March 2008 CMC ComMuniCator Page 35

Prevention Trumps Intervention

by Renee Hill, Riversideteamteach@sbcglobal.net

The urgency for supporting studentlearning in mathematics often comestoo late. Our need for intervention

would be virtually eliminated if we couldsystematically provide the opportunity tolearn, prevent misunderstanding, teach diag-nostically, analyze errors, and offer immediatecorrective feedback.◆ Provide the opportunity to learn. Schools

must deliver grade level content to eachand every student. If students do not havethe opportunity to learn grade level con-tent, they will never demonstrate profi-ciency in an assessment of the standards.

◆ Prevent misunderstandings. This marks thedifference between a good teacher and anexpert teacher. The expert works with herprofessional learning community (PLC) togo beyond calendaring lessons to planningthe lesson delivery. The expert relies onpersonal and PLC experience to knowwhere students typically stumble and todesign lessons to prevent those typicalmisunderstandings. The expert makes surethat the topic is sequenced properly andalerts students to problem areas.

◆ Teach diagnostically. The Mathematics Frame-work for California Public Schools states:Most challenges to learning can be corrected withgood diagnostic teaching that combines repetitionof instruction, focus on the key skills and under-standings, and practice. For some students modifi-cation of curriculum or instruction (or both) may berequired to accommodate differences in communi-cation modes, physical skills, or learning abilities.(p. 230)

The diagnostic teacher asks deliberatequestions, evaluates student responses,and moves the students forward duringthe interactions.

◆ Analyze errors. This work extends diagnos-tic teaching to the guided and independentpractice times as well as assessment andhomework sessions. Error analysis has tobe paired with immediate, correctivefeedback.

◆ Give immediate, corrective feedback. In theDecember issue of Educational Leadership,assessment expert Thomas Guskey re-minds us that “35 years ago. . . [Benjamin]Bloom and his colleagues stressed that toimprove student learning, . . . progresschecks must provide feedback. . . and befollowed up with correctives.”

Strategies That Work for MathematicsInterventionThe first step in supporting student learning ishigh quality instruction as a preventive mea-sure; then intervention strategies can be ap-plied. The strategies below are some of thestrategies that I have found to be effective inclassrooms are listed in alphabetical order,rather than order of importance. Teachersshould not attempt to try all of them butshould instead incorporate those that they feelwould work best in their individual situa-tions.Answer KeysStudents solve problems, then check. Establisha routine and be adamant that students ad-here to it. If a norm of integrity has beenestablished in the student learning commu-nity, students will not copy the answers. Onemethod is that students complete the assign-ment and take only that page to a correctingtable where the teacher’s edition or answerkey and highlighters can be found. Studentsmark the incorrect answers with a highlighter,

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Our need for intervention would bevirtually eliminated if we could

systematically provide theopportunity to learn, prevent

misunderstanding, teachdiagnostically, analyze errors, andoffer immediate corrective feedback.

Page 36 CMC ComMuniCator Volume 32, Number 3

then rework those problems. The highlightinglets the teacher know how the student did onthe first pass and the rework allows the stu-dent to persist in solving correctly. Anothermethod is “Scrambled Answers” where stu-dents receive assigned problems and a sheetwith answers scrambled. They first solve, thenmatch their solutions.Ballpark AnswersStudents use their mental math, known opera-tions, compatible numbers, visual images, andpart-part-whole knowledge (see van de Walle,Ch 6) to “ballpark” the answers before findingsolutions. I use the term “ballpark” to indicatea preference for sense-making over textbook-style estimation or rounding, which is tooformulaic and often not useful. I have stu-dents write down the homework problems,then we ballpark all the solutions. Studentsuse the ballpark estimates to judge the reason-ableness of their calculated solutions.Build, Draw, WriteStudents model with concrete objects, recordtheir work in a drawing, and then record thesame work with numbers and symbols. Somepeople refer to this as concrete, representa-tional, and abstract; but I have found thatteachers and students more easily rememberbuild, draw, write. Build, draw, write is notnecessarily sequential. Draw doesn’t literallymean draw. It might mean visualize, imagine,or rehearse, and uses the numbers, symbols,and notation of mathematics. If the writing isprose, it is about the how and why of themathematics—not a step-by-step regurgitationof how to carry out a procedure.Cues, Clues, and CrutchesCues. Students are prompted to be wary oftheir own areas of vulnerability. My thirdgrade student, Lewis, dove directly into sub-traction problems. I asked what would makehim remember to check for regrouping and hedrew a small cartoon figure he called RoosterMan.

