEîˆF™

ÞìI™¬ô.

èEî è†ì¬ñŠð£ù¶ «ï˜ˆF ñŸÁ‹ ¶™Lòñ£ù¶ .

èEîñ£ù¶ èŸðõ˜è¬÷ èEî ÃŸÁè¬÷ ¶™Lòñ£è ªîK‰¶

ªè£œ÷ î°Fò£ùõ˜è÷£‚°A;ø¶. èEî‹ èŸðõ˜èO¡ Üµ°º¬ø

ªõOŠð£ì£ù¶ ªîOõ£è¾‹, Iè„êKò£è¾‹ Þ¼‚°‹.

64



å¼õ¡ èEîˆF¡ àœ÷ èí‚°è¬÷ b˜¾ ªêŒ»‹ ªð£¿¶

Þîù¶ õ£›M™ ãŸð´‹ Hó„ê¬ùè¬÷»‹ âO¬ñò£è b˜¾ è£í

º®»‹.

èEî‹ èŸðõ˜ î£‹ Ü¬ìò G¬ù‚°‹ °P‚«è£œè¬÷ î£«ñ

«î˜‰ªî´ˆ¶‚ ªè£œõ˜ .ªî£ì‚è G¬ôèO™ Ã´ñ£ùõ¬ó

ð¼Šªð£¼œè¬÷Þ õ£›‚¬è àî£óíƒè¬÷ ²†®‚è£†® è¼ˆ¶‚èœ

ï¡° ªîOõ£è ñùF™ ðF»‹ õ‡í‹ èŸè„ ªêŒAø£˜èœ.

åšªõ£¼ ñQîQ¡ ªêò™ð£´èO½‹ èEî ªè£œ¬èèœ

Üõù¶ Ü¡ø£ì õ£›‚¬èJ™ ðò¡ðÁø¶.

èEî‚è¼ˆ¶è¬÷ êKð£˜‚°‹ ªð£¿¶ ïñ¶ ñŸø C‚è™èÀ‚°‹

«î¬õò£ù õNº¬øè¬÷ î¼A¡øù. èEî‚ ÃŸÁ å¼

ªêò™º¬øJ™ îõø£ùî£è Ü¬ñ‰î£™ ñŸø ªêò™ð£´èÀ‹ îõø£è

Ü¬ñ‰¶ M´‹.

èEî C‰î¬ùò£ù¶ Hó„ê¬ùèÀ‚è£ù b˜¾ è£í™ ñŸÁ‹

ÜîŸè£ù Mù£M¬ù â¿Š¹î™ «ð£¡øõŸ¬ø‚ ªè£‡´œ÷¶.

èEî ÜPõ£ù¶ ñQîQ¡ Ü¡ø£ì õ£›M™ ãŸð´‹

C‚è™èÀ‚°ˆ b˜¾ è£í ðò¡ðÁø¶. èEî ªêò™ð£ì£ù¶ å¼

ªð£¼¬÷ŠðŸP àŸÁ«ï£‚è™ ÞÜÂñ£Qˆî™Þ åŠH´î™Þ

H¡ðŸÁî™Þ «ð£¡øõŸ¬ø êKð£˜ˆî™Þ H¬öÞ HKˆ¶ ÜPî™

Ü†ìõ¬í¬ò ðò¡ð´î¶‹ ð®G¬ôèO¡ Íô‹ èEî‹

èŸðõ˜èÀ‚° àî¾A¡ø¶.

1 . º¡ùP «ê£î¬ùèÀ‚è£ù M¬ìèœ :

E1 : MF M÷‚è º¬ø

E2 : Ý‹/ èEîˆ¬î èŸ°‹«ð£¶ ïñ¶ ðö‚èõö‚èñ£ù¶

ªîOõ£è¾‹ Þ²¼‚èñ£è¾‹/ êKò£è¾‹/ Ièˆ¶™Lòñ£è¾‹

â¿î¾‹ ªê£™õîŸ°‹ èEî‹ õ½×†Áø¶.

èEî‚è¼ˆ¶‚èÀ‹ èEî °lf´èÀ‹ èEî õ£˜ˆ¬îè¬÷

²¼‚èñ£è ÃP MKõ£ù M¬ì î¼õîŸ° ðò¡ðÁø¶.

65



E3 : å¼ õ¬óðìˆî£O™ ðôîóŠð†ì º‚«è£íƒè¬÷ õ¬ó‰¶å¼ °ö‰¬îJì‹ îó«õ‡´‹. åšªõ£¼ º‚«è£íˆ¬îŠðŸP»‹ Ü‰î °ö‰¬î¬ò MõK‚è ªê£™ô «õ‡´‹.Ü‚°ö‰¬î‚° º‚«è£íˆF¡ ªð£¶Š ð‡¹è÷Þ; õ®õ‹Þð‚èƒèO¡ â‡E‚¬èÞ «è£íƒèO¡ â‡E‚¬èÞà„CèO¡ â‡E‚¬è¬òŠ ðŸP ÃP àîõ «õ‡´‹. Hø°Ü‰î õ¬óðìˆî£O¬ùŠ ªðŸÁ‚ªè£‡´ Ü‚°ö‰¬î¬ò å¼º‚«è£í‹ ðìˆ¬î õ¬óò„ ªêŒ¶ Üî¬ùŠ ðŸPM÷‚è„ªê£™ô «õ‡´‹.

E 4 : ¹ˆîèˆF™ å¼ â´ˆ¶‚è£†¬ì‚ ÃP ÜîŸ° M¬ì â¿î ¬õ‚è«õ‡´‹.

E 5 : å¼ °ö‰¬î¬ò 18 ñŸÁ‹ 17 ä Ã†ì„ªêŒò «õ‡´‹.Ü‚°ö‰¬î ; ºîL™ 18 °„Cè¬÷ â´ˆ¶ ÜF™ 10 °„Cè¬÷«ê˜ˆ¶ å¼ è†ì£è¾‹ eF â†´ °„Cè¬÷ ¬õˆ¶œ÷¶.Ü´ˆ¶ 17 °„CèO™ 10 °„Cè¬÷ «ê˜ˆî å¼ è†ì£è¾‹ eF 7°„Cè¬÷»‹ ¬õˆ¶œ÷¶. îQˆîQò£è ¬õˆ¶œ÷ 8ñŸÁ‹7 °„Cè¬÷ «ê˜ˆ¶ ªñ£ˆî‹ 15 °„CèO™ 10 °„Cè¬÷ å¼è†ì£è¾‹ eF 5 °„Cè¬÷ îQò£è¾‹ ¬õˆ¶œ÷¶. ªñ£ˆî‹3 è†´èO™ 30 °„CèÀ‹ eF 5 °„Cè¬÷»‹ «ê˜ˆ¶ ªñ£ˆî‹35 °„Cèœ âù‚ ÃÁA¡ø¶.ÞˆFøQù£™ °ö‰¬î‚°

ðôîóŠð†ì C‰F‚°‹ Fø¡ à¼õ£‚èŠð´Aø¶.

2.6 «ñŸ«è£œ Ë™èœ

NCERT (2008) : Source Book on assessment for classes I-V : Mathematics ,New Delhi –NCERT.

NCERT (2008) : National Curriculum frame work 2005 . New Delhi – NCERT.

Cruikshanl , D.E., Fitzgerald, D.L , jensen ,L.r.(1941)Young Children Learn9ingMathematics , Boston.

CBSE (2010) Continuous and Comprehensive evaluation : manual for teachers ofclasses Vi to VII –New Delhi . CBSE

IGNOU (1997) Teaching of Primary School Mathematics Block I – Aspects ofTeaching Mathematics , New Delhi : IGNOU

66



2.7 Üô°ˆ«î˜¾

1. “èEîˆF™ î˜‚è C‰î¬ù” â¡ø èEîˆF¡ î¡¬ñJ¬ù

ªî£ì‚è G¬ô ñ£íõ˜èÀ‚° âšõ£Á ðò¡ð´ˆ¶õ£Œ?

2. èEîˆF¡; è†ì¬ñŠ¹ ðŸP ÃÁ ?

3. èEîMò™ è™M èEî C‰î¬ù FøQ¡ õ÷˜„C‚°

º‚Aòˆ¶õ‹ î¼A¡ø¶ â¡ð¬î â´ˆ¶‚è£†´ î‰¶ MõK.

67



Üô° - 3

èEî‚ è™MJ¡°P‚«è£œèÀ‹ ªî£¬ô«ï£‚°‹.

(Goals and vision of Mathematics Education)

fl;likg; ;G:

3.0 Kd;Diu.

3.1 fw;wypd; Nehf;fq;fs;.

3.2 fzpjf; fy;tpapd; Fwpf;Nfhs;fs.;

3.2.1 tphpthd kw;Wk; Kiwahd Nehf;fq;fs;.

3.2.2. rpwg;G Nehf;fq;fs;.

3.3. gs;spf;fzpjj;jpd; jd;ikfs;.

3.3.1. Foe;ijfSk; fzpjKk;.

3.3.2. tFg;giwr; #oypy; fzpjk;.

3.3.3. fzpjj;ij Mh;tKld; fw;gjw;fhd #o;epiyfs;.

3.3.4. fzpjk; fw;wYf;fhd #oiy cUthf;Fjy;

3.4. njhFj;jy;

3.5. jd;dwpTr;Nrhjid.

3.6 Nkw;Nfhs; E}y;fs;

3.7. myFj;Njh;T

Kd;Diu

fzpjk; ,e;j cyfj;jpy; mf;fhyj;jpYk; ,f;fhyj;jpYk; Kf;fpa gq;F

tfpf;fpwJ. Ke;ija myFfspy; fz;lJ Nghy; mwpTrhh;e;j vy;yh ,lq;fspYk

md;whl tho;f;ifapYk;; fzpjk; Kf;fpa gq;F tfpf;fpwJ. ,e;j cyfj;fpd;

68



eilngWk; midj;J mwptpay; njhopy;El;gj;jpd; Kd;Ndw;wj;jpw;Fk;

EZf;fj;jpw;Fk; fzpjNk Kf;fpa topfhl;Ljyhf jpfo;fpwJ. ,e;j cyfk;

KOtJk; fzpjk; epuk;gpAs;sJ vd;gij Sir James Jeans Gfongw;w.

Mq;fpNya N[hjplh; ‘flTs; vd;gth; rpwe;j fzpj Nkij mth; ,g;gpugQ;rj;ij

Kiwahd rpwe;j mbg;gilapy; cUthf;fpAs;shh;.

kw;w ghlj;jpdUf;F fzpjk; Ghpahj Gjpuha; czug;gLfpd;wJ. Vndd;why;

mjd; vjhh;j;jkhd jd;ikNa MFk;. nghJthf kf;fs; fzpjk; vd;gJ

fbdkhd ghlk; vd vz;Zfpd;wdh;. kpFjpahd kf;fs; fy;tp mwptpy;

rpwe;jth;fshf tpsq;fpdhYk; fzpjj;ijf; fz;L mr;rg;gLtJ xd;Wk;

mjprakpy;iy. Kjy; ,aypNyNa fzpj mr;rj;ijg;gw;wpAk; fzpjk; fw;gthpd;

ftiyiag; gw;wpAk; fz;Nlhk;. rpy Mrphpahfs;; fzpjj;jpd; jd;ikia

khzth;fSf;F fbdkhdjhfNt fw;gpf;fpwhh;fs;. khzth;fSk; fzpjj;ij

fbdkhdjhfNt fUJfpwhh;fs;;. ek; kdjpy; gy tpjkhd Nfs;tpfs; vOfpd;wd.

fzpjj;ij ,d;iwa khzth;fs; fw;Fk;; nghOJ vjw;fhf xUth; fbdkhd

fzpjj;ijf; fw;Wf;nfhs;sNtz;Lk;? vd vz;Zfpd;wdh;.md;whl tho;f;ifapy;

fzpjj;jpy; cs;s FwpaPLfs;, nray;Kiwfs;, EZf;fq;fs; Nghd;wit miktJ

vg;gb?fzpjj;ij fw;Fk;NghJ ehk; clNd vijg; gw;wp mwpe;J nfhs;fpNwhk;?

fzpjk; fw;gpj;jypYk; fw;wypYk; Ntbf;if cz;lh? ,t;thwhd

Nfs;tpfSf;F tpilaspf;Fk;NghJ fzpjk; gw;wpa njspthd fUj;Jf;fs;

tpsq;Fk;. fzpjk; kPJ cs;s mr;rKk;; tpyFk;.

,e;j ghlj;jpd; %yk; fzpjj;ijg;gw;wp tphpthfTk; njspthfTk;

mwpe;Jnfhs;syhk; . ,j;jifa Nfhl;ghl;bd; %yk; njhlf;fg; gs;spapy;

fzpjk; fw;Wf;nfhLf;fg;gLfpwJ.

3.1. fw;wypd; Nehf;fq;fs; : (Learning Objectives)

,dptUk;; ,e;jg;ghlj;jpy; ehk; fw;Wf;nfhs;git

,ilepiyg;gs;spfspy; fzpjk;; fw;gpj;jypd; Nehf;fk; FWfyhf cs;sJ.

,d;iwa fy;tpapy; fzpjk; gaDs;sjhf mikfpwJ.

69



gs;spfspy; fzpjj;ij tho;f;ifr; #oNyhL njhlh;G gLj;jp fw;gpf;f

Ntz;Lk;.

3.2.fzpjf; fy;tpapd; Fwpf;Nfhs; :(Aims of mathematics Education )

fzpjj;jpw;F Ra fy;tp Nehf;fq;fs; cs;sJ. David Wheeler vd;ghh;;

“fzpjj;ij ehk; vt;thW nray;gLj;JfpNwhNkh me;j mstpw;F fzpjj;ij

fw;Wf;nfhs;fpNwhk;.”vdf; $Wfpwhh;. Njrpa fiyj;jpl;l tbtikg;G 2005 d;

fUj;Jg;gb fzpjf; fy;tpapd; Kf;fpa ,yf;F Foe;ijfspd; jpwikfis

Nkk;gLj;JtjhFk;;.George Polya tpd; fUj;jpw;fpzq;f ,uz;L tpjkhd Nehf;fq;fs;

gs;spf; fy;tpapy; cs;sd. mit gue;j kw;Wk; FWfyhd Nehf;fkhFk;.

3.2.1. tpupthd kw;Wk; Kiwahd Nehf;fq;fs;:

(Broader and Narrower Aims )

fzpjf; fy;tpapy; gue;j kw;Wk; FWfyhd Nehf;fk; gw;wp fhZk; Kd;G

nray;Kiwiar; nra;aTk;.

nray;Kiw-1

fzpjk; Kf;fpa gq;Ftfpf;Fk; ,lq;fisAk; njhopy;fisAk; gl;baypLf.

nray;Kiw -2

Foe;ijfs; Vd; fzpjk; fw;f Ntz;Lk; vd ehk; epidf;fpNwhk;?

Nkw;$wpathW fzpjj; jpwikfis Nkk;gLj;j fzpjk; fw;wy; Kf;fpa

Nehf;fkhFk;.Mdhy; ‘Mathematize’ vd;gjd; nghUs; vspikahf;Fjy;; my;yJ

fzpj #j;jpuq;fis gad;gLj;JtJ. nghJthf‘Mathematization’ vd;w nrhy;

fUj;Jf;fis gad;gLj;Jk; topKiwfs; kw;Wk; fzpj nray;ghl;L Kiwfs;

Mfpatw;iw Fwpf;fpwJ. ,j;Jiw kl;Lky;yhky; gpw Jiwfspd;

Kd;Ndw;wj;jp;w;Fk; fzpj mwpT gad;gLfpwJ. xUth; fzpjj;jpy; Kiwg;gb

70



nray;tpsf;fk; mspj;jy;, Kiwahd FwpaPLfis mwpjy; ,kw;Wk; Jy;ypakhd

nray;Kiwfis Nkw;nfhz;L ,Uj;jy; ,Muk;g epiyapy; fzpjk; fw;gjd;

cah;e;j Nehf;fk; vd;gJ cUthf;Fjy; kl;Lky;yhky; mjid eilKiwapy;

gad;gLj;JtjhFk;.

fzpj nray;ghl;L Kiwfis Nkk;gLj;Jjy; kw;Wk; Kd;Ndw;wk; mila

jPh;tha;T Kiw , Ragapw;rp Kiw, Njhuhag;gLj;Jjy; , kw;Wk; kjpg;gpLjy;

kdf;fhl;rpfis cUthf;Fjy; , njhlh;GgLj;Jjy;, fiy czh;r;rp ,t;thwhd

jpwikfs; tsu fzpjk; cjtp GhpfpwJ.” Foe;ijfspd; jdpj;jpwid tsh;j;jy;

, fzpj Kiwapy; Nahrpj;J tpilaspj;jy;, jh;f;f Kiwapy; Cfpj;J jPh;T

fhZjy;, fl;likg;ig ifahSjy;” Mfpad fzpjj;jpd; Kf;fpa Fwpf;Nfhs;fs;

MFk;.

jPh;tha;T Kiw: (Problem Solving)

khzth;fs; kdg;ghlk; nra;tij jtph;j;J, fUj;Jfis Ghpe;J nfhz;L,

njhpe;j kw;Wk; njhpahj #o;epiyfs; , md;whl tho;f;if gpur;ridfs; ,

my;yJ ghlj;jpYs;s gapw;rpfSf;F jPh;T fhz;gjw;Fk; jPh;tha;T Kiw

Kf;fpag;gq;fpid Mw;WfpwJ.

jPh;tha;T Kiwapy; Ghpe;J nfhs;Sjy;,nray;gLj;Jjy; ,kjpg;gpLjy, ;

fhuzg;gLj;Jjy; kw;Wk; rhpghh;j;jy; Mfpait mlq;Fk;. tbtikj;jy;,

mstpLjy;, xg;Gikg;gLj;Jjy;, vspikg;gLj;Jjy;, njhFj;Jmwpjy;, Mfpait

ghlE}y; fzf;FfSf;F jPh;Tfhz cjtpGhpfpwJ. (v.L): khzth;fspilNa

“xU tPl;il vt;thW msg;gha;?” vd;W Nfs;tp vOg;gg;gbd; mth;fs; jd;

iffs; , fhy;fs; ,Fr;rp; , fapW , msTNfhy; kw;Wk; msTehlh nfhz;L

mstp;;lyhk; vd;ghh;fs;. gpd;G rpy Nrhjidapd; mbg;gilapy; kpfg;nghpa

,lj;ij msTehlh nfhz;L msf;f Ntz;Lk; vd;gjid Ghpe;J nfhs;thh;fs;.

