> HJG} ,bjbj55 ;<__#JFF8$_)+(SSS...(((((((k+
.(0..00(SS(<\\\0SS(\0(\\rl'Tl(S0gҧ'()H_)'.v|.l(.l(D.0\|...((j..._)0000..........F O: Focus on the Curriculum Focal Points: Part 2
Implementation of the NCTM Curriculum Focal Points: Concept vs. Content
Jane F. Schielack, Texas A&M University
and Cathy Seeley, Charles A. Dana Center, The University of Texas at Austin
On September 12, 2006, the National Council of Teachers of Mathematics released a new lens through which to view the content presented in Principles and Standards for School Mathematics (NCTM, 2000). In Principles and Standards, the Curriculum Principle calls for a mathematics curriculum to be coherent, focused on important mathematics, and well articulated across the grades. This means that an effective curriculum gives teachers guidance regarding important ideas or major themes, which receive special attention at different points in time. It also gives guidance about the depth of study warranted at particular times and when closure is expected for particular skills or concepts (NCTM, 2000, p. 16). With Curriculum Focal Points for Prekindergarten Through Grade 8A Quest for Coherence (NCTM, 2006), NCTM has taken a major step toward making this principle real for defining the mathematics that every student should know and be able to do. Where the Principles and Standards presented content across multiple-grade bands, this new publication, which can be purchased or downloaded free from HYPERLINK "http://www.nctm.org/" www.nctm.org, applies the Curriculum Principle to create a specific example of a focused and coherent curriculum, prekindergarten through Grade 8.
Curriculum Focal Points can be viewed from two perspectives: the concept of a focal point and the actual content delineated within the focal points at particular grade levels. From the concept perspective, NCTMs focal points present a way to think about how curriculum is organized and presented, regardless of the grade-level placement of topics. From the content perspective, NCTMs focal points identify particular content for particular grade levels that link together into a cohesive whole across grade levels, but that may or may not match directly to a states or districts existing mathematics curriculum. Each of these perspectives can be useful in the development and refinement of curriculum and instruction.
The Concept of Curriculum Focal Points
Many state, provincial and school district curriculum documents have evolved into lists of specific skills that are designed to be used more as test specifications for accountability purposes than for instructional guidance. A mathematics curriculum organized around focal pointsthat is, a focused mathematics curriculumhighlights the most important mathematical ideas for each grade and presents these key ideas as interconnected packages of related concepts, skills, and procedures. A mathematical topic such as area measure may begin in the background of a particular grade as a context for exploration, then emerge into the foreground at a later grade level for intense instruction, and then recede into the background again. Using the focal points model, such a topic might appear in the Connections section in one grade, then as a Focal Point in the next grade. Since the Curriculum Focal Points target the most important or prominent topics for each grade, this same important topic might not be evident in the focal points description at the next grade level because it has become a context for problem solving.
NCTMs Curriculum Focal Points publication represents the next generation of curriculum documents that present mathematical content in a way that will help teachers better manage their instructional time to support the development of deep understanding of key ideas that grow and connect over time. Although it is tempting to begin a review of an existing mathematics curriculum by matching up pieces of content to their locations in NCTMs Curriculum Focal Points, the concept of a focused curriculum is more about highlighting a continuum of key ideas across grades and identifying particular grade levels where the emphasis of instruction will be placed for each of these key ideas. The advantage of such an organizational structure is that teachers, materials developers, test developers, administrators and the public can get a clearer and more accurate sense of what will be happening in Ms. Robinsons seventh-grade math classroom this year than they might be able to when presented with a long, unprioritized list of topics or textbook chapters. Knowing where the emphasis will be placed and what the most important outcomes should be can help all of these individuals support students in learning the critical foundation of mathematics knowledge and skills for the year.
To promote the use of NCTMs Curriculum Focal Points to support the implementation of the concept of a focused curriculum, consider the following questions:
What key ideas can be seen in our existing mathematics curriculum? Are there any key ideas in NCTMs set of focal points that dont appear somewhere in our curriculum, or vice versa? If so, how do we address that?
