With the introduction oftheCommon Core StateStandardsforMathematics,teachers and students will have the opportunityto engage in mathematicswithfocus, coherence, andunderstanding.Thoughthird gradeclassrooms in Hillsborough Countywillnotbe usingthe Common Core contentstandardsfor instructionthis year,the Common Core StandardsforMathematicalPracticeshould beimplemented in all math classrooms.The practice standards describe ways thatstudents willengage in subjectmatter and also ways whichteachers shouldfacilitate instructionalopportunities forstudents’developmentof understanding.
Acommunitycan be thoughtofasa group ofpeople,interactingwith each other,sharing common interests orgoals,and who can worktogetherto help each otherachieve. Applyingthistoa mathematicsclassroom, this would producea group ofstudents, with common learning goals(math contentstandards), cooperatingwitheach othertolearn thesegoalswith understanding(standardsfor math practice).In otherwords, amath classroomcommunitywillbebuiltand defined bythe opportunities,structures, and supporttheteacherfacilitatesto helpstudents workand learntogether.
In the 2013-2014thirdgradeinstructionalguidelines, thefirst9 days oftheschoolyearhave been devoted to buildingcommunityinthe mathematicsclassroom. Buildingcommunitywillnotbe donein 9 days, butthefoundation can be developed. Communitywillcontinueto develop and buildthroughout the remainderoftheschoolyear; however, withoutasolidfoundation in the beginningof the year,communitywillnotblossom. Thesefirst9 days ofcommunitybuildingshould place anemphasison collaborationand problemsolvingthrough the useofthe practicestandardsand prioryears’content standards.
First, most students should alreadyhavemastered thesestandards priorto the current gradelevel, butmaybenefit from exploringwith thesestandardscomingback to school in anew gradelevel, as wellasconnectingprior contentand strategies to gradelevelcontent.Secondly, some students maynot havemastered priorcontent standards.Forthesestudents, teachers will havethe opportunityto informallyassess and provide reteachingstructures. Finally, students willbe able to use content theyarefamiliar with to focus upon buildingcommunityand usingthe practicestandards.
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As you work throughtheFirst 9 Days ofMathematics, besureto practiceclassroom norms and classroomroutines. Theselearning tasks aredesigned to help students practice mathematical classroom norms and routines. Incorporatenumber sense contentfromthe previous grade level to use familiar concepts while learning structures tobeused year round. The suggestions below includea focus ontwo-digitadditionand subtractioninproblem solving.
Day 1-Chooseand SolveProblems Using Manipulatives:
During Day1,teachers should establish classroom norms to promoteapositive classroom environment. The followingare someexamples of norms thatyoucan utilizeto promote productive communication in the classroom:
1. Explainyour thinking
2. Ask questions
3. Challenge ideas, notclassmates (peers)
4. Say when I don’t understandoragree
5. Actively participateinalllearning tasks
These arejustexamples, work withyour students and /or school to establish thesenorms.
Task 1:Students chooseamanipulative from awideselection oftools. Havemanipulativeseasilyaccessible, so students don’t haveto askyou for them.Oncestudents selecttheirmanipulative, havethemdraw arepresentation of it in theirjournals and brainstorm at least 5 math concepts that theycould learn moreabout byusingthem. (SMP 5)
Task 2:Present students with the followingproblem:Thereare32 animals at the zoo. 14of the animals weremonkeys. Howmany animals werenotmonkeys?Students will work individuallyor in pairs atthis time and choose anappropriate manipulativeto help them solvethe problem. Students will record theirsolution in their math journal.Remind students that they willbe expected to share theirthinking. Students should directly model the problem situation referenced in bold using manipulatives and or pictures (SMP 1, 2, 4). Direct modeling is a strategy that helps to build students’ understanding of what is happening in a problem solving scenario and what operations might be useful in finding a solution. The more opportunities students have to directly model situations, the better they will conceptually understand the operations.