,jd;%yk; xU gpur;ridf;F vt;thW jPh;Tfhz Ntz;Lk; vd;gjid mwpe;J

nfhs;th;.

fz;lwpKiwapd; nrayghLfs; :(Use of Heuristics )

fzpjj;jpy; fz;lwpKiwiag; gad;gLj;jpr; rpe;jidf; fzf;Ffisr;

nra;AkhW khzhf;fh;fsplk; $wyhk;. mg;NghJ xt;;nthUtUk; jdpj;jdpNa

71



gpur;ridia Muha;e;J ghh;g;ghh;fs;. khzhf;fh;fs; jq;fSf;F vof;$ba

re;Njfq;fisf; Nfl;Lj;njhpe;J nfhs;s chpikaspf;fg;glNtz;Lk;. Vnddpy;

,e;j KiwahdJ khzf;fh;fisf; Nfs;tpfs; Nfl;fj;J}z;lTk; mth;fs;

mtw;wpw;Fg; gjpyspg;gijANk Fwpf;Nfhshff; nfhz;bUf;fpwJ. khzf;fh;fsplk;

RWRWg;igAk; Mh;tj;ijAk; Cl;lty;y Nfs;tpfis Mrphpah; mbf;fb Nfl;f

Ntz;Lk;. vdNt tpdhf;fis Nfl;Fk; fiyia Mrphpah; mwpe;jpUj;jy; mtrpak;.

NkYk; xU tpjpNah my;yJ #j;jpuNkh epUgpf;fg;gl;l gpd; mjid khzf;fh;fs;

Ghpe;J nfhz;lhh;fsh? vd;W fhz;gjw;fhf vLj;Jf;fhl;LfSk; , gapw;rpfSk;

nfhLf;fg;gLtJk; mtw;wpd; tuk;G vd;d? tpjptpyf;Ffs; ahit? vd;W

Nrhjpg;gjw;fhf khzth;fis Mrphpah; Nfs;tpfs; Nfl;gJk; fz;lwp Kiwapd;

Nehf;fq;fshFk;. tpjptUKiwapYk; tpjp tpsf;fKiwapYk; rpf;fy;fis

tpLtpg;gJ fz;lwp Kiwiar; rhh;e;jNjahFk;.

kjpg;gpLjy; kw;Wk; Njhuhag;gLjy; :(Estimation and approximation)

mwptpay; Muha;r;rpapy; Fwpg;gplj;jf;f jPh;T fpilf;fhj NghJ mstPLfis

khzth;fs; fbdkhd fzf;FfSf;F vspikahf jPh;T fhz;fpwhh;fs;.

ghlg;Gj;jfq;fspy; ,y;yhjijAk; , tFg;giwapd; ghpkhw;wq;fspy; Fwpg;gplhjijAk; , gs;spf; fzf;Ffs; ekf;F fw;Wj;jUfpd;wd.

Njh;T Kiw:(optimization)

Njh;T Kiw vd;gJ ngwg;gl;l tsq;fis gad;gLj;jp nfhLf;fg;gl;l

fzf;FfSf;F KOikahd jPh;T fhz;gJ MFk;. ,k;Kiwia gs;spf;

fzpj fiyj;jpl;lj;jpy; Nrh;f;ftpy;iy. nfhLf;fg;gl;l epge;jidfis fUj;jpy;

nfhz;L mjw;Nfw;wthW jPh;T fhZjNy Njh;T KiwahFk;.

1. m[apd; tUl tUkhdk; &gha;. 3.5 yl;rk;. mth; &gha;.15 yl;rk;

kjpg;Gs;s tPl;il nrhe;jkhf thq;fNtz;Lk; vd;why; vj;jid Mz;Lfs;

MFk;? ve;j tpj flDkpd;wp Nkw;fz;l fzf;if gy;NtW Kiwfisg;

gad;gLj;jp jPh;T fhz KbAk;. mtUila tUkhdk; Nrkpg;G nryTfs;

kw;Wk; tPl;bd; Nja;khdk; Mfpatw;iw fzf;fpl nfhLf;fg;gl;l ,U

epge;jidfshy; jPh;T fhzKbahJ.vy;yh #o;epiyfspYk; Njh;TKiwia

gad;gLj;jp fzf;if jPh;T fhz;gJ vd;gJ vspjy;y. ,jid

Jtf;fg;gs;spfspy; fw;Wj;jUtJ cz;L.

72



khjphpapd; gad;fs;:(Use of Patterns)

gad;fs;:

khjphpfs; khzth;fSf;F fzpjj;jpy; jdpj;Jtj;ij ngwTk;,tpsf;fk;

ngwTk;(k)#j;jpuq;is njhFf;fTk; gad;gLfpwJ. Foe;ijfs; khjphpia

gad;gLj;jp tbtq;fs;, epfo;Tfs;, vz;fspd; njhFg;G Mfpatw;iw mwpe;J

nfhs;s ,aYk;NghJ fzpjj;jpd; Rit ntspg;gLfpwJ. fz;ftUk; khjphpfis

gad;gLj;Jk;NghJ Foe;ijfs; Mh;tk; nfhs;fpd;wdh;. NkYk; ,J mth;fis

kfpo;T+l;Lk; nray;fspy; <LgLj;JfpwJ.

cUtikg;G : (Representation)

mstPLfSf;Fk; tbtq;fSf;Fk; nray;gLk; khjphpfis cUthf;FjNy

fzpjj;jpd; rpwe;j gad;ghlhFk.; cUtikg;ghdJ fhl;rpg;gLj;Jjy;,

njspTgLj;Jjy; , jtwhd jfty;fis xJf;fTk; gad;gLfpwJ. NkYk; cUt

mikg;gpd; gy gupzhkq;fs; %yk; gy vLj;Jf;fhl;Lfs; nfhz;L ,J

njhlh;ghd jd;ikfisAk; mwpa ,aYk;. ,jw;F vLj;Jf;fhl;lhf tpfpj

vz;iz xU nghUspd; %ykhfNth mjd; ghfq;fs; %ykhfNth, vz;Nfhl;bd;

%ykhfNth Fwpj;Jf;fhl;lyhk;. jFgpd;dq;fis fhl;bYk; jfhgpd;dq;fSf;F

,k;Kiw gaDs;sjhf ,Uf;Fk;.

fhuzk; kw;Wk; jPh;T :(Reasoning and Proof)

fhuzk; kw;Wk; jPh;thdJ fzpjj;jpd; mbg;gilahFk;. xU tpdhtpw;F

,UNtW egh;fs; xNu tpilia ntt;NtW Nfhzq;fspy; mspf;fyhk;. ,jw;fhd

vLj;Jf;fhl;il fhzyhk.; mLj;JtUk; vz;izf; fhz;f. ; 3 15 35 63

99………?

,jd; jPh;it fPo;tUkhW fhzyhk;.

22 , 42 –1, 62 -1 ,82 – 1 ,102 – 1, 122- 1 = 143

kw;nwhU jPh;T :

3,3+12,15+12+8,35+12+8+8, 63+12+8+8+8,

99+12+8+8+8+8=143.

73



fhuzk; kw;Wk; jPh;tpd; nray;KiwahdJ fzpjj;jpy; Kf;fpa gq;F

Mw;WfpwJ. gs;spf;fy;tpfspy; tpthjq;fspd; %yk; jPh;T fhz;gJ fzpjj;ij

nghpJk; Cf;fg;gLj;JfpwJ. tpthjq;fis Cf;Ftpj;jYk; , kjpg;gplYk; ,

fUj;JUthf;fq;fis Muha;jYk; fhuzk; fz;lwpAk; Kiwfis

Ghpe;Jnfhs;S;jYk; ,jd; Kf;fpa Nehf;fk; MFk;.

njhlh;G cUthf;Fjy; : (Making Connection )

fzpjkhdJ fzpjk; kl;Lk; my;yhky; fzpjk; njhlh;ghd NtW gy

ghlq;fisAk; fw;f cjTfpwJ. fzpj tFg;Gfspy; khzth;fs; tiugl

gapw;rp Nkw;nfhs;fpd;wdh;. vdpDk; jdJ jpl;l Ntiyfspy; gad;gLj;JtJ

,y;iy. ,Ug;gpDk; ,aw;gpay; kw;Wk; mJ njhlh;ghd ghlq;fspy; vOk;

gpur;ridfSf;F ,jd; %yk; jPh;T fhz;fpd;wd. mwptpay; kw;Wk; mjd;

tpjpfspy; vOk; gpur;ridfis jPh;f;f jh;f;fKiw kw;Wk; fzpjf;FwpaPl;L

Kiw gad;gLfpwJ. fiyj;jpl;lj;jpy; NtW gFjpfspy; fzpj mwpthdJ

jd;id njhlh;GgLj;jpf;nfhs;s cjTfpwJ. NkYk; md;whl tho;tpy; vOk;

gpur;ridfSf;F mbg;gil fzpj mwpT ,d;wpaikahjJ MFk;.

fzpj njhlh;G: (Mathematical Communication)

nkhopia Jy;ypakhd ntspg;ghL kw;Wk; Fog;gkw;w #oypy;

gad;gLj;JtJ fzpjf; fy;tpapd; gz;ghFk;.fzpjf;FwpaPLfs; ,nkhop

nray;ghLfs; Nghd;wtw;iw gad;gLj;Jtjd; %yk; fzpjk; mh;j;jKs;sjhfTk;

KiwahdjhfTk; jpfo;fpwJ.

Y – d; ,Ulkq;Fld; 52 ia $l;bdhy; X fpilf;Fk;. Y – d; kjpg;ghdJ

75 vdpy; X – d; kjpg;G vd;d?

X=2y+52

2(75)+52 = 150+52=202

jdpeghpd; mDgtk; kw;Wk; Jy;ypakhd Kiwfis gfph;e;Jnfhs;s

cjtp GhpfpwJ. ekJ gs;spfspy; ghlq;fSk;; ghl ,iznray;ghLfSk;

gue;j Nehf;fj;jpw;F Kf;fpaj;Jtk; nfhLg;gjpy;iy.

74



gs;sp fzpjj;jpy; jpwd; Nkk;ghlila fw;gpj;jy; gue;j Nehf;fk; nfhz;Ljhf

mika Ntz;Lk;.,ay; vz;fs;, fzpj nray;ghLfs;, ; mstPLfs;, jrk vz;fs;

kw;Wk; rjtpfpjk; Kjypatw;wpy; Fwpg;gplj;jf;f njhlh;G nfhz;Ls;sJ. vd

NCF 2005 Neubahf ntspg;gLj;jpAs;sJ.; mbg;gil mwpit ntspg;gLj;Jjy;,;

Cf;fg;gLj;Jjy;, Ghpe;Jnfhs;Sjy; Nkk;gLj;Jjy; Mfpa jpwd;fisf; nfhz;L

fzpj fy;tpapd; gue;j Nehf;fj;ij mila ,aYk.; fiyjpl;lkhdJ tFg;giw

ghpkhw;wq;fspd; %yk; ;xj;jpiritAk; ,UNtW mk;rq;fis mspg;gjhf mika

Ntz;Lk;.

fPo;tUtd fzpj fy;tpapd; Kf;fpa Nehf;fkhFk;

Nahrpj;jy; kw;Wk; fhuzk; fz;lwpAk; jpwid Nkk;gLj;Jjy;

md;whl tho;tpy; Vw;gLk; fzpj gpur;ridfSf;F jPu;T fhZjy;.

#w;W#oy; kw;Wk; gz;ghl;bid Gupe;J nfhz;L mwpKfkhjy;

Foe;ijfis NtWgl;l njhopy;El;gk; kw;Wk; vjpu;fhy nghJ

NtiyfSf;fhf jhahh;nra;jy;.

cah;epiy fy;tpf;F Foe;ijfis jahh; nra;jy;.

Foe;ijfspd; fz;Lgpbf;Fk; jpwid Nkk;gLj;jy;.

E 1 : fzpjk; fw;wy; - fw;gpj;jypy; Nehf;fq;fs; VNjDk; Ie;J fUj;Jf;fis

$Wf.

3.2.2 rpwg;G Nehf;fk; :

fzpjf; fy;tpapd; rpwg;G Nehf;fkhdJ fPo;tUtdtw;iw tiuaWf;f

gad;gLfpwJ. nghUj;jkhd tFg;giw fw;wy; topKiwia tbtikf;f

,fiyj;jpl;lk; ,TLM –ia tbtikf;f , topfhl;l, kjpg;gPl;L topKiwfis

cUthf;f, NkYk; rpwg;G Nehf;fkhdJ tpUk;gj;jf;fthW nray;tpid,

Rl;bfhl;lg;gl;l, FWfpa, milaf;$ba gy $Wfis jd;dfj;Nj nfhz;Ls;sJ.

75



fPo;tUtd fzpjf;fy;tpapd; rpy rpwg;G Nehf;fq;fshFk;;;.

khzth;fspilNa fzpjk; fw;wy; cWjpahd ey;y Jtf;fj;ij

Vw;gLj;Jfp;wJ.

ghlj;jpd; mbg;gil $WfisAk; topKiwfisAk; njspTgLj;j

gad;gLfpwJ

fzpjk; fw;wyhdJ Foe;ijfSf;F fw;wypy; Mh;t%l;lTk; <h;g;ig

Vw;gLj;jTk; Ntz;Lk;.

fzpjj;jpd; kPJ xUtpj ek;gpf;ifia Nkk;gLj;j Ntz;Lk;.

ghuhl;Lfs; %yk; njspTgLj;Jjiy Nkk;gLj;j Ntz;Lk;.

epfo; kw;Wk; vjph;fhy tho;f;ifapy; fzpjk; vt;thW njhlh;G nfhs;fpwJ

vd;gij tpsf;f Ntz;Lk;.

fzpjj;jpd; mofpaiy ghh;itapl Ntz;Lk;.

KiwgLj;Jjy;, gapw;rp, nray;gLj;Jjy; , Raek;gpf;if, kw;Wk; fbd

ciog;G Mfpa gof;fq;fis Nkk;gLj;Jjy;.

fzpjj;ij gpw ghlq;fNshL nghUj;Jjy;

fzpj nkhop kw;Wk; FwpaPLfis mwpe;J nfhs;s Ntz;Lk;.

cau;tFg;Gfspy; fzpjk; fw;wiy cUthf;f Ntz;Lk;.

fzpj fz;fhl;rpfSf;F jahh;gLj;j Ntz;Lk;.

xw;iw kw;Wk; ,ul;il vz;fis fw;gpj;jypy; cs;s mwpTWj;jypd;

Nehf;fq;fs;:

khzth;fs;

nghUl;fis ,U rkghfq;fshf gpupf;f ,aYk;.

%d;W ,yf;f vz;fspy; xw;iw kw;Wk; ,ul;il vz;fshf milahsk;

fhz ,aYk;.

Xw;iw kw;Wk; ,ul;il vz;fis NtWgLj;j ,aYk;.

76



Xw;iw kw;Wk; ,ul;il vz;fSf;F vLj;Jf;fhl;L ju ,aYk;.

tho;tpay; #o;epiyfspy; xw;iw kw;Wk; ,ul;il vz;fis gad;gLj;jp

mwpa ,aYk;;.

E:2 fzpjf; $Wfis fw;wy; kw;Wk; fw;gpj;jypy; VNjDk; ,U fhuzpfis

$Wf.

E :3 fPo;tUtd fzpjj;jpd; mwpTWj;jy; Nehf;fq;fsh?

