Whole Class Discussions
Facilitating instead of Evaluating:
I believe that students should not only create their own mathematical ideas but also gain the confidence to evaluate for themselves whether their reasoning is correct or not. It is my job to support them in gaining that confidence instead of evaluating their thinking myself.
This semester we are video taping our IBL classes and as I am watching the videos I am reflecting (again) on all the pieces necessary for a productive whole class discussion. My goal for a discussion is to make the “Big Mathematical Ideas” visible by having students construct connections between different solution strategies or attempts. (For more goals of whole class discussions, download Goals for Whole Class Discussions)
I will try to organize the context and some of the key pedagogical ideas in a list so that it is easier to see the structure:
- Group Work: Groups of about 4 students work on the investigation(s) and find examples, conjectures and proof ideas (see the blog Our Inquiry-Based Classroom).
- During the group work I walk around, observe and support the groups using careful questioning (see the blog The Art of Asking Good Questions). My hope is to have groups present different solutions and so I will use my questions to support the groups in pursuing and completing their different strategies. It is amazing to see that for every problem there are at least two very different approaches in the class!
- While I am observing the groups I am already thinking about which groups I want to ask to share out and which order would work best. Sometimes it helps to start with a strategy that did not succeed and the other groups can add on. Sometimes it is helpful to compare and contrast two very different approaches.
Sometimes I let only one group share out and ask the other students to decide if they agree or disagree. Sometimes I ask two groups to share out that have contradicting results.
- The word share is a bit misleading since there is way more happening than just one group sharing their thoughts while the other students are listening (or not?). The goal is to get students to discuss the strategy that was shared, and compare it to their own thinking. During the discussion I am often using talk moves that I learned from Suzanne Chapin’s book “Math Talk” (Five Talk Moves).
- Wait time. Doing nothing is often the most difficult for me. It also has the most amazing results.
- Revoicing. I repeat what a student said in my own words or I have another student repeat what a student said. I do this all time; it is fascinating to see how much better students listen to each other if I make clear that listening is important to us.
- Agree or Disagree. After a student or group make a statement I can ask the other students if they agree or disagree and why. Rather than making someone “right” or “wrong” this provides more neutral language for discussing a mathematical idea. I want the whole class to decide if something is correct or not (instead of me). I only jump in (with a question) if all the students agree with an incorrect statement.
- Adding on. When a discussion stops or seems finished I like to ask if anyone wants to add on. This opens the conversation up to either more mathematical ideas or a reflection on what just happened. It certainly makes the discussion richer for me.
- After a bigger investigation (for instance a proof) is completed, all students write up their thinking as graded homework. These are real essays, not just lists of computations. See Proof as Sense-Making for some examples of student work.
Videos: Talk Moves in the Classroom
In practice, several talk moves and other pedagogical moves can be used together. The following videos from our classrooms illustrate some of our moves:
Video 1: "Revoicing & Recording"
The students are trying to prove why the finger trick of determining the multiples of 9 works.
We see one student explain her thinking. I ask if someone can repeat her reasoning. As the second student revoices, I silently represent her thinking on the board. Notice that I do not evaluate in either case whether the thinking is correct or complete. Instead I continue with a new question that connects this idea back to the original problem.
Video 2: "Repeat or Ask Questions"
Students are investigating Salsa Rueda dancing. The question is how to predict if you will dance with every leader/follower given $n$ pairs and only dancing dame $k$ (Salsa Rueda chapter in the Dance Book).
Prior to the scene in the video, I asked a student to share her conjecture with the class because she had done the greatest number of examples and I knew that her work would be a great resource for the class. We see the student explain her conjecture. I then ask whether somebody in the class can repeat her conjecture or ask her a question, which leads to an interaction between the students.
Notice that I am not involved in this interaction other than gently suggesting a particular example to look at.
Video 3: "Wait Time & Agree/Disagree"
For homework, students were trying to find an equation that connects two whole numbers, their least common multiple and their greatest common factor.
The video begins with one student sharing her findings. I revoice and represent her thinking. In her explanation, the student has made a distinction between the larger and the smaller number. Even though I know that that is unnecessary, I go along with her thinking. Another student asks "Does it matter?" and again I give that question back to the class without evaluating.
Then I'm asking other groups to agree or disagree with the first conjecture. I gave some wait time, then ask for questions and examples. The next student shares his equation, which is a minor (from my point of view) algebraic variation of the first equation. Again I go along with the student's suggestion. Later on, comparing the two versions leads to the students understanding (independently of me) that the distinction between larger and smaller number does not matter.
Video 4: "Show of Hands"
Students have been exploring the $3a + 5b$ problem: Determine all possible values generated by the Diophantine equation $3a+5b$ when $a,b ≥ 0$
(see the blog $3a+5b$ Proofs which contains a complete video of her explanation).
