Week 2: Introducing the Five Practices

14 Jan 2014 10:50 AM | Anonymous member (Administrator)

Week Two:  Introduction and Chapter 1:  Introducing the Five Practices

Read pp 1-12  


Telling would appear to be a more efficient means of communicating to students what they need to know.  What are the costs and benefits of learning through discussion of student generated solutions versus learning from carefully constructed teacher explanations?


  • 15 Jan 2014 1:08 PM | Anonymous member
    New research in Mathematics Content Knowledge [MCK] shows that teachers need to know not only content, but how to communicate and reason with the content, use models to help with the communication, know when a student will find content difficult and know how to redirect learning through discussion when errors are made.

    The CCSS also has mathematical practices that emphasize this same content knowledge and puts it in terms of student behavior that will result from such MCK in teachers.

    We are now more aware that math is a literacy and should be learned the same way a language is learned. This takes looking at connections, seeing patterns, using the math in a problem and discussing and explaining what is happening. This process will be varied for students and the same path to knowledge will not fit all. So, the 'well constructed problem' will only be illustrative to certain students.
    The benefits to this MCK approach is a richer understanding for students and one that they own on their own terms.
    The difficulties are helping teachers to acquire MCK and manage instruction to reflect this knowledge. This instructional approach is more individualized and will require varied groupings of students working at different levels.
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  • 15 Jan 2014 1:18 PM | Anonymous member
    The most obvious benefit of learning from carefully constructed teacher explanations is that it takes much less time. Of course, we all know this is a very poor reason to choose this option. On the first page of the introduction to "5 Practices for Orchestrating Productive Mathematics Discussions" we are told "Research tells us that complex knowledge and skills are learned through social interaction (Vygotsky 1978; Lave and Wegner 1991). In order to have meaningful social interaction about mathematical ideas students would need to have worked with a problem and generated some ideas, strategies, and solutions. It would be hard to lead a good discussion in which students are discussing ideas that they are not invested in, that have been given to them directly by the teacher.

    In a recent keynote speech Steve Leinwand shared that the idea that "Teaching by “showing and saying” is just another way of doing “In one ear and out the other”. If I teach by solely lecturing I do believe many of my students would be able to do well on an assessment given shortly after the lecture, however, they would not retain the material, and the questions that needed a deep understanding of the material would be much more challenging for them. Students need to wrestle with the mathematics of a problem.

    I am drawn to the idea that balance between student authorship and accountability for mathematical ideas is hugely important in instruction. While direct telling eliminates student authorship, I do need to be very aware of the goals for my lessons in order to direct the work and conversations to help students be accountable for the mathematics.
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  • 17 Jan 2014 2:04 PM | Deleted user
    This is a very interesting question as I have conducted 6 grade level workshops this week on the 8 Mathematical Practices. MP3- Construct viable arguments & critique the reasoning of others created quite a discussion amongst the groups. Some felt that there wasn't enough time for this to happen while others found it quite valuable.

    In my 3rd & 4th grade classes, I quickly modeled a strategy to my students, and then asked them to try using it on one of their problems. Later during whole class discussions, I found that the students listened to each other and LEARNED from each other during this time. They were able to critique each other's reasoning, and often found their own mistakes while explaining a strategy to others. I chose several strategies for students to model to the class, and later we would discuss what the most efficient strategy used. I would guide my students to the most efficient strategy if it was not chosen. They felt some ownership in the process.

    This was time consuming, but I feel that the students were able to reason more, and many students were more willing to take risks and try new strategies because their classmates did. The classroom teacher has to set up a safe-risk free environment so that students would be willing to share their thoughts or ideas- right or wrong. I often asked why or how come questions whether a student was right or wrong. In the beginning, students immediately thought they were wrong & would change their answer. My next statement was the either why or how come & I told them that I would always ask that and it did not mean they were wrong- I just wanted to understand their reasoning. As the year progressed, the students were able to not only orally explain their reasoning, but also write it. I think learning through discussions is a very powerful tool that is underutilized.
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  • 17 Jan 2014 9:04 PM | Deleted user
    Learning through discussion changes the role of the student from receiver to maker, and it changes the role of the teacher from giver to guide. The cost of this is the time it takes to carefully orchestrate discussion-based instruction, as well as the time it takes to build a community of learners who feel safe enough to speak up and take risks, and who have learned to trust the guide who is encouraging them to go bushwhacking through ideas that are new them. 