Day 2-Journaling aboutProblemSolving andCreating Rubrics:
Task 1:Students reviewtheirsolution and manipulative choicefrom Day1.Students will write in their journals about their manipulative choiceand explain how it helped them to solvethe problem. (SMP 3)
Task 2: Journalingis an importantpart ofthe learningprocess in mathematics.Students should know and be comfortablewith the expectations forjournaling.In order to facilitatethis, workwith yourstudents to createaclass rubricforevaluatingjournalresponses. (SMP 3, 6)
ExampleRubric:
4Thestudentdemonstratesanin-depthunderstandingofthemathematics.Information iscorrectand complete.
Explanation,description,or justification is clear.
Writingmayinclude details and/orexamples.
3-Thestudentdemonstratesconsiderableunderstandingof themathematics.
Information/explanationiscorrect.
Minorcomputationalerrorsmaybe present.
Explanation, description, orjustification maylackclarityand details.
2-Thestudentdemonstrateslimitedunderstandingofthemathematics.
Informationand/orexplanationare incomplete. Significantcomputationalerrors mayoccur.
1-Thestudentdemonstrateslittleunderstandingof themathematics.
Informationisincorrect.
Explanation, description, orjustificationis unclearorincorrect.
0-Noattemptwasmade.
Day 3-Beresponsible foryourlearning andwhat to do when you are done journaling.
Task 1:Usingthe journalresponse and the rubricthatyoucreated on Day2, havestudents self assess their workusingthe rubric. Then, have students swap their journal with a peer for evaluation. Use SMP 3 (construct viable arguments and critique the reasoning of others) to promote discourse in the classroom.Model how to have appropriate conversations when critiquing others’ work.
Additional activities forbeingresponsible fortheirlearningmight include:
Concept Cards-studentsgenerateadefinition ofthe concept, pictoriallyrepresent the concept andcreate examples and non-examples of the concept to create adeeper understandingof it.
Task 2:Students createword problemsbased onthe following expressions:
143 + 29and 100 – 63
Record the word problems in theirjournal to beused on Day4.
Day 4-ShareandCompareyourSolution andAccountableTalk
Task 1:Usingthe word problems created on Day3, students will pairup with a partner,swap questionsandsolve them. (SMP 1, 2, 3, 4)
- Strategies to pair students:
ParallelLines-
Students number off 1-22 or however manyareinyourclass
Even numbers form oneline
Odd numbers form another line,so that theyarefacingtheeven lineand each student hasa facepartner
- Highfiveyourfacepartner
- Havepairsfind aspacein the classroom to solveeach other’s problem.
- Afterstudents havesolved the problem, switch back papersand have students evaluatehow theproblem was solved.Was itsolved correctlyor incorrectly?
Stand Up, Hand Up, PairUp-
Instruct students to stand up and push in their chairs
Havethem placetheirhand in the air
Students walk around theroom with theirhand up
Whenthe teacher says PairUp, students find the closest partner
Highfiveyour partner
Havepairsfind aspacein the classroom to solveeach other’s problem
Afterstudents havesolved the problem, switch back papersand have students evaluatehow theproblem was solved. Was itsolved correctlyor incorrectly.
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Usethesequestion stemsto promoteaccountabletalk solving each other’s problems.
So whatI heard you sayis…. / I agreewithyou
because…
I disagree with you
because… / My strategyis likeyours
because…
Couldyou explain that to me anotherway? / What is another strategy we could use to check?
How doyouknow your answer isright? / I don’tunderstand…
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Day 5-SmallGroupInstruction:
Explain to students thatsometimes in themathematics classroom theymayhavedifferenttasks to completethan their peers. Engagethe class with the followingproblem to determineyour small groups.Mrs. Short bakedsomeyummychocolatebrownies. She placed7 plates on the kitchentable andput 2 brownies on eachplate. Howmany brownies didshe put on plates? Bepreparedtojustify youranswer. As students solvethis problem, sort students into 3 differentgroups; Reteach, Core, Enrich (You may not always have 3 groups during small group instruction).Usethefollowingquestionswithyour 3groups. (SMP 2, 3,4)
Journal Prompts: (SMP 1, 3, 4)
Reteach:Colin had 2 baskets with 3 apples in each basket. Colin said he had about 20 apples altogether. Is this a reasonable estimate? Explain why or why not?
How manyapples did hehave?
