Assignment 1 TJ - Assignment Essays

Assignment 1

Following the description in the reading assignment “Common Core: Solve Math Problems,” select one end-of-lesson assessment problem that is rich and representative of the bulk of the concepts in the standard you are targeting. Develop three different stations in three different ways—concretely, representationally, and abstractly. Include a description of what will be happening in each station.

��
��
�
��
��
��
��
��
��
�� !”

#��$��

%&'()*+,-.*/0&1′)23
45678697:;<<;87:;=67>4?8@?=@>�
��
��
��

��
��
�
��
�A��
�� BC��D��E�
��

��
��
��F��
� ��
��
��

�
��
��
� ��G��F��H�� I��

�I�� J��
�� K��
��
�

�� L�� E��MH�NOPQRSTUVOWUXYZ[��M� ��

��
��
��I��E��
��

�D��
�
��
��
��H��
��

��H��
��
��
H��
��
��

\��
��M��
��F��E��
��
�]��
�
��]

�E
��
��
H��$̂ _H��
��
��
�� ̀�
�
��
��

L
� ��
��H��
��I��
��
��
��
��
�

M��
��
��
��
��F��
�
��
��

��$��H��
��F��
��
��
��
��

��_ ��
��
a�bc�
��L��
��H�L��
��

M��
� ��
��
��
� ��d�

efggfhiefjkli

mnopqrmstnutvmswxnspwuuopqtyrzu{|}{~��uuu

��~��~
��

��

��
�� ¡��¢�
��£

¤¥
¦
§¦̈
©ª
«
¬
®ª̄
°
±
²®
³

��
�

�
��
��
��
�� !�”�#”$�%&’�()”*+)’!�(&�,&&-�”(�(+)�

.&/),!�!(0/)1(!�0!)�(&�

!&,2)�3’&4,).!5�

6+)�!(0/)1(!�#&’-�&1�! . ,”‘�3’&4,).!�

“(�(+’))�/ %%)’)1(�!(“( &1!� 1�(+’))�

/ %%)’)1(�#”$!7*&1*’)(),$8�’)3′)!)1(“9

( &1″,,$8�”1/�”4!(‘”*(,$5�:(;!�3’ .”‘ ,$�”�

!*’))1 1<�"*( 2 ($8�!&�!*'))1 1<�"*( 2 ($8�!&�$&0;')�,&&- 1<�%&'� 0;')�,&&- 1<�%&'�

3″(()’1!� 1�(+)�*,”!!5

�(�(+)�% ‘!(�!(“( &18�!(0/)1(!�0!)�

3+$! *”,�.”()’ “,!73,”*)92”,0)�4,&*-!8�

=1 % >�*04)!7(&�!&,2)�”�3’&4,).5��(�

(+)�’)3′)!)1(“( &1”,�,)2),8�(+)�’)3′)!)1(“( &1”,�,)2),8�$&0;’)�(“,-9
0;’)�(“,-9

1<�"4&0(�/'"# 1

(&�!&3+ !( *”()/�.&/),!�, -)�*+”‘(!8�

&’<"1 ?)/�, !(!8�/ "<'".!8�"1/�<'"3+!5�

@ 1″,,$8�”(�(+)�”4!(‘”*(�,)2),8�@ 1″,,$8�”(�(+)�”4!(‘”*(�,)2),8�$&0;’)�
0;’)�

“!- 1<�- /!�(&�#&'-�# (+�)A0"( &1!�(&�

!&,2)�3’&4,).!5�

:1��.)’ *”8�3″‘( *0,”‘,$8�#)�+”2)�”1�

&2)’).3+”! !�&1�3’&*)/0′”,�01/)’9

!(“1/ 1<5�B&!(�()"*+)'!�!))�(+)�)1/�

<&",�"!�-1&# 1<�(+)�3'&*)/0')� 1!()"/�

&%�01/)’!(“1/ 1<�(+)�."(+)."( *",�

/)”5�C&�:�, -)�(&�(),,�()”*+)’!8�DE&�

.&/),8�1&�01/)’!(“1/ 1<5F�

G��H��I�J�K��K�I�L�M�
L��J��N��O�K�MP��
��
�
��J��K�MM��Q�J�
�4!&,0(),$R�DS))3)’8�1&(�# /)’5F�:%�G��H��I�J�K��K�I�L�M�
L��J��N��O�K�MP��
��
�
��J��K�MM��Q�J�
�4!&,0(),$R�DS))3)’8�1&(�# /)’5F�:%�$&0�
0�

,&&-�”(�(+)�3′”*( *)�!(“1/”‘/!8�,&&-�”(�(+)�3′”*( *)�!(“1/”‘/!8�$&0�
0�

‘)”, ?)�(+”(�T0!(�(+)�#”$�� !�!)(�

03� !�<& 1<�(&�!033&'(�.&!(�&%�(+).5�

6+)’);!�”�#+&,)�!(“1/”‘/�”4&0(�.&/),9

1<5�U&0�#"1(�!(0/)1(!�(&�4)�"4,)�(&�

(+ 1-�&%�.0,( 3,)�#”$!�(&�!&,2)�3’&49

,).!5�V0(�(+)�’)”,�*&11)*( &1� !�(+)�

/)3(+5��+),3!�()”*+)’!�,&&-�/))3,$�

1(&�!(0/)1(!;�*&1*)3(0″, ?”( &18�!&�

(+)$�01/)’!(“1/�+&#�!(0/)1(!�01/)’9

!(“1/�”�.”(+).”( *”,�*&1*)3(5�

W)(( 1<�"�' <+(�"1!#)'�&1�"�()!(�

‘)3’)!)1(!�(+)�3&()1( “,�%&’�01/)’9

!(“1/ 1<8�40(� (�*&0,/�4)�"(�(+)�,)2),�&%�

D:�/ /�(+)�’ <+(�!()3!7:�%&,,&#)/�(+)�

3’&*)/0′)5F�:1�%'”*( &1!8�%&’�)>”.3,)8�

#+)’)�(+)’);!�+)”2$�).3+”! !�&1�

3’&*)/0′)8� (;!�1&(�01*&..&1�%&’�”�

!(0/)1(�(&�4)�”4,)�(&�#’ ()�”1�)A0″9

( &1�(&�!&,2)�”�3’&4,).�40(�1&(�4)�

“4,)�(&�*’)”()�”1$�.&/),�”(�”,,5�6+ !� !�

#+)’)� 1!(‘0*( &1�%”,,!�/&#15�X+)1�

$&0�,)”‘1�3’&*)/0’)!8� (;!�”�(‘ *-�$&0�
0�,)”‘1�3’&*)/0’)!8� (;!�”�(‘ *-�$&0�,)”‘1�3’&*)/0’)!8� (;!�”�(‘ *-�$&0�
0�

/&�%&’�/&�%&’�$&0’�.”(+�()”*+)’5
0’�.”(+�()”*+)’5

�
��
M��Y�M�Y��K��
6+)$�”‘)�0!0″,,$�< 2)1�"(�(+)�4)< 11 1<�

&%�”�,”‘<)�01 (8�"!�"�1))/!�"!!)!!.)1(5�

6)”*+)’!�”,!&�0!)�(+).�”(�(+)�)1/�&%�

01 (!�”1/�!&.)( .)!�# (+� 1/ 2 /0″,�

!(0/)1(!5�W)1)'”,,$8�()”*+)’!�<&�(&�(+)�

)1/9&%901 (�”!!)!!.)1(�”1/�3 *-�&1)�

3’&4,).�(+”(� !�’ *+�”1/�’)3′)!)1(“( 2)�

&%�(+)�40,-�&%�(+)�#&’-5�:1�(+ ‘/�<'"/)8�

%&’� 1!(“1*)8�%&’� 1!(“1*)8�$&0;’)�%&*0!)/�&1�.0,( 3, 9
0;’)�%&*0!)/�&1�.0,( 3, 9