Does our sequence of key ideas make sense mathematically? Does it build developmentally from grade to grade without unnecessary repetition? If not, what needs to be changed to make the key ideas flow from grade to grade?
Can we determine from our curriculum where to place the emphasis at each grade? Can we tell how a topic (like fraction operations) should be treated differently at Grades 5, 6, 7 and 8 to help the student learn for the long term without doing the same thing each year?
At what point do we expect the most focused instruction on a particular key idea to occur? Is there an appropriate amount of time for instruction in relation to the whole set of key ideas at that time? If not, how can the key ideas be shifted in a coordinated way to provide for that time?
Can we follow the development of important concepts as they move through the grades from background to foreground to background? Does that movement make sense?
The Content of NCTMs Curriculum Focal Points
As mentioned in the previous section, the actual focal point statements for each grade level in the NCTM publication can be used to help guide thinking about ways to refine existing mathematics curricula. Beyond simply looking at which grade level specific topics might appear in the focal points document, the following specific content components of NCTMs Curriculum Focal Points suggest guidelines for designing implementation of a focused curriculum:
Many of NCTMs focal points in Grades 5-8 are labeled with combinations of content strands, e.g. Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes.
The paragraphs explaining the focal points highlight the depth of the ideas and give a sense of the range of knowledge, skills, contexts and problems that might accompany the development of the focal point.
The topics listed in the sections labeled Connections to the Focal Points provide meaningful contexts for the focal points, identify connections between strands and across grade levels, and round out a well-balanced curriculum. They help support the importance of building the relationships in mathematics while distinguishing between which topics are in the foreground and which are in the background at each grade level.
The introductory statement at each grade level regarding mathematical processes emphasizes the central role that these processes must play in a coherent curriculum and remind us that instructional design in mathematic must support the development of key mathematical ideas in ways that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations.
Implementation of a Focused Mathematics Curriculum
By considering both the organizational concept and the mathematical content of NCTMs Curriculum Focal Points, mathematics teachers and instructional leaders can design effective instruction based on focused content supported by appropriate connections, contexts, and tools and addressed through the development and application of mathematical processes. In the next three articles in this series, we will look at specific content threads from the perspective of a focused curriculum in Grades 5-8. We hope that sharing some specific examples of possible implementations of NCTMs focal points from both the concept and content perspectives will lead you to ideas for moving toward a more focused curriculum in your own setting. We challenge you to discuss with your colleagues these two aspects of focusing your mathematics curriculum. Consider using this series of articles as a tool for productive professional discussions or for reflection about your curriculum and how you teach it.
References
National Council of Teachers of Mathematics. (2000). Principles and Standards of School Mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (2006) Curriculum Focal Points for Prekindergarten through Grade 8: A Quest for Coherence. Reston, VA: NCTM.
MTMS Focus on CFPs: Part 2
PAGE 2
PAGE 1
-Dx{
E
F
h
i
u
v
*[fn-C'0ce *+01h*qhhhqV3h6hh6hkWh6
h0JjhUh/hhhh6h h6h h5hW
h5?./xy
dhd`gdIdEƀ;=fgdgddgd$a$gd!"X#%&&&*L
&FdEƀd=fgd
&Fdgddgdgd
&Fdgd
abfg./Ax#y########$$$$<$%%&&&&)****+`+++++++++++++++++++++++
h
0Jjh
0JUh?dh
h6?djh6?dU h/6hh6hqV3hhpEchh"%hX7h h6h*qh*qh*qh=***.+++++++++++++++++,,,,,$a$gdd^`gd
$da$gddgd++,,,,,,,,
,,,,,,,hh6?d
h
0Jh?dh
jh
0JUhQz*0JmHnHu,,,,$a$gd9
0&P1h:p
/ =!"#$%^2 0@P`p2( 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p8XV~_HmH nH sH tH @`@NormalCJ_HaJmH sH tH DA`DDefault Paragraph FontRiRTable Normal4
l4a(k (No List4U@4kW Hyperlink >*phDDBalloon TextCJOJQJaJ>'>Comment ReferenceCJ4"4Comment Text:j!":Comment Subject4@B4
Header
!4 @R4
Footer
!.)@a.