    The opportunity cost appears high if we say that more content could be covered in the time that designing and implementing inquiry/discussion-based instruction requires with its attendant strategies and practices. For example, the authors note the critical absence of filtering, highlighting, drawing connections, and weighing in David Crane's lesson (5). A direct, telling kind of instruction doesn't require those practices in order to be effective, though the results will not be the same. The benefit of discussing student generated solutions is the likelihood of teaching students how to be creatively responsive to the "unstructured, messy challenges of today’s world" (1). The cost of telling is the risk of setting students into a rigid posture from which they will have difficulty reacting to new and unexpected problems, having never learned to follow a teacher's footsteps into that wild before.
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    • 20 Jan 2014 2:01 PM | Deleted user
      Barry, I totally agree with your response and like the way you state it. "Teacher from giver to guide....is a good description of a teacher's role.
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  • 19 Jan 2014 9:51 AM | Deleted user
    Because finding the time to include discussions that support critical thinking and to adequately cover content is a very real challenge to classroom teachers, guidance on how to find balance would be valuable. For example, weaving in brief "turn and talk" opportunities and asking for evidence of thinking throughout a lesson can support reasoning while moving through a lesson. There are times when a teacher needs to give room for exploration and "messy" discussions; how often? in what situations? with what ideas? for how long? I really like Smith and Stein's suggestion in the "monitoring" section (p. 9) for teachers to "create a list of solutions that he or she anticipates students will produce."
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    • 19 Jan 2014 10:10 AM | Deleted user
      I agree with you, Anne, about the benefits of teachers anticipating and monitoring of student solutions. Doing so can make the discussions more efficient. Teachers planning to use discussion will need to be in charge of the flow of ideas during the discussion, and having thought about the ideas students are likely to bring to the discussion can save teachers from having to process those ideas during the discussion if teachers have processed the ideas beforehand.
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  • 19 Jan 2014 10:04 AM | Deleted user
    Telling students what the teacher wants them to know fails to acknowledge what students already know. Telling almost assumes that the students are blank slates. But students are not blank slates. Some know already what the teacher might tell. Others might not have the required prerequisite knowledge that would help them to understand what the teacher might tell.

    Discussion has advantages. Teachers can assess student understanding by listening to student explanations. Other students can benefit from discussions by listening to and thinking about others' explanations. Students can benefit from generating their own explanations since formulating and expressing thoughts aids reasoning and memory. Students also gain language concepts through discussion. Social skills also improve.

    Costs of using discussion might relate to the time it takes to develop and deepen learning of concepts. More time is likely to be spent on discussing, but the benefit should be deeper student learning. This might result in less time being needed for review and reteaching, saving time in the end.
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    • 20 Jan 2014 2:15 PM | Deleted user
      More time up front teaching and using discussion, can save time down the road, especially for auditory learners.
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    • 21 Jan 2014 5:27 PM | Anonymous member (Administrator)
      Students are not blank slates, and will hold on to misconceptions if we don't know they have them, and can't address them. Whatever students bring, the learning evolves from there. We need to be careful to elicit what they know/believe and use that understanding to build deeper understanding.
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  • 19 Jan 2014 12:55 PM | Anonymous member
    "Without expert guidance, discussions in mathematics classrooms can easily devolve into the teacher taking over the lesson providing 'lecture' on the one hand, or, on the other, the students presenting an unconnected series of show-and-tell demonstrations, all of which treated equally and together illuminate little about the mathematical ideas that are the goals of the lesson." This statement on page 2 jumped out at me. There has been a huge push in the last few years to encourage student led discussions and teacher many teachers are eagerly jumping in. However, most are doing so without little thought about what the students should take away from the discussions. It seems so obvious that teacher should still be planning and preparing to help students take away some mathematical understanding from the lessons yet, in many cases, as was the case with Mr. Crane's class, it appears that the practice of discussing and and making mathematical arguments has become the focus of the lesson. I think the goal should be to use the rich discussions to bring out the particular mathematical understanding that is intended for the lesson.
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  • 19 Jan 2014 7:26 PM | Anonymous member
    I agree that students learn math best when they're allowed to construct the knowledge for themselves, rather than being told. And I know that when I've taught a lesson that's deeply familiar to me, I know what possible responses will come up and can indeed anticipate those responses and monitor for them. But what I've been seeing this year as we implement a new math program is that it's really hard to do this when every lesson is new. Teachers have to really understand the math in order to teach this way, and that can be a tall order. You have to know the point of the task, the possible responses from students, and where you want to go with the math in future lessons as well. And--then there's always the issue of classroom management: how do you create the climate that gets all students listening, as opposed to just a select few? Thoughts on this?
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    • 21 Jan 2014 1:42 PM | Deleted user

      I so agree with you in that teachers really need to know their math to understand the point of tasks and to also anticipate various student responses.