*”( 2)�’)”!&1 1<5�:(;!�(+ !� /)"�&%�*&01(9

1<�# (+�<'&03!5�U&0;,,�#"1(�"�3'&4,).�

(+”(�!”$!�!&.)(+ 1<�, -)8�D6+)')�"')�

Z[�*+ ,/’)1� 1�”�W ‘,�C*&0(�(‘&&35�6+)�

(‘&&3�,)”/)’�#”1(!�(&�< 2)�)"*+�&%�(+).�

\�])’!+)$;!�̂ !!)!5�C+)�+”!�”�4″<�# (+�

_̀�̂ !!)!5�X ,,�!+)�+”2)�)1&0<+a��1/�

%�1&(8�+&#�.”1$�*”1�)2)’ %�1&(8�+&#�.”1$�*”1�)2)’$&1)�<)(aF� 1)�<)(aF�

6+)1�6+)1�$&0�”,()’�(+”(�!, <+(,$�%&'�)"*+� 0�",()'�(+"(�!, <+(,$�%&'�)"*+�

&%�(+)�(+’))�!(“( &1!8�4$�*+”1< 1<�(+)�

10.4)’!�&’�*&1()>(� 1�!&.)�#”$5�

�(�(+)�*&1*’)()�!(“( &18�(+)’)�# ,,�

4)�,&(!�&%�.”()’ “,!�”2” ,”4,)7( ,)!�&’�

=1 % >�*04)!5��(�(+)�’)3′)!)1(“( &1”,�

!(“( &18�!(“( &18�$&0�1))/�/ %%)’)1(�- 1/!�&%�
0�1))/�/ %%)’)1(�- 1/!�&%�

3″3)’8�3)1* ,!8�”1/�/'”# 1<�."()' ",!8�

“1/�(+)�”4!(‘”*(�!(“( &1�+”!�3)1* ,!�

“1/�3″3)’�”!�#),,5

6)”*+)’!�’)*&..)1/�,)(( 1<�(+)�- /!�

.&2)�%’&.�!(“( &1�(&�!(“( &1�&1�(+) ‘�

&#15�B”-)�(+)�!(“( &1!�,”‘<)8�!&�(+)$�

*”1�”**&..&/”()�”�10.4)’�&%�- /!�

“(�&1*)5�6+)�� (!),%�(“-)!�”1�+&0’5�

�!!)!! 1<�(+)�')!0,(!8�&1*)�()"*+)'!�<)(�

0!)/�(&� (8� !�”�Zb9�(&�[̀9. 10()�/)”,5

�
��
�II��
��K
��Q��
��
�(�(+)�*&1*’)()�!(“( &18�(&�<&�4"*-�(&�

&0’�)>”.3,)�3’&4,).8�&0’�)>”.3,)�3’&4,).8�$&0;’)�+&3 1<� 0;')�+&3 1<�

!(0/)1(!�.”-)�<'&03!�&%�!)2)1�# (+�

(+)�&4T)*(!�”1/�/&�!&.)�*&01( 1<5�6+)�

4)!(�#”$�(&�’)*&’/�(+)�*&1*’)()�#&’-� !�

# (+�”�*”.)'”5�B”1$�()”*+)’!�0!)�(+) ‘�

3+&1)!5��(�(+)�’)3′)!)1(“( &1”,�!(“( &18�

!(0/)1(!�# ,,�.”-)�* ‘*,)!�&1�(+)�3″<)c�

(+)$;,,�.”-)�!,”!+�.”‘-!�&’�/&(!�&’�

!&.)(+ 1<� 1! /)�(+)�* '*,)!�"1/�(+)$;,,�

!(“‘(�(&�*&01(5�6+)$;,,�/&�”1$�&%�”�10.9

4)’�&%�(+ 1

��
��
��
��

��

��!�”��#�� $��

��%&#��

‘()*+�,)(-�./0123451678
91:46242394;<679=>4?<3//@4 AB4C1622@9D/1/E4F91=/B2E40:94 =6BAG0@62AH9:46242394E4
K4L4MNOP4

Q394:262A/B4716I:4/024I3923914
:2079B2:461946D@942/423ABR4AB4
291=:4/S451/0GAB5E464R9>4

T39B4239>U1947/B94IA234239A14
I/1RE4239>4716I464GA<20194/S4A2E4 /142394296<3914

V(WXYZ[�\)ZZ]̂_Y)Z̀�a919E4
2394:2079B2:461949b0AGG974IA234
H61A/0:4I1A2AB546B74716IAB54
2//@:E4516G34G6G91E46B740B@AB974
G6G91P4Q39>46194AB:210<29742/4 0:94I/17:46B74GA<2019:42/4:3/I4 239A1423ABRAB5P4Q394I/174G1/D8 @9=436:4D99B4=/7ASA97464DA24 J23A:42A=9E4KN4c4dOP

e9:G/B:9:4/S29B40:94=61R:46B74
4B0=D914
/S4=923/7:4S/14

Q3A:4:262A/B4A:4f2394196@@>41A<34 G612Eg4I39194RA7:4=6R94

h*ij](̀�’Wi]�.AB6@@>E4:208
79B2:461946:R9742/4:/@H9464
:A=A@614G1/D@9=4J3919E4Nk4L4MlO4
0:AB549b062A/B:46B74IA234m0:24
G6G9146B74G9B

Q>GA<6@@>E4:2079B2:4@961BAB54
=0@2AG@A<62A/B4IA@@419G19:9B24 2394G1/D@9=46:46B49b062A/B4 0:AB5419G962974677A2A/BP4

Q394I/1R462423A:4:262A/B46@:/4
:397:4@A5324/B42394H61A92>4/S4
:216295A9:4:2079B2:40:942/4:/@H94
9b062A/B:P4

��
�
��
��

��

� ��!��!� “�#$��%�

��&��

“��’��&�(��)�*� “+��

+��”,��-.��/,�0��1�� $��

!� �’� ��2��’��

��”�� (��&��0��

�!�3��’�� 1��4��

�� $� ��)�5″�� 3�(3 “�3��

&$��$��6

785��-931��0��(� “��-.��/�

:��!�;��1��<%� �!�

�� &��)

78*� “+�� +��”�;=�

;��35�!�� “��”��

��%�$��>�� ?@@�

� $�� $�� $&��1 ��
��1 ��

ABBCDEFCGHIJKLMNC
OCGLKPCQORSTUQV
�WX��

YZ[[\]̂__̀\aâbcd_ce\
fghi\jd_cka

lmnnopq
rsstnq

uvwxntyqz{q
|}~�|~}�|}

��

� ¡¢ £ ¤¥¦¥§̈¤¦©ª« ¢̈¬ª£ £ª
¥̈ª¢¦¤® ª̄¢̈ª°̈¤°¢ ¥ ±£¥²© ª
¦¢¬ ¢£ª¥̈ª¤¦£° ¤¥ª̄̈ ¢³©¦£́
µ¶·̧¹º»¼¹½¾�¿À¼�ÁÂÃÃ¹¼�Â·�»À¼�Ä¼Å¹¹½�

