DOCUMENT RESUME SE 064 363 ED 449 015 Carpenter, Thomas P.; Levi, Linda; Farnsworth, Valerie AUTHOR Building a Foundation for Learning Algebra in the Elementary TITLE Grades. Wisconsin Univ., Madison. National Center for Improving INSTITUTION Student Learning and Achievement in Mathematics and Science. Office of Educational Research and Improvement (ED), SPONS AGENCY Washington, DC. 2000-00-00 PUB DATE 8p.; Published quarterly. NOTE R305A960007 CONTRACT _NCISLA, University of Wisconsin, Wisconsin Center for AVAILABLE FROM Education Research, 1025 W. Johnson Street, Madison, WI 53706. For full text: http://www.wcer.wisc.edu/ncisla. Non-Classroom Collected Works - Serials (022) -- Guides PUB TYPE Descriptive (141) -- Reports _(055) In Brief; vl n2 Fall 2000 JOURNAL CIT MF01/PC01 Plus Postage. EDRS PRICE Abstract Reasoning; *Algebra; *Arithmetic; Educational DESCRIPTORS Change; Elementary Education; *Elementary School Mathematics; *Mathematics Instruction; *Professional Development; Teaching Methods ABSTRACT "In Brief" focuses on K-12 mathematics and science research seeking to and implications for policymakers, educators, and researchers highlights the learning improve student learning and achievement This brief study in Madison, gains of 240 elementary students involved in a long-term arithmetic in ways that Wisconsin and their remarkable ability to reason with summarized in this build their capacity for algebraic reasoning. The research development, article shows that with appropriate teacher professional in ways that both teachers can learn to help students learn mathematics foundation for enhance their understanding of arithmetic and provide a learning algebra in learning algebra. Teaching and policy considerations for the elementary grades are attached. (ASK) Reproductions supplied by EDRS are the best that can be made from the original document. Building a Foundation for Learning Algebra in the Elementary Grades Thomas P. Carpenter, Linda Levi, Valerie Farnsworth In Brief, Volume 1, Number 2, Fall 2000 U.S. DEPARTMENT OF EDUCATION PERMISSION TO REPRODUCE AND Office of Educational Research and Improvement DISSEMINATE THIS MATERIAL HAS EDUCATIONAL RESOURCES INFORMATION BEEN GRANTED BY CENTER (ERIC) This document has been reproduced as received from the person or organization S. Anderson originating it. O Minor changes have been made to improve TO THE EDUCATIONAL RESOURCES reproduction quality INFORMATION CENTER (ERIC) Points of view or opinions stated in this document do not necessarily represent official OERI position or policy. 1 AVAILABLE BEST COPY MATHEMATICS & SCIENCE NATIONAL CENTER FOR IMPROVING STUDENT LEARNING & ACHIEVEMENT IN K-12 Mathematics & Science ,nBtriet RESEARCH & IMPLICATIONS FOR POLICYMAKERS, EDUCATORS & RESEARCHERS SEEKING TO IMPROVE STUDENT LEARNING & ACHIEVEMENT Va. k A -7 -7 \. ABOVE: Elementary students-wojik-onsolving mathematics probleins and developing generalizations. BUILDING l'As A research,1 Carpenter and Levi support %):116 study led by researchers Thomas teachers' professional development by sys- ter and Linda. Levi citthe 'National .Center tematically helping them to,,focus on students' g and for.'Impiiiving: Student mathematical thinking; in particular students' inMathernatics,,-arid FOR LEARNING , 7r abilities to .articulate, represent, and justif)F1 Science ( NCISLA), found that . generalizationS about the underlying structure innovatiire, teacher ; and properties of arithmetic: sionallievelopinent and refb- , rnsed-7;mathernatitVinstinc- ,.,,-- Building on student thinking.,The researchers tion paved the way for even and teachers have:found that students have a 7.: "first- and cfsuec-aridlgiaders to great deal of implicit knowledge about-basic begin to reason algebraically. tementctry properties of arithmetic. For example, even - --Research results': show :that 1N- :THE first-grade students: will' choose to count on young students can learn to . from the larger number to find a sum like 3+ make and justify:generalilations 9: These students implicitly recognize they can ahout the underlying structure and interchange the order from 3'4: 9 .to 9 + 3 to properties_of.arithmetic tV easier: _When these basic make the tions that form-the basis for much Ofilge- , braterit-l with goals in the ,p_r_op_erties are not made .explicit, however, in many students are unsurewhether the proper - -Nationalcouncil of Teachers of Mathematics rr eachers.,and-;researcheg have long recog _..ties apply-in-new or unfamiliar problem con- nized that the-transition' from learning .'