      Creating a climate of listening and questioning became a fun time in my 3rd grade classroom. It started out with students sharing at the end of the day while we were waiting for announcements. I used it as an oral speaking skills lesson. The speaker commanded his/her audience by waiting and looking around the room for all to settle. The audience looked at the speaker & didn't play with anything. I modeled this, and helped students during the first weeks of school.

      When we had whole class discussions in math, the person explaining his/her strategy commanded the audience- waiting for them to be ready to listen. After explaining the strategy, the speaker asked if their were any questions or comments. For the most part, I said nothing. Sometimes I had to guide the speaker to stay on track, but overall this was a very effective way of having discussions in math.
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    • 21 Jan 2014 5:24 PM | Anonymous member (Administrator)
      I hope that after reading this book and participating in the study I will know the Math more, and be more able to anticipate responses. I agree it sounds hard!
      Tips for getting students to listen:
      I point out my mistakes and make light of little ones, and apologize for big ones. If I can readily admit where I need to work, that is a role model for students.
      I celebrate risk-taking. I notice it and talk about students having guts and confidence, and generally build it up.
      I also insist on listening. I talk about how each person here is valuable and important. To show how important each one is, we listen. We respond to his/her ideas. We often repeat what the other one has said, and then agree or disagree, and say why.
      Building a culture of listening takes work, for sure.
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  • 19 Jan 2014 8:52 PM | Anonymous member
    Student discussions about how they solve problems have a multitude of benefits. Besides letting me know what they are thinking (correct or incorrect)they are practicing proving. Constructing viable arguments and critiquing the reasoning of others are skills we are developing in language arts, science and social studies as well as math. Right answers are only part of the class culture; supporting thinking with evidence and being open to more than one method of solving problems is another part.

    Discussion promotes use of and understanding of math language and vocabulary (with questioning and clarification by the teacher sometimes!)

    Another benefit is social interaction. Students connect with and learn from their peers through discussion. And, at the risk of sounding less than academic, discussions can be fun. We get to share those "ah-ha!" moments and help others understand.

    Costs are the investment in setting a safe tone in the class from the very beginning of the year and helping students (I work with 9 and 10 year olds!) learn how to have productive discussions. I'm beginning to see the benefit of investing more time in planning individual lesson discussions as well.
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    • 21 Jan 2014 5:15 PM | Anonymous member (Administrator)
      I agree that laying the ground work for a safe enviornment is huge. It is hard for kids to share.
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  • 19 Jan 2014 9:43 PM | Deleted user
    DONNA HARDT writes: I chose several strategies for students to model to the class, and later we would discuss what the most efficient strategy used. I would guide my students to the most efficient strategy if it was not chosen. They felt some ownership in the process.

    Inviting your students to evaluate the efficiency of different strategies supports development of their metacognitive processes. That kind of invitation is not usually a feature of direct instruction. Modeling the varieties that you did IS the DI part of your lesson and the blueprint for the discussion that the students built. When students examine and talk about different perspectives on ideas, they begin to see that mathematics is a creative and dynamic way of thinking.
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  • 19 Jan 2014 9:44 PM | Deleted user
    SARA JONES writes: Students need to wrestle with the mathematics of a problem.

    I like this imagery. Aside from the romantic notion of heated debates among artists and philosophers in Greek academies and Parisian cafes, it reminds me that while private struggle is always a part of learning, social struggle, of the sort that carefully guided discussion can yield, can inspire and reward collaborative effort. Open mathematics wrestling should be instigated as much as possible.
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  • 19 Jan 2014 9:46 PM | Deleted user
    JIM COOK writes: This might result in less time being needed for review and reteaching, saving time in the end.

    Yes. Especially if, as you point out, feedback from guided discussion is used to draw an accurate picture of where students are with their thinking. This is a lot of real-time analysis to conduct, which, as VICTORIA COHEN points out, will draw a lot of power from a teacher's well of understanding of both the material and her students' dispositions.
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    • 19 Jan 2014 9:52 PM | Deleted user
      I find that many teachers are afraid to allow the students to stray from the "how to" method because they do not feel that they have a good enough knowledge of the material themselves to know if there are other methods that work. I like Jim's words "guided" discussion -- allowing the students time to wrap their heads around different methods may save time in the long run as they will remember what was talked about and give them other methods to work with if they forget the "formula."
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  • 19 Jan 2014 9:48 PM | Deleted user
    "Telling" how to do a particular skill or answer a particular problem is fast, to the point, and will allow the students to memorize the method and move on. This does not allow the student to take ownership of his/her own learning...the student knows that as soon as there is a question the answer will be forth coming.