ÄÂÆÀ�ÇÅÄ»¾�È»É·�ÊÀ¼Ä¼�ËÂÃ·�ÅÄ¼�Ã¼Ì¼¹̧ÇÂÍÎ�

·»ÄÅ»¼ÎÂ¼·�ÅÍÃ�ÁÅËÂÍÎ�»À¼�¹ÅÄÎ¼�Æ̧ÍÍ¼ÆÏ

»Â̧Í·¾�¿À¼�Á ·̧»�ÐÄ¼Ñº¼Í»�Ä¼·Ç̧Í·¼�Å»�

»À¼�Å¶·»ÄÅÆ»�·»Å»Â̧Í�Ð̧Ä�»ÀÂ·�·̧Ä»�̧Ð�ÇÄ̧¶Ï

¹¼Á�ÀÅ·�»̧�Ã̧�ÊÂ»À�Ä¼Ç¼Å»¼Ã�ÅÃÃÂ»Â̧Í¾�

¿À¼½É¹¹�ÊÄÂ»¼�ÒÓÔ�ÓÔ�ÓÔ�ÓÕ�»Ê¼¹Ì¼�»ÂÁ¼·�ÅÍÃ�

»À¼Í�ÅÃÃ�»À¼Á�Å¹¹�»̧Î¼»À¼Ä¾�

Ö̈ «ª×̈ª²̈³ª¦££ ££ª¥Ø ªÙ�Ú£Ûª
¿À¼�¶¼·»�ÇÄ̧Æ¼··�ÈÉÌ¼�·¼¼Í�Â·�Ð̧Ä�»Ê �̧

»¼ÅÆÀ¼Ä·�»̧�Ã̧�Â»�»̧Î¼»À¼Ä¾�¿À¼½�Î̧�

»ÀÄ̧ºÎÀ�»À¼�ÇÂ¹¼�ÅÍÃ�ÁÅË¼�Æ̧ÁÁ¼Í»·�

Í̧�ÊÀÅ»�»À¼½�·¼¼Ü�ÒÝÀÔ�Å�¹̧»�̧Ð�ËÂÃ·�ÅÄ¼�

Ã̧ÂÍÎ�Ä¼Ç¼Å»¼Ã�ÅÃÃÂ»Â̧Í�ÂÍ�»À¼�Å¶·»ÄÅÆ»�

ÇÂ¹¼ÔÕ�̧Ä�Òµ�¹̧»�̧Ð�ËÂÃ·�º·¼�ÆÂÄÆ¹¼·�ÅÍÃ�

·¹Å·À¼·�»̧�·À̧Ê�ÎÄ̧ºÇÂÍÎ·ÔÕ�̧Ä�ÒÞÂÃ·�Å»�

»À¼�Æ̧ÍÆÄ¼»¼�·»Å»Â̧Í�ÅÄ¼�ÁÅËÂÍÎ�¹̧ÍÎ�

·»ÅÆË·�̧Ð�ßÍÂÐÂà�Æº¶¼·á»À¼½ÉÄ¼�Í̧»�

ÎÄ̧ºÇÂÍÎ¾Õ�¿ÀÅ»�»ÅË¼·�Å¶̧º»�ÐÂÌ¼�ÁÂÍÏ

º»¼·¾�¿À¼Í�½̧º�Î̧�»ÀÄ̧ºÎÀ�»À¼�Ê Ä̧Ë�Å�

·¼Æ̧ÍÃ�»ÂÁ¼�ÅÍÃ�·̧Ä»�¶½�»À̧·¼�Ð¼Å»ºÄ¼·¾�

ß·ºÅ¹¹½Ô�½̧º�¼ÍÃ�ºÇ�ÊÂ»À�Ð̧ºÄ�̧Ä�ÐÂÌ¼�

ÇÂ¹¼·�ÅÍÃ�ÀÅÌ¼�Å�Î̧ Ã̧�·¼Í·¼�̧Ð�ÊÀÅ»É·�

Î̧ÂÍÎ�̧Í�ÊÂ»À�»À¼�Æ¹Å··�Å·�Å�ÊÀ̧¹¼¾�

¿À¼Ä¼�ÅÄ¼�Å¹ÊÅ½·�ÇÂ¼Æ¼·�̧Ð�Ê Ä̧Ë�

ÊÀ¼Ä¼�½̧º�·Å½Ô�ÒÈ�ÀÅÌ¼�Í̧�ÂÃ¼Å�ÊÀÅ»�»ÀÂ·�

ËÂÃ�ÊÅ·�»ÀÂÍËÂÍÎ¾Õ�È»É·�Å�ÃÄÅÊÂÍÎ�½̧º�

ÆÅÍÉ»�ÁÅË¼�·¼Í·¼�̧ÐÔ�̧Ä�»À¼Ä¼�ÅÄ¼�ÍºÁÏ

¶¼Ä·�̧Í�»À¼�ÇÅÎ¼�ÊÂ»À̧º»�ÅÍ½�Æ̧Í»¼à»¾�

¿À¼·¼�Î̧�ÂÍ»̧�»À¼�Ñº¼·»Â̧Í·�ÇÂ¹¼Ô�Å¹·̧�

ËÍ̧ÊÍ�Å·�»À¼�ÒâÀÅ»�»À¼�À¼ÆËãÕ�ÇÂ¹¼¾�

äØ¦¥ª×̈ª²̈³ª×̈ª«§¥Øª¥Ø̈ £ Û
¿ÀÅ»É·�ÊÀ¼Ä¼�»À¼�Ð¹¼àÂ¶¹¼�ÂÍ»¼ÄÌÂ¼Ê�

Æ̧Á¼·�ÂÍ¾�å̧º�·Â»�Ã̧ÊÍÔ�ÇÄ¼·¼Í»�»À¼�

·ÅÁ¼�ÇÄ̧¶¹¼ÁÔ�ÅÍÃ�Å·Ë�»À¼�·»ºÃ¼Í»�»̧�

»ÀÂÍË�Å¹̧ºÃ¾

Ö̈ «ª×̈ª²̈³ª³£ ª«Ø¦¥ª²̈³æç ª
© ¦¢¤ ×ª¥̈ª§¤̄ ¢̈ª§¤£¥¢³°¥§̈¤Ûª
è¼»É·�·ºÇÇ̧·¼�È�ÀÅÌ¼�ÐÂÌ¼�ËÂÃ·�ÊÀ̧�

ÁÅÃ¼�»À̧·¼�º¹»ÄÅ¹̧ÍÎ�·»ÅÆË·�̧Ð�ßÍÂÐÂà�

Æº¶¼·¾�µÍÃ�»À¼�̧»À¼Ä�ÇÅÄ»·�̧Ð�»À¼ÂÄ�

éêµ·�Å¹·̧�ÂÍÃÂÆÅ»¼�»À¼�Í̧»Â̧Í�̧Ð�

ÎÄ̧ºÇÂÍÎ�Â·�¹ÅÆËÂÍÎ¾�ëºÄÂÍÎ�»À¼�ÂÍÃÂÏ

ÌÂÃºÅ¹Âì¼ÃÔ�ÃÂÐÐ¼Ä¼Í»ÂÅ»¼Ã�ÇÅÄ»�̧Ð�»À¼�

¹¼··̧ÍáÒ»À¼�Á¼ÍºÔÕ�Å·�È�ÆÅ¹¹�Â»áÈÉ¹¹�

Çº¹¹�»ÀÅ»�ÎÄ̧ºÇ�»̧Î¼»À¼ÄÔ�ÅÍÃ�È�ÁÂÎÀ»�

Ç¹Å½�»À¼�ÎÅÁ¼�éÂÄÆ¹¼·�ÅÍÃ�í»ÅÄ·�ÊÂ»À�

»À¼Á¾�¿À¼½�Ä̧¹¹�»À¼�ÃÂÆ¼�ÅÍÃ�ÁÅË¼�Å�

Æ¼Ä»ÅÂÍ�ÍºÁ¶¼Ä�̧Ð�ÆÂÄÆ¹¼·¾�¿À¼½�Ä̧¹¹�»À¼�

ÃÂÆ¼�ÅÎÅÂÍ�ÅÍÃ�ÁÅË¼�»ÀÅ»�ÍºÁ¶¼Ä�̧Ð�

·»ÅÄ·�ÂÍ�¼ÅÆÀ�ÆÂÄÆ¹¼¾�µÍÃ�»À¼Í�»À¼½�ÐÂÍÃ�

»À¼�»̧»Å¹¾�µÐ»¼Ä�Ç¹Å½ÂÍÎ�»À¼�ÎÅÁ¼�Ð̧Ä�Å�

Æ̧ºÇ¹¼�̧Ð�ÃÅ½·Ô�Ê¼�¶¼ÎÂÍ�»̧�»Å¹Ë�Å¶̧º»Ô�

Òȩ́ º¹Ã�½̧º�Ã̧�»ÀÂ·�ÊÂ»À̧º»�ÅÆ»ºÅ¹¹½�

ÃÄÅÊÂÍÎ�»À¼�·»ÅÄ·ãÕ�ÅÍÃ�Òî̧ Ê�Ê º̧¹Ã�

½̧º�Î̧�Å¶̧º»�Ã̧ÂÍÎ�»ÀÅ»ãÕ�

Ú¤×ªÙ�Úª°Ø¦©© ¤® £ª¥ ¦°Ø ¢£ª
¥̈ª¥Ø§¤¬ª°̈¤° ¡¥³¦©©²ïª¥̈΅
È»É·�º·¼Ã�Ð̧Ä�ÇÄ̧Ð¼··Â̧ÍÅ¹�Ã¼Ì¼¹̧ÇÁ¼Í»�

Ð̧Ä�ðº·»�»ÀÅ»�Ä¼Å·̧Í¾�È�»¼¹¹�Á½�·»ºÃ¼Í»·Ô�

È�ËÍ̧Ê�Ì¼Ä½�Ð¼Ê�̧Ð�»À¼Á�ÅÄ¼�¶¼Æ̧ÁÂÍÎ�

¼¹¼Á¼Í»ÅÄ½�·ÆÀ̧ ¹̧�»¼ÅÆÀ¼Ä·�¶¼ÆÅº·¼�

»À¼½�ÊÅÍ»�»̧�»¼ÅÆÀ�ÁÅ»À¾�¿À¼½�ÊÅÍ»�»̧�

»¼ÅÆÀ�Ä¼ÅÃÂÍÎ¾�ñº»�Å¹¹�̧Ð�»À¼Á�ÅÄ¼�Î̧ÂÍÎ�

»̧�¶¼�ÁÅ»À�»¼ÅÆÀ¼Ä·Ô�·̧�»À¼½�Í¼¼Ã�»̧�

Ã¼Ì¼¹̧Ç�Å�Ã¼¼Ç�ËÍ̧Ê¹¼ÃÎ¼�̧Ð�»À¼�ÁÅ»À�

Â»·¼¹Ð¾�ÝÍ¼�Ä¼Å¹¹½�Æ̧ÁÁ Í̧�»ÀÂÍÎ�È�À¼ÅÄ�

Â·Ô�Ò¿À¼�·¼Æ̧ÍÃ�ÅÍÃ�»ÀÂÄÃ�»ÂÁ¼�È�º·¼Ã�

éêµÔ�È�ËÍ¼Ê�·̧�ÁºÆÀ�Á Ä̧¼¾Õ��

�òòó�ó��ô��õ�ö��õ
÷øøøø�ùúúûüùýùþ��ÿÿ�üû��ÿ�ø��ø��
�ø
�
��
�ø�
��ø� �� ø��
��
��ø� ��
��ø��ø�
ø��ø��ø� ��ø��
�
�ø��
�ø

��ø��
��
ø��ø �
��ø��ø ø�� ø
��
��ø�
�ø �
ø �ø
��ø�
�ø��
ø!”##$!”%”&'(()!