(NCTM) Principles,And-Stindards (2000),. this texts. For example, .students may not under- arithmetic hi' learning: algebra is one of-the'' NCISLA study.shOws_that stand that they can similarly change the order major hurdles studerifs*einlearning math- begin bniging.a_fon_ndation.inalgebra much- of the numbers if they are adding very large .earligi than typical curricula allow. .eniaticsAtesearchers,3are finding, however, numbers, or fractionS, or expressions involv- ir4c well/ thatelementary stUdehts 'can learn to think - ":.=. r . . ing variables in, algebra. Other students may ,both enhance Focus abchitaffifunetic iin ,RESEARCH overgeneralize:.They may for example, inter- their early learningof ar*ietic_ancl provide Reieciith--Project ''.-The Tiifiy-.-Algebra change the order of numbers when subtract- forleargiiiialgebra. This in Brief a foundation . ing or dividing. ird year, the early alge- highlights learning gains of 240 elementary iirrefitly. in its. students' involved in a long-term- study in is led by NCISLA researchers %;,,,:ibia:piciject Algebra builds on the same fundamental prop- Carpenter and Levi and 12 Madison Metro- Madison,' Wisconsifif,1 and their remarkable . erties that form the-basis for arithmetic. The Politan School District teachers. Building, on ability to reason-about arithmetic in ways that abstract nature of algebra makes it even more 15 years 'of Cognitively Guided Instruction build their capacity for algebraic reasoning. 1 The research in Madison is peridding a foundation for teacher professional development in schools in Phoenix, Los Angeles, and San Diego. own content 2 The Cognitively Guided Instruction Professional Development Program engages teachers in learning about the development of children's mathematical thinking in particular content areas, building their 1999. knowledge, and refining instructional practice. For more information about CGI, see in Brief reference: Children's Mathematics: Cognitively Guided Instruction (a book and two multimedia CDs), . 1 eralizations from students in the class. The Yeah. 49.78 49 + 49 = 78. important that students understand precisely MIKE: class then discussed these generalizations and when and why properties of arithmetic can be Do you all think that is true? TEACHER: WOW. whether they were always true for all num- [All but one child answered yes.] applied. The goal of the early algebra project is bers, which led to an extended classroom to help students make explicit and understand I do, because you took the 49 away, JENNY: analysis of what is required to justify a gener- the underlying structure and properties of and it's just like getting it back. alization. This form of instruction built on as they are learning arithmetic arithmetic [Emphasis added to lift out teacher's question strategy.! student thinking and supported their under- so that they will have a solid base to build on as standing of basic properties of arithmetic they go on to learn algebra with understanding. As the discussion above continued, the group required for algebra. The following classroom excerpts illustrate how these goals were collectively came up with the generalization: Supporting professional development. A criti- "Zero added to another number equals that accomplished. cal component of the early algebra project is other number:' They also came up with the teacher professional development. At summer First- and Second-Graders' generalizations: "Zero subtracted from another workshops and meetings held throughout the number equals that number," and "Any num- Capacity for Algebraic Reasoning year, the teachers and researchers analyzed the ber minus the same number equals zero." The structure and basic properties of arithmetic students not only applied these generaliza- tudents from a first- and second-grade and considered learning contexts that could S tions to solve problems involving zeros, they class were given a set of problems to guide encourage students to explicitly articulate gen- also came up with number sentences that them into expressing a generalization about eralizations about these properties. The group embodied more complex ideas (e.g., 78 49+ 49 what happens when zero is added to a number. considered how students might think about The children were not only able to articulate = 78).3 specific problems and ways students might jus- the generalization; they were able to take the tify (or prove) their proposed generalizations The study shows that even first- and second- discussion to a higher level mathematically. bet- were true. These sessions helped teachers grade children are able to argue about mathe- ter