Foe;ijfspd; Nahrpj;jy; kw;Wk; fhuzk; fz;lwpAk; jpwd;fis

Nkk;gLj;Jk;

Foe;ijfsplk; mwptpay; kw;Wk; ,ay;ghd nray;Kiwfis

Cf;Ftpj;jy;.

md;whl tho;tpy; Vw;gLk; gpur;ridfis ,U ,yf;f vz;fspy; $Ljy;

%yk; jPu;T fhZjy;.

NtWgl;l ehzaq;fis njspthf milahsk; fhZjy;.

NtfkhfTk; tpNtfkhfTk; fzf;fpl cjTk;.

gjpide;J gj;ij tpl Ie;J $Ljyhf cs;sJ vd;gij mjw;fhd

FwPaPLfs; %yk; tpsf;fyhk;.

mstPLfis kjpg;gplNth (m) Njhuhakhd kjpg;ig fz;lwpaNth

,aYk;.

ek; md;whl tho;tpy; Vw;gLk; rpW fzpjg; gpur;ridfis #j;jpuj;ij

nfhz;L tpsf;fyhk;.

thpir kw;Wk; Kiwia milahsk; fz;L nfhs;s

%d;W ,yf;f vz;fspy; xw;iw kw;Wk; ,ul;il vz;fis milahsk;

fhz.

77



nray;ghL – 3

fw;wy; kw;Wk; fw;gpj;jypd; mwpTWj;jypd; Nehf;fq;fis vOJf?

I. %d;W ,yf;f vz;zpy; cs;s ,yf;fj;jpd; kjpg;G

II.jdp tl;b

——————————————————————————————

————————————————————————————————————————————————————————————

3.3 gs;spf; fzpjk; xU ghh;it (Vision For School Mathematics)

xU fzpj Mrpupah; jd;dplk; fw;Fk; fzpjk; fw;Fk; Foe;ijfSf;F

fzpjj;jpd; kPJ Mh;tj;ijAk; , <h;g;igAk; Vw;gLj;j Ntz;Lk;.tFg;giwapy;

Mrphpah; fw;gpf;Fk; Kiw , khzth; Mrphpah; ,ilNaahd ey;YwT, ,iltpid,

,it khzth;fSf;F fzpjk; fw;gjpy; Mh;tj;ijAk; ek;gpf;ifiaAk;

Vw;gLfpwJ.fzpjk; xU rpf;fyhd ghlkh? fzpjf; fUj;Jf;fis kddk;

nra;a KbAkh ? fzpjj;ij khzth;fSf;F tpUg;gkhdjhfTk; ,

Ghpe;Jnfhs;SkhW vt;thW fw;gpf;f Ntz;Lk; vd;gij Mrpupah; Mfpa ehk;

njupe;J nfhs;s Ntz;Lk;.

Njrpa fy;tpf; nfhs;if (1986) ,d; fUj;J :

fzpjk; vd;gJ xU Foe;ijia gapw;Wtpf;Fk; mikg;ghfTk;,

epidg;gjw;Fk; , fhuzq;fs; fz;lwpaTk; , gFj;jwpaTk; jh;f;fhPjpahf

tpthjpf;fTk; gad;gLfpwJ.

kw;w midj;J ghlq;fSf;fk; gFj;jwpjy; ,fhuzk; fw;gpj;jy; Nghd;wtw;iw

<Lghl;Lld; nra;a fzpjk; gad;gLfpwJ.fzpjf; fy;tpahdJ Njrpa

tsh;r;rpf;Fk; Foe;ijfspd; tsh;r;rpf;Fk; xU fUtpahf mikfpwJ.vdNt,fzpjf;

fUj;Jf;fis Kjd;ikahff; nfhz;L Njrpa fiyj;jpl;l tbtikg;G 2005

cUthf;fg;gl;Ls;sJ. “Nfl;lwpjy; ,ntspg;gLj;Jjy; , tpdh vOg;Gjy; ,

tpthjpj;jy; , gad;gLj;Jjy; , fUj;Jf;fis gpujpgypj;jy, ;Gjpa fUj;Jf;fisf;

78



fl;likj;jy”Nghd;w fUj;Jf;fis NCF -2005 K;jd;ikahff; nfhz;L

Foe;ijfSf;F mwpTWj;JfpwJ.

Nkw;$wpa cah;e;j fUj;Jf;fis mbg;gilahff; nfhz;L NCF -2005

Foe;ijfspd; ghh;itapy; gs;spf; fzpjk; vt;thW fw;gpf;fg;gl Ntz;Lk;

vd;gjid tiuaWj;Js;sJ.

mbg;gil fUj;ij czh;jy;.

tupirg;gLj;Jjy;.

gpur;ridia kjpg;gpLjy;.

gpur;ridfSf;F jPh;T nra;jy;.

Cfpj;jg; gpd; rupah vdg;ghh;f;Fk; gapw;rpfs; Kjypad MFk;.

[,j;jpwikfis tsh;g;gjw;fhd ghlg;gFjpfis cs;slf;fpaNjhL

fw;gpf;Fk; KiwfisAk; Gjpa mZFKiwfisAk; ]

fzpjk; vd;why; Foe;ijfs; jhq;fNs rpe;jpj;J Muha;e;J fz;Lgpbf;Fk;

topKiwfSf;F gof;fg;gLj;Jtjw;fhd ;nray;ghLfs; vd;gJ xU nghUs;.

3.3.1 : Foe;ijfSk; fzpjKk; ( Children and Mathematics Education )

njhlf;fg;gs;sp Mrphpah; tFg;giwapy; fzpjf; fUj;ij fw;gpg;gjhf

epidtpy; nfhs;Nthk;.mLj;jJ ehk; ghh;f;f NghtJ vd;dntd;why; Mrphpau;

fUk;gyifapd; Kd; epw;gJk; Rz;zf; fl;bfis ifapy; vLj;Jf;nfhz;Lk;

fzf;Ffis tpsf;fp jPu;T fz;Lk; fUk;gyifapy; cs;sij ghh;j;J Gj;jfj;jpy;

vLj;Jr;nrhy;tJk; fzf;Ffis gapw;rp mspg;gJk; Mrphpau; NflFk; tpdhtpw;F

khzth;fs; gjpy; mspg;gijAk; ehk; fhzyhk;. Mrpupah; tpupTiuahsh;

MfTk; khzth;fs; fw;gtuhfTk; ,Uf;fpd;wdh;.

79



fzpjk; fw;gJ & fw;gpg;gjpy; Vw;gLk; gpur;ridfs; :

( Problems in Teaching and Learning of Mathematics )

gak; kw;Wk; Njhy;tp. (Fear and Failure)

mNef khzth;fs; rf khzth; FO,Mrpupah;fs;,ngw;Nwhh;fs, ; midtUk;

.., vd;w NghjpYk; Muk;g epiyapNyNa fzpjk; fw;wy; kw;Wk; fw;gpj;jYf;Nf

Kf;fpaj;Jtk; mspj;Js;sdh;. gak; kw;Wk; Njhy;tpf;F tpopg;Gzh;Tapd;ikNa

kw;nwhU fhuzpahf mikfpwJ. ,yf;fq;fspd; ,lkjpg;ig gw;wp mwpa

Kbahjjhy; ehd;F mbg;gil nray;fisAk; fw;fj; jtWfpd;wdh;.

fiyj;jpl;lj;jpy; Vkhw;wk; (Disappointing Curriculum)

fzpj fiyj;jpl;lk; Rik kw;Wk; fth;r;rpaw;wjhf ,Ug;gjhy; khzth;fs;

Vkhw;wk; milfpd;wdh;. mNef fzpj fiyj;jpl;lj;jpy; gytopKiwfs;,

#j;jpuq;fs;, ; Njw;wq;fs;, fUj;Jf;fs; Mfpait mOj;jkhf $wg;gLfpd;wd.

ghlg ; Gj ;jfq ;fs ; kw ; wk ; ghlj ;j p l ; lq ; fs ; fbdkhditahf

gupe;Jiuf;fg;gl;Ls;sJ. fzpjf; fiyj;jpl;lk; tho;f;iff;F mg;ghw; gl;Ls;sJ.

fw;gpj;jy; nghUs; gw;whf;Fiw : (Inadequate Learning Materials)

mjpfgl;r khzth;fs; njhlf;fg;gs;spapy; ghlg;Gj;jfj;ijNa tskhf

Vw;Wf; fw;Wf; nfhs;fpd;wdh;. fzpjg; Gj;jfq;fs; ghlr;Rik kw;Wk; tpjpKiwf;F

cl;gl;Nl ,Uf;fpd;wd. ,jdhy; khzth;fs; kfpo;r;rpahfTk; fspg;GlDk;

fw;f ,aytpy;iy. fpuhkq;fs, ; Ff;fpuhkq;fs; cs;s gs;spfspy; fzpjg;

Gj;jfj;ij jtpu NtW VNjDk; fw;gpj;jy; nghUs;fisf; nfhz;L fw;gpj;jhy;

rpwg;ghf ,Uf;Fk;.

xOq;fw;w kjpg;gPL : (Crude Assessment )

fzpjf; fiyj;jpl;lk; #j;jpuq;fis kdg;ghlk; nra;tijNa

typAWj;Jfpd;wJ. tFg;giwf; fw;wy; Nju;it Kd;Ndhf;fpNa mike;Js;sJ.

ek; gs;spapy; gy tpjkhd Njh;Tfs; khzth;fspd; mwpTj;jpwid kjpg;gPL

nra;a cjTfpwJ. tpdhj;jhs;fs; khzthh;fspd; mDgtj;jpw;fhf my;y:

khzth;fspd; rupahd tpilia Nju;T nra;tjw;F MFk;. (v.fh) 2+6 =?

80



vd;gij tpl 2 + 6 = 8 vd;gjw;F tpilaspf;f Kw;gLth.; NkYk; ,J khjpupahd

kjpg;gPLfs; gFj;jwp kw;Wk; njhFj;jwpKiwapy; gad;gLj;jg;gLfpd;wd.

,k;khjpupahd kjpg;gPl;L Kiw xOq;fw;w ,ae;jpuj;jdj;ij typAWj;JfpwJ.

Mrphpah; jahupj;jypy; gw;whf;Fiw : (Inadequate Teacher Preparation)

Muk;gg;gs;sp khzth;fs; midtUk; mt;thrphpah; fw;gpf;Fk; ghlj;ij

rhh;e;Nj cs;sdh;.Mrphpah; gw;whf;Fiw fhuzkhf, gpw Mrphpah;fs; fzpjk;

fw;gpg;gjw;F tw;GWj;jg;gLfpd;wdh;. mth;fs; ghlg;Gj;jfj;ij rhh;e;Nj

,Uf;fpd;wdh;. mNef Muk;gg;gs;sp Mrpupah;fs;, jq;fsplk; midj;Jf; fw;gpj;jy;

nghUs;fSk; cs;sd vd epidj;jf; nfhs;fpd;wdh;. Mjyhy; MrphpaH

jahupj;jypy; gw;whf;Fiw cs;sJ.

fw;wy;, fw;gpj;jy; Kiw :( Teaching Learning Process)

njhlf;ff; fy;tpapy; fw;wy; , fw;gpj;jy; fth;r;rpaw;w Kiwapy; cs;;sJ.

Vnddpy;

(i) Gj;jf mwpT NghJkhdjhf ,y;iy.

(ii) gs;sp fzpjf; fw;wy; Nrhh;TlDk; , tpUg;gkw;wjhfTk; mike;jpUf;fpwJ

(iii) FUl;L kdg;ghlj;ij typAWj;JfpwJ.

(iv) fw;gpj;jiy tpl fw;wiy typAWj;Jfpd;wd.

(v) Ghpe;J nfhs;Sjy;, gad;gLj;jy; ,jpwd;fs; Mfpatw;iw Gwf;fzpf;fpd;wd.

Mh;tg; gw;whf;Fiw : [ Lack of interest ]

ngUk;ghyhd gs;spf; Foe;ijfs; fzpjk; fw;wiy fbdk; vd;W vz;zp

ek;gpf;ifia ,of;fpd;wdh;. fzpj fw;wy; kw;Wk; fw;gpj;jy; kfpo;r;rpahfTk;,

fth;r;rpahfTk; miktJ ,y;iy. khzth;fs; fzpjk; fw;gjd; gaid

mwptjpy;iy. Mifahy; fzpjk; fw;gjpy; tpUg;gkpd;ikNa fhzg;gLfpd;wJ..

81



3.3.2 tFg;giwr; #oYf;F mg;ghy; fzpjk;;:

(Mathematics Education Beyond Class room)

Muk;gg; gs;sp$lj;jpy; ghlg;Gj;jfq;fs;; kl;LNk fw;gjw;fhf ,Uf;fpd;wd.

gy tpjq;fspy; khzth;fs; ghlg;Gj;jfq;fis jtpu NtW Gj;jfq;fis

fUj;jpy; nfhs;tjw;F topapy;iy. Vnddpy; Njh;tpy; ghlg;Gj;jfq;fspy; cs;s

Nfs;tpfis kl;LNk Nfl;fpd;wdh;. rpy Nfs;tpfs; kdjpy; vOfpd;wd. mit,ghlg;Gj;jfq;fs; kl;LNk KOmwpit jUtJ ,y;iy. Foe;ijfspd; mwpitg;

gw;wp vOj;jhshh;fSf;F KOikahf njupe;jpUf;fpd;wjh? Foe;ijfspd; #oiy

fUj;jpy; nfhs;fpwhh;fsh ? ek;kplk; ,f;Nfs;tpf;fhd Vw;Wf;nfhs;s jf;f

gjpy;fs; ,y;iy. Mdhy; ghlg; Gj;jfk; kl;LNk fw;gpg;gjw;fhf NghJkhdjhf

,Uf;fpwJ.

Foe;ijfs; Mrphpah;fsplk; kl;Lk; fw;Wf;nfhs;tjpy;iy jd;Dld;

cs;s Foe;ijfsplKk;,#w;Wr;#oypypUe;Jk; fw;wf;nfhs;fpd;wd.mth;fs;

czh;T %ykhf Efh;jy;, njhL;jy;, Nfl;ly;, ghh;j;jy; kw;Wk; Ritj;jy; %yKk;

Foe;ijfs; vspjhf fw;Wf;nfhs;fpd;wdh;. xU Foe;ij vspikahf fw;wy;

fw;gpj;jy; jpwid Mu;tj;JlDk; KO <Lghl;LlDk; tsh;j;;Jf; nfhs;fp;wJ.

Foe;ijfSila gq;fspg;G, rpe;jpf;fk; msT, vspikahd kw;Wk; ,itfis

rhh;e;J Foe;ijfSf;F fw;wy; #oy; mikfpwJ. mjpfg;gbahd Foe;ijfs;

vy;yh Neuq;fspYk; vy;yh ,lq;fspy; fw;Wf; nfhs;fpd;wdh;. mjhtJ

tPL,tpisahl;L ikjhdk, ; re;ij, kw;Wk; gy. mjdhy; ek;Kila

Fwpf;NfhshdJ gs;spf; fy;tpia kfpo;r;rpahd mDgtkhfTk;, tFg;giwapy;

fw;Fk; fy;tpf;Fk; tFg;giwf;F ntspapy; fw;Fk; fy;tpf;Fk; jilfis

jtph;f;f Ntz;Lk; vd;gjhFk;.

gpd;tUk; #o;epiyfisf; ftdpg;Nghk;.

#o;epiy - 1

xU $ilapy; 7 nghk;ikfs; cs;sd.mjpy; %d;W cile;J

tpl;lJ.Foe;ijfs; mjpy;kPjp vj;jid nghk;ikfs; cs;sJ vd;gij

$w Ntz;Lk;?ud;gPu; : $ilapy; 7 nghk;ikfs; cs;sd.ah]; :

82



$ilapy; 4 nghk;ikfs; cs;sd.ulhdh : $ilapy; 10 nghk;ikfs;

cs;sd. m\p\; : vd;dhy; nrhy;y Kbatpy;iy.vz;fs; xUtUf;nfhUth;

khWfpwJ.nrskpah: ,e;jf; Nfs;tpapy; rpy jpUj;jq;fs; Njit.ahUila

$w;W rhp? mJ Vd;?$w;wpDila tpupthf;fk; :ud;gPu; : nghk;ikfs;

cile;Js;sJ. Mdhy; midj;J nghk;ikfSk; $ilf;Fs;

cs;sJ..ah]; : 7 – 3 =4ulhdh : %d;W nghk;ikfSk;

cile;Js;sd.mjdhy; MW cile;Js;sJ. mjdhy; 7 –

3+6 =10m\p\;: xUtUf;nfhUth; vz;fs; khWk; vd;W vd;dhy; nrhy;y

KbahJ.nghk;ikfs; 2/3/4….Jz;Lfshf cile;Js;sJ.nrskpah :

,f;Nfs;tpapy; jpUj;jk; cz;L.mjhtJ ,g;NghJ cilahj nghk;ikfs;

$ilapy; vt;tsT cs;sJ? mjDila tpil 7-3 = 4 xU $w;iw

tpupthf $Wk;NghJ mJ rupah? jtwh? vd;W ghh;j;jy; mtrpak;.

xUth;kl;Lk; KOkjpg;ngz; vLf;fpwhh; mth; ah]; Mdhy; kw;wth;fsJ

$w;iwAk; fhz;gJmtrpak;.fPof;;fhZk; Nfs;tpfis fhz;f:,f;Nfs;tpapd;

Kf;fpa Nehf;fk; vd;d?ah]; me;Nehf;fj;ij mile;jhuh? ulhdh,

m\p\;, ah]; kw;Wk; nrskpah ,th;fs; me;Nehf;fj;ij mile;jhh;fsh?

ah]; ,d; $w;iw tFg;giwapy; fye;Jiuahbdh;. ulhdh, m\p\;, ah];

kw;Wk;nrskpah Nfs;tpia NtW tpjj;jpy; fz;ldh;.mth;fSila mDgtk;

tho;f;if#o;epiyfs; Nahrpj;jy; kw;Wk; gy ,it midj;Jk; tFg;giwr;

#o;epiyf;F mg;ghy;cs;sJ.