÷øøøø-��.ý�ÿ��/ùü��û�0�ø1
�2��ø��
ø��ø

��ø��
�
�ø�� ø��ø �ø
��ø
�ø��ø
�
��
�ø�
��ø �
ø
��ø
��ø��
ø��ø
��ø �ø� ��ø��
�ø��ø �ø��ø �ø ø

��ø
�ø��33ø�
��
��ø��ø��
��ø�
�ø
� ��ø��
�
�ø��
4��
ø ��ø
�
ø�
�� ø5+)5%+(+6″!’*$)7,!)&ø

÷ø��8�õ�ýù�û�0ÿ�ø9��ø��ø��ø��ø

�ø�� ø��
��
ø�
��ø
�
ø:;ø�
��ø<� ��2�ø=)%>$7?@A)#@BCD@
�
��ø��E�øF ��ø3
��
��ø� �ø� ��ø

��ø��
�ø�
ø��ø��ø��
�
� ø
&’*C()%”*$)7(,!)&

÷øøøø��8��þþ�0��ú��ýû�û�0�ø��ø
G��Fø �
ø��ø� ��ø� ��ø ��
ø

ø��
��ø
�ø�
��
ø��ø:;�ø
�
ø ��ø��
�� ø&’*C!!!,)#?

Hõ�þ.ý��ÿ
;
��

ø�I ��ø
�ø��ø �
ø

��
�ø��J�ø �øÿü8�ý�ÿ�ûü�ü�þ
Kû0ÿ�úùü��ú�

*+(,!)&

÷øøøø-��.ý�ÿ��/ùü��û�0�ø1
�2��ø��
ø��ø

��ø
�ø��33ø�
��
��ø��ø��
��ø�
�ø
� ��ø��
�
�ø��
4��
ø ��ø
�
ø�
�� ø5+)5%+(+6″!’*$)7,!)&ø

÷ø��8�õ�ýù�û�0ÿ�ø9��ø��ø��ø��ø

�ø�� ø��
��
ø�
��ø
�
ø:;ø�
��ø<� ��2�ø=)%>$7?@A)#@BCD@
�
��ø��E�øF ��ø3
��
��ø� �ø� ��ø

��ø��
�ø�
ø��ø��ø��
�
� ø
&’*C()%”*$)7(,!)&

÷øøøø��8��þþ�0��ú��ýû�û�0�ø��ø
G��Fø �
ø��ø� ��ø� ��ø ��
ø

ø��
��ø
�ø�
��
ø��ø:;�ø
�
ø ��ø��
�� ø&’*C!!!,)#?

Hõ�þ.ý��ÿ
;
��

ø�I ��ø
�ø��ø �
ø

��
�ø��J�ø �øÿü8�ý�ÿ�ûü�ü�þ
Kû0ÿ�úùü��ú�

1 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS

FALL 1999

THE TITLE of this article is also the title of a remark-
able new book written by Liping Ma.1 The basic

format of the book is simple. Each of the first four
chapters opens with a standard topic in elementary
school mathematics, presented as a part of a situation
that would arise naturally in a classroom. These scenar-
ios are followed by extensive discussion by teachers
regarding how they would handle each problem, and
this discussion is interspersed with commentary by
Liping Ma.

Here are the four scenarios:

Scenario 1: Subtraction with Regrouping

Let’s spend some time thinking about one particular
topic that you may work with when you teach: sub-
traction with regrouping. Look at these questions:

52 91
– 25 – 79

How would you approach these problems if you
were teaching second grade? What would you say
pupils would need to understand or be able to do be-
fore they could start learning subtraction with re-
grouping?

* * *
Scenario 2: Multidigit Multiplication

Some sixth-grade teachers noticed that several of
their students were making the same mistake in multi-
plying large numbers. In trying to calculate

123
3 645

the students seemed to be forgetting to “move the
numbers” (i.e., the partial products) over on each line.
They were doing this:

123
3 645

615
492

738

1845

instead of this:

123
3 645

615
492

738
79335

While these teachers agreed that this was a problem,
they did not agree on what to do about it. What would
you do if you were teaching sixth grade and you no-
ticed that several of your students were doing this?

* * *
Scenario 3: Division by Fractions
People seem to have different approaches to solving
problems involving division with fractions. How do
you solve a problem like this one?

1³⁄₄ ÷ ¹⁄₂

Imagine that you are teaching division with fractions.
To make this meaningful for kids, something that many
teachers try to do is relate mathematics to other
things. Sometimes they try to come up with real-world
situations or story problems to show the application of
some particular piece of content. What would you say
would be a good story or model for 1³⁄₄ ÷ ¹⁄₂?

* * *
Scenario 4: The Relationship Between
Perimeter and Area
Imagine that one of your students comes to class very
excited. She tells you that she has figured out a theory
that you never told the class. She explains that she has
discovered that as the perimeter of a closed figure in-
creases, the area also increases. She shows you this

KNOWING
AND TEACHING

ELEMENTARY
MATHEMATICS

BY RICHARD ASKEY

Richard Askey is John Bascom Professor of Mathe-
matics at the University of Wisconsin-Madison. In
addition to work on special functions, he has a long
term interest in the history of mathematics, and of
the life and work of the great Indian mathematician
Srinivasa Ramanujan.

Cindy
Typewritten Text

Cindy
Typewritten Text
Retrieved on August 19 from http://achievethecore.org/content/upload/
askey_knowing_and_teaching_elementary_mathematics_math

picture to prove what she is doing:

How would you respond to this student?

* * *
The 20- to 30-page discussions that follow each of

these four problems are the richest examples I have
encountered of teachers explaining what it means to
really know and be able to teach elementary school
mathematics. As the word “understanding” continues
to be bandied about loosely in the debates over math
education, this book provides a much-needed ground-
ing. It disabuses people of the notion that elementary
school mathematics is simple—or easy to teach. It cau-
tions us, as Ma says in her conclusion, that “the key to
reform…[is to] focus on substantive mathematics.” And
at the book’s heart is the idea that student understand-
ing is heavily dependent on teacher understanding. We
can all learn from this book.

The problem that best illustrates the insights in this
book is the one about the division of fractions. For
that reason and because of space limitations, I will
confine my comments in this article to that problem.

The teachers Ma interviewed composed numerous
story problems to illustrate fractional division. They
also explained the mathematical reasoning that under-
lies the calculation of division of fractions. And they
provided mathematical proofs for their calculation pro-
cedures.

Before giving examples of story problems composed
by the teachers Ma interviewed, it is worthwhile to
give a general picture of different types of division
problems, using whole numbers:

• 8 feet / 2 feet = 4 (measurement model)
• 8 feet / 2 = 4 feet (partitive model)
• 8 square feet / 2 feet = 4 feet (product and factors)

Now if we substitute fractions, using 1³⁄₄ in place of
8 and ¹⁄₂ in place of 2, these categories can be illus-
trated by the following examples:

• How many ¹⁄₂ foot lengths are there in something
that is 1 and ³⁄₄ feet long?

• If half a length is 1 and ³⁄₄ feet, how long is the
whole?

• If one side of a 1³⁄₄ square foot rectangle is ¹⁄₂ feet,
how long is the other side?

Many other examples are given in Ma’s book to rep-
resent this division problem. Here are two examples
that use the measurement model:

Given that a team of workers construct ¹⁄₂ km of road
each day, how many days will it take them to con-
struct a road 1³⁄₄ km long?

Given that ¹⁄₂ apple will be a serving, how many serv-
ings can we get from 1³⁄₄ apples? (p. 73)

Many of the teachers favored the partitive model of

division. Here are some of the story problems they
composed based on that model:

Yesterday I rode a bicycle from town A to town B. I
spent 1³⁄₄ hours for ¹⁄₂ of my journey; how much time
did I take for the whole journey?

