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�%”,,!�/�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�(+) ‘�

�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

/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�� �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�

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.

1

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/

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

2

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/

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

3

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/

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

4

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

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