#o;epiy 2 :

Mrphpah; khzth;fsplk; ,e;jf; fzf;if jPu;T fhzr; nrhd;dhh;.18 ;

vOJ Nfhs pd ; Jiza pd ;w p t pil fhz ;f . u hz p : 1 8

;uh[; : 12 ;uhFy; : 18

;Mfh]; : 20 ; NkNy fhZk; ehd;F

khzth;fspd; gjpiy MuhaNtz;Lk; kw;Wk; gs;spf;F ntspNa

cs;sNjhL mwpitAk; ,izj;J nray;gl;lth; ahh;? vg;gb ?

Foe;ijfis fzpj fzf;if mth;fSila Ranray; %yk; nra;a

tpl Ntz;Lk; , ,jd; %yk; ehk; gy fUj;Jf;fis mwpayhk;.

83



gs;spapy; ngWk; mwpit eilKiw tho;f;ifAld; njhlh;GgLj;Jjy;:

(Scope of connecting Knowledge to life out side the School)

mq;fhb (Market)

khzth;fs; jq;fs; ngw;Nwhh;fSld; mq;fhbf;Fr; nrd;W mq;F nghUl;fs;

tpw;gid nra;tijAk;, thq;FtijAk; mth;fspd; mZFKiwfisAk;,

cw;WNehf;fpapUg;gh;. ,yhgk;, el;lk; fzf;fpLjy;> tpiyg;gl;bay;jahhpj;jy;>

vilfz;lwpjy;> gzk; vz;Zjy; Nghd;witfspy; khzth;fs; fzpjj;ijg;

gad;gLj;jpf; nfhs;syhk;.

Njhl;lk; (Garden)

rpWtaJ Foe;ijfs; xg;ghh; FOTld; tpisahLk; NghJ tPL> Njhl;lk;

Nghd;w mikg;Gfis cUthf;fp tpisahLk; NghJk; Nfhzq;fs;> tbtq;fs;>

NfhLfs;> gug;Gfs;> ruhrhp Nghd;w fzpjf; fUj;Jf;fis mit fzpj

fUj;Jf;fs;jhd; vd;gij mwpahkNyNa gad;gLj;Jfpd;wdh;. ,e;j

mDgtq;fis fzpj fUj;Jf;fisg; Ghpe;J nfhs;sTk;. fzpjj; njhlh;ghd

Kiwahd mwptpidg; ngw;Wf;nfhs;sTk; gad;gLj;jpf;nfhs;syhk;.

eilKiw tho;f;if

xU jtis xU fk;gj;jpy; gfypy; 30 kP Vwp efh;e;J ,utpy; 20 kP

rWf;FfpwJ. me;j fk;gj;jpd; cauk; 70 kP vd;why; vj;jid ehl;fspy;

jtis fk;gj;jpd; cr;rpia milAk;? cah; njhlf;ff;fy;tp gapYk; mNef

khzth;fs; VO ehl;fspy; vd;gijNa tpilahf $Wth;. Mdhy; kpfr;

rpyNu 5 ehl;fspy; vd;w tpilia mspg;gh;. (Ie;jhtJ ehspy; jtis

vOgJ kPl;liu vl;Lfpw) Foe;ijfs; ,aw;ifAld; ,iltpidahw;Wk; NghJ

(,aw;if #oypy;) Gyd;fhl;rp njhlh;ghd mwptpid gad;gLj;Jfpd;wdh;

vdNt Foe;ijfspd; eilKiw tho;f;if mDgtq;fis fUj;jpy;

nfhs;sNtz;Lk;.

cUtq;fs; tbtikj;jy; :

khzth;fs; jq;fs; Gj;jfq;fSf;F ciwapLjy;, ; glq;fSf;F tz;zk;

jPl;Ljy;, tPLfis myq;fhpj;jy;, Njhl;lq;fis guhkhpj;jy;, ; nrbfis

84



ghJfhj;jy;> tpisahl;Lg; nghUl;fis tbtikj;jy; Nghd;w nray;fspy;

<LgLfpd;wdh;. mt;Ntisfspy; mth;fs; fzpjj;ijg; gad;gLj;Jfpd;whh;fsh?

cq;fsJ ngaiu tbtikf;f vj;jid jPf;Fr;rpfs; gad;gLj;jg;gl;ld?

,j;jifa nray;Kiwfis Mrphpah; cw;W Nehf;fp tFg;giwapy;

gad;gLj;jNtz;Lk;.

tpohf;fs;

ek; tPLfspy; kw;Wk; gs;spfspy; epiwa tpohf;fis nfhz;lhLfpNwhk;.

khzth;fs; Rje;jpu jpdtpoh> FbauR jpdtpoh> Mrphpah; jpdk;> Foe;ijfs;

jpdk;> ru];tjp G+i[> tpehafh; rJh;j;jp> <ifj;jpUehs;> fpwp];Jk];

Nghd;wtw;iw KOkdJld; nfhz;lhLfpwhh;fs;. ,j;jifa ehl;fs; epidit

tpl;L ePq;fhky; ,Ug;gjw;fhf gy nray;ghLfspy; <LgLfpd;wdh;. mth;fs;

mq;fhbf;Fr; nrd;W gytifg; nghUl;fis thq;Fjy;> gs;spia myq;fhpj;jy;>

njUf;fis myq;fhpj;jy;> ,dpg;Gfs; toq;Fjy; Nghd;w rkaq;fspy; fzpjj;ijf;

fw;Wf; nfhs;fpd;wdh;. (fzpj mwptpidg; ngWfpd;wdh;.)

tpisahl;L ikjhdk;

khzth;fs; fgb> fhy;ge;J> fphpf;nfl;> ifg;ge;J> $ilg;ge;J kw;Wk;

gy cs;suq;f tpisahl;LfisAk;> tpisahLfpd;wdh;. jq;fSf;Fhpa tpjpfis

jhq;fNs cUthf;Ffpd;wdh;. mth;fSf;Fhpa tpisahLk; fsj;ij mth;fNs

cUthf;fpf; nfhs;fpd;wdh;. khzth;fs; tl;lq;fs; , rJuq;fs; nrt;tfq;fs;>

Kf;Nfhzk; Nghd;w tbtq;fis tiuAk; tpjpKiwfis mwpahkNyNa

tiufpd;wdh;. jdp kw;Wk; FOtpw;fhd Gs;spfis (kjpg;ngz;fis)

fzf;fpl mth;fspd; nrhe;j topKiwfis ifahSfpd;wdh;. uNk\; ,uz;L

ehd;FfisAk;> ,uz;L ,uz;LfisAk;> xU xw;iwg; Gs;spiaAk; ngw;Ws;shh;.

ngUf;fy; njhpahky; mtuhy; vt;thW nkhj;j kjpg;ngz;izf; fzf;fpl

KbAk;. E4 ghlg;Gj;jfq;fSf;F mg;ghw;gl;L fzpj fUj;Jf;fisf; fw;fTk;>

fw;gpf;fTk; ,aYk;.

E4 fzpjj;jpy; fw;wy; , fw;gpj;jy; fUj;Jf;fis fzpj Gj;jfq;fs;

thapyhf tpsf;Ftha;.

85



3.3.3 fzpjk; fw;wiy ,dpikahf;Fjy ;

(Making Mathematics Learning Joyful)

gs;spfspy; fzpjk; fw;gJ mNef Neuq;fspy; fbdkhdjhf>

Nrhh;tspg;gjhf> rypg;ig Vw;gLj;Jtjhf cs;sJ. Mrphpah;fs; kw;Wk;

khzth;fspd; re;Njhrkw;w mDgtq;fNs ,jw;fhd Kf;fpakhd fhuzkhFk;.

ehk; fzpjk; fw;gpf;Fk; NghJ fw;Wf;nfhs;s Ntz;bait>khzth;fSf;F

fzpjj;ij tpUg;gkhf fw;Wf; nfhLg;gJ vg;gb? mth;fSila NjitfisAk;

Mh;tj;ijAk; vd;dntd;W mwpaNtz;Lk;.kw;Wk; gy.

nray;Kiw :5

jq;fs; khzth;fspd; tpUg;gKs;s gFjpia tiuKiwg;gLj;Jf kw;Wk;

ve;jg; gFjpfs; fzpjg;ghlj;jpw;F Vw;g cs;sJ vd;gij fz;Lgpbf;f

______________________________________________________________________________________________

______________________________________________________________________________________________

______________________________________________________________________________________________

fzpjk; fw;gpj;jiy re;Njhrkhdjhf khw;Wjy; vg;gb?

Xt;nthU Foe;ijapd; fw;gpj;jy; mDgtj;ij fUj;jpy; nfhs;sNtz;Lk;;.

];J}ykhd nghUSf;Fk; RUf;fkhd fUj;Jf;Fk; cs;s tpj;jpahrj;ij

fzpjk; fw;gpj;jypy; NtWgLj;jpfhl;Lf.

khzth;fspilNa fzpjk; rk;ge;jkhd tpisahl;L fijfisf; $wp

Mh;tj;ij J}z;lNtz;Lk;.

fzpjk; fw;Fk; Mh;tj;ij fzpj Gjpu;fshy; Nkk;gLj;jyhk;.

fzpj Nkijfspd; glq;fis Nrfhpf;f Ntz;Lk;.

md;whl tho;f;ifAld; fzpjj;ij ,izf;fNtz;Lk;.

86



khzth;fsplk; tpisahl;Lf;fs;>ghly;fs;> ehlfq;fs; Nghd;wtw;iw

Nrfhpf;fr; nrhy;y Ntz;Lk;.

khzth;fis Rakhf KbTfs; vLf;f mDkjpf;f Ntz;Lk;.

khzth;fSila cUthf;Fk; jpwid jLj;jy; $lhJ.

vLj;Jf;fhl;L

,g;glj;jpy; vj;jid nrq;Nfhzq;fs; cs;sJ?

E5: fzpjj;ij ,dpikahf fw;f VNjDk; vLj;Jf;fhl;L jUf.

3.3.4 fzpjk; fw;wypd; #oiy cUthf;Fjy ; :

(Creating conducive Learning environment for Mathematization)

Muk;gg; gs;spfspy; fzpjk; fw;gpf;Fk; #o;epiyfs; khzth;fspd;

Mh;tj;ijj; J}z;Lk; tifapy; miktJ ,y;iy. gy Foe;ijfSf;F

fzpjj;jpd; kPJ cs;s gaKk; mjid Vw;Wf;nfhs;Sk; kdg;gf;FtKk;

,y;iy. ,jd; %ykhf fzpjj;jpd; kPJ cs;s gpur;ridahf mikfpwJ..

ngUk;ghyhd Mrphpah;fs; khzth;fSf;F Vw;wthW fzpjj;ij fw;f Ntz;Lk;

vd;gjw;F topfhl;l Ntz;Lk;. Mdhy; Mrphpah;fNs ,t;thwhd Kiwfspy;

jLkhWfpd;wdh;. ngUk;ghyhd Muk;gg; gs;spfspy; Mrphpah;fSf;F fw;gpf;Fk;

#o;epiyfs; rupahf miktJ ,y;iy..

E6. cdJ gs;spr;#oy; / fpuhkg;Gwr;#oy; fzpjk; fw;f Vw;wthW

cs;sjh?tptup.

87



fw;gth;fSf;F ew;#oiy mwpTWj;jy;; :

khzth;fis mwpjy ; :

khzth;fSf;F fzpjk; fw;gpg;gjpy; kl;Lk; Kf;fpaj;Jtk; juhky; kw;w

ghlq;fSf;Fk; Kf;fpaj;Jtk; juNtz;Lk;. fzpjk; fw;gpf;Fk; Mrphpah;fs;

mwpaNtz;bait>

tFg;gpy; cs;s xt;nthU khztiug; gw;wpAk; mwpe;J nfhs;s Ntz;Lk;.

fzpj fzf;F NghLk; NghJ khzth;fis Cf;Ftpf;f Ntz;Lk; .

ve;j xU Foe;ijapd; jpwid mwpAk; Kd;G Mrphpah;fs; mth;fisg;gw;wp

vt;tpj Kbtpw;Fk; tuf;$lhJ.

fzpjj;jpy; xU fzf;if Kaw;rp nra;Ak; NghJ Foe;ijapd; jpwdpd;

epiw , Fiwfis mwpjy; Ntz;Lk;.

fzpj fzf;if Kaw;rp nra;Ak; NghJ mth;fSf;F NghJkhd Neuj;ij

Vw;gLj;jp ju Ntz;Lk;.

fw;wypYk; fw;gpj;jypYk; elf;Fk; epfo;Tfs;

gy khzth;fs; fzpjj;ij Rikahd ghlkhf fUJfpd;wdh; Mrphpah;

fw;wypYk; fw;gpj;jypYk; elf;Fk; epfo;tpd; %ykhf ey;y elj;ijapd; %yk;

khzth;fSf;F fzpjj;ij fw;gpf;fyhk;. fzpj Mrphpah;fs; nra;a Ntz;bait.

ehs;NjhWk; Foe;ijfs; fzpj rk;ge;jkhd eifr;Rit> fijfs;> Gjph;fs;

Nghd;wt;iw Mrphpah;fs; fw;Wf;nfhs;s Cf;Ftpf;f nfhLf;f Ntz;Lk;.

fzpjg; ghlq;fSf;F mg;ghy; kw;w tp\aq;fisAk; fw;gpf;fNtz;Lk;.

xU khzth; gapw;rpia Kbg;gjw;F NghJkhd fhy mtfhrk; toq;f

Ntz;Lk;.

fzpjf; fUj;Jf;fis kpd; ml;il> glq;fs;> tiuglq;fs;> nghUs;fs;

kw;Wk; gpw nghUl;fspd; %ykhf khzth;fs; mwpe;Jnfhs;s

gad;gLj;jyhk;.

88



fw;wy; fw;gpj;jy; fUtpfs;:

fzpjg;ghlg;Gj;jfk;> fzpjNkw;Nfhs; Gj;jfk;> fijGj;jfk;> fzpj je;jpuf;

fijfs;> Gjph;fs; Gj;jfk; > jpl;l Gj;jfk;> fzpj tuyhw;W Gj;jfk;

,tw;iwnay;yhk; fzpj Mrph pah ; Nrfhpf ;f Ntz;Lk; . Mrph pah ;

khzth;fsplj;jpYk;> ngw;Nwhh;fsplj;jpYk; kw;wth;fsplj;jpYk; ciuahLtjd;

%ykhf fzpj fUtpfis Nrfhpj;Jf; nfhs;syhk;.

gs;spr; #o;epiy :

fzpjk; fw;gjw;F gs;sp#o;epiy Kf;fpa gq;F tfpf;fpwJ. xU khzth;

fzpjk; fw;gjw;F cWJizahf miktJ #o;epiyNa MFk;. tFg;giwr;

RtUk; gs;spfSk; fzpjj;jpd; %yk; tiuaWf;fg;gl;lJ. fzpjj;jpd; Jy;ypakhd

fUj;ij Rtw;wpy; mikf;f Ntz;Lk;. tFg;giwapy; topghl;L Neuj;jpy; rpy

fzpj Nkijfspd; tuyhw;iw gbj;jy; Ntz;Lk;. rpwe;j tFg;giwapy; fz;bg;ghf

khzth;fSila nraYk; nghUl;fSk; ,Uj;jy; mtrpak;.

fw;wy; ikak; :

mbg;gilj; Njitfs;: kpd;dl;il> fw;fs;> Nfhy;fs;> nghUs;fs;> glq;fs;>

tiuglq;fs;> fhyz;lh; tpisahl;L ml;ilfs; kw;Wk; gy ,it midj;Jk;

tFg;giwapy; ,Uj;jy; mtrpak;. Mrphpah; fw;wypd; ikaj;jpw;Fr; nry;Yk;

NghJ, fzpj nray;ghl;L Jizf;fUtpfisAk; nfhz;L nry;Yjy; Ntz;Lk;.