A factory that produces machine tools now uses 1³⁄₄
tons of steel to make one machine tool, ¹⁄₂ of what
they used to use. How much steel did they used to
use for producing one machine tool?

We want to know how much vegetable oil there is in
a big bottle, but we only have a small scale. We draw
¹⁄₂ of the oil from the bottle, weigh it, and find that it
is 1³⁄₄ kg. Can you tell me how much all the oil in the
bottle originally weighed? (p. 79)

These are illuminating examples. They show the
teachers’ deep mathematical knowledge and their abil-
ity to represent mathematical problems to students.
The latter has been called “pedagogical content knowl-
edge.”

It is important for students to learn both how to
translate mathematical expressions into verbal prob-
lems and how to translate verbal problems into mathe-
matical expressions that can be worked with. It is also
important for students to understand how to do the
calculation of division of fractions, and why this calcu-
lation works. Just telling students to “invert and multi-
ply” is not enough. The following quotation from one
of the teachers Ma interviewed starts with a brief state-
ment about the relationship between division and mul-
tiplication. This statement provides a background for
the story problem that follows.

Division is the inverse of multiplication. Multiplying
by a fraction means that we know a number that rep-
resents a whole and want to find a number that rep-
resents a certain fraction of that. For example, given
that we want to know what number represents ¹⁄₂ of
1³⁄₄, we multiply 1³⁄₄ by ¹⁄₂ and get ⁷⁄₈. In other
words, the whole is 1³⁄₄ and ¹⁄₂ of it is ⁷⁄₈. In division
by a fraction, on the other hand, the number that
represents the whole becomes the unknown to be
found. We know a fractional part of it and want to
find the number that represents the whole. For ex-
ample, if ¹⁄₂ of a jump rope is 1³⁄₄ meters, what is the
length of the whole rope? We know that a part of the
rope is 1³⁄₄ meters, and we also know that this part is
¹⁄₂ of the rope. When we divide the number of the
part, 1³⁄₄ meters, by the corresponding fraction of
the whole, ¹⁄₂, we get the number representing the
whole, 3¹⁄₂ meters.… But I prefer not to use dividing
by ¹⁄₂ to illustrate the meaning of division by frac-
tions. Because one can easily see the answer without
really doing division by fractions. If we say ⁴⁄₅ of a
jump rope is 1³⁄₄ meters, how long is the whole
rope? The division operation will be more significant
because then you can’t see the answer immediately.
The best way to calculate it is to divide 1³⁄₄ by ⁴⁄₅ and
get 2 ³⁄₁₆ meters. (p. 74)

This is a rich passage. The teacher begins by remind-
ing her students that division is the inverse of multipli-
cation. She then reviews what it means to multiply
fractions, a topic that her students have already stud-
ied. Then building on their previous knowledge, the
teacher offers an example that moves her class
smoothly and logically to the division of fractions.

2 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS FALL 1999

Perimeter = 16 cm
Area = 16 square cm

Perimeter = 2

4 cm

Area = 32 square cm

4 cm

4 cm4 cm

8 cm

But this teacher is not content with the problem the
interviewer gave her, 1³⁄₄ ÷ ¹⁄₂. She fears it will allow
her students to “see the answer without really doing
division by fractions.” She substitutes a different prob-
lem—1³⁄₄ ÷ ⁴⁄₅— one that her students cannot easily vi-
sualize, thus forcing them deeper into the mathematics
of the division of fractions.

This teacher is telling us something important about
the level of knowledge needed if that knowledge is to
be stable rather than fragile. If all that is expected of
students is that they have a picture of how to deal
with simple fractions like ¹⁄₂ and 3¹⁄₂ , their knowledge
will not be deep enough to build on. Likewise, if their
knowledge is limited to the computational procedure
without any idea why the procedure works, this is also
not enough to build on. Students need both.

Like the teacher quoted above, many of the other
teachers Liping Ma interviewed used the explanation
that division is the inverse of multiplication. However,
Ma points out that the teachers who used this explana-
tion preferred the phrase “dividing by a number is
equivalent to multiplying by its reciprocal.” That is,

one can do division by multiplying by the reciprocal
(or inverse) of the number being divided by. This is
the mathematical reasoning that lies behind the “invert
and multiply” computation.

Some of the teachers interviewed offered a formal
mathematical proof to show why the algorithm for di-
vision of fractions works:

OK, fifth-grade students know the rule of “maintain-
ing the value of a quotient.” That is, when we multi-
ply both the dividend and the divisor with the same
number, the quotient will remain unchanged. For ex-
ample, dividing 10 by 2 the quotient is 5. Given that
we multiply both 10 and 2 by a number, let’s say 6,
we will get 60 divided by 12, and the quotient will
remain the same, 5. Now if both the dividend and
the divisor are multiplied by the reciprocal of the di-
visor, the divisor will become 1. Since dividing by 1
does not change a number, it can be omitted. So the
equation will become that of multiplying the divi-
dend by the reciprocal of the divisor. Let me show
you the procedure:

1³⁄₄ ÷ ¹⁄₂ = (1³⁄₄ 3 ²⁄₁) ÷ (¹⁄₂ 3 ²⁄₁)
= (1³⁄₄ 3 ²⁄₁) ÷1
= 1³⁄₄ 3 ²⁄₁
= 3¹⁄₂

With this procedure we can explain to students that
this seemingly arbitrary algorithm is reasonable. (p.
60)

This is what Ma said of the teachers who offered
proofs: “Their performance is mathematician-like in
the sense that to convince someone of a truth one
needs to prove it, not just assert it.”

Many of the teachers Ma interviewed emphasized
the necessity of thorough mastery of a topic before
moving on to the next. In this instance, solid com-
mand of the multiplication of fractions was considered
a “necessary basis” for approaching the division of
fractions.

The meaning of multiplication with fractions is par-
ticularly important because it is where the concepts
of division by fractions are derived…. Given that our
students understand very well that multiplying by a
fraction means finding a fractional part of a unit, they
will follow this logic to understand how the models
of its inverse operation work. On the other hand,
given that they do not have a clear idea of what mul-
tiplication with fractions means, concepts of division
by a fraction will be arbitrary for them and very diffi-
cult to understand. Therefore, in order to let our stu-
dents grasp the meaning of division by fractions, we
should first of all devote significant time and effort
when teaching multiplication with fractions to make
sure students understand thoroughly the meaning of
this operation…. Usually, my teaching of the meaning
of division of fractions starts with a review of the
meaning of multiplication with fractions. (p. 77)

This description shows an appreciation of how new
knowledge is built on old knowledge. The insistence
on mastery of a topic before moving on to the next
stands in sharp contrast to the curriculum organization
known as the “spiral curriculum.” In the “spiral” ap-
proach to learning, mastery is not expected the first

3 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS FALL 1999

As the word
‘understanding’
continues to be bandied
about loosely in the
debates over math
education, this book
provides a much-needed
grounding.

(Continued on page 6)

For too many people, mathematics stopped mak-
ing sense somewhere along the way. Either slowly or
dramatically, they gave up on the field as hopelessly
baffling and difficult, and they grew up to be adults
who—confident that others share their experience—
nonchalantly announce, “Math was just not for me”
or “I was never good at it.”

Usually the process is gradual, but for Ruth
McNeill, the turning point was clearly defined. In an
article in the Journal of Mathematical Behavior, she
described how it happened:1

What did me in was the idea that a negative number
times a negative number comes out to a positive
number. This seemed (and still seems) inherently
unlikely—counterintuitive, as mathematicians say.
I wrestled with the idea for what I imagine to be
several weeks, trying to get a sensible explanation
from my teacher, my classmates, my parents, any-
body. Whatever explanations they offered could not
overcome my strong sense that multiplying intensi-
fies something, and thus two negative numbers
multiplied together should properly produce a very
negative result. I have since been offered a moder-
ately convincing explanation that features a film of a
swimming pool being drained that gets run back-
wards through the projector. At the time, however,
nothing convinced me. The most commonsense of
all school subjects had abandoned common sense;
I was indignant and baffled.

Meanwhile, the curriculum kept rolling on, and I
could see that I couldn’t stay behind, stuck on nega-
tive times negative. I would have to pay attention to
the next topic, and the only practical course open to
me was to pretend to agree that negative times nega-
tive equals positive. The book and the teacher and
the general consensus of the algebra survivors of so-
ciety were clearly more powerful than I was. I capitu-
lated. I did the rest of algebra, and geometry, and
trigonometry; I did them in the advanced sections,
and I often had that nice sense of “aha!” when I
could suddenly see how a proof was going to come
out. Underneath, however, a kind of resentment and
betrayal lurked, and I was not surprised or dismayed
by any further foolishness my math teachers had up
their sleeves…. Intellectually, I was disengaged, and
when math was no longer required, I took German
instead.