At first, I reminded Lewis to have RoosterMan help him remember to regroup. Eventu-ally, he drew the character on his own andtogether they checked for regrouping. Eventu-ally, Lewis did not need Rooster Man.

Clues. Students learn a rhyme, mnemonicdevice, or other learning support that helpsthem over a rough spot. For example, at thebottom of flashcards for adding six to a single

digit number, a student might add the clue“five and one more.”

Crutches. Students learn a coping strategy.For example, whenever my sixth grade stu-dent Ronisha got an assignment that requiredmultiplication or division, she had to turn herpaper over and make her own multiplicationtable for 3 through 8 times 6, 7, 8. Since shehad not committed those facts to memory, Ipreferred to hold her responsible for the gradelevel learning rather than spending a gener-ous portion of time remediating her factknowledge.Danger Zone AlertsWhen delivering instruction on the topic, alertstudents to areas that are commonly misun-derstood. For example, when teaching deci-mal place value, alert students that decimalplace value and whole number place value arevery specific. For instance, 0.12 might bemistaken as greater then 0.3. Demonstrate,using base ten models, that 0.12 has a singletenth and two hundredths whereas 0.3 has 3tenths pieces.Double PointsStudents receive one point for the correctanswer and one point (or more) for the correctsolution. This places grading emphasis on thesolving rather than the answering.Focus FridayOn Fridays, students meet in a group that isfocused on their specific learning need. Forexample, a team of three teachers who arecurrently teaching multi-digit division mightorganize a group to review 6, 7, 8, and 9 facts;one group for review of procedure; and achallenge group. The majority of students ineach teacher’s classroom would complete anindependent assignment. Each teacher wouldwork with the small focus group made up ofstudents from all three classrooms. Groupscould meet weekly, every other week, or onceper month. This can also be done acrossgrades when there are overlapping learningneeds.Hot Spot AssignmentsStudent assignments are specific to areas ofdifficulty that arise when teachers monitorindependent practice, analyze homework, andask questions. In this strategy, teachers makenote of the specific areas of need, and thengenerate work that untangles the difficulty.

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For example, simplifying fraction solutionsmight be overlooked. I recommend giving apage of fractions and asking students to iden-tify whether they could be simplified. Then, asfraction operation assignments are completed,ask students to review solutions to verify thatall fractions have been simplified.Immediate FeedbackStudents get an immediate response as theywork each problem. Feedback could be pro-vided by the teacher or a learning partner.My MistakeStudents earn back credit for incorrect home-work problems by analyzing and noting theirown mistakes (“I added incorrectly,” or “Imultiplied instead of divided”). Reworkingthe problem earns back points or fractions ofpoints.

Nontraditional Computation MethodsStudents learn methods other than the stan-dard United States computation methods, forexample, partial products for multi-digitmultiplication or compensation methods foraddition and subtraction.PreviewAll students can benefit from participating ina preview of the lesson. For struggling stu-dents, this can take place during a before-school session, in a small group a day aheadof the whole-class lesson, or in supportclasses. This front-loading allows strugglingstudents to then participate in the whole-classinstruction.Repeat, Review, Reteach, RemediateStudents require various levels of interventionsupport and these typical intervention strate-gies. Repeat works best when the student has agrasp of the topic, but might be unclear oncertain areas. Review suits students who donot need a full repeat. They do not need adifferent approach; they just need a review ofportions of the lesson. Reteach is for studentswho need a different approach. For example,if the teacher presented the topic abstractly,build and draw might help the student.Remediate is for students needing prerequisiteskills. Their missing skills prevent them fromsucceeding in the current topic and there areno cues, clues, or crutches to provide adequatesupport.