Even though the proof is still incomplete, I comment that the proof idea is beautiful. Next, I ask: Does it make sense to you? Are there questions? Normally I would give wait time here but since class is almost done, I use a show of hands to monitor understanding.
References and Resources
Chapin, S. H., O'Connor, M. C., & Anderson, N. C. (2009). Classroom discussions: Using math talk to help students learn, grades K-6. Sausalito, Calif: Math Solutions.
A summary of Chapin's "Talk Moves" is included here for reference.
"""Kusababisha badala ya kutathmini: Ninaamini kwamba wanafunzi wanapaswa si tu kuunda mawazo yao wenyewe ya hisabati, lakini pia kupata ujasiri wa kutathmini wenyewe kama hoja zao ni sahihi au la." Ni kazi yangu kuwasaidia kupata uhakika huo badala ya kuchunguza mawazo yao mimi mwenyewe. Katika kipindi hiki cha semesters, tunajifunza video za kila siku na tunatumia video za kila siku kuelezea mambo muhimu ambayo yanahitajika ili kuunda mazungumzo mazuri. Lengo langu kwa ajili ya majadiliano ni kufanya <unk>Big Mathematical Mawazo<unk> inayoonekana kwa kuwa na wanafunzi kujenga uhusiano kati ya mikakati tofauti ya ufumbuzi au majaribio. "Kwa sababu ya kuvutia kwa wanafunzi wa darasa la kwanza, ""Mafunzo ya Darasa la Kwanza"" ni orodha ya masomo ya msingi ya masomo ya darasa la kwanza, ambayo inaonyesha jinsi ya kuunda na kuunda masomo ya darasa la kwanza, na jinsi ya kuunda masomo ya darasa la kwanza." - Wakati wa kazi ya kikundi mimi kutembea kuzunguka, kuchunguza na kusaidia makundi kwa kutumia maswali ya makini (tazama blog Sanaa ya kuuliza maswali mazuri). Natumaini kuwa makundi yatatoa suluhisho tofauti na hivyo nitatumia maswali yangu kuunga mkono makundi katika kufuatilia na kukamilisha mikakati yao tofauti. Ni jambo la kushangaza kuona kwamba kwa kila tatizo kuna angalau mbinu mbili tofauti sana katika darasa! - Wakati mimi ni kuchunguza makundi mimi tayari kufikiri kuhusu makundi ambayo nataka kuuliza kugawanya na utaratibu gani kazi bora. Wakati mwingine ni muhimu kuanza na mkakati ambao haukufanikiwa na makundi mengine yanaweza kuongeza. Nyakati nyingine ni muhimu kulinganisha na kupinganisha mbinu mbili tofauti sana. Wakati mwingine mimi huwa na kikundi kimoja tu cha wanafunzi na kuwauliza wengine waamue kama wanakubaliana au hawapatani. Wakati mwingine mimi huwauliza vikundi viwili kushiriki kwamba kuwa na matokeo yanayopingana. "Neno ""share"" ni la kupotosha kwa kuwa kuna mengi yanayofanyika kuliko kikundi kimoja tu cha kushiriki mawazo yao wakati wanafunzi wengine wanasikiliza." Lengo ni kuwafanya wanafunzi wazungumze juu ya mkakati ulioshirikiwa, na kulinganisha na mawazo yao wenyewe. Wakati wa mazungumzo mimi ni mara nyingi kutumia mazungumzo hatua kwamba nilijifunza kutoka Suzanne Chapin ya kitabu <unk>Math Majadiliano<unk> (Five Majadiliano Hatua). - Kusubiri wakati. Mara nyingi siwezi kufanya chochote. Pia ina matokeo ya kushangaza zaidi. - Kurejelea. Ninakiri kile mwanafunzi alisema kwa maneno yangu mwenyewe au nina mwanafunzi mwingine kurudia kile mwanafunzi alisema. "Ninafanya hivyo wakati wote; ni ya kuvutia kuona jinsi wanafunzi wanavyosikiliza zaidi ikiwa ninaeleza wazi kwamba kusikiliza ni muhimu kwetu.""" - Kukubaliana au kukataa. Baada ya mwanafunzi au kikundi kutoa taarifa naweza kuwauliza wanafunzi wengine kama wanakubali au hawakubali na kwa nini. Badala ya kufanya mtu <unk> haki <unk> au <unk> makosa <unk> hii hutoa lugha zaidi neutral kwa ajili ya kujadili wazo hisabati. Nataka darasa lote kuamua kama kitu ni sahihi au si (badala yangu). Mimi huingia tu (kwa swali) ikiwa wanafunzi wote wanakubaliana na taarifa isiyo sahihi. - Kuongeza juu. Wakati mazungumzo yanapokoma au yanaonekana kuwa yamekwisha napenda kuuliza