    Benefit - get through the material quickly and efficiently. The cost to the student is the lack of opportunity to see different ways that might also work for him/her. The child may forget the one particular method and then can not move on as the chance to explore possibilities, discuss methods, see what else is out there, and figure it out for him/her self has not been provided.

    Through discussion of problems the student is taught several of the math practices: perseverance, reason abstractly, construct viable arguments, and model with math. Through conversation a student can use his own ideas yet listen to others and refine and realize that the thinking is the most important. We do not solve problems in a vacuum in the "real world" - our students should be allowed to guide their peers and be guided by their peers.

    Jo Boaler's course "How to learn Math" (Stanford University) stresses the need to allow students to question, hypothesize, and discuss mathematics - even at the lower grades.
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  • 20 Jan 2014 2:12 PM | Deleted user
    Vygotsky states "after children master mental tools, they become in charge of their own learning..." Discussion is a mental tool needed by all. Teaching students to become effective at discussion takes time and practice, but is well worth the time. Some benefits of learning through discussion are students understand the concepts more deeply. They also take ownership of their learning.
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  • 20 Jan 2014 4:11 PM | Deleted user
    Telling would appear to be a more efficient means of communicating to students what they need to know. What are the costs and benefits of learning through discussion of student generated solutions versus learning from carefully constructed teacher explanations?
    Telling is always an efficient way to share knowledge. A benefit to this is learning to pass on information in a way that students can understand and remember. Telling is not always oral/auditory. It may use another of the senses such as a cartoon explaining a concept or an email reminder. Telling students information may create a comfortable structure in which they can prepare for a task such as telling when to begin and when to end etc.
    Telling is often the ground floor of beginning a class or a task. To be clear and succinct as well as mutli-modal is a good thing when delivering information.
    Telling is not the whole class but a way of making a framework for the class. As a class progresses, information cannot be internalized or understood as fast as it can be said. Telling is like putting eggs, flour and sugar on the table and expect it to become a cake. A whole lot more needs to be done to the materials you have laid out for it to become the desired product. Students use discussion and thoughtfulness to blend these ingredients and make something that is uniquely theirs. Of course sometimes they end up with cupcakes and then our job is to guide them towards the desired outcome.
    Guiding a discussion is a lot more work than delivering one. If the only ideas delivered are the ones from the teacher then they can be sure of what they are getting. It is a lot more demanding to try to shape discussions than to lead them. This is a challenge many teachers do not feel comfortable doing if they are not math majors themselves. The use of mathematical pushes everyone to put their toe in the water and get started.
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    • 21 Jan 2014 5:13 PM | Anonymous member (Administrator)
      I agree that the internalization is crested when students have an opportunity to create the recipe (as per your example). The challenge is how to guide the discussion.
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    • 26 Jan 2014 6:30 PM | Anonymous member (Administrator)
      I agree with Mary about the fact that guiding a discussion is a lot of work and definitely takes more time and perhaps thoughtful use of class time, than a telling lesson. I think there can be some good "telling" lessons but I am convinced through my years of learning and teaching, that learning through doing and discussing of work will be retained longer by the students.
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  • 21 Jan 2014 7:50 AM | Deleted user
    I think that both methods present pros and cons and that one method may not be best for every group of students. Having students construct their own learning is a great way to teach math. It allows the students to grapple with a problem and, as a group, find a solution that works. It allows students to understand that there are multiple ways to find an answer and what works best for Johnny may not be what works best for Susie. On the other hand it is very time consuming, and scary for teachers that are not comfortable with math concepts. It can leave very little room for students to practice the concepts in class with the teacher available to offer help to those that may still be struggling and could really slow the pace of the class.
    On the other hand teacher directed instruction also has it's pros and cons. Being able to help students construct meanings as a whole class has it's benefits. The teacher can control the pace, the teacher can make sure that all students are participating, the teacher can make sure that there is time available in class to practice. The cons are that it's mostly the teacher asking the questions and the students responding with their ideas, there isn't much opportunity for the kids to grapple with the concept and come up with their own strategies, and it can offer one way of finding the solution to a question instead of multiple.
    I think it is important to note that no one strategy works for every group of students. The dynamics of a class my dictate one method or another. The reality of education is that as much as we would all like to have our math students learn through struggle and self realization of math concepts it isn’t always the case.
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    • 21 Jan 2014 11:53 AM | Deleted user
      One of the great tasks for every teacher each year is to "learn" their students. Who retains best from modeled example, who figures it out from prior experience, who needs to feel it with manipulatives, who understands better when he/she talks it out, who needs more time to think? One of the central tenets of Smith & Stein's book is to study the students as they process a mathematical problem and design specific strategies and questions to reflect each child's thinking/problem solving; and then strengthen those attributes through discussion. As Erik points out, "no one strategy works for every group"; and Donna says she uses several strategies for students to model. These reflect understanding of the children in those classes. For the teacher who uses "telling" as the vehicle for most of the instruction, although we all agree that it is efficient, it tends to leave little room for dissecting and using the individual learning attributes of each child. I do agree with Mary Belisle that telling can be the floor of beginning a task - students need to know where they're headed and why. The power of Smith & Stein's model lies in collective genius - the way that each child thinks through a problem; or the way that several students think through a problem together - and then discuss it, model it, critique it, connect it )to previous learning or experience in the real world.
      I would share with all that there are two excellent books which are companion pieces to this one - Kassia Wedekind's, Math Exchanges and Tom Crpenter's, Children's Mathematics. Both have excellent examples of question types, strategies for discussion that lead to rich learning, and the rationale for student discussion to develop deep reasoning.
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  • 21 Jan 2014 5:09 PM | Anonymous member (Administrator)
    The cost is time for the discussion. At first it seems like discussion takes a long time. It is also a lot of work for students to learn to truly listen to one another, and honor each other's contributions. It takes time to create a community of risk takers. Discussion also can take teachers off course, and put students in control of the lesson, unless teachers have strategies to guide the discussion. I am looking forward to continued reading, because I hope to learn more of those strategies.