Happily, Ruth McNeill’s story doesn’t end there.
Thanks to some friendships she formed in college,
her interest in math was rekindled. For most of our
students, there is no rekindling. This is a tragedy,
both for our students and for our country. Part of the
reason students give up on math can be attributed
to the poor quality of most of the math textbooks
used in the United States. Many texts are written
with the premise that if they end a problem with the
words, “Explain your answer,” they are engendering
“understanding.” However, because these texts do not
give students what they would need to enable them
to “explain,” the books only add to students’ mystifi-

cation and frustration.
Here is an example of how a widely acclaimed

contemporary math series handles the topic that baf-
fled Ruth McNeill: After a short set of problems deal-
ing with patterns in multiplication of integers from
5 to 0 times (–4), the student is asked to continue the
pattern to predict what (–1)(–4) is and then to give
the next four equations in this pattern. There are
then four problems, one of them being the product of
two negative numbers. In the follow-up problems
given next, there are four problems dealing with neg-
ative numbers, the last of which is the only one treat-
ing multiplication of negative numbers. This is how
it reads: “When you add two negative numbers, you
get a negative result. Is the same true when you mul-
tiply two negative numbers? Explain.”

The suggested answer to the “explain” part is: “The
product of two negative numbers is a positive.” This
is not an explanation, but a claim that the stated an-
swer is correct.

Simply asking students to explain something isn’t
sufficient. They need to be taught enough so that
they can explain. And they need to learn what an ex-
planation is and when a statement is not an expla-
nation.

The excerpt that follows is taken from a serious but
lively volume entitled Algebra by I.M.Gelfand and A.
Shen, which was originally written to be used in a cor-
respondence school that Gelfand had established. Con-
trast the inadequate treatment of the multiplication of
negative numbers described above to the way Gelfand
and Shen handle the topic.2 Although their presenta-
tion would need to be fleshed out more if it’s being pre-
sented to students for the first time, it provides us with
a much better model for what “explain” might entail,
offering as it does both an accessible explanation and
a formal proof.

—Richard Askey

The multiplication of negative numbers
To find how much three times five is, you add three

numbers equal to five:

5 + 5 + 5 = 15.

The same explanation may be used for the product
1 . 5 if we agree that a sum having only one term is
equal to this term. But it is evidently not applicable to
the product 0 . 5 or (–3) . 5: Can you imagine a sum
with a zero or with minus three terms?

However, we may exchange the factors:

5 . 0 = 0 + 0 + 0 + 0 + 0 = 0,

5 . (–3) = (–3) + (–3) + (–3) + (–3) + (–3) = –15.

So if we want the product to be independent of the
order of factors (as it was for positive numbers) we
must agree that

0 . 5 = 0, (–3) . 5 = –15.

4 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS FALL 1999

Why does a negative 3 a negative = a positive?
(including how to explain it to your younger brother or sister)

Now let us consider the product (–3) . (–5). Is
it equal to –15 or to +15? Both answers may
have advocates. From one point of view, even
one negative factor makes the product nega-
tive—so if both factors are negative the product
has a very strong reason to be negative. From
the other point of view, in the table

we already have two minuses and only one plus; so the
“equal opportunities” policy requires one more plus.
So what?

Of course, these “arguments” are not convincing to
you. School education says very definitely that minus
times minus is plus. But imagine that your small
brother or sister asks you, “Why?” (Is it a caprice of
the teacher, a law adopted by Congress, or a theorem
that can be proved?) You may try to answer this ques-
tion using the following example:

Another explanation. Let us write the numbers

1, 2, 3, 4, 5,…

and the same numbers multiplied by three:

3, 6, 9, 12, 15,…

Each number is bigger than the preceding one by
three. Let us write the same numbers in the reverse
order (starting, for example, with 5 and 15):

5, 4, 3, 2, 1
15, 12, 9, 6, 3

Now let us continue both sequences:

5, 4, 3, 2, 1, 0, –1, –2, –3, –4, –5,…
15, 12, 9, 6, 3, 0, –3, –6, –9, –12, –15,…

Here –15 is under –5, so 3 . (–5) = –15; plus times
minus is minus.

Now repeat the same procedure multiplying 1, 2, 3,
4, 5,… by –3 (we know already that plus times minus is
minus):

1, 2, 3, 4, 5
–3,–6, –9, –12, –15

Each number is three units less than the pre-
ceding one. Now write the same numbers in
the reverse order:

5, 4, 3, 2, 1
–15, –12, –9, –6, –3

and continue:

5, 4, 3, 2, 1, 0, –1, –2, –3, –4, –5,…
–15,–12, –9, –6, –3, 0, 3, 6, 9, 12, 15,…

Now 15 is under –5; therefore (–3) . (–5) = 15.

Probably this argument would be convincing for
your younger brother or sister. But you have the right
to ask: So what? Is it possible to prove that (–3) . (–5)
= 15?

Let us tell the whole truth now. Yes, it is possible to
prove that (–3) . (–5) must be 15 if we want the usual
properties of addition, subtraction, and multiplication
that are true for positive numbers to remain true for
any integers (including negative ones).

Here is the outline of this proof: Let us prove first
that 3 . (–5) = –15. What is –15? It is a number oppo-
site to 15, that is, a number that produces zero when
added to 15. So we must prove that

3 . (–5) + 15 = 0.

Indeed,

3 . (–5) + 15 = 3 . (–5) + 3 . 5 = 3 . (–5 + 5) = 3 . 0 = 0.

(When taking 3 out of the parentheses we use the law
ab + ac = a(b + c) for a = 3, b = –5, c = 5; we assume
that it is true for all numbers, including negative ones.)
So 3 . (–5) = –15. (The careful reader will ask why 3 . 0
= 0. To tell you the truth, this step of the proof is omit-
ted—as well as the whole discussion of what zero is.)

Now we are ready to prove that (–3) . (–5) = 15. Let
us start with

(–3) + 3 = 0

and multiply both sides of this equality by –5:

((–3) + 3) . (–5) = 0 . (–5) = 0.

Now removing the parentheses in the left-hand side
we get

(–3) . (–5) + 3 . (–5) = 0,

that is, (–3) . (–5) + (–15) = 0. Therefore, the number
(–3) . (–5) is opposite to –15, that is, is equal to 15.
(This argument also has gaps. We should prove first
that 0 . (–5) = 0 and that there is only one number op-
posite to –15.) l
______
1 Ruth McNeill, “A Reflection on When I Loved Math and How I

Stopped.” Journal of Mathematical Behavior, vol. 7 (1988)
pp. 45-50.

2 Algebra by I. M. Gelfand and A. Shen. Birkhauser Boston
(1995, Second Printing): Cambridge, Mass. © 1993 by I. M.
Gelfand. Reprinted with permission.

FALL 1999

3 . 5 = 15 Getting five dollars three times is
getting fifteen dollars.

3 . (–5) = –15 Paying a five-dollar penalty three
times is a fifteen-dollar penalty.

(–3) . 5 = –15 Not getting five dollars three times
is not getting fifteen dollars.

(–3) . (–5) = 15 Not paying a five-dollar penalty
three times is getting fifteen dollars.

5 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS

3 . 5 = +15 3 . (–5) = –15

(–3) . 5 = –15 (–3) . (–5) = ?

time, and the same topics are revisited in two, three,
and even four successive years.

In her discussion of the division of fractions, Ma
mentions other methods of doing the calculation, in-
cluding changing the problem to decimals, and dealing
with numerators and denominators separately. The
teachers who suggested these methods also noted that
they were not always easier than the standard text-
book method of multiplying by the reciprocal. The
level of knowledge expected is illustrated by the fol-
lowing quotation:

The teachers argued that not only should students
know various ways of calculating a problem but they
should also be able to evaluate these ways and to de-
termine which would be the most reasonable to use.
(p.64)

Throughout her book, Ma provides illustrations that
help show how the topic under consideration fits into
the larger picture of elementary mathematics. For ex-
ample:

The learning of mathematical concepts is not a unidi-
rectional journey. Even though the concept of divi-
sion by fractions is logically built on the previous
learning of various concepts, it, in turn, plays a role
in reinforcing and deepening that previous learning.
For example, work on the meaning of division by
fractions will intensify previous concepts of rational
number multiplication. Similarly, by developing ratio-
nal number versions of the two division models,
one’s original understanding of the two whole num-
ber models will become more comprehensive. (p.
76)

AS THE reader might have suspected from the mea-
surement units used in some of the story prob-

lems, the teachers who were quoted are from a coun-
try that uses the metric system. The country is China,
and these teachers live in Shanghai and neighboring
areas of China. Liping Ma grew up in Shanghai until
she was in the eighth grade, when China’s “Cultural
Revolution” sent her to the countryside for “re-educa-
tion” by the peasants. In the poor rural village in South
China where she was sent, the mostly illiterate vil-
lagers wanted their children to get an education. Ma
was asked to teach, which she did for seven years, and
later became elementary school superintendent for the
county. Later she returned to Shanghai and started to
read the classical works in the field of education. This
eventually led her to Michigan State University (MSU)
where she began working on a doctoral degree.