kama kuna mtu yeyote anayetaka kuongeza. Hii inafungua mazungumzo hadi ama mawazo zaidi ya hisabati au tafakari juu ya kile kilichotokea tu. Kwa hakika hilo hufanya mazungumzo yawe yenye kufurahisha zaidi kwangu. - Baada ya uchunguzi mkubwa (kwa mfano uthibitisho) kukamilika, wanafunzi wote kuandika juu ya kufikiri yao kama kazi ya nyumbani graded. Hizi ni insha halisi, si orodha tu ya hesabu. Angalia Uthibitisho kama Sense-Making kwa baadhi ya mifano ya kazi ya mwanafunzi. Video: Mazungumzo Hatua katika darasa Katika mazoezi, kadhaa mazungumzo hatua na hatua nyingine za kielimu inaweza kutumika pamoja. "Kama ilivyoelezwa katika video ya ""Kuchunguza na Kuandika"" wanafunzi wanajaribu kuthibitisha kwa nini ujanja wa vidole wa kuamua mara nyingi za 9 hufanya kazi." Mmoja wa wanafunzi hao anaeleza jinsi alivyofikiri. Nauliza ikiwa mtu yeyote aweza kurudia hoja yake. Mwanafunzi wa pili anaporudi, mimi huwakilisha kimya-kimya mawazo yake kwenye ubao. Angalia kwamba mimi si kutathmini katika kesi yoyote kama kufikiri ni sahihi au kamili. Badala yake mimi kuendelea na swali jipya kwamba unaunganisha wazo hili nyuma ya tatizo la awali. "Mfano wa ""Soma na Ujifunze"" ni mchezo wa kuigiza wa Salsa Rueda." "Kama unavyojua, kuna aina ya dansi ambayo hufanywa na ""mwisho wa kila kiongozi na mfuasi"" (kwa mfano, katika sura ya Salsa Rueda katika Kitabu cha Dansi)." Kabla ya tukio katika video, niliuliza mwanafunzi kushiriki conjecture yake na darasa kwa sababu alikuwa amefanya idadi kubwa ya mifano na nilijua kwamba kazi yake itakuwa rasilimali kubwa kwa darasa. Tunaona mwanafunzi akieleza dhana yake. Kisha nauliza kama mtu yeyote katika darasa anaweza kurudia dhana yake au kumwuliza swali, ambalo husababisha mwingiliano kati ya wanafunzi. Angalia kwamba mimi si kushiriki katika maingiliano haya zaidi ya polepole kupendekeza mfano maalum kuangalia. "Kama ""Kufanya kazi ya nyumbani"" wanafunzi walijaribu kupata equation ambayo inaunganisha namba mbili kamili, ndogo yao ya kawaida mara nyingi na kubwa yao ya kawaida sababu." Video hiyo huanza na mwanafunzi mmoja akishiriki matokeo ya utafiti wake. Mimi huonyesha tena na kuwakilisha mawazo yake. Katika ufafanuzi wake, mwanafunzi huyo amefanya tofauti kati ya nambari kubwa na ndogo. Ingawa najua kwamba hilo si la lazima, mimi hufuata maoni yake. Mwanafunzi mwingine anauliza "Je, hilo lina maana?" Na tena, nitarudi na swali hilo kwa darasa bila kuchunguza. Kisha ninauliza makundi mengine kukubali au kukataa dhana ya kwanza. Nilimpa muda wa kungojea, kisha nikauliza maswali na mifano. Mwanafunzi anayefuata anashiriki equation yake, ambayo ni ndogo (kutoka mtazamo wangu) algebraic tofauti ya equation ya kwanza. Kwa mara nyingine tena ninakubaliana na pendekezo la mwanafunzi huyo. Baadaye, kulinganisha matoleo mawili inaongoza kwa wanafunzi kuelewa (bila mimi) kwamba tofauti kati ya idadi kubwa na ndogo haina maana. "Kama ""Mfano wa Diophantine"" (kwa Kiingereza: Diophantine equation) ni mfano wa mfano wa Diophantine, unaoonyesha jinsi ya kuunda hesabu ya thamani ya 3a + 5b, ambayo inahusu thamani ya jumla ya 3a + 5b, na kwa hivyo inahusu thamani ya jumla ya 3a + 5b." Hata ingawa uthibitisho bado haujakamilika, mimi maoni kwamba uthibitisho wazo ni nzuri. Halafu, nauliza: Je, unaelewa? Je, kuna maswali? Kwa kawaida ningewapa muda wa kungojea hapa lakini kwa kuwa darasa linakaribia kumalizika, ninatumia kuinua mikono ili kudhibiti uelewa. Maelezo na Rasilimali Chapin, S. H., O'Connor, M. C., & Anderson, N. C. (2009). Mazoezi ya darasa: Kutumia mazungumzo ya hisabati kusaidia wanafunzi kujifunza, madarasa ya K-6. Sausalito, Calif: Suluhisho za Hisabati. muhtasari wa Chapin ya "Talk Moves" ni pamoja na hapa kwa ajili ya kumbukumbu.