understand their students' thinking and matical concepts and operations in ways not The teacher asked the students if "78 49 = 78" build their own mathematics knowledge. In generally expected of students at this age. was true or false. The students immediately addition, the meetings supported the evolu- These students were able to express generaliza- responded: tion of a committed and growing professional tions and reason about them. community. False! No, no false! No way! CHILDREN: Why is that false? Third- Through Fitth-Graders' TEACHER: Enhancing classroom instruction. True-false Because it is the same number as Introduction to Mathematical Proof and open number sentences were the primary JENNY: in the beginning, and you already means of eliciting generalizations from students. took away some, so it would have hird- through fifth-grade students partici- (See "Number Sentences Used to Generate Gen- to be lower than the number you pating in the early algebra project not eralizations".) Teachers used true-false sentences started with. only identified more complex generalizations, to initiate discussions that focused on how stu- Unless it was 78 0 = 78. That they also were challenged to justify their gen- MIKE: dents knew a sentence was true or false. This would be right. eralizations using arguments that helped them method generally was sufficient to elicit gen- to gain an appreciation for mathematical Is that true? Why is that true? We TEACHER: NUMBER SENTENCES USED took something away. proof. They learned that justification went Generalizations to Generate beyond proposing an endless supply of exam- that something is, there is, But STEVE: that like, nothing. Zero is nothing. ples for which the generalization applied. Below are examples of number sentences teachers used to help students articulate gener- Is that always going to work? TEACHER: alizations about zero and multiplication. For example, Mary Bostrom's third- and fourth- If you want to start with a number LYNN: Examples: 78 + 0= 78; 23 + 7 = 23' grade students were asked to justify the gener- and end with a number, and you "When you add zero to a number, alization: "When you multiply two numbers, do a number sentence, you should with." you get the number you started you can change the order of the numbers" (ax t always put a zero. Since you wrote =74 Examples: 96 96 =0; 74 = tx a). The students initially calculated a lot of 78 49 = 78, you have to change a 49 to a zero to equal 78, because if examples, such as 8 x 5 = 5 x 8, but Ms. Bostrom "When you subtract a number from itself, you get zero.' you want the same answer as the pressed them to show that the generalization first number and the last number, was true for all numbers, not just some. Examples: 96 x 0 =0; 43 x 0=43' you have to make a zero in "When you multiply a number times zero, between. To prove the generalization was always true, you get zero.' do you think that will always one pair of students went back to the basic def- TEACHER: So Examples: 65 x 54 = 54 x 65; 94 x 71 = 71 x work with zero? inition of multiplication using linking cubes "When multiplying two numbers, you can Oh, no. Unless you 78 minus, to illustrate a specific example (8 x 5 = 5 x 8, see MIKE: change the order of the numbers." umm, 49, plus something. Figure 1). After some discussion, the students denotes a false number sentence provided a concrete justification, demonstrat- Plus 49. ELLEN: After students were introduced to variables and open number sentets, they were able to express generalizations in open number sentences that 3 Students at first used everyday language to express generalizations. were always true, such as b-b=0,andb-a.a=b. BEST COPY AVAILABLE algebra can be a part of elementary instruction calculation or that they should just add all the students' FIGURE 1. Proving 8 x5 = 5 x8: Two that builds on students' implicit mathematical numbers together. Furthermore, teachers par- proof using the rotation of cube arrays. knowledge and increases their ability to ticipating in the research found that explain- understand, reason, and engage in challenging 8x5 ing the equal sign was not sufficient to ensure problem solving. that students understood its meaning. Build- ing off students' different conceptions of what 5x8 The instructional strategies outlined here have the equal sign meant, teachers engaged stu- significant implications for teacher learning dents in mathematical discussions that helped and professional development. Through their them confront their misunderstandings and ongoing work with fellow teachers and achieve an