While at MSU, Liping Ma worked on a project run
by Deborah Ball, which was a study to find out more
about the mathematical knowledge of elementary
school teachers in the United States. The four ques-
tions Ma used in her interviews with Chinese teachers
were originally developed by Ball as part of the MSU
study, and first used to interview U.S. teachers.2 In her
book, Ma draws on this database of U.S. teacher inter-
views as a point of comparison to the Chinese teach-
ers.

The U.S. teachers fared poorly when compared to
their Chinese counterparts. For the division of frac-

tions problem discussed in this article, some of the
U.S. teachers had difficulties with the calculations.
None of them could adequately explain the mathemati-
cal reasoning embedded in the algorithm, provide ap-
propriate real-world applications, or offer proofs.

It was not surprising to find that our elementary
teachers’ mathematical knowledge is not nearly as ro-
bust as that of the Chinese teachers. How could it be
otherwise? Where could our teachers possibly have ac-
quired the knowledge base that the Shanghai teachers
demonstrated? Not from their own K-12 schooling,
which focused mainly on developing a little skill on
routine problems. Not from the math methods courses
U.S. colleges offer, since these are light on math con-
tent. And—what may be surprising to many people—
not even from the math courses they might have taken
from a university mathematics department. At most
colleges and universitites, there is a major disconnect
between what is taught in these courses and the kind
of math elementary school teachers need. As H. Wu
has written: “There is an alarming irrelevance in the
present preservice professional development in mathe-
matics.”3

A high school teacher who took a course from the
well-known mathematician George Polya put it an-
other way:

The prospective teacher is badly treated both by the
mathematics department and by the school of educa-
tion. The mathematics department offers us tough
steak which we cannot chew and the school of edu-
cation vapid soup with no meat in it.4

It is not just the courses for high school math teach-
ers that are problematic. Courses for prospective ele-
mentary school teachers, for example, frequently
slight material dealing with fractions since whole num-
ber arithmetic is the main focus in our elementary
schools. Middle school teachers frequently fall be-
tween the cracks. The material they will be teaching is
not taught in detail to either prospective elementary
school teachers or to prospective high school teach-
ers; there are no courses specifically for middle school
teachers.

If not from their pre-college education and not from
their college education, where else might a U.S.
teacher have acquired a deep understanding of mathe-
matics? Perhaps from the textbooks and teachers’
guides they use in their teaching. Liping Ma reports
that Chinese teachers spend considerable time study-
ing the textbooks:

6 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS FALL 1999

Ordering Information
Knowing and Teaching Elementary
Mathematics by Liping Ma is available
in paperback from Lawrence Erlbaum
Associates for $19.95 plus $2 han-
dling charges by sending a check to
Lawrence Erlbaum Associates Inc.,
10 Industrial Avenue, Mahwah, NJ
07430-2262. For more information, call
800/926-6579 or e-mail: orders@erlbaum.com.

(Continued from page 3)

Teachers study textbooks very carefully; they investi-
gate them individually and in groups, they talk about
what textbooks mean, they do the problems to-
gether, and they have conversations about them.
Teachers’ manuals provide information about con-
tent and pedagogy, student thinking, and longitudinal
coherence. (p. 149)

Unfortunately, there are very few of our textbooks that
a teacher would profit much from studying.

The U.S. Department of Education has just an-
nounced the results of an exercise to identify “exem-
plary” and “promising” texts. Connected Mathematics,
a series for grades 6-8, is one the department has
deemed exemplary. I do not understand why it de-
serves that rating. I am quite familiar with this series,
as I reviewed it as part of a textbook adoption process.
Regarding fractions, for example, Connected Math has
some material on the addition and subtraction of frac-
tions, but nothing as systematic as described by the
Chinese teachers interviewed by Ma. There is less on
multiplication of fractions, and nothing on the division
of fractions. If our students go through grade 8 with-
out having studied the division of fractions, where are
our future primary teachers going to learn this? The
criteria used by the Department of Education review

should be rewritten now that Liping Ma’s book has
provided us with a model of what school mathematics
should look like.

Another recent development that leaves me less
than encouraged is the way fractions are addressed in
the draft of the revised K-12 mathematics standards re-
leased last year by the National Council of Teachers of
Mathematics (Principles and Standards for School
Mathematics: Discussion Draft 5). Most of the work on
fractions has been put in the grades 6 to 8 band. Stu-
dents are to “develop a deep understanding of rational
number concepts and reasonable proficiency in ratio-
nal-number computation.” It is the adjective “reason-
able” that bothers me. Proficiency should be the
goal. It is hard to imagine the Chinese teachers that
Ma interviewed settling for “reasonable” proficiency
with fractions for their students. These lower expec-
tations show in every international comparison.

Furthermore, the only problem used to illustrate di-
vision of fractions in NCTM’s draft revision is how
many pieces of ribbon ³⁄₄ yards long can be cut from
4¹⁄₂ yards of ribbon. The text continues with: “The
image is of repeatedly cutting off ³⁄₄ of a yard of rib-
bon. Having students work with concrete objects or
drawings is helpful as students develop and deepen
their understanding of operations.” It seems that we
are back again to simple fractions and concrete ob-
jects that students can visualize. Contrast this with
what Liping Ma observed:

The concept of fractions as well as the operations
with fractions taught in China and the U.S. seem dif-
ferent. U.S. teachers tend to deal with “real” and
“concrete” wholes (usually circular or rectangular
shapes) and their fractions. Although Chinese teach-
ers also use these shapes when they introduce the
concept of a fraction, when they teach operations
with fractions they tend to use “abstract” and “invisi-
ble” wholes (e.g., the length of a particular stretch of
road, the length of time it takes to complete a
task…). (p. 76)

THE LAST three chapters in Liping Ma’s book deal
with when the Chinese teachers acquired the

knowledge they showed, and a description of what Ma
calls “Profound Understanding of Fundamental Mathe-
matics,” or PUFM. Here is part of her description:

A teacher with PUFM is aware of the “simple but
powerful” basic ideas of mathematics and tends to re-
visit and reinforce them. He or she has a fundamental
understanding of the whole elementary mathematics
curriculum, thus is ready to exploit an opportunity
to review concepts that students have previously
studied or to lay the groundwork for a concept to be
studied later. (p. 124)

From their pre-collegiate studies, the Chinese teach-
ers Ma interviewed had a firm base of knowledge on
which to build. However, PUFM did not come directly
from their studies in school, but from the work they
did as teachers. These teachers did not specialize in
mathematics in “normal” school, which is what their
teacher preparation schools are called. But after they
started teaching, most of them taught only mathemat-
ics or mathematics and one other subject. This al-
lowed them to specialize in ways that few of our ele-

7 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS FALL 1999

Unfortunately, there are
very few of our textbooks
that a teacher would profit
much from studying.

8 AMERICAN EDUCATOR/AMERICAN FEDERATION OF TEACHERS FALL 1999

mentary school teachers can. Quite a few regularly
changed the level at which they taught. They might go
through a cycle of three grades, then repeat the same
cycle, or change and teach a different age group. This
allows them to see the development of mathematics
from the perspective of a teacher, something too few
of our elementary school teachers are able to do.

Recently, the Learning First Alliance—an organiza-
tion composed of many of the major national educa-
tion organizations—recommended that beginning in
the fifth grade, every student should be taught by a
mathematics specialist.6 This is a hopeful develop-
ment, and for many teachers it would mean unburden-
ing themselves from something they now find difficult
and unpleasant.

There is more we can do. Our teachers need good
textbooks. They need much better teachers’ manuals.
As noted before, our college math courses for future
teachers at all levels need to be improved. And just ask
any teacher who has sat through mindless “work-
shops” whether our in-service “professional develop-
ment” isn’t long overdue for major overhaul.