    The benefits far out weigh the costs. The time it takes for discussion, is time of real student learning. Most of us have to speak to learn. We need to verbalize- and when we give children the opportunity, much deeper learning occurs. The time spent, is really time saved in reteaching and reteaching. Students also grapple with each others ideas more readily, than with a teacher's declaration.
    Teacher explanations are likely to only rest a short time between ears, but student's speaking for themselves requires thinking, constructing and communication.
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    • 21 Jan 2014 5:57 PM | Anonymous member
      Good point about the cost being time used in the classroom. I think this book will provide a philosophical shift in our thinking as adults, that it may take time to engage students in discussions, but the long-term benefits will be seen in our students knowledge of math skills and concepts.
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  • 21 Jan 2014 5:54 PM | Anonymous member
    When I think of telling, I think of an image of a cartoon where the teacher is talking and all of her students have a different image in their thought bubble. As teachers we want to build students math conceptions alongside their prior knowledge. When we tell students it may enter their short-term memory, but students are less apt to be able to use the skills weeks from now and certainly do not have much ability to apply it to real life math situations. Engaging in discussions can feel dangerous and I find myself continually asking are the students engaged and are they learning. Our students benefit from teachers providing high level tasks and student generated solutions either proving or disproving the math.
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  • 26 Jan 2014 6:27 PM | Anonymous member (Administrator)
    What are the costs and benefits of learning through discussion of student generated solutions versus learning from carefully constructed teacher explanations?
    The benefits of learning through discussion greatly outweigh the costs. I see the major cost of students discussing their solutions and critiquing that of their classmates is the time it takes to do this well. The way a teacher solves a problem may well not be the way several students approach and solve a problem. It chapter 1 Smith and Stein present the Five Practices and provide support for doing each practice. Their reference back to the Leaves and Caterpillar task was helpful as a way to understand each of their descriptions of the practices. By having students share their thinking they are practicing the CCSS - Math Practice: Construct viable arguments and critique the reasoning of others. The teacher's role is to make sure that he/she has completed the task prior to the lesson and to use the student work time to develop a plan for which students will share, how they will share, and the order in which they'll share. Keeping in mind the end goal of the lesson.
    The major cost is that this type of mathematics learning takes more time than the teacher explaining his/her reasoning. This type of strategy does not allow for the students to make sense of their own reasoning and to question the reasoning of others.
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  • 05 Feb 2014 6:58 PM | Anonymous member
    I feel the costs would be the time it takes to go through discussions of student generated solutions. It is also more difficult to manage, in my case, middle schoolers if they are left to "discuss" with others in small groups. Not every student's learning style lends itself to this type of process. Some students process by themselves, others do not do well with groups, and not all students are kind to others during a process like this. Granted, the teacher is responsible for creating a culture of respect, but this can be problematics depending on the make up of the class. Silent students who are being told what to do are much easier to manage.
    The obvious benefit is that if students are thinking critically, then they are truly learning and not just regurgitating material told to them by the teacher. True learning also creates long term knowledge that can be built upon.
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