Mrphpah; fl;lhakhf fw;wypd; ikaj;ij Njitf;F Vw;wNghJ gad;gLj;j

Ntz;Lk;.

nghOJNghf;Fr; rhh ;e;j nray;Kiwfs;

,d;iwa gs;spf;$lq;fspy; nghOJ Nghf;F rhh;e;j nray;Kiwfis

jtph;f;fg;gLtjhy; ey;y Neh;kiw vz;zq;fs;; cUthtjpy;iy.khzth;fspilNa

cs;s ey;y Fzhjpraq;fs; Nkk;gLj;j gLfpwJ. mjdhy; nghOJNghf;F

rhh;e;j nray;Kiwfs; ,Uj;jy; mtrpak.; fzpj kd;w nray;ghLfis

xUq;fpizj;jy;> fzpj tpdhbtpdh> mwpT fzf;F Nghl;bfs; kw;Wk; gy

Nfs;tp jpl;L tsh;r;rp (vOj;J > tha;nkhop nray;gLj;J Kiw) nray;

Kiw jpl;lj;jpy; cs;s jpwd;fs;> Nfs;tpf;F tpilasp> Njitahd nghUl;fs;.

89



kjpg;gPL

fw;Wf;nfhs;gth;fs; tha;g;G fpilf;Fk; NghJ jd; Fwpf;Nfhs;fis

jPh;khdpj;Jf; nfhs;fpd;wdh; xU Foe;ij jtwhf gjpy; mspj;jhy; mf;Foe;ijia

mtkhdg; gLj;jf;$lhJ. xUtUila jFjpia mwpe;J nray;glNtz;Lk;.

nray;ghl;L fy;tpg;gphpT xUthpd; fw;Fk; jpwid jLg;gjpy;iy fzpjj;ij

nray;ghl;L gphpT %yk; tpUg;gj;Jld; fw;gNj xUtUila ew;gz;ig tsh;;fpwJ.

gs;spfspy; cliy tUj;Jk; jz;lidia jtph;f;f Ntz;Lk;. kw;Wk; gs;sp

vd;gJ “jz;lid ,y;yhj ,lkhf ,Uf;f Ntz;Lk;;”.

3.4 njhFj;jy; (Let us sum up )

fzpjf; Nfhl;ghL vd;gJ gue;j, tpupe;j Fwpf;Nfhs;fshFk;. Fzpj

Nfhl;ghLfspd;; tsh;r;rpfs;> gpur;ridfSf;F jPh;T fhZjy;>fz;lwpjypd;

gad;ghL;fs;> ; Njhuhakhf kjpg;gPLfs;> Njh;T Kiwfs;> cUtikg;G>

fhl;rpg;gLj;Jjy;> tbtq;fspd; ; gad;ghLfs;;> fhuzq;fs; kw;Wk; Mjhuq;fs;

njhlh;Gfis cUthf;Fjy; kw;Wk; fzpjk; rhh;e;j jfty;fs; Nghd;witfis

nfhz;l Nfhl;ghLfs; mlq;fpAs;sd.

xU Foe;ij fzpjj;ij fw;Fk; NghJ cahpa Nehf;fk; tYtilfpwJ.

mjd; %yk; jh;f;f Kiwapyhd KbTk; fUj;jpay; rpe;jid jpwDk;

tsh;fpd;wJ.xUth; nra;Ak; nray; jpwd;fs;;,gpur;ridia jPu;f;Fk; khdg;ghd;ikNghd;wtw;iw cs;slf;fpaJ. .

gs;spf; fzpjk; xU rpwg;G Nehf;fj;Jld; tsh;e;J tUfpwJ. mit

vz;fs; ,vz; nray;ghLfs; , mstPL, jrk kw;Wk; rjtPjk; MFk;. fzpjf;

fy;tpahdJ mwpjy;,Gupjy;; ,czh;jy;,Gyk;,cly; ,cs ,af;fk;,Nghd;witfspd;cjtpNahL ghlj;jpl;lk; ,fiyj;jpl;lk; fw;wy; fw;gpj;jy; Jizf;fUtpfis

cUthf;Ftjw;F topfhl;ly;,kjpg;gha;T Nfs;tpfs; Nghd;wtw;iw cWjpg;gLj;jp

$w gad;gLfpwJ.Nkw;$upait fzpjj;jpd; nray; tpidfs; Fwpg;Gr;nrhy;

kw;Wk; rhjidfhfhyq;fs; gw;wp vOJtjw;F gad;gLfpwJ.

NCF -2005 vd;gJ xt;nthU Foe;ijAk; fzpjj;jpid tpUg;gj;JlDk;

mbg;gil jd;ikfs; %yKk; njhlh;GwTlDk; gy topj; njhlh;Gfs; %yk;

ghlj ;j p l ; lj ; j pid cUthf ;f Ntz;Lk ; . vd ; w Nehf ; fj ;Jld ;

mikf;fg;gl;lJ.Foe;ijfs; Mrpupauplk; kl;Lk; fw;Wf; nfhs;tjpy;iy. kw;w

90



Foe;ijfsplkpUe;Jk; fw;Wf; nfhs;s Ntz;Lk;. Rw;W#oy; ,aw;if nghUl;fs;

,it midj;Jk; nray;fs; %ykhfTk; nkhopfs; %ykhfTk; gad;gLj;jg;gl

Ntz;Lk;.,d;iwa gs;spfspd; Nehf;fk; kfpo;r;rpAld; fw;Wf; nfhs;s Ntz;Lk;.

gy nghUl;fis nfhz;L tFg;giwf;Fs;Sk; tFg;giwf;F ntspAk; gad;gLj;j

Ntz;Lk;.fzpj ghlj;jpw;Fk; tho;f;if #o;epiyf;Fk; cs;s njhlu;ig Fiwf;f

Ntz;Lk;.fzpj ghlj;jpw;Fk; fzpj ghl nray;KiwfSf;Fk; cs;s njhlh;G

,ilntspiaf; Fiwf;f Ntz;Lk;.

3.5 : jd;dwpTr; Nrhjid :

E.1 : tsh;j;Jf;nfhs;s Ntz;ba gof;ftof;fj;jpypUe;J khwhky; Ra

ek;gpf;if ,fz;Lgpbj;jy; ,kw;Wk; ntspf;fhl;Lk; jpwd; md;whl

tho;f;ifapy; fzpjk; xUtUila rpe;jidiaAk; gpur;ridia

vjpu;nfhs;Sk; jpwd; tsu gad;gLfpwJ.

#oiyAk; gz;ghl;ilAk; Ghpe;J nfhs;sTk; gad;gLfpwJ.

gytpj njhopy; El;gq;fis gad;gLj;jp jq;fsJ tUq;fhy

njho py ;Kiwia tsh ;j ;Jf ; nfhs ;tjw ;F Foe ;ijfis

jahu;nra;aNtz;Lk;.

GJtpjkhd fz;Lgpbg;Gfis njupe;J nfhs;tjw;fhd Mw;wiy

Foe;ijfsplk; tsh;f;f Ntz;Lk;.

E -2 : fzpjf;fy;tpapd; Kf;fpa Fwpf;NfhshdJ Fwpg;gpl;l nray;Kiwfis

mikg;gjw;F cjTfpwJ. mJkl;Lkpy;yhky; fw;wy; fw;gpj;jy;

Jizf;fUtpfs; jahh; nra;aTk; kjpg;gPl;L Nfs;tpfis jahh;

nra;aTk; gad;gLfpwJ. kw;Wk; gy.

E-3 . iii , iv , vi ,vii kw;Wk; xi Mfpait fzpjj;jpw;F topfhl;Lk; Nehf;fq;fs;

MFk;.

E-4 : fsg;gazk;,fzpj tpisahl;Lfs;,fzpj Gjph;fs; kw;Wk; tpLfijfs;,

fzpj ke;jpu[hyk;,fzpjj;jpw;Fk; md;whl tho;f;iff;Fkhd njhlh;Gfs;,

gFj;jwpjy; gpujpepjpj;Jtk; , milahsk; fhzy;, tpupTgLj;Jjy; ,

Njhuhakhf kjpg;gpLjy; kw;Wk; gpur;ridia jPu;j;jy;.

91



E-5 : xNu khjpupahd Ie;J vz;fisf; nfhz;L ia tpilahfj;

jUtp.

3.6:-SUGGESTED READINGS AND REFERENCES

1. Ediger ; merlow and rao, D.B(2004)Teaching Mathematics in elemandry schools

‘NEW DELHI”, Discovery puplishing house .

2. Gagne R.M(1985) the Contions of learning and theory of instruction New York;

CBS Colleage Schuster .

3. joyce, bruce and weil marsha (1996)Models of Teaching.Need ham Heighcs,

MA Simon and Schuster .

4. NCERT(1997) The Primary Years ;A curriculum frame work (Part II) NEW DELHI.

5. NCERT (2005) National Criculum Frame work 2005 NEW DELHI.

6. NCERT (2008) Source Book for learning assessment in Mathemetics for Primary

Schools – New Delhi.NCERT

7. NCERT (1995)Self Instructional package for special oriendation . Programme for

Primary Schools Teachers NEW DELHI .NCERT

8. NCTE (1999)Evamplar modules in Mathematics New Delhi.NCTE.Sindhu

Kulpirsingh (1989) the teaching of Meathematics NEW DELHI Sterling.

3.7 : UNIT – END EXERCISES

1. xU Foe;ijf;F A,B,C njupatpy;iy vdpy; ;mf;Foe;ij fzpjj;jpy;

vjpu; nfhs;Sk; fbd epiy vd;d ?;

2. fy;tp Nehf;fq;fSf;F Nghjid Nehf;fq;fSf;Fk; ,ilNa cs;s

NtWghl;il mwpf.

3. njhlf;f epiyf;fy;tpapy; fzpjk; fw;gpf;fg;gLtjd; Nehf;fj;ijg; gw;wp

tpthjp.

4. njhlf;fg;gs;spapy; cs;s fzpjg;ghlj;jpid gw;wp vOJf.

5. cdJ tFg;giwapy; fzpjk; fw;gpg;gjw;fhd gj;J vLj;Jf;fhl;Lfis

cdJ Rw;Wr;#oypy; ,Ue;J vLj;J gad;gLj;J.

92



mikg;G

4.0 mwpKfk;

4.1 fw;wypd; Nehf;fq;fs;

4.2 fzpjj;jpy; fw;wy; - fw;gpj;jy; Kiwfs; /

4.2.1 tpjptUKiw kw;Wk; tpjptpsf;f Kiw.

4.2.2 gFj;jwp Kiw kw;Wk; njhFj;jwp Kiw.

4.2.3 nray;jpl;l Kiw

4.2.4 rpf;fy; jPu;f;Fk; Kiw kw;Wk; rpf;fiy tpLj;jy;.

4.3 fzpjk; fw;gpj;jypy; fw;gth; ika Kiwfs;.

4.3.1 5E apd; fw;wy; khjphp.

4.3.2 Mf;fG+h;tkhd khjpup tiug;glq;fis tpsf;Fjy;.

4.3.3 fUj;J tiuglk;

4.3.4 nray;topf;fw;wy;

4.4 fzpjj;jpy; rthyhd kw;Wk; jpUg;jpfukhd fw;wy; Kiw:

4.4.1 fw;gth; ika jpwd;fis Nkk;gLj;Jjy;

4.4.2 fzpj – E}yfk; kw;Wk; fzpj Ma;tfj;jpd; gad;ghL.

4.5 njhFj;jy;

4.6 khjpup tpilfis rupghh;j;jy;

4.7 gbj;jy; kw;Wk; Fwpg;Gfs; ghpe;Jiuj;jy;

4.8 myFj; Njh;T

Üô° - 4

ªî£ì‚è‚ è™MJ™ èŸø™ ñŸÁ‹èŸHˆî™ ¬ñò º¬øèœ

(LEARNER AND LEARNING – CENTREDMETHODOLOGIES AT ELEMENTARY LEVEL )

93



4.0 mwpKfk; :-

fzpjf; fy;tpia fw;gpf;f fzpj MrphpaUf;F Vuhskhd KiwfSk;

EZf;fSk; cs;sd. khzth;fspd; Njit, ghlj;jpd; jd;ik kw;Wk; ghlj;jpd;

Nehf;fq;fisg; nghWj;J Njitahd fw;gpj;jy; Kiwfis Mrphpah;

Nju;e;njLf;f Ntz;Lk;. rpy Kiwfs; FO fw;gpj;jYf;F Vw;wjhf mike;jpUf;Fk;.

rpy Kiwfs; jdp fw;gpj;jYf;Fk; Vw;wjhf mike;jpUf;Fk;. fzpjk; Ghpe;J

nfhs;tJ kl;Lk; fpilahJ, gy;NtW Kiwfis gad;gLj;jp fw;gpg;gJ vd;gJ

kl;Lk; jhd; fbdk;. gs;spfspy; midj;J fzpj ghlq;fSk; fl;likg;Gld;

eilKiwg ;gLj ;jg ;gLf pwJ. md ;whl tho ;f ;ifapy ; ngwg ;gLk ;

mDgtk;,njhlf;ff;fy;tpapy; fw;gpf;fg;gLfpwJ. ,g;ghl jiyg;gpy;

njhlf;ff;fy;tpapy; fzpjk; fw;wy; kw;Wk; fw;gpj;jy; Kiwfisg; gw;wp

fye;JiuahLNthk;.

4.1 fw;wypd; Nehf;fk; :

njhlf;ff;fy;tpapy; fw;wy; kw;Wk; fw;gpj;jypy; Mrphpah;fs; kw;Wk;

khzth;fs; Mfpa ,UtUf;Fk; ghl jpl;lj;ij jahhpf;Fk; epiyapy;;

tFg;giwf;Fr; nry;Yk; NghJ jdf;Fj; jhNd ,e;j tpdhit Nfl;Lf; nfhs;s

Ntz;Lk;. “,g;ghlj;jpd; %yk; khzth;fsplk; ehd; vt;tifahd khw;wj;ijf;

nfhz;L tUNtd;?” ,t;tifahd khw;wq;fs; Mrphpahpd; fy;tp Nehf;fq;fshFk;.

4.2 fzpjj;jpy; fw;wy; - fw;gpj;jy; Kiwfs; :-

khzth;fSf;F gy;NtW fw;gpj;jy; Kiwfis gad;gLj;jp fw;gpf;Fk;

nghOJ, khzth;fspd; ftdkhdJ Mrphpahpd;ghy; <h;f;fg;gLfpwJ. Mrphpah;

gy;NtW Neuq;fspy;, gy;NtW fw;gpj;jy; Kiwfisg; gad;gLj;jp fw;gpf;Fk;

nghOJ khzth;fSf;F fw;gjpy; rypg;G Vw;glhJ. khzth;fspd; ftdk;

Mrphpah;fsplk; ,Uf;Fk;.

4.2.1 tpjptU Kiw – tpjptpsf;f Kiw :

[INDUCTIVE AND DEDUCTIVE METHOD ]

fw;gpj;jy; Kiw khwhj jd;ik nfhz;lJ. mJ Njhd;wpa fhyj;jpy;

,Ue;J gad;ghl;by; cs;stiu fhyj;jhYk;, eguhYk;, #o;epiyahYk;

94



khWtjpy;iy.fw;gpj;jy; KiwAk;, fw;gpj;jy; cj;jpAk; fy;tpr; nray;ghLfSld;

xd;Nwhnlhd;W njhlh;GilajhFk;. gy;NtW fw;gpj;jy; Kiwfspy; tpjptUKiw,

tpjptpsf;f Kiw mbg;gilahd KiwahFk;.

tpjptU Kiw-tpsf;fk; :

gy vLj;Jf;fhl;LfisAk;,njhpe;j cz;ikfisAk; khzth;fsplk;

vLj;Jf;$wp mtw;wpypUe;J nghJ tpjpia tUtpf;Fk; Kiw tpjptU KiwahFk;.

,k;Kiwapd; Nfhl;ghLfs; :

1. njhpe;jjpypUe;J , njhpahjjw;Fr; nry;Yjy;.

2. gUg;nghUspypUe;J fUg;nghUSf;F nry;Yjy;.

3. vspikapypUe;J fbdj;jpw;F nry;Yjy;.

tpjptU Kiwapd; Ie;J gbepiyfs; :

1 . Maj;jk; : Gjpa mwpit ngWtjw;F khzth;fis Cf;fg;gLj;JtJ

kw;Wk; #o;epiyia cUthf;FtJ.

2 . vLj;Jf;$wy; : ,q;F njspthd vLj;Jf;fhl;Lfs; %yk; fw;gpj;jy;

Kf;fpa ,lj;ij ngWfpwJ.

3 . xg;gpLjYk; RUf;FjYk; : ,q;F gpur;ridia njspthf Ghpe;J

nfhz;L nfhLf;fg;gl;l tptuq;fis xg;gpl;L gpur;ridapd; KbTf;F

Njitahd tptuq;fs; Muha;e;J ghh;f;fg;gLfpwJ.Vw;fdNt fw;w mwpTld;

Gjpa mwpT njhlh;GgLj;jg;gLfpwJ.

4 . nghJikg;gLj;Jjy; : khzf;fh; Gjpa #j;jpuk; , tpjp Mfpatw;iw

gy;NtW vLj;Jf;fhl;LfspypUe;J jhNk fz;lwpa Kw;gLfpwhh;fs;.

5 . tpjpia nray;gLj;Jjy; : nghJ tpjpia Gjpa #o;epiyapy; gad;gLj;jp

rpf;fiy jPh;g;gJ.

