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Chapter 1 – Option 1:

01/04/2019 8:07 PM | Anonymous

Choose a phrase, sentence, or passage that you found meaningful or thought-provoking and tell why.

Comments

  • 01/05/2019 8:21 AM | Alison Foley
    On page 10, in the "Attend to Precision" chapter, I found the following sentence thought provoking: "Just as students need to think strategically about the tools they use when problem solving, students must build the habit of considering the appropriate level of precision for their math-doing." I agree with the authors that we tend to think about precision as accurate computations and perfect use of vocabulary, but it is interesting to think of different levels of precision needed depending on the task. Sometimes if students attend to precision too much in the wrong context, they can miss some of the big ideas.
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    • 01/06/2019 7:52 PM | Michele Hanly
      I agree that we do tend to think of "Attending to Precision" in connection with accurate computations and there certainly is a time and place for that type of accuracy. However, as many high school students are often under "time constraints" learning how precise they need to be for a particular task may save them valuable time.
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      • 01/09/2019 8:11 PM | Jami Packer
        This is especially true when dealing with standardized tests like the ACT. If the format of the question is multiple choice, sometimes a rough estimate gets you close enough to determine an answer. Even in basic life functions, I often estimate the costs of things before/after discounts rather than formally calculate them. Students should realize that this flexibility in precision is valuable.
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  • 01/05/2019 2:03 PM | Cortni Muir
    On page 17 in the section on growth mindset the line “instruction in which the teacher does the thinking by creating and choosing the problem-solving tools and in which the student does the calculating can result in students becoming dependent on the teacher to suggest the right tool for the job, putting the teach in the role of the GPS, telling the student what to do at every turn.” This really struck a cord for me. As I coach I often try to help teachers understand the importance of allowing students to work through problems with out so much teacher direction. Allow some productive struggle and often you will witness things you didn’t expect, this also goes along with wait time, which is something I struggle with myself. Eli Luberoff, CEO of Desmos, said it best at the ATMNE conference in Rhode Island “if you let students surprise you, they will!” We need to allow students to shine and be the star of the class, while the teacher takes on a bit more of a supporting role. I am a strong believer in growth mindset and helping students (and teachers) see that everyone can be a mathematician, which is why this part really spoke to me.
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    • 01/05/2019 6:08 PM | Nicole Gilson
      I totally agree with your thoughts and ideas. I also want to add that many of us get caught in "covering" the material for the "test" and we loose sight that we are building life long learners and mathematical thinkers. I love the quote from Eli Luberoff - very inspiring.
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      • 01/07/2019 8:21 AM | Anonymous
        Absoultely! I was there was more time (I know every teacher's dream). But unfortunately that pressure from testing is very real.
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        • 01/11/2019 9:29 AM | Cortni Muir
          The above comment was me (not sure why it was anonymous). It should have said.. Absolutely! I wish there was more time (I know every teacher's dream). But unfortunately that pressure from testing is very real. (original post was typed on my phone, that could have been the problem)
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    • 01/09/2019 8:24 PM | Jami Packer
      Hi, Cortni!
      I think we all struggle a bit with the "be less helpful" stuff, but I totally agree with you (and Eli) on its value. I think it will also helpful with SMP 1 if students have grown accustomed to operating in math spaces where they feel safe to try new things but aren't necessarily hand-held and guided through the process. The challenge for me is often when students are completely unaccustomed to that sort of environment and shift into learned helplessness mode quickly. How do we help to scaffold them into learning situations where they are going to have to carry the weight of thinking...it takes some adjusting...gradual norm-setting...constant rebooting. It's a definite learning process for me as well.
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    • 01/10/2019 9:11 PM | Anonymous
      This metaphor also struck me -- teacher in the role of a compass, pointing students in a fruitful direction, rather than acting as a GPS. I have observed that when students are allowed to interact with & reflect on problem situations freely first (e.g. "what do you notice? what do you wonder?) they might take a "scenic route" and uncover other mathematics than what the teacher intended... but still worthwhile.
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    • 01/11/2019 4:52 PM | Anonymous
      Cortni, I so agree with your post. I work with teacher candidates and strive to get through the math phobia right from the start. Most candidates are anxious about teaching even the most elementary concepts. They have had negative experiences that have resulted in a fixed mindset around mathematical practices. Their knowledge base of rote recall and repeating a procedure, traditionally has left them with low efficacy.