Response Cards or BoardsStudents individually respond to a question orprompt by using a white board, scratch paper,voting device, or response card. The teacherscans all responses, providing scaffolding,cues, or questions where necessary. Theteacher gives specific feedback and providesimmediate correctives as appropriate. Youmight also use a preprinted card. For example,an index card with the letters A, B, C, or Dwritten on each edge would allow students torespond to multiple choice questions. It iscritical for every student to respond and forscaffolding, correctives, and feedback to beoffered.Right-sized ProblemsStudents receive problems selected for theircurrent level of understanding plus a littlepush. Struggling students get less complexproblems and gifted students or fast finishersget more complex problems. For example,when subtracting fractions, struggling stu-dents might get problems with simple de-nominators or easily computed commondenominators or fewer problems needingsimplified solutions. Gifted students wouldget mixed numbers with difficult to computecommon denominators and solutions requir-ing simplifying.Sequenced ProblemsStudents receive specially designed problemsthat progress from easy to hard in order tofacilitate error analysis. For example, multi-digit multiplication problems can progressfrom no regrouping needed to multiples oftens to regrouping in one place with easy factsto regrouping with more than one place witheasy facts to regrouping in one place withhard facts, and so on.Solve, Match, and ChallengeStudents are partnered. They solve two orthree problems independently and then com-pare solutions. If solutions do not match,students challenge the response of the otherand rework together until they both knowhow to solve for the correct solution.Strategy InstructionStudents receive instruction in strategies thatsupport computation. John van de Walle’sElementary and Middle School Mathematicsoffers a wealth of strategies. Addition ex-

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amples include making a ten, doubles, andnear doubles. Multiplication examples aredoubles, squares, and times five plus onemore set or times ten less one set.Technology AssistanceThere are a multitude of computer softwareprograms, applets, virtual courses, and websites that claim to support students. My per-sonal opinion is that it is rare to find a technol-ogy solution that adequately supports a strug-gling student. I have had more success havingproficient and advanced students utilizetechnology while I work with the strugglingstudents. I have used Excel and SchoolhouseTechnologies to generate hot spot assign-ments. Multiflyer allows students to work ontheir facts-to-learn and occasionally refreshtheir facts-I-know. This site can be found atgdbdp.com/multiflyer/play_online. html.The National Council of Teachers of Math-ematics’ Illuminations web site illuminations.nctm.org/ActivitySearch.aspx includes manyuseful applets as does the National Library ofVirtual Manipulatives at nlvm.usu.edu/en/nav/vlibrary.html. I have worked withschools that use Get Ahead Math, AcceleratedMath, and publisher intervention sites, but allmust be monitored in order to get the besteffect for student learning. Math Forum has asearchable site called Math Tools atmathforum.org/mathtools/.TextbookOnce, when providing a model lesson in afourth grade classroom, I made a study guidefor the geometry chapter. Not a single studentin the class knew how to use the textbook tofind information or learning supports. Stu-dents must use textbooks to support theirlearning. They need to learn the features oftheir text that support learning: the examples,glossary, index, and review pages. Textbooksare sent home as learning support tools andparents should familiarize themselves withthe text. Many teachers also have alternatetextbooks on hand for reteaching purposes.Tune-UpsStudents receive four to ten problems inaddition to the current homework. The tune-ups address specific areas of need. During thetime of the year I call “division season” Iusually recommend that teachers cease devot-ing full lesson sessions to practice divisionand instead add three to four division prob-

lems to all homework assignments for thenext two or three weeks.Warm-UpsStudents solve warm-up problems specificallyselected to clarify areas of misunderstanding.Use this method when you do not have thetime or the need to dedicate a full lesson toclarification. Warm-ups can be used in con-junction with preview or review to its besteffect.Workstations and GamesStudents are provided with memorable expe-riences where they explore mathematicalideas, connect to prior learning, and engage inpractice. I have seen Marilyn Burns’ AboutTeaching Mathematics, Kathy Richardson’sDeveloping Math Concepts, Patsy Kanter’sPartner Games, and Frog Publications’ LearningGames all used to improve learning. I oftenrecommend that teachers place repeatableactivities in the workstation area. For ex-ample, give students three index cards. Havethem write the decimal form on one card,fraction form on another, and a shaded deci-mal grid or fraction square on the third card.Once students ensure that the three represen-tations are correct, the cards can be used at aworkstation for a matching or concentrationgame.

ReferencesBurns, Marilyn. About Teaching Mathematics: A K–8

Resource. Sausalito, CA: Math Solutions Publications,2000.

California Department of Education (CDE). MathematicsFramework for California Public Schools: KindergartenThrough Grade Twelve. Sacramento, CA: CDE, 2006.

Guskey, Thomas. “The Rest of the Story.” EducationalLeadership 65 (December 2007/January 2008): 28–35.

Marzano, Robert J., Debra Pickering, and Jane E.Pollock. Classroom Instruction That Works. Alexandria,VA: Association for the Supervision of CurriculumDevelopment, 2001.

Richardson, Kathy. Developing Number Concepts (series).White Plains, NY: Wesley Longman, 1999.

van de Walle, John A. Elementary and Middle SchoolMathematics: Teaching Developmentally. White Plains,NY: Longman, 2001.

© 2008 Renee Hill