95



vLj;Jf;fhl;L : 1

(a) 12 =1 , 32 = 9 , 52 = 25, 72=49,........1,3,5,kw;Wk; 7 MdJ xw;iwg;gil

vz;fs; kw;Wk; xw;iwg;gil vz;fspd; th;f;f%yk; 1,9,25,49,.....

(b) 22=4 , 42=16 , 62=36 . 82 =64 .......................2,4,6,8,....... vd;gJ ,ul;ilg;gil

vz;fs; ,ul;ilg;gil vz;fspd; th;f;f%yk; 4,6,36,64,.......

KbT (a) : xw;iwg;gil vz;fspd; th;f;f%yk; xw;iwg;gil vz;zhFk;.

KbT (b) : ,ul;ilg;gil vz;fspd; th;f;f%yk; ,ul;ilg;gil vz;zhFk;.

vLj;Jf;fhl;L : 2

1+1=2 , 1+3=4 , 1+5=6 : 3+5=8...... 1,3,5,.... MdJ xw;iwg;gil vz;zhFk;

,uz;L xw;iwg;gil vz;fspd; $Ljy; ,ul;ilg;gil vz;zhFk;.

nray;ghL :1

gpd;tUk; $w;Wf;fs; rhpah vd Muha;f.

1 . %d;W xw;ig;gil vz;fspd; $Ljy; 2. ,uz;L xw;iw / ,ul;ilvz;fspd; $Ljy; xw;iwg;gil vz;zhFk;.

vLj;Jf;fhl;L : 3

1. a2 a3 = (aa) (aaa) = a5=a 2+3

2. a3a4 = (aaa)(aaaa) = a7 = a3+4

3. a3a6 = (aaa) aaaaaa) = a9 =a3+6

ngwg;gl;l KbTfs; = aman = (aa.....m times) (aa.....n times)

= a....(m+n)times

= am+n

/ amn = m+n

96



tpjptpsf;f Kiw : [ DEDUCTIVE MEDHOD]

Muk;gj;jpNyNa nghJtpjp my;yJ #j;jpuj;ijf; $wp mjpYs;s

tptuq;fSf;F tpsf;fkspj;J fw;gpg;gJ tpjp tpsf;f KiwahFk;.,J tpjptU

Kiwf;F Neu; vjpuhdJ.

,k;Kiwapd; Nfhl;ghLfs; :

1. nghJ tpjpapypUe;J Fwpg;gpl;l tptuj;jpw;F nry;Yjy;

2. fUg;nghUspypUe;J gUg;nghUSf;F nry;Yjy;.

3. tpjpapypUe;J vLj;Jf;fhl;LfSf;F nry;Yjy;.

,k;Kiwapy; ghlj;jiyg;ig mwptpj;jTld; mjw;F cz;lhd tpjpia

my;yJ #j;jpuj;ij Mrphpah; jUthh;..

mr;#j;jpuj;ij tpsf;fpa gpwF mtw;iw gad;gLj;jp tpdhit jPh;f;Fk;

topKiwfis mspg;ghh;.#j;jpuj;ij kdg;ghlk; nra;J nfhz;L NkYk; gy

tpdhf;fSf;F khzhf;fh; jPh;T fhz;gh;.

tpjptpsf;f Kiw fw;gpj;jYf;fhd mZFKiwfs; :

rpf;fiy jPu;f;f Kiwahd mZFKiwfspy; xd;W tpjptpsf;f Kiwfs;

MFk;.

jw;fhypfkhd fUJNfhs;fis Nju;e;njLj;jy;.

jw;fhypfkhf Njh;njLf;fg;gl;l fUJNfhs;fSf;F rhpahd #j;jpuj;ijg;

gad;gLj;jp jPu;T fhZjy;.

gpur;ridia jPu;j;jy;.

gpur;ridf;fhd jPu;Tfis rhpghh;j;jy;.

vLj;Jf;fhl;L : 1

a2a10 = ? fz;Lgpb.

[ #j;jpuk; : am an = am+n gad;gLj;jTk; ]

97



MfNt , a2a10 = a 2+10

= a12 ( ,q;Nf m=2 , n =10 )

vLj;Jf;fhl;L : 2

(102)2 = ? fz;Lgpb.

[ #j;jpuk; (a+b)2 = a2+b2+2ab gad;gLj;jTk; ]

(100+2)2 = 1002 + 22 + (2 (a =100, b=2)

= 10000 + 4 +400 = 10404.

E3 : ,k;Kiw nghJikg;gLj;Jjy; kw;Wk; #j;jpuj;ij gad;gLj;Jk;

mbg;gilapyhd fw;wy;.E4.,t;tpjk; Neubahf #j;jpuj;ij

gad;gLj;jp rpf;fiy jPu;j;jy;.

4.2.2 gFj;jwp Kiw kw;Wk; njhFj;jwpKiw :

[Analytic and synthetic Method ]

gy;NtW mDgtq;fs; gFj;jwp fw;gpj ;jy; Kiwapy; ,Ue;J

ngwg;gLfpwJ.xU fzpjf;fUj;ij rpWrpW gFjpfshfg; gphpj;J Muha;e;J

mjid njspthf fw;Wf;nfhs;tjw;F gFj;jwp Kiw vd;W ngah;.xU njupahj

fzpjf; fUj;ij cz;ik vd vLj;Jf; nfhz;L mij ep&gpf;f Ntz;Lk;.

Nfhl;ghL; :

gFj;jwp Kiw vd;gJ njupahjjpypUe;J njupe;jjw;F nry;Yjy;.

vLj;Jf;fhl;L:

dc

ba

vdpy; d

bdcb

bac 22 22

vd epWTf.

98



ep&gzk; :

dbdc

bbac 22 22

vdj ; nju pa hj fUj ;ij cz;iknad

vLj;Jf;nfhs;Nthk;.

d(ac-2b2) = b(c2-2bd)= (FWf;Fg; ngUf;fy; ngUf;f.]

dac - 2b2d = bc2-2b2d = vd;gJ cz;ikahFk;.

,UGwKk; (-2b2d) I ePf;f.

dac = bc2 vd;gJ cz;ikahFk;.

,UGwKk; C I ePf;f ,

da = bc vd;gJ cz;ikahFk;.

mjhtJ dc

ba = vd;gJ cz;ikahFk;.

njhFj;jwp Kiw : [ Synthetic Method ]

nfhLf;fg;gl;l njupe;j jfty;fspypUe;J njupahj jfty;fisf;

fz;Lgpbf;f top nra;Ak; Kiw njhFj;jwp Kiw vdg;gLk;.,J tpjp

tUKiwiag; Nghd;wJ, gFj;jwp Kiwf;F Neh; vjpuhdJ.

Nfhl;ghL :

njhFj;jwp Kiw vd;gJ njupe;jjpypUe;J njupahjjw;F nry;Yjy;.

vLj;Jf;fhl;L :

gFj;jwp Kiwapy; nfhLf;fg;gl;l vLj;Jf;fhl;ilNa vLj;Jf; nfhs;Nthk;.

dc

ba vdpy;

dbdc

bbac 22 22

vd epWTf.

99



ep&gzk; :

dc

ba = vd;f.

,UGwKk; cb2ia fopf;f.

cb

dc

cb

ba 22

,UGwKk; c1 I ePf;f.

dbdc

bbac 22 22

vd epUgpf;fg;gl;lJ.

4.2.3 nray;jpl;l Kiw : [PROJECT METHOD]

,ay;ghd #o;epiyapy; gs;spapNyh gs;spf;F ntspapNyh xU Fwpg;gpl;l

Nehf;fj;Jld; nra;Ak; nraiy nray;jpl;l Kiw vd;fpNwhk;.,k;Kiw mnkhpf;f

fy;tpahsh; lhf;lh;.fpy;gh;l; vd;gtuhy; mwpKfg;gLj;jg;gl;lJ.,k;Kiwapy;

cz;ikahd tho;f;ifr; #oypy; ghlr; nra;jpfis mwpe;J nfhs;fpd;wdh;.

vLj;Jf;fhl;lhf, fzpjk; fw;gpf;Fk; NghJ ntWkNd Gj;jfj;jpy; cs;s

fzf;Ffis nfhLf;fhky; nray;fs; rpytw;iw mikj;Jf;nfhz;L mtw;wpd;

thapyhf Gj;jf mwpitg; ngWfpNwhk;.

nray;jpl;l Ntiyia jdpahfNth my;yJ rpy FOf;fshfNth my;yJ

vy;yh khzth;fSk; Nru;e;Njh nra;th;.nray;jpl;lk; ntw;wpfukhf mika

mJ eilKiwf;F Vw;wjhfTk;. tho;f;iff;Fg; gad;juj;jf;fjhfTk; mika

Ntz;Lk;. mg;NghJ jhd; khzth;fs; Mh;tj;NjhL kdk; tpUk;gp nra;th;.

nray;jpl;lj;jpd; Nfhl;ghLfs; :

1. nray; topf;fw;wy;

2. tho;f;ifr; #oypy; fw;wy;

3. r%f gz;Gfis fw;wy;

100



nray; jpl;lq;fspd; nray;ghLfs;:

1. gs;spapy; khjpup tq;fp elj;Jjy;.

2. gs;spapy; $l;LwT gz;lf rhiyia elj;Jjy;.

3. gs;spapy; Njhl;lk; mikj;jy;.

4. gs;spapy; rhiy mikj;jy;.

5. fzpj rhh;ghd Rw;Wyhf;fSf;F Vw;ghL nra;jy; Kjypad.

nray;jpl;l Ntiyia jdpahfNth my;yJ xU FOthfNth vLj;Jf;

nfhz;L nra;ayhk;.FOtpy; cs;s xt;nthUtUk; nray;fspd; rpWrpW

gFjpfis vLj;Jf;nfhz;L nghWg;ghf nra;J Kbg;gh;.

nray; jpl;l Kiwapd; gbfs; :

1. #o;epiyia mikj;jy;

2. nraiy Nju;e;njLj;jy;

3. jpl;lkpLjy;

4. nray;gLj;jy;

5. kjpg;gpLjy;

6. gjpT nra;jy;

4:2:4 rpf;fy; jPu;f;Fk; Kiw kw;Wk; rpf;fiy tpLj;jy; :[ Problem solving and Problem Posing ]

nfhLf;fg;gl;l fbdkhd tpdhit Mf;fg;G+h;tkhf rpe;jpj;J, tptuq;fisNrfhpj;J kjpg;gPL nra;J jPu;it vl;Lk; Kiwf;F rpf;fy; jPu;f;Fk; Kiw

vdg;gLk;.

njhlf;fg;gs;spapy; fzpjj;jpy; $l;ly;,fopj;jy;,ngUf;fy;,tFj;jy; Nghd;wnray;fs; %yk;; fzpjj;jpd; mbg;gilf; nfhs;iffisf; fw;Wf;nfhs;fpd;wdh;.

cjhuzk; : ,uz;lhk; tFg;G khzth;fs; eilKiw tho;tpy;

jq;fSila fzpjj;jpwd;fisAk; El;gq;fisAk; Nkd;NkYk; ngUf;fpf;nfhs;s

KbfpwJ.

101



cjhuzk; :-

$l;liyg; nghWj;J 75+29 jPu;T fhZk; Kiw.

i) Neupil Kiw : 75 + 29= 104

ii) 75+29= 75+(30-1)=(75+30)-1=105-1=104

iii) 75+29= 74+1+29=74+30=104

iv) 75+ 29=75+25+4 = 100+4 =104

NkNy fz;l topKiwfs; midj;Jk; rhpahdit.njhlf;fg; gs;spapy;

,uz;lhk; tFg;G khzth;fSf;F’ $l;ly; fzf;fpd; jPh;it fz;lwpa nra;ayhk;

rpf;fy; jPh;f;Fk; Kiwapd; cs;s gbepiyfs; :

a) rpf;fiy milahsk; fhZjy. ;

b) fzf;if tiuaWj;jy;.

c) Gs;sp tptuq;fis Nrfhpj;jy. ;

d) fUJNfhs;fis mikj;jy;.

e) fUJNfhs;fis Nrhjpj;J ghu;j;jy;.

f) khjpup tbtq;fis cUthf;Fjy;.

g) KbTfis gpw #o;epiyapy; gad;gLj;Jjy; .

h) tpiliar; rhpghh;j;jy;.

a) rpf;fiy milahsk; fhZjy; :

khzth; rpf;fypd; jd;ikia Ghpe;J nfhz;L mjid tiuaWf;f

Ntz;Lk;. rpf;fyhdJ khzthpd; Mh;tj;jpw;Fk; Cf;fj;jpw;Fk; toptFg;gNjhL

khzt rpf;fiy fz;Lgpbf;f J}z;Lk; tifapy; mike;jpUf;Fk;.

b) fzf;if tiuaWj;jy; :

khzth;fspd; nrhe;j thh;j;ijfs; %yk; fzpjj;jpw;F jPu;T mspf;ff;

$bajhf mikfpwJ.

102



c) Gs;sp tptuq;fis Nrfhpj;jy; :

rpf;fypy; vd;d nfhLf;fg;gl;Ls;sJ? vd;d Nfl;fg;gl;Ls;sJ.?

nfhLf;fg;gl;l tptuq;fs; NghJkhditah ? vd ftdkhf gFj;jha;jy;

nra;a Ntz;Lk ;gpur;ridf;F jPu;T fhz cjTk; jfty;fis Nrfhpf;f

Ntz;Lk;.

cjhuzk; :

i) mLj;jLj;j %d;W xw;iw vz;fspd; $Ljy; 51

ii) jphpNfhzkpjp tpfpjk; vd;w fUj;ij Ghpe;J nfhs;tjw;fhf

khzth;fsplkpUe;J cauk; kw;Wk; J}uk; rk;ge;jkhd jfty;fis jpul;b

mjd; %yk; jPu;T fhzy;.

d) fUJNfhs;fis mikj;jy; :

nfhLf;fg;gl;l tptuq;fis Muha;e;J mt;tptuq;fSf;fpilNa

njhlh;Gfis Vw;gLj;jp jw;fhypfkhd fUJNfhs;fis mikf;f Ntz;Lk;.

cjhuzk; :

miuf;Nfhsj;jpd; nkhj;jg;gug;ig fzf;fpLtjw;F miuf;Nfhsj;jpd;

tisgug;igAk; miuf;Nfhsj;jpd; mbg;gf;f gug;igAk; $l;l Ntz;Lk;.

miuf;Nfhsj;jpd; nkhj;j Gwg;gug;G = 2r2+r2

= 2r2

,q;F r miuf;Nfhsj;jpd; Muk; MFk;.

e) fUJNfhs;fis Nrhjpj;J ghu;j;jy;

jw;fhypfkhf fUJNfhs;fis rpf;fypy; Nrhjpj;J ghh;f;f Ntz;Lk;.

rpf;fypd; jPu;Tf;F jw;fhypf fUJNfhs;fs; nghUe;jtpy;iy vdpy; khw;W

fUJNfhs;fis Muha;e;J rpf;fYf;F jPu;T fhz Ntz;Lk;.

103



f) khjpup tbtq;fis cUthf;Fjy;

khzth;fSf;F md;whl tho;f;ifapy; cs;s khjpup tbtq;fis

cUthf;fpj;jUjy;.

(v.fh) khzth;fSf;F rJug;gyifia nfhLj;J mjpy; vj;jid rJuq;fs;

;,Uf;fpwJ vd;gij fz;Lgpbf;f nra;jy;.

g) KbTfis gpw #o;epiyapy; gad;gLj;Jjy; ; :

rpf;fypd; jPu;it cWjp nra;j gpd; , mjid nghJikg;gLj;jp Gjpa

#o;epiyapy; gad;gLj;j Ntz;Lk;.

h) tpiliar; rhpghh;j;jy;:

khzth;fs; mspf;Fk; gjpy;fs; rhpahdjhfTk; , #j;jpuj;ij gad;gLj;jp

tiuaiwapd; gb tpilaspj;jpUe;jhy; me;j tpil rhpah ?jtwh? vd;W

Mrphpah; rhpghh;g;ghh;.

rpf;fiy tpLj;jy; : (Problem Posing)

rpf;fiy tpLj;jy; vd;gJ rpf;fy; jPu;f;Fk; KiwAld; neUq;fpa

njhlh;GilaJ.nfhLf;fg;gl;l fbdkhd tpdhit Mf;fG+u;tkhf rpe;jpj;J,

tptuq;fis Nrfhpj;J, jpl;lkpl;L, jpwhdha;T Kiwapy; jpUg;jpfukhd jPu;it

vl;Lk; Kiwf;F rpf;fy; tpLj;jy;; Kiw vdg;gLk;.

Mrphpah;fs; khzth;fSf;F gy;NtW tifahd tpdhf;fs; nfhLj;J

tpilfis fhzr; nra;jy;. mjdhy; khzth;fsplk; rpf;fy; jPu;f;Fk; jpwd;fs;

tsh;fpd;wd.

rpf;fy; tpLj;jy; kw;Wk; rpf;fy; jPu;f;Fk; topKiwfis khzth;fs;

fw ; wf ; nfhs ;tjhy ; fy ;t pa py ; gy ; NtW mZFKiwfis

ngWfpd;wdh;.mJkl;Lky;yhky; tplhKaw;rp ,rpe;jpf;Fk; jpwd; , jPu;T fhZk;

jpwd; Nghd;w jpwd;fisg; ngWfpd;wdh;.