Page NumberPK![Content_Types].xmlj0Eжr(Iw},-j4 wP-t#bΙ{UTU^hd}㨫)*1P' ^W0)T9<l#$yi};~@(Hu*Dנz/0ǰ$X3aZ,D0j~3߶b~i>3\`?/[G\!-Rk.sԻ..a濭?PK!֧6_rels/.relsj0}Q%v/C/}(h"O
= C?hv=Ʌ%[xp{۵_Pѣ<1H0ORBdJE4b$q_6LR7`0̞O,En7Lib/SeеPK!kytheme/theme/themeManager.xmlM
@}w7c(EbˮCAǠҟ7՛K
Y,
e.|,H,lxɴIsQ}#Ր ֵ+!,^$j=GW)E+&
8PK!Ptheme/theme/theme1.xmlYOo6w toc'vuر-MniP@I}úama[إ4:lЯGRX^6؊>$!)O^rC$y@/yH*)UDb`}"qۋJחX^)I`nEp)liV[]1M<OP6r=zgbIguSebORD۫qu gZo~ٺlAplxpT0+[}`jzAV2Fi@qv֬5\|ʜ̭NleXdsjcs7f
W+Ն7`gȘJj|h(KD-
dXiJ؇(x$(:;˹!I_TS1?E??ZBΪmU/?~xY'y5g&/ɋ>GMGeD3Vq%'#q$8K)fw9:ĵ
x}rxwr:\TZaG*y8IjbRc|XŻǿI
u3KGnD1NIBs
RuK>V.EL+M2#'fi~Vvl{u8zH
*:(W☕
~JTe\O*tHGHY}KNP*ݾ˦TѼ9/#A7qZ$*c?qUnwN%Oi4=3ڗP
1Pm\\9Mؓ2aD];Yt\[x]}Wr|]g-
eW
)6-rCSj
id DЇAΜIqbJ#x꺃6k#ASh&ʌt(Q%p%m&]caSl=X\P1Mh9MVdDAaVB[݈fJíP|8քAV^f
Hn-"d>znǊ ة>b&2vKyϼD:,AGm\nziÙ.uχYC6OMf3or$5NHT[XF64T,ќM0E)`#5XY`פ;%1U٥m;R>QDDcpU'&LE/pm%]8firS4d7y\`JnίIR3U~7+#mqBiDi*L69mY&iHE=(K&N!V.KeLDĕ{D vEꦚdeNƟe(MN9ߜR6&3(a/DUz<{ˊYȳV)9Z[4^n5!J?Q3eBoCMm<.vpIYfZY_p[=al-Y}Nc͙ŋ4vfavl'SA8|*u{-ߟ0%M07%<ҍPK!
ѐ'theme/theme/_rels/themeManager.xml.relsM
0wooӺ&݈Э5
6?$Q
,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-![Content_Types].xmlPK-!֧6+_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!Ptheme/theme/theme1.xmlPK-!
ѐ' theme/theme/_rels/themeManager.xml.relsPK]
$< +-:<IL+,*,,Ehu$X.57=DFL!!8@0(
B
S ?P>dgP>!P>THP>\P>TP>P>P>P>4P>ԟP>|P>P>P>gP>4P>$P>l%#######$
!#%#%####$
9*urn:schemas-microsoft-com:office:smarttagsState8*urn:schemas-microsoft-com:office:smarttagsCity9*urn:schemas-microsoft-com:office:smarttagsplace=*urn:schemas-microsoft-com:office:smarttags PlaceType=*urn:schemas-microsoft-com:office:smarttags PlaceName6##################$$$$$""""#############$$33333333################$$###############$$N@ABCDEFIRoot Entry FNҧK1Table.WordDocument;<SummaryInformation(7DocumentSummaryInformation8?CompObjy
F'Microsoft Office Word 97-2003 Document
MSWordDocWord.Document.89q