accurate understanding of the mean- researchers, the teachers gained essential ing of the equal sign. insight into their students' mathematical rea- (b) (a) soning; they also forged a community through NUMBER SENTENCES THAT Equality Ccncepticn which they gained mathematical knowledge Challenge Students' . . and crafted problems that benefited their stu- ing that they could rotate one group of cubes 7=3+4 a) dents' learning. Although it requires a signifi- (Fig. la) 90 degrees and place the rotated array 8=8 schools cant commitment and support from b) "Look, they are over the other group (Fig. lb). and policymakers (see insert on Policy Consid- 5+8=8+5 the same," exclaimed one student. The other c) erations), participants in both CGI and the student added, "That works, but you don't 8=5+13* d) early algebra project have seen this form of [Fig. lb). You even have to have the other one denotes a false number sentence exciting professional development yield and see the can just turn this one [Fig. la] results in students' learning of mathematics. other groups." Interested educators should also refer to Across several class periods, the teachers con- the Teaching Considerations insert, which tinued to provide examples of true and false Although the students used a specific example specifies some resources and strategies number sentences that challenged students' they to justify the generalization, the way teachers might adopt with their students and conceptions of the meaning of the equal sign explained the example showed that they colleagues in their educational community. and reinforced their learning. These discus- understood that they could do the same thing sions resulted in significant changes in stu- with any numbers. These students were begin- For More Intormation dent understanding and problem-solving ning to learn what was required to justify that improvements (see Figure 2). all numbers. a generalization was true for A research report about the early algebra proj- listed ect and other relevant publications are Equality of IMPLICATIONS: Students' Understanding in the reference section of this in Brief and are Elementary Retiorm available at the NCISLA website at http:// Mathematics inatruction and equality and the An understanding of Teacher Proteadional Development www.wcer.wisc. edu/ncisla/. appropriate use of the equal sign is criti- cal for expressing generalizations and for Researchers Tom Carpenter and Linda Levi T his study showed that young students can developing algebraic reasoning. For example, National Center can be reached through the learn arithmetic in ways that provide a the concept of equality (indicating a relation- for Improving Student Learning and Achieve- foundation for learning algebra. Recognizing ship between different parts of an equation ment in Mathematics and Science at the Uni- young students' ability to reason algebraically and meaning "the same as") is embedded in a versity of Wisconsin-Madison, 1025 W. John- does not suggest that elementary students 8. Unfortu- 5= 5 number sentence like 8 x x 263-3605; son St., Madison, WI 53706; (608) should learn high school algebra. Rather, this nately, many students typically hold miscon- E-mail: [email protected] study showed that a broader conception of ceptions about the meaning of the equal sign (see Kieran, 1981; Matz, 1982) and tend to think FIGURE 2. Students' Increase in Understanding of the Meaning of the Equal Sign it means merely to carry out an operation. In order to assess students' understanding of ; ooZ got the meaning of the equal sign, the researchers Percent of students correctly answering and teachers gave the students a seemingly 8oR. 8+4. +5 228 -66Z 70Z- simple number sentence: before and after teachers 6oZ adjusted instruction. 8+4=0 +5 soZ. 40X Consistent with previous research (Kieran, 30% Before Adjusting 1981; Saenz-Ludlow & Walgamuth, 1998), 20Z Instruction ...... most students responded that either 12 or 17, 9% loZ sz ox After Adjusting rather than 7, should fill in the box. They oX Instruction Grade 6 Grades 3 & 4 Grades ; & 2 thought either that the number immediately (N78) (N.19) (N -/13) after the equal sign had to be the answer to the 5 BEST COPY AVAILABLE References ABOUT in Brief Carpenter, T. P., & Levi, L. (2000). Developing con- and This publication and the research reported herein are supported under the Educational Research ceptions of algebraic reasoning in the primary of Development Centers Program, PR/Award Number R305A960007, as administered by the Office grades (Res. Rep. 00-2). Madison, WI: National Center for Improving Student Learning and Educational Research and Improvement, U.S. Department of Education. The opinions expressed the Achievement in Mathematics and Science. [Avail- herein are those of the featured researchers and authors and do not necessarily reflect the views of able at www.wcer.wisc.edu/ncisla] supporting agencies. Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., Empson, S. B. (1999). Children's mathematics: Permission to copy is not necessary. This publi- 111ARIgki-4.4 fp:al Cognitively Guided Instruction. Portsmouth, NH: cation is free on request. K-I2 Mathematics & Science Heinemann. inBriely RESEARCH & IMPLICATIONS in Brief Volume 1, Number 2 is a publication of RESEARCHERS (LOS FOR POLICTORAKERS. EDUCATORS Carpenter, T. P., & Franke, M. L. (1998, fall). Teachers A0.1.11110, SEA SURMA 10 01111010 STUOINT as learners. Principled Practice, 2 (2). [Available at the National Center for Improving Student www.wcer.wisc.edu/ncisla] Learning and Achievement in Mathematics and Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Science, at the Wisconsin Center for Education Children's understanding of equality: A founda- Research, University of Wisconsin-Madison. tion for algebra. Teaching Children Mathematics, 6 10. * (4), 232-6. [Available at www.wcer.wisc.edu/ ... BUILD MS A 1 Foundation 1 Kieran, Carolyn. (1981). Concepts associated with the sImmatolool . &Hamm l Whom/. FON IMG air NCISLA Director: Thomas P. Carpenter equality symbol. Educational Studies in Mathe- kims 01CUL/111 Oa. exA re, 104.0* matics 12,317-26. ............. NCISLA Associate Director: Richard Lehrer .1 ad um.= Ilo.1. De ...A.. ai . a =Oa Matz, Marilyn. (1982). Towards a process model for Ws b............: Elementary osigs O.. ua h amp.. REIgal Relmea IN TNN rnim school algebra errors. In D. Sleeman, & J. S. Brown Amanse,...1 cm Imo le NCISLA Communication Director: win aelltiksowIlmo Grade a De mama web. Room. a.. (Eds.), Intelligent tutoring systems (pp. 25-50). New Mil.. NW 11.- e Mom as woe Susan Smetzer Anderson .1,1* bat b.. al. wel York: Arademic Press. Mama or no Nob .1.ser mu, 41.1 sara. ena=or. aaa. .1 1.1/1..0116 Ns... mem 1 awe um. gius INCTIO ca Writers: Thomas P. Carpenter, Linda Levi, National Council of Teachers of Mathematics. (2000). Orimla tb arbes Saly Maw shalom blessma armpit dana.* Principles and standards for school mathematics. maaSL Mambas WO ray rata. ORA imOre b WI. Valerie Farnsworth a tram. mo saws 4.1. m.o. um WAN Oil.* Vegada Yob ahet Am. andrams in won .1, 99 KOlf all....easia ad*, Reston, VA: Author. =la ."."*"... an.. is Ion. irskr. be. mid* a . 1 so *moan Designer: Kristen Roderick Saenz-Ludlow, A., & Walgamuth, C. (1998). Third 14/1 Km.; AY !I. /. a,/ I! Man 1.28. otal.*111% Gam. graders' interpretations of equality and the equal Av.. Rm... Icamaa 716. 'ALM Arbal [VIM *en Cos.. Caled Wawa. lama alt or. .1HDRame, ix *um Ammo if wan symbol. Educational Studies in Mathematics, 35q(2), 153-87. ED Nonprofit Organization NATIONAL CENTER FOR IMPROVING STUDENT U.S. POSTAGE LEARNING & ACHIEVEMENT IN MATHEMATICS & SCIENCE PAID Wisconsin Center for Education Research Madison,Wisconsin School of Education, University of Wisconsin-Madison Permit No. 658 1025 West Johnson Street, Madison, WI 53706 H1OU1 BECKRUM DATABASE COORD ER1C/CSMEE 1929 KENNY RD 43218-1015 COLUMBUS OH 6 BEST COPY AVAILABLE .c7.25 77Z.'1* STUDENT LEARNING & ACHIEVEMENT IN MATHEMA NATIONAL CENTER FOR IMPROVING K-12 Mathematics & Science inBriet POLICY CONSIDERATIONS Elementary Students Can Reason Algebraically dents' ways of solving group-selected mathe- he research summarized in this in Brief. The matics problems and number sentences. 1 shows that with appropriate teacher pro- learn- teachers themselves were learners" learn to fessional development, teachers can _learning ing about students' thinking and help students learn mathematics in ways that 'eiders can mathematics. School and district both .eilhance :their:understanding of arith- foster the development of these kinds of com- metic and provide a foundation for learning teachers munities by making time available for algebra. Policymakers and administrators can for teach- to meet, by arranging opportunities kind ._.of support the development of thiS planning time_as they ers to have common instruction in_ their schools by and by work to develop such a community, being active participants in the communities. -e Supporting sustained, long -term programs Of professional .deirelopment that focUs on thedevelopment of t-uderits' mathematical Focusing on Students' - ABOVE:--- Elementary students reason together Shared Vision thinking; as well as teachers' understanding Mathematical Thinking about mathematics problems. .'