Core: Meg’s mom puts 5 cupcakes on each plate. Shehas 8 plates foraparty. Meg said she had about 50 cupcakes altogether. Is this a reasonable estimate? Explain why or why not?
How many cupcakes does she haveforthe party?
Enrich:Ladybugs in Kim’syard have10 spots.She catches 4 ladybugs on Tuesday, 5 ladybugs on Wednesday,and 6 ladybugs on Thursday.Kim said she there were about 1,000 spots on the ladybugs. Is this a reasonable estimate? Explain why or why not?
Howmanytotal spots did she catch? Explain howyou know.
Allow students to work with their groups or in partners tosolve their assigned problem. Encourage the use of direct modeling.Have the question stems available from Day4, so that students can haveproductivemathematical conversations to help them makesenseof theproblem. Circulate and monitor as neededand pull asmall groupas needed to provideadditional support. As students aresolvingtheirproblem remind them to explain their thinkingin their journals. Once students solved the problems through direct modeling have them complete the following journaling prompts. Have students provide evidence for their response.
Day 6-HOT Questions:
One goal of rigorouseducation isto increasestudent’s higher order thinkingskills. This can be accomplished through theuse of Higher Order Thinking (HOT)questions to extend students thinking. HOTquestions should bemorethan just reproducingaskill or fact. Theyare NOTyes or no questions, they are open ended.Forthis mini-lesson, students need to haveaccess to theirmath journals, the rubric, and avarietyofmanipulatives to help in theproblem solvingprocess. Students will receivetheirHOT question and willwork withapartner to solve theproblem and providejustification for their solution method. Themostimportant part of this process is forstudents to provide justification with precision (vocabulary, explanation etc.)(SMP 3, 6)Students deepentheir understandingbecause in order to explain an idea,astudent must conceptually understand it. If students aremathematically proficient, theywillbe able to explain theirthinking.
Task 1: Usethe followingquestion withyour students:
The third andfourth grade classes wereonthe playgroundfor recess.18 ofthemwent backto their classroomsto work ona special project.Now there are 37 on the playground. How many students were on the playground to begin with?
Havethe question stems availableto help with productivemathematical conversations. Purposefully select and sequencestudents to share their solutions. Start with the student who used a concrete representation, then move to a student who usepictorial representation andfinallyhaveastudent who used an abstract representation to share. Usethe question stems to connect thestudent strategies.
Day 7: ProblemSolving Process
During Day7, studentswill engage in the problem solving process,understand what thequestion is asking in a problem situationand formulate aplan for solvingthe problem. Agood strategyto useto help students makesenseofthe problem is aK-W-P-Lchart. K-What doIknowabout theproblem,W-What doIWantto Know about the problem, P-What is myplanforsolvingthe problem,L-What haveIlearnedbysolvingthis problem. (SMP 2)
Task 1-Present thefollowingproblem to the students inyourclass:
TheBoy Scouts wereplanning abreakfast intheschool gym. Therewere5 roundtables and4 square tables. 6 people cansit at each roundtable and4 peoplecansit at each square table. Howmany people cansit atallof the tables?(SMP 1,4)
In pairs, have students restate the problem in their own words to ensure comprehension of the question.Students then complete the K-W-Pportions of theKWPLchart in their journal. Afterstudents have completed theKWP sections of their chart, havestudents turn and talkto a neighbor about theirwork. Remind students that duringaturn/talkyou aresharingyourmathematical thinking and activelylistening toyour partner’s thinkingto help makesenseof theproblem (SMP 3). Theteacher’s role is to circulate and monitor thesediscussionsthroughout the classroom.Once students have shared their KWP,students will try theirplan. Havestudents directly model the action in the problemto find a solution. (SMP 1, 2, 4)Students will individually solve the problem first, then with a partnerand finallywith theirgroup. Students complete theLportion of theKWPLChart at thistime. A good strategyto use is 3-3-3.Students spend 3 minutes workingindividually, 3 minutes workingwith a partner, and 3 minutes working with theirgroup. This allows individual think timeas wellas support of theirteam to help students work through the problem solvingprocess.
After3-3-3, bringthe class back togetherwholegroup.Facilitate wholegroup discussion that should focus around howstudents mayhaveuseddifferent plans to solvetheproblem, how did the smallgroupdiscussion help justifycorrect solutions and make connections to other concepts. Inaddition, look formisconceptions students mayhavewhile solvingthis problem. You willuse this information on Day9.