104



cjhuzkhf : 4X5=20

,UgJ vd;w vz;Zf;F 4 vd;w ,ul;ilg;gil vz;Zk; 5 vd;w

xw;iwg;gil vz;Zk; ngUf;fpfshf cs;sJ.

4 , 5 d; kPg;ngU nghJtFj;jp 1 MFk; .

,jpy; xd;W xw;iwgil vz; kw;nwhU vz; ,ul;;ilgil vz;.

Kjy; ngUf;fy; vz; 4 kw;nwhU ngUf;fy; vz; 5 .

4 d; th;f;fk; 16 , 5 d; th;f;fk; 25

,e;j ngUf;fy; vz; njhlh; vz;.

vLj;Jf;fhl;Lfs; :

1. Xw;iw gil vz;izAk; ,ul;ilgil vz;izAk; ngUf;Ftjhy;

fpilf;Fk; tpil vd;d?

2. Xw;iw gil vz;izAk; Xw;iw gil vz;izAk; ngUf;fpdhy; vd;d

fpilf;fk;?

3. ,ul;ilgil vz;izAk; ,ul;ilgil vz;izAk; ngUf;fpdhy; vd;d

fpilf;fk;?

4. %d;W Xw;iw gil vz;izAk; my;yJ %d;W ,ul;ilgil vz;izAk;

ngUf;fpdhy; vd;d fpilf;fk;?

5. NtW ,U vz;fis ngUf;Fk; NghJ 20 tpil fpilf;f $ba vz;

vd;d? ,e;j jPu;T rhpah?

6. rpf;fiy tpLtpj;jy; MdJ fUJNfhs;fis mikf;Fk; jd;ik nfhz;lJ

kw;Wk; mjw;fhd jPu;T ngwg;gLfpwJ.

E.7 : nrq;Nfhz Kf;Nfhzj;jpd ; ,uz;L gf;fq;fspd; $Ljy;

%d;whtJ gf;fj;jpw;Frkk;vd;w tbtpay; $w;Wf;F rhh;e;j

rpf;fiy ngwg;gLfpd;wd.

105



4.3 : fzpjk; fw;gpj;jypy; fw;gth; - ika mZFKiwfs; :

( Learning – Centred Approaches of Teaching Mathematics)

fzpjj;ij kw;w ghlq;fNshL xg;gpLifapy; jdpj;J epw;Fk; xU ghlkhfTk;

mNj tifapy; kpfTk; Kf;fpakhd xU ghlkhfTk; jpfo;fpwJ. tFg;giwapy;

fzpjk; fw;gpj;jypd; NghJ ghlj;jpid tpsf;Fjy;> #j;jpuk; kw;Wk;

tiuglq;fis ifahSjy;> jPh;T fhZjy;> tpdhf;fs; Nfl;ly; Nghd;w

epfo;Tfs; eilngWfpd;wd.

ghlj;jpid fw;gpf;Fk; NghJ mjw;F rk;ge;jkhd fw;gpj;jy; fUtpfs;

nfhz;L fw;gpj;jy; Ntz;Lk;. Mrphpah; fw;gpf;Fk; nghOJ mij khzth;fs;

cw;WNehf;fp Fwpg;Gfs; vLj;Jf;nfhs;sNtz;Lk;.

Mrphpah; khzth;fsplk; tpdhf;fs; Nfl;Fk; NghJ kpfTk; vspikahd

Nfs;tpfis kl;Lk; Nfl;f Ntz;Lk;. khzth;fs; tFg;giw tpthjj;jpy;

<LgLtjw;F kpf Fiwe;j tha;g;G cs;sJ. ghlj;jpid fw;gpf;Fk; nghOJ

mjw;F Ke;ija tFg;gpy; vd;d elj;jpNdhk; vd;W xU Kiw $wpa gpd;

ghlj;ij njhluNtz;Lk;. ,jpypUe;J khzth;fs; jq;fSila Ke;ija

mDgtq;fis ngWfpd;wdh;.

4.3.1 5E – apd; fw;wy; khjphpfs; :(5 E’s Larning Model )

fw;wy; Kiwapy; 5 khjphpfis gad;gLj;jp khzth;fs; fw;fpd;wdh;.

mwpKfg;gLj;Jk; Kiw

fz;lwpjy; Kiw

tpsf;Fjy; Kiw

tpdhthhpahf fw;gpf;Fk; Kiw

kjpg;gpLjy; Kiw

I. mwpKfg;gLj;Jk; Kiw :

khzth;fSf;F fw;Wj;jUk;NghJ gy tifahd fw;gpj;jy; Kiwfis

ifahSfpd;wdh;. ,t;tif fw;gpj;jy; KiwahdJ nray;ghl;L Kiw>

vjph;ghuhj epfo;Tfis ifahSjy;> jdpr;rpwg;Gtha;e;j vLj;Jf;fhl;Lfis

ifahSjy; Nghd;witahFk;.

106



khzth;fspd; jpwikfis ntspf;nfhz;L tu toptif nra;jy; Ntz;Lk;.

Mrphpahpd; gzp vd;dntd;why; khzth;fspd; mwpT jpwikia Ghpe;J nfhz;L

mjw;Nfw;g fw;Wf;nfhLj;jy; Ntz;Lk;. Mrphpah; khzth;fis ftdj;jpw;F

nfhz;L tu Ie;J topfis gpd;gw;w Ntz;Lk;. fzpjk; rk;ge;jkhd jfty;fis

khzth;fspd; kdjpy; nfhz;L tu Ntz;Lk.; khzth;fspd; kdjpy; Nfs;tpfis

vOg;g Ntz;Lk; kw;Wk; Cf;Ftpj;jy; Ntz;Lk;.

(v.fh) Mrphpah; Mwhk; tFg;G gpd;dq;iis gw;wp vLj;Jf; $Wfpd;wdh;.

khzth;fSf;F fw;W jUk; nghOJ Kf;fpakhf mth;fspd; nray;ghLfis

Cf;Ftpf;Fk; tifapy; khzth;fSf;F gy;NtW vLj;Jfhl;Lfis nfhLf;f

Ntz;Lk;.

nray;ghL : 1

khzth;fSf;F Ngg;giu nfhLj;J tl;lk; kw;Wk; nrt;tf tbtkhf

nra;a itf;fTk;. gpd;G ,U mzpfshf gphpf;f Ntz;Lk;.

nray;ghL :2

tl;lk; kw;Wk; nrt;tf tbtj;jhs;fis tz;zk; jPl;b itf;fNtz;Lk;.

mjpy; xU gFjpapy; cs;s xt;nthU glj;ijAk; rkkhfTk;> rkkw;witahfTk;

tz;zk; jPl;l Ntz;Lk;.

107



Mrphpah; glj;jpid itj;J fw;Wj;jUk; nghOJ khzth;fSf;F

vspikahfTk;> Ghpe;J nfhs;Sk; gbahfTk; fw;W juNtz;Lk;. ,g;glj;jpd;

%yk; khzth;fs; ghlj;jpid vspikahd Kiwapy; fw;W nfhs;thh;fs;.

II .fz;lwpjy; Kiw :

Kjypy; Mrphpah; xU nray;ghl;bid nra;J fhl;b gpd;G me;j

nra;Kiwia khzth;fSf;F nra;J fhl;LkhW toptFf;fNtz;Lk; ,t;tpj

nray;Kiwia jdpahfNth my;yJ FOthfNth nra;a itf;fyhk;. mt;thW

nra;Ak; nghOJ khzth;fs; mtuth; Mh;tq;fis gfph;e;J nfhs;s Kbfpd;wJ.

khzth;fs; FOthf nra;Ak; NghJ jdpj;jpwik ntspg;gLk;.

III.tpsf;Fjy; Kiw :

Vw;fdNt fw;w mwpTld; Gjpa mwpT njhlh;G gLj;jgLfpwJ. fzpjk;

fw;gpf;Fk; NghJ khzth;fs; fUj;Jf;fs;> Njw;wq;fs; Mfpatw;iw Vw;fdNt

ngw;w mwpTld; njhlh;Gg;gLj;jp fw;Wf;nfhs;fpd;wdh;.

mt;thW Mrphpah; tpsf;Fk; NghJ khzth;fs; Ghpe;J nfhs;fpd;wdh;.

tpsf;Fjy; Kiwapy; Mrph pah ; nray;ghL KOikahf cs;sJ.

khzth;nray;ghL vJTk; ,y;iy. tpsf;Fjy; jpwd; Kiwahd Ghpe;J

nfhs;Sjiy rhh;e;Js;sJ.

IV.tpdhthhpahf fw;gpf;Fk; Kiw

khzth;fspd; Kd;dwpit gad;gLj;jp Gjpa Nfs;tpfis Nfl;f itf;f

Ntz;Lk; Nfs;tpia vOg;gp rpe;jidiaj; J}z;lNtz;Lk;.

108



NkNy fw;Wf;nfhz;l %d;W topKiwfspypUe;J mij rk;ge;jg;gLj;jp

cyf eilKiwfSld; xg;Gikg; gLj;jpf;fhl;l Ntz;Lk;.

V.kjpg;gpLjy; Kiw

filrpahf khzth;fs; me;j ghlj;jpid KOikahf Ghpe;J

nfhz;lhh;fsh vd;gjid Njh;tpd;%yk; kjpg;gPL nra;jy; Ntz;Lk;.

khzth;fSf;F tFg;giwapy; GJtpjkhd nray;ghLfis nra;a itf;f

Ntz;Lk;.

cjhuzkhf – xg;gilg;G> cw;W Nehf;Fjy;. FO xg;gilg;G .....etc

4.3.2 Mf;fG+h;tkhd fl;likg;G tpsf;f khjphp tbtk; :

(Interpretation Construction (Icon)design Model)

1. cw;WNehf;fy;

fw;wypy; khzth;fs; ghlq;fis cw;WNehf;f Ntz;Lk;;. ghlj; njhlh;Gila

fUj;Jf;fs; kw;Wk; epiyikfis cw;WNehf;f Ntz;Lk;.

2. #o;epiyfis cUthf;Fjy ;

cw;WNehf;FjYf;F gpd;G Mrphpah; khzth;fSf;F xU jPh;tpid

fzf;fpLtjw;fhd #o;epiyia cUthf;fp ju Ntz;Lk;. khzth;fs; mj;jPh;tpid

fhz;gjw;F Kd; mDgtq;fis gad;gLj;jp jPh;f;f Ntz;Lk.;

3. mwpthw;wy; gapw;rp

khzth;fis Cf;Ftpf;Fk; tifahf Mrphpah; khzth;fSf;F

gy;tifahd gapw;rp ju Ntz;Lk;. xU Mrphpauhf khzth;fis me;j

#o;epiyf;F Vw;wthW vt;thW NgrNtz;Lk;> vt;thW Nahrpf;f Ntz;Lk;

vd;gjid fw;Wf; nfhLg;gJ kpf Kf;fpak;. mt;thW nra;jy; %yk;

khzth;fspd; GJtpjkhd Nahridfis mwpa toptFf;Fk;.

109



4. ,izj;jy ;

nfhLf;fg;gLk; nray;ghLfis nra;tjw;F khzth;fis FOthf

,izf;f Ntz;Lk;. gpd;dh; khzth;fs; FOthf tpthjpf;f Ntz;Lk;. FO

fw;wYf;F Mrphpah; topfhl;bahf ,Ug;ghh;. khzth;fspd; Fog;gq;fis jPh;j;J

itg;ghh;.

5. fl;likg;G kw;Wk; tpsf;fk;

,jd; %yk; khzth;fs; mwpe;J nfhs;tJ vd;dntd;why; mth;fspd;

Gjpjhd vz;zq;fis ntspf;nfhz;L tuyhk;.

cjhuzk; : fye;Jiuahly;> tpthjk; Mfpatw;wpd; %yk; Gjpa

fUj;Jf;fis mwpe;J nfhs;th;.

6. gy;tifahd tpsf;fk ;

khzth;fs; cw;WNehf;Fk; NghJ mth;fsplk; gy;NtW nefpo;T jd;ikfs;

cUthfpd ;wd. mjd; %yk ; khzth ;fs ; jq ;fs ; czh ;Tfis

ntspg;gLj;Jfpd;wdh;. xU gpur;ridf;fhd jPh;tpid nfhz;L tuf; $ba

jpwd; tsh;fpwJ.

7. gy;tifahd ntspg;ghL

fw;gth; gytifahd tpsf;fq;fis xU gpur;ridf;F gad;gLj;jp

jPh;Tfis ngWfpd;wdh;. ,jd; %yk; xNu Neuj;jpy; epiwa mDgtq;fis

khzth;fs; ngWfpd;wdh;. xU tFg;giw #oypy; Mrphpah; - khzth;

gytifahd tpsf;fq;fs; %yk; vLj;J $Wk; Mw;wiy ngWfpd;wdh;. Mrphpah;

fw;Fk; khzth;fSf;F topfhl;bahf nray;gLfpd;whh;.

4.3.3 fUj;J tiuglk;. (Concept Mapping )

xU fUj;jpDs; Mokhf nrd;W mjd; cl;fUj;Jf;fis milahsk;

fz;L mitfspd; njhlh;Gfis tiuglk; thapyhf myrp Muha;e;J Gjpa

nghUs; NjLtJ fUj;J tiuglj;jpd; Nehf;fkhFk;.

110



tpj;jpahrkhd topfspy;

tpj;jpahrkhd topKiwfspy; fzpjj;ij mwptpay; rhh;e;j tuyhw;W

Kiwapy; nfhz;L tUjy;.

gy;NtW tifahd tpj;jpahrkhd gf;f vz;fis nfhz;L ntt;NtW

tpjkhd fUj;J tiuglq;fis nfhz;L tuyhk;.

cjhuzkhf ehd ;F gf ;fq ;fis nfhz;l tiuglk ; f P No

nfhLf;fg;gl;Ls;sJ.

4.3.4 nray; topfw;wy; (Activity Based)

tFg;giwapy; khzth;fis cw;WNehf;Fk; NghJ> nray;ghL rhh;e;j

eltbf;ifapy; Mh;tj;ij fhl;Lfpwhh ;fs; .mjdhy; khzth ;fSf;F

nray;topf;fw;wypd; %yk; tFg;giwapy; KOikahd <LghL Vw;gLfpd;wJ.

nray;top fw;wypd; %yk; Foe;ijahdJ gy;NtW czh;T cWg;Gfis

gad;gLj;jp fw;Wf; nfhs;fpd;wjJ. ,t;tifahd fw;wyhdJ Nrhjid %yk;

nra;J fw;Fk; jd;ik kw;Wk; gy;NtW nray;fis iffspy; nra;J mjd;%yk;

jq;fSila fw;wiy Nkk;gLj;Jfpd;wdh;. ,t;tpjkhd nray;fSf;F Mrphpah;

111



cjtpfukhf khzth;fSf;F ,Ug;ghh;.nray;top fw;wypd; %yk; Foe;ijfs;

jq;fs; gbf;f ,Uf;Fk; jfty;fis Neubahf ngw;W gad;ngWfpd;wd ,jdhy;

khzth;fspd; fw;wypd; jd;ik mjpfhpg;gJ kl;Lkpy;yhky; Mh;tj;ijAk;

gbg;gbahd Kd;Ndw;wj;ijAk; ngwg;gLfpd;wd. Foe;ijahdJ jdJ nrhe;j

Ma;T kw;Wk; xU cfe;j fw;wy; #oiy toq;f tha;g;G Vw;gLfpd;wJ.

,jdhy; khzth ;fsplk ; e Pz ;l fhykhf fUj;Jf ;fs ; kdjpy ;

epiyepWj;jg;gLfpd;wd.gy;NtW nghUl;fis ifahz;L mjd; nghUs;> gad;ghL

kw;Wk; Ra fw;wy; Nghd;wit Cf;Ftpf;fg;gLfpd;wd. ,jd;%yk; khzth;fspd;

jpwikfs; kw;Wk; jpwd;fs; tsh;fpd;wd.njhlf;ff;fy;tpapy; Foe;ijfspd;

nray;top fw;wyhdJ tpisahl;L;> Gjph;> fhfpjk; klf;Fjy; / fhfpjk; ntl;Ljy;>fzpj khjphp... etc., Mfpatw;wpd; %yk; mwpfpd;wdh;.

,aw;fzpj Kw;nwhUikfis gw;wp khzth;fSf;F fw;gpf;fg;gLfpwJ.

#j;jpuk; ( a+b )2= a2 +2ab+b2 ,t;tpj Kw;nwhUikfis Mrphpah; tpsf;Fk;

NghJ fUk;gyif kw;Wk; gy;NtW jpwd;fis gad;gLj;jpAk; khzth;fSf;F

vLj;Jf;$wyhk;. khzth;fs; ,t;tpj Kw;nwhUikfis njh;khNfhy; fhfpjk;

gprpd;> njh;khNfhy; ntl;Ljy;> fhfpjk; fyh;ngd;rpy; Mfpatw;iw gad;gLj;jp

fzpj khjphpfs; cUthf;fg;gLfpd;wd.