.'ofihe related mathematical ideas: chool and district leaders can help teachers .profes- e Supporting the development.of S focus on developing students' mathemati- ing and how it is reflected in their classroom sional communities that help teachers in cal thinking by making that focus a priority -interactioni-iiith-sludents, as well as time to ;make their Classiocimi_placet 'for both their the school. By talking to students about reflect on their teaching and their students' dents and teachers to engage in inquiry and mathematics and by talking to teachers about understandings andmisconceptions. Teachers learn with understanding. princi- their students' mathematical thinking, also need access to resources; including people ez.gncouragnig.teachers to conceive of teach- which pals can help create an atmosphere in and materials, that can help them develop ing in terms of understanding and develop- One such conversations become the norm. their-understanding. ing students' Mathematical thinking. principal in the project asked teachers to regu- work to larly bring examples of their students' School. and district-leaders can also support students her and discuss with her what the professional development by participating in Suatained,Long- Supporting. understood and were learning. Teachers' it themselves. Leaders need to understand the Term Protesaional Development analysis of their students' thinking was ideas that the teachers are learning so that they included as a significant part of the teachers' can understand whatds going on in the teach- here are no simple solutions foithe prob- candidates' annual 'review. Questions about ers! classrooms and-provide the necessary sup- a- lems facing our schOok Simply introduc- understanding of and interest in understand- port and assistance when called on. Further- iiig-Tiiiiiitirticulunrorproviding teachers a might ing students' mathematical. thinking more, the participation of all interested parties one-week workshOp. i..9iotIgoing to resultin of hiring also be an important component communicates..to. the -teachers that their pro- sustainable improvement_ in teaching or sig- interviews. fessional development: is important and . val- nificant gains-in student achievement. Fonda- -- ued by the school_ and district.-- change requiresthat teachers develop a - mental deep understanding ofthe matheinatics they For More Intormation Supporting Prote&sional Communities teach and the ways their students think about and learn that mathematics. Developing that Policymakers who would like more informa- iacheis- participating in the NCISLA T understanding takes time and requires sus- sustainable tion about student learning and early algebra project entered into the tained, long-term prafesSional development. teacher professional development can contact project at varying levels of knowledge and School and district leaders need to find ways to Student the National Center for Improving matEematics 'confidence; they also came into provide resources to support those kinds of long- Mathematics Learning and Achievement in it from different types of schools and class- term professional development opportunities. and Science (NCISLA) at (608) 263-3605 or _ student rooms (in terms of grade level and . www.wcer. refer to the NCISLA web site: diversity). Together with the researchers, these One of the most critical resources is time. wisc.edu/ncisla. Researchers Tom Carpenter teachers committed to forming a professional Teachers need time to participate in the pro- the and Linda Levi can also be reached at development community. They met every fessional development, time to meet with NCISLA. stu- month and focused their discussions other teachers to discuss what they are learn- . I . k LEARNING SI ACHIEVEMENT IN al/NATIONAL CENTER FoR IMPROVING STUDENT K-i2 Mathematics & Science inBriet TEACHING CONSIDERATIONS Algebraically Elementary Students Can Reason ,------- ; early-algebra-projea committed-to . NCISLA NUMBER SENTENCES Brief rr he research, summarized in this in forming a professional development corn- to Elicit Generalizations 1 showed that elementary students can miuulk.Thei met every month and focused false? learn to think:about arithmetic in ways that Is this number sentence true or mathematical '- their discussions on students' both enhance their learning of arithmetiC and 48x0=48* how 97 97 = 0 thinking, ways to elicit that thinking, 56 +0 =0" provide a foundation for learning algebra. If 37+58=58+37 to figurnnupshatstutilents ideas in your class- you want to applk..these What number can you put in the box IR discussing math- ways to engage students to make this a true sentence? following: room, consider the ematicif ideas. The teachers themselves 35 x0=8x35 0+74=74 were.