Teachers also need time to prepare their lessons and
further their study of mathematics. Recall Ma’s com-
ments that it is during their teaching careers that Chi-
nese teachers perfect their knowledge of mathematics.
Listen to this Shanghai teacher describe his class
preparation:

I always spend more time on preparing a class than
on teaching, sometimes three, even four, times the
latter. I spend the time in studying the teaching mate-
rials: What is it that I am going to teach in this les-
son? How should I introduce the topic? What con-
cepts or skills have the students learned that I should
draw on? Is it a key piece on which other pieces of
knowledge will build, or is it built on other knowl-
edge? If it is a key piece of knowledge, how can I
teach it so students can grasp it solidly enough to
support their later learning? If it is not a key piece,
what is the concept or the procedure it is built on?
How am I going to pull out that knowledge and make
sure my students are aware of it and the relation be-
tween the old knowledge and the new topic? What
kind of review will my students need? How should I
present the topic step-by-step? How will students re-
spond after I raise a certain question? Where should I
explain it at length, and where should I leave it to
students to learn it by themselves? What are the top-
ics that the students will learn which are built di-
rectly or indirectly on this topic? How can my lesson
set a basis for their learning of the next topic, and for
related topics that they will learn in their future?
What do I expect the advanced students to learn
from the lesson? What do I expect the slow students
to learn? How can I reach these goals? etc. In a word,
one thing is to study whom you are teaching, the
other thing is to study the knowledge you are teach-
ing. If you can interweave the two things together
nicely, you will succeed. We think about these two
things over and over in studying teaching materials.
Believe me, it seems to be simple when I talk about
it, but when you really do it, it is very complicated,
subtle, and takes a lot of time. It is easy to be an ele-
mentary school teacher, but it is difficult to be a
good elementary school teacher. (p. 135)

LIPING MA’S book provides a start to what I hope
will be a continuing study of fundamental mathe-

matics and the connections between different parts of
it. We need many more commentaries on the teaching
of mathematics like those contained in Ma’s book. We
also need more detailed lesson plans, as are frequently
provided in Japan.7 There are a few places where one
can read comments by U.S. teachers or by mathemat-
ics education researchers. However, these comments
almost all deal with the initial steps of an idea, which
typically means using pictures or manipulatives to try
to get across the basic concept. Almost never is there
elaboration of what should be done next, to help de-
velop a deeper view of the subject, which will be nec-
essary for later work.

And elementary school mathematics is much
deeper, more profound, than almost everyone has
thought it to be. As Ma comments, toward the end of
her book:

In the United States, it is widely accepted that ele-
mentary mathematics is “basic,” superficial, and com-
monly understood. The data in this book explode
this myth. Elementary mathematics is not superficial
at all, and any one who teaches it has to study it hard
in order to understand it in a comprehensive way. (p.
146)

But, she concludes:
The factors that support Chinese teachers’ develop-
ment of their mathematical knowledge are not pre-
sent in the United States. Even worse, conditions in
the United States militate against the development of
elementary teachers’ mathematical knowledge….
(p.xxv)

This must change. We cannot continue to abandon
teachers at every critical stage of their development
and then send them into the classroom with a mandate
to “teach for understanding.” This is dishonest and irre-
sponsible. As things stand now, we are asking teachers
to do the impossible. They and the students they teach
deserve better. l
REFERENCES
1Liping Ma, Knowing and Teaching Elementary Mathematics,

Lawrence Erlbaum Associates, Mahwah, NJ, 1999.
2Deborah Ball, “Knowledge and reasoning in mathematical

pedagogy: Examining what prospective teachers bring to
teacher education.” Unpublished Ph.D. thesis, Michigan State
University, 1988 and published papers on this topic.

3H.Wu, “Preservice Professional Development of Mathematics
Teachers.” Unpublished manuscript (1999). Available at
www.math.berkeley.edu/~wu

4George Polya, “Ten Commandments for Teachers,” Journal of
Education of the Faculty and College of Education of the
University of British Columbia, (3) 1959, pp. 61-69.

5National Council of Teachers of Mathematics, Principles and
Standards for School Mathematics: Discussion Draft, NCTM,
Reston, Va., 1998.

6Learning First Alliance, An Action Plan for Changing School
Mathematics, Washington, D.C., 1998.

7Alice Gill and Liz McPike, “What We Can Learn from Japanese
Teachers’ Manuals,” American Educator, vol. 19, no. 1,
Spring 1995, pp. 14-24.

Writing Assignment

Writing Assignment (50 points) Following the description in the reading assignment “Common Core: Solve Math Problems,” select
one
end-of-lesson assessment problem that is rich and
representative of the bulk of the concepts in the
standard 5.MD.5 . Develop three different stations in three different ways—concretely, representationally, and abstractly. Include a description of what will be happening in each station.

Standard 5.MD.5

Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems

Create a math assessment problem that you would use to measure mastery of a lesson on this standard (all of your math work will be based on this standard). *This problem will most likely be the final station in this assignment.

Design three stations/tasks that the students will complete to gain understanding and mastery of the concept of measuring volume:

·
·

· Concrete (16 points)–what will you have students do here? Describe the materials, what the students’ tasks will be, etc.

· Representational (17 points)–what will the students do at this station? Describe the materials, what the students’ task/s will be, etc.

· Abstract (17 points)–what problem will be solved here? Describe the materials, problem, etc.

Unit 1: Writing Assignments

Lah’Qiana Fain

Belhaven University

EDU622: Teaching Reading and Math Skills

Dr. Garmon

January 14, 2021

This study source was downloaded by 100000806408568 from CourseHero.com on 01-14-2023 10:29:15 GMT -06:00

https://www.coursehero.com/file/82472919/edu622-unit1docx/

Unit 1: Writing Assignments

Implementing the CRA Instructional Approach into mathematic interventions is

beneficial in improving the performance of students. Using the 3 part instructional strategy

promote student learning and retentions. The CRA instructional sequence is consist of three

stages: concrete, representation and abstract. In using CRA sequence of instruction it proves a

graduated, scaffold foundation. Example of the mathematical concept (CRA method) is

illustrated below using standard 3.NBT.2, Fluently add and subtract (including subtracting

across zeros) within 1000 using strategies and algorithms based on place value, properties of

operations, and/or the relationship between addition and subtraction. Include problems with

whole dollar amounts (Mississippi College and Career Readiness Standards for Mathematics

Scaffolding Document, 2021).

Question

A farmer was planting vegetables in a garden. He planted 623 corn seeds, 519 turnip seeds and

81 potato seeds. How many seeds did he plant total?

Station 1

At this station we will focus on the concrete phase of the CRA instructional approach .

Students will have access to hands on manipulative. In using the manipulative students will be

able to move objects to represent their thinking. I reference to the question student should be able

to demonstrate their thinking with base ten blocks and hundred tiles to represent the addition

expression. Example: to show 623 students will use 6 hundreds block, 2 tens blocks and 3 ones

blocks.

This study source was downloaded by 100000806408568 from CourseHero.com on 01-14-2023 10:29:15 GMT -06:00

https://www.coursehero.com/file/82472919/edu622-unit1docx/

Station 2

At this station students will focus on the representation phase of the CRA instructional

approach. While at this station students will be able draw their representation. I will also

introduce the hundred grid and have students color their representation from station one on the

grid. Students should be able to make connection to the concrete phase at this station.

Station 3

At this station students will focus on the abstract phase of the CRA instructional approach.

Students will use the prior stations to close the gap by writing out equations to demonstrate the

question. I will show students to cue students the place value of each number. For example I will

show students 6 hundred tiles and cue them to make the connection between the name of the tile

and how many tiles we actually have , leading the students to write six with two zeros ( 600). At

this station I will continue to practice with several examples until the child connects the concrete

blocks to place value and addition.

This study source was downloaded by 100000806408568 from CourseHero.com on 01-14-2023 10:29:15 GMT -06:00

https://www.coursehero.com/file/82472919/edu622-unit1docx/

References

Common Core: Solve Math Problems | Scholastic.

(2021). Retrieved 15 January 2021, from

https://www.scholastic.com/teachers/articles/teaching-content/common-core-solve-math-

problems/

Mississippi College and Career Readiness Standards for Mathematics Scaffolding Document

(2021). Retrieved 15 January 2021, from

https://www.mdek12.org/sites/default/files/Offices/Secondary

%20Ed/ELA/ccr/Math/03.Grade-3-Math-Scaffolding-Doc

This study source was downloaded by 100000806408568 from CourseHero.com on 01-14-2023 10:29:15 GMT -06:00

https://www.coursehero.com/file/82472919/edu622-unit1docx/
Powered by TCPDF (www.tcpdf.org)

https://www.coursehero.com/file/82472919/edu622-unit1docx/

http://www.tcpdf.org

Turn in your highest-quality paper
Get a qualified writer to help you with

“ Assignment 1 TJ ”

Get high-quality paper

NEW! AI matching with writer

Order an Essay Now & Get These Features For Free:

Turnitin Report

Formatting

Title Page

Citation

Outline

Place an Order