P

112



mDgt fw;wy; : (Experiential Learning )

gq;Nfw;ghsh;fs; gaDs;s Ez;zwpT kw;Wk; fw;wy; eltbf;ifapy;

<LgLfpd;wdh;. khzth;fs; Nrhjidfspd; %yk; nra;J fw;wy; vd;w gof;fq;fis

mwpa KbfpwJ. ,jd; %yk; khzth;fs; nrhe;jkhf rpe;jpf;Fk; jpwid

ngWfpd;wdh;. ,jdhy; khzth;fsplk; elj;ij khw;wk; kw;Wk; xUq;fpize;j

mDgtq;fs; ngwg;gLfpd;wdh;. jpwd;fspd; %yk; mDgtf; fw;wypy;

ngwg;gLfpd;wd.

mDgtf; fw;wyhdJ 5 epiyfis ngWfpd;wd.

mDgtk;

nra;J fw;wypd; %yk; khzth;fs; mDgtq;fis ngWfpd;wdh; ,jdhy;

khzth;fs; fy;tpapd; Nehf;fj;ij milfpd;wdh;. nra;J fw;wy; %yk;

xUtUf;nfhUth; njhlh;G nfhz;L fUj;Jf;fis Ghpe;J nfhs;s Kbfpd;wJ.

ntspaPL

mDgtq;fspd; %yk; ngwg;gl;l KbTfis khzth;fs; xUtUf;nfhUth;

fye;Jiuahb jd; ntspaPLfis nfhLf;fpd;wdh;. ,jdhy; khzth;fspd;

Ra rpe;jid ntspg;gLfpd;wJ.

113



nray;ghL

FOtpy; khzth;fs; xUtUf;nfhUth; fUj;Jf;fis ghpkhwpf;nfhs;fpd;wdh;.

khzth;fspd; nray;ghL rhh;e;j tptuq;fs; ngwg;gLfpd;wdh;. ,f;FOtpd;

nray;ghl;by; khw;wk; ,Ug;gpd; tpthjk; njhlq;Fk;.

nghJikg;gLj;Jjy;

FOtpd; ftdkhdJ ghlj;jiyg;ig xj;J fhzg;gLfpd;wd.

nray;ghLfspd; %yk; ghlepiyik gw;wpa tpopg;Gzh;T kw;Wk; mDgtq;fs;

Vw;gLfpd;wd.

gad;gLj;Jjy;:

fw;w mwpthdJ Gjpa #o;epiyapy; rpf;fy; jPh;f;f gad;gLj;jg;gLfpd;wJ.

,jd; %yk; nghJik gLj;Jjypd; kjpg;Gilikia ghpNrhjpj;J rhpghh;f;f

gLfpd;wJ. ,jdhy; fw;w mwpT mh;j;jKs;sjhf khwp khzth; kdjpy; ePz;l

fhyj;jpw;F jf;f itf;f KbfpwJ.

4.4 fzpjj;jpd; rthyhd kw;Wk; jpUg;jpfukhd fw;wy; :

(Making Mathematics Learning more challenging and satisfying)

fzpjk; fw;gpj;jyhdJ eilKiw tho;f;ifapy; rthyhd kw;Wk;

jpUg;jpfukhd fw;wyhf cs;sJ.fzpj fw;wy; khzth;fSf;F Mf;fg;G+u;tkhd

Mh;tj;ijAk;,rthyhd #o;epiyAk;, mtdJ milT epiyia njupe;J

nfhs;s cjTfpwJ.Xa;T Neuj;jpy; cs;s khzth;fSf;F fzpjk;

fw;gpf;fg;gLfpwJ. khzth;fspd;; kdjpy; cs;s ftiy kw;Wk; kd mOj;jkhdJ

Fiwf;fg;gLfpd;wd.,jdhy; khzth;fs; kdjpy; Njhy;tp kw;Wk; gak; Mfpait

KOtJkhf Nghf;FfpwJ.,jd; %yk; khzth;fSf;F ,ilNa Neh;kiw

mZFKiw kw;Wk; gilg;G jpwd;fs; fzpjk; fw;wypdhy; cUthfpd;wd.

4.4.1 : fw;gtupd; gilg;G jpwd;fis Nkk;gLj;Jjy; :

(Development of Learners’ creative abilities).

tFg;giwapy; khzth;fs; cw;W Nehf;Fk; nghOJ mth;fsJ

nray;ghLfshdJ khzth;f;F khzth; NtWgLfpd;wd.fzpjg; gpur;ridfSf;F

114



jPu;T fhZjy; kw;Wk; gy;NtW nray;ghLfis jPu;f;Fk; jpwd;fshdJ khzth;fs;

kdjpy; Gjpad gilf;Fk; jpwd;fs; xNu khjpupahf cUthfpd;wJ.

,t;tpjkhd nray;ghLfNs Mf;fj;jpwd; vdg;gLk;.fzpjk; fw;gpj;jy;

nray;ghLfspdhy; gy;NtW jpwd;fs; khzth;fs; kdjpy; cz;lhf;Ffpd;wJ.

fw;gthpd; nray;ghl;Lj;jd;ik kw;Wk; mjdhy; tpisaf;$ba

khzth;fspd; gilg;Gj;jpwid milahsk; fhZjy;.

nray;topf; fw;wy; :

nray;topf; fw;wy; vd;gJ nray;ghLfspd; mbg;gilapyhd xU fw;wy;

KiwahFk;.

jpwd;fis jpl;lkpl;L cUthf;fg;gl;l nray;ghLfspd; %yk; khzth;fis

milTngw itg;gNj ,jd; Kjd;ikahd Nehf;fkhFk;.

mDkhdk; :

fzpj nray;fshdJ vspikahfTk; kw;Wk; khWghL mile;j

rthy;fshf fw;gtUf;F ,Uf;f Ntz;Lk;.mt;thW ,Uf;Fk; nghOJ fzpj

nray;ghLfis jPu;f;Fk; jpwd;fis ngWfpd;wdh;.

jPu;T khWghL :

fzpj rpf;fy;fis khzth;fs; tpUg;gg;glhj #o;epiyapy; jPu;f;Fk;

nghOJ rupahd jPu;Tfs; rpy Neuq;fspy; mike;J tpLfpd;wd.mt;thW

rhpahd jPu ;Tfs; khzth;fs; fz;Lgpbf;Fk; Neuj;jpy; mth ;fsJ

gilg;Gjpwd;fshdJ tsh;fpd;wd.mLj;jjhf fzpj jPu;Tfis fz;Lgpbf;Fk;

jpwd;fs; tsh;fpd;wd.

jUf;f Kiw :

xU rpf;fiy jPh;f;f KaYk;NghJ nfhLf;fg;gl;l tptuq;fis thpirgLj;jp

fhuz , fhhpaq;fspd; mbg;gilapy; njhlh;G gLj;jp jPu;T fhZk; Kiwf;F

jUf;f Kiw vd;W ngah;. fzpjj;jpy; ve;j xU rpf;fiyAk; vLj;Jf;

115



nfhz;lhYk; xU epiyapypUe;J mLj;j epiyf;F nry;tjw;F jUf;f

Kiwapyhd xU njhlh;r;rp ,Uf;Fk;.,J jUf;f Kiwapy; rpe;jpf;f fzpjk;

xU Mw;wy; kpF Jizf;fUtpahFk;.

tiufiyKiwapy; tptuq;fis mspj;jy; :

tiufiy mikg;Gfs; vd;gJ xU ghlg;nghUis tiufiy Kiwiag;

gad;gLj;jp Kiwg;gLj;jg;gl;L khzth;fs; vspjhf fw;Wf;nfhs;s cjTk;

KiwahFk;.fzf;ifj; jPh ;g;gjw;Fj; Njitahd fUj;Jf;fs; kw;Wk;

nrhw;nwhlh;fisg; ghlg; nghUNshL ,izj;J khzth;fs; mf;fzf;if

nghUNshL ed;F Gupe;J nfhs;s ,t;tiufiy mikg;Gfs; kpfTk;

gad;gLfpd;wd.

fw;gtupd; mZFKiwfspd; Kd;Ndw;wk; gpd;tUkhW :-

khzth;fSf;F toq;fg;gl;l jPu;Tf;fhd khw;W Nahrid; topKiwfis

mq;fPfupg;gJ.

khzth; kw;Wk; Mrpupah;fs; ,UtUk; ,ize;J jPu;Tfhd khw;W

topKiwfis NjLjy;.

,jd; %yk; khzth;fspd; rpe;jid kw;Wk; %is njhlh;Gila

nray;ghLfSf;F tha;g;G Vw;gLfpd;wd.

khWgl;l epidTfis Cf;fg;gLj;Jjy;

gpur;ridfis tpLj;jy; kw;Wk; gpur;ridfis jPu;j;jy; ,jd; Kf;fpa

Nehf;fkhFk;.

khzth;fSf;F Nfs;tp Nfl;l Rje;jpukhd #o;epiyia cUthf;Fjy;

nray;topf; fw;wypd;; mbg;gil xg;gilg;G MFk;.

4.4.2 : fzpj Ma;tfk; kw;Wk; E}yfj;jpd; gad;ghL :

(Use of Mathematics Laboratory and Library)

cq;fs; gs;spapy; Mrphpah; fzpjk; fw;gpf;Fk;NghJ gug;gsT , tl;lk;

Mfpaitfis #j;jpuq;fis gad;gLj;jp vz;fspd; %ykhf jPu;T fhz;gh;.

116



,t;top fw;wy; vd;gJ fzf;fPl;Lj; jpwikahd fw;wy; MFk;.khzth;fs;

#j;jpuj;ijg; gad;gLj;jp r2 vd;gij fzf;fpl ,ay;ghf cs;sJ.Mjyhy;

fzpjk; fw;gpj;jy; Kiw vd;gJ kpfTk; Kf;fpakhdJ.,jdhy; fl;Lkhdk; ,

kw;Wk; mwpT ; gad; vd;gJ fzpj fUj;Jfis njspthf Ghpe;J nfhs;s

cjTfpwJ.gs;sp khzth;fSf;F rpwpa tha;g;Gfs; kw;Wk; mDgtq;fspdhy;

fw;wYf;Fz;lhd Mh;tj;ij cz;lhf;Ftjhy; Ma;tfj;jpd; %ykhf khzth;fs;

cz;ikahd mwpit ngWfpd;wdh;.

fzpjk; fw;Wf;nfhs;Sjy; vd;gJ fzpj Ma;tfj;jpd; gad;ghl;bid

ngWfpd;w nrayghL; MFk;. fzpj Ma;tfj;jpid gad;gLj;jp khzth;fSf;F

fzpj nray;ghLfis nfhLg;gjd; %yk; mDgtq;fSk; Gjpa rpe;jidfSk;

ngWfpwhh;fs;.,jdhy; fzpj Ma;tfj;jpy; fzpj tha;g;G tpopg;Gzh;T

jpwikahd mbj;jsk; Neh;ikahd kdg;ghq;F gy;NtW tifahd fzpjk;

fw;wypdhy; Vw;gLfpwJ.khzth;fSf; fzpj cz;ikfis tiuglq;fs;

%ykhfTk;, mstpLjy; %ykhfTk;, fzpj khjpupfs;, fzpj Ma;tfj;jpd;

thapyhfTk; ngwyhk;.

cq;fs; gs;spapy; cs;s midj;J khzth;fSk; ghlg;Gj;jfq;fis

gad;gLj;Jfpd;wdh; ,mNjhL ,y;yhky; gj;jpu pf ;iffs; , fzpj thu

,jo;fs;,Ma;Tf; cz;lhdg; Gj;jfq;fs;, FWe;jfLfs;, tPbNah,Nlg;Gfs; fzpj

Ma;tfj;jpy;; fpilf;fpd;wd.,e;jg; nghUl;fisg; gad;gLj;jp tpj;jpahrkhd

Nahridfs; %yk; fzpj mwpit ,e;j cyfj;jpy; ngw KbfpwJ Mjyhy;

fzpj E}yfk; vd;gJ kpfTk; Kf;fpakhf fw;wYf;F gad;gLfpwJ.

E}yfj;ij gad;gLj;j khzth;fSf;F vLj;Jf; $WtjhYk; khzth;fspd;

mwpit Nkk;ghL milar;nra;ayhk;.

fzpj E}yfk; vd;gJ fw;wYf;F kpfTk; Kf;fpakhd xd;W. ,J

fw;wYf ;fhd Xu ; ,lj;ijg ; ngWfpwJ. NkYk ; fij , fl ;Liu ,

tpisahl;L,Muha;r;rp, Gjpu;fs;, ghly;fs;, fzpj fUj;Jfs; vd;W

midj;Jtpjkhd Gj;jfq;fSk; E}yfj;jpy; fpilf;fpd;wd.,it midj;Jk;

jdJ Njitf;Nfw;g vLj;J Ma;tjd; %ykhf ehk; ekJ mwpit Nkk;gLj;j

KbAk;.

117



4.5 njhFj;Jf;$wy; : ( Let us sum up )

gy;NtW ghlj;jiyg;GfSf;F fw;gpj;jy;khjpupfis gad;gLj;jp

khzth;fSf;F fw;gpj;jy;.

t pj ptUKiw mbg;gilahdJ gy vLj ;Jf ;fhl ;LfisAk ; ,

njupe;jcz;ikfisAk; khzth;fSf;F vLj;Jf; $wp mtw;wpypUe;J

nghJtpjpfhZjy;.

tpjptpsf;fKiwapd; mbg;gil nghJ tpjpapypUe;J Fwpg;gpl;l tptuj;jpw;F

nry;Yjy;.

gpur;ridid tpLj;jy; kw;Wk; gpur;ridia jPu;j;jy; vd;gJ rpf;fiy

tpLj;J mjw;fhd jP;h;Tfis khzth;fspd; Ra rpe;jid %yk; tpdhf;fSf;F

tpilia ngwr;nra;jy;.

5’E khjpu p fw;wypy; Ie;J tifahd fw;wy; cs;sJ.mit

mwpKfg;gLj;Jjy;, fz;lw;pjy;, tpsf;fkspj;jy;,tpdhthhpahf, kjpg;gpLjy; MFk;.

fUj;J tiuglk; xd;Wf;nfhd;W njhlh;GilaJ ,jd; %yk; khzth;fs;

jq;fSila mwpit xUq;fpizj;jy; kw;Wk; xd;Nwhnlhd;W ,izj;Jg;

ngWfpd;wdh;. nray;topf; fw;wy; MdJ gy czh;Tfis gad;gLj;jp if

Ntiyg;ghL jpwd;fis nfhz;L nra;fpd;wdh;.,jdhy; epidT Mw;wy;

mjpfhpf;fpd;wJ.

fzpj Ma;tfj;jpd; %yk; khzth;fs; Gjpad gilf;Fk; jpwd;fis

ngWfpd;wdh;.fzpj E}yfj;jpd; %yk; khzth;fs; fUj;Jf;fs;, fzpjGjph;fs;,

fijfs; , fzpj tpisahl ;L , fzpj tiuglq ;fs ; Mfpatw;iw

njupe;Jnfhs;fpd;wdh;..

4.6 khjpup tpilfis rhpghh;j;jy; :(Model Answer to check your progress)

,uz;L ,ay; vz;fspd; $Ljy; ,ay; vz; MFk;.

tpjptUKiw

118



tpjptpsf;fKiw

nray;jpl;lKiw

gFj;jwpKiw

njhFj;jwp Kiw

fzpj $w;Wfis vz;fshfTk; FwPaPLfshfTk; khw;Wjy;

tpdhthhpahf tpsf;Fjy;

#o;epiyia cUthf;Fjy;

fUj;Jf;fis ,izj;jy;.

4.7 Suggested Readings and References

Bransfrod .J.D.Brown ,A.L.S cocking,R.R [2000] How people Learn Washington

Dc;National Academy press.

Wood,T.cobb, P.S Yackel, E.[1995] Reflections learning and teaching Mathematics

in Elementary school . In L.P. steffe & J. Glade [Eds] constructivism in

education (Hillsdale ! Lawrence Erlbaum Associates .

4.8 myFj; Nju;T (Unit end Exercises )

1 . njhlf;ff;fy;tpapy; tbtpay; ghlq;fis vLj;J tpjptUKiwapy;

fw;gpj;jy;.

2 . njhlf;ff;fy;tp fzpjg;ghlj;jpy; VNjDk; xU jiyg;ig vLj;J mjw;F

tpjptUKiw kw;Wk; tpjptpsf;fKiwiag; gad;gLj;jp fw;gpj;jy;.

3 . nray;topf;fw;wypd; mbg;gilapy; Kf;Nfhzj;ij fw;gpj;jy;.

4 . Vohk; tFg;G fzpj ghlj;jpy; VNjDk; xU jiyg;Gf;F 5 E khjpupapy;

ghlj;jpl;lk; jahhpj;jy;.

ªî£ì‚è‚ è™m g¬ôj™ èeî‹...

Documents