- learners - Build a Foundation thizikingand-4eamini about mathematiis- denotes a false number sentence or Learning Algebra window into ei Ask questions that provide a For More bitormatiOn bei from itself, you get zero." When-they do children's-understanding .of important &p. state a generalizationlilsOlics ask, for exam- mathematical ideas. For example, students:. publications Elementary teacher resources and -pie, Is that true for alfnumbesr.,.:! sentence responses to the number + www.wcer.viise.eduincislal:Sier are available at their.iinder- e Have-students justify generalizationithey:or 0 + 8" tells -a. great deal about 1 4e-vilot)- standing of the meaning of the equal-sign, Teacher learpial and. pzbfesSi anal theiL'peerse.projiise (see-page 3 for an- exam -. 1998 newsletter Prcibe itudenl's -feisito for: their answers. ple). Justification of-generalizations re* ment:- "-The---NCISLA Principled. Practice: Teaikeis -as-Lek-ruffs,- at Ask studentsohy they answered as they did. (e.g., .more than providing.a lot Of examples www.wcer.wisc.eddicisla. By expecting-children to justify x 5= 5 8 x 8). discuss e Provide students opportunity to Ckik. Children's understanding of equality: stalls, in you can help them gain their claims, and resolve different conceptions of mathe- Foundation pre-senting mathematical arguments and dren's Ilndemndirs ofAquality: 7,A, matical ideas. For example, the different Linda Levi iad Karen-P: proofs. Use the:questions "Will that be true for Algebra, conceptionS.Of 'the equal sign that emerge Children ThomaS..P. Carpenter,: in-:Teaching for alLnumbersMantF"How do you know from students' solntions to the open num- DeC..1999.- (Also-at Mathematics, .vol. 6,- that is true for all numbers?"--repeatedly ber sentence '9 + 6 = 0 + 8!. can provide the www.wcermisc.ecluinclila/teichers) they encourage students to recognize that basis for a productive discussion._ dithstiiii's Cognitively Guided-CrisI:ruction: in mathematics. need to justify their claim's that help e Provide students with equations Instruction Mathematics - Cognidiely Guided Car: them understand that the equal sign repre- (with t*Itiultimedia CDs),.% Thomas. P.. Form a TeaChing.TCommunity Focused Feiineina, . Megan Loef snits a relation between numbers, not a sig- penter,. Elizabeth on Students' Mathematical Thinking Franke, Linda Levi, -and'sSusan B. Empson. nal to carry out' the preieding calculation. :1_ Publishers;1999. e Make classrooms a place where both you Heinemann Examples include ff0= 8 +9"'"8 +6 =6 +0," and your students are learning. Engaging 87.Varythe format of number sen- "9 + 6= 0 + in regular inquiry about students' mathe- tences: Include sentences in which the answer matical thinking can be one of the most does not come right after the equal sign. Mt powerful forms of professional develop- false num- e Provide students with true and ment. Such -inquiry' does not,. Jaowever, ber sentences that challenge their miscon- thrive inisolation. Seek out other teachers ceptions about the equal sign (e.g., 8 = 5 + 3, who share your-interest in talking about stu- 9=9, 7-4=7111). dents' thinking and share with one another s Provide students problems that encourage the interesting observations you are making them to make generalizations about basic about students' mathematical thinking in number properties (see "Number Sentences your classes. they pro- to Elicit Generalizations".) When Form a community with teachers and e vide an answer to one of the problems, ask other resource people. You will find the them how they know their answer is correct. the Together . with support invaluable. elementary student explains his rea- ABOVE: An That often will result in their stating a gener- soning to his teacher and fellow students. researchers, the teachers participating in the alization such as "When you subtract a num- 8 I I I 002 NCISLA MON 14:05 FAX 263 3406 02/05/01 S(o'S _a5 U.S. Department of Education Office of Educational Research and Improvement (OERI) National Library of Education (NLE) Educational Resources information Center (ERIC) Reproduction Release (Specific Document) I. DOCUMENT IDENTIFICATION: in the Elementary Grades Title: Building a FOundation for Learning Algebra Valerie Farnsworth Author(s): Thomas P. 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