Prior to students actually solving the problem, have students respond to the following prompts:
The troop leader had 50 seats. He says that is enough for the breakfast. Do you agree with his thinking? Why or why not?
Day 8: ProblemSolving Process
During Day8, studentswill engage in the problem solving process,understand what thequestion is asking in a problem situationand formulate aplan for solvingthe problem. Agood strategyto useto help students makesenseofthe problem is aK-W-P-Lchart.K-What doIknowabout theproblem,W-What doIWantto Know about the problem, P-What is myplanforsolvingthe problem,L-What haveIlearnedbysolvingthis problem. (SMP 2)
Task 1-Present thefollowingproblems to the students inyourclass:
In pairs, have students restate the problem in their own words to ensure comprehension of the question.Students then complete the K-W-Pportions of theKWPLchart in their journal. Afterstudents have completed theKWP sections of their chart, havestudents turn and talkto a neighbor about theirwork. Remind students that duringaturn/talkyou aresharingyourmathematical thinking and activelylistening toyour partner’s thinkingto help makesenseof theproblem. (SMP 3) Theteacher’s role is to circulate and monitor thesediscussionsthroughout the classroom. Once students have shared their KWP, students will try theirplan. Havestudents directly model the action in the problemto find a solution. (SMP 1, 2, 4)
Students will individually solve the problem first, then with a partnerand finallywith theirgroup. Students complete theLportion of theKWPLchart at thistime. A good strategyto use is 3-3-3.Students spend 3 minutes workingindividually, 3 minutes workingwith a partner, and 3 minutes working with theirgroup.This allows individual think timeas wellas support of theirteam to help students work through the problem solvingprocess.
After3-3-3, bringthe class back togetherwholegroup.Facilitate wholegroup discussion that should focus around howstudents mayhaveuseddifferent plans to solvetheproblem, how did the smallgroupdiscussion help justifycorrect solutions and make connections to other concepts. Inaddition, look formisconceptions students mayhavewhile solvingthis problem. You willuse this information on Day9.
Day 9-Reflect andAnalyze Misconceptions
Based on the observations from Day 7 and8, highlightcommon misconceptions thatyou saw whilemonitoring the classroom. Rewrite themisconception thatyousaw multiplestudents makingeither onapieceofchart paper ordisplayedunder theElmo. Have students work in pairs to analyzewhat themisconceptionwas andreflect and record the errors in theirjournal. Bringpairsback to their respectivesmall groups and havestudents sharetheirreflections byusingtheKaganStructureRound Robin.
Randomlychooseateammateto go first (example: student with the longest hairgoes first).
In teams, participantsrespond orallyidentifying possible misconceptions.
Each student takesabout 1 minute to share.
Once allsmall groups haveshared, allow onerepresentative from eachgroupto sharethemisconceptions theirgroup noticed to theclass. Facilitatediscussion byusing AccountableTalk structures. “Doyouagreewith that misconception and why?”“What other misconceptions might thestudents have?”
After students have identified possible misconceptions, work as a class to generate an anchor chart on “Ways to prevent misconceptions.”
Ways to Avoid MisconceptionsReread the problem
Use tools to model the action
Ask myself, “Does this make sense?”
Make a KWPL
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Days 10, 11…
Continuingto buildmathcommunity
9 dayswillnot be enough to fullyestablish themath communityin a classroom, it willtake much longer. Continueto usethe skills presentedhereand thestandardsformathematical practiceto build acommunityof problem solving,collaborative, mathematicallyinclined thinkers.Also continueto add new tasks for students such as creating goals, chartingscores, flexiblethinking, Shareand Compare, Kagan strategies andotherideas forbuildingcommunity. As new students join the class,theywillalso haveto be introduced into what is expected in this particular math class’ community. Continueto refer to theGlobal Concept Guides in the instructional guideline for ideas.
****Asyou work through the First 9 Days of Mathematics, besureto practice classroom norms and classroom routines. Theselearningtasks aredesigned to help students practice mathematical classroomnormsand routines.****
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