      We work all year to break down those walls so future teachers can experience and therefore provide opportunities to explain, reason and justify one's own thinking. I think by providing opportunities to verbalize, illustrate, create and build solutions in several different ways will provide "access and opportunity" in an equitable way. Hopefully, by experiencing various tools, strategies and manipulatives in class the self efficacy will become more positive and they will enjoy teaching mathematics to their own class someday.
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  • 01/05/2019 6:05 PM | Nicole Gilson
    The sentence I thought was most thought provoking was
    "the practices actually provide access and opportunity for all students to engage successfully with mathematics". I really identified with Ms. King's reaction in the beginning of the chapter. The reason why we teach is to try to reach all our students needs. I have a few students SWLDs and this statement gave me focus as to how to better utilize the mathematical practices. I was also encouraged after reading the following statement: "SWLDs benefit most from explicit instruction, opportunities to verbalize their mathematical reasoning, visual representations, and a range and sequence of mathematics examples".
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    • 01/10/2019 8:11 PM | Anonymous
      I also noted that sentence about "access and opportunity"; I hadn't previously thought about the MPs specifically (or mathematical reasoning and sense making in general) as an equity issue. Framing the MPs as an opportunity for all students and a lever for special populations (p. 14) was a new insight for me.

      Karen
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  • 01/05/2019 8:33 PM | Anonymous
    The diagram on page 3 caught my attention as I’ve not seen the mathematical practices organized in this way. Have MP 1 as the overarching practice is powerful. Comprehension and perseverance are key to all learning. Seeing how the other Math Practices fit together to support the learner and learning was easy to conceptualize.
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    • 01/12/2019 2:31 PM | Lisa Cosgrove
      I also think this visual representation is a powerful tool for students as it shows the interactions between the practices, instead of viewing them as separate entities.
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  • 01/06/2019 7:37 PM | Michele Hanly
    As a visual learner, I agree that "All students can benefit from building the capacity to explore and communicate mathematics using a range of modalities -- visual, auditory, kinesthetic and tactile (Page 14)." A diagram is often the missing key to understanding what the word problem is asking.
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    • 01/14/2019 2:20 PM | David Wees
      One thing I really like about Grace and Amy's work is that they don't try to tailor instruction to different learning styles but instead incorporate elements of support through different modalities and this then results in supports for all students.

      For example, we don't offer diagrams to some students and words to others; we offer both to all students. Over time, this helps students build on both their strengths and weaknesses.
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  • 01/09/2019 8:48 AM | Dave Johnston
    p. 14: "In service of making mathematics accessible to students who struggle, it is sometimes common for teachers to focus almost exclusively on skills development and break instruction down into small discrete steps to be memorized and repeated...it is a disservice to these students to not also provide them with opportunities to engage in higher-demand thinking"

    This idea shows up a few other places in this chapter as well. My role is to support and coach middle school teachers of who teach classes of 4-10 students with learning disabilities who are typically performing two or more grade levels below their peers. Some of my teachers are strong advocates for procedural instruction, key words, and memorization because they see immediate impact. The research and long-term results do not bear this out. It is so refreshing to see the authors beginning with the expectation that all students can learn math to high levels. Our struggling learners need rich instruction, let's explore ways to create those opportunities!
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    • 01/09/2019 8:30 PM | Jami Packer
      I am curious as to the background of the teachers you are describing. In my county, the teachers that work with those populations are often trained in learning disabilities but not in math pedagogy per se. It results in a default preference for procedural instruction much like what you are describing. I often suspect this is due to a lack of confidence in teaching strategies that are more rich and problem-based.

      I also agree, as you pointed out, that they often see (misleading) short-term success from such an approach perpetuating their assumptions about its benefits. [Or maybe I'm just projecting because I was once one of those teachers.]
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    • 01/10/2019 9:21 PM | Anonymous
      It can be surprising when you first read the cognitive science research about how immediate results from procedural instruction don't last.

      Here are some resources for this research if you wish to share with your teachers (or explore yourself):
      ---book "Make It Stick: The Science of Successful Learning" by Brown, Roediger, McDaniel
      ---website www.learningscientists.org/
      ---book and podcasts by Craig Barton "How I Wish I'd Taught Maths" and www.mrbartonmaths.com/teachers/ and www.mrbartonmaths.com/podcast/

      And, to Jami's point, lack of confidence is definitely a contributing factor when teachers are trying (or avoiding) new/unfamiliar teaching strategies.
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    • 01/10/2019 9:25 PM | Luke
      In my experience, I too see this immediate benefits of direct teaching and leading kids through the math. I agree more time needs to be spent accessing higher level math and connecting its purpose in a meaningful way for kids. In some cases, the struggle comes from the language surrounding the math. I tell my kids all the time that they can all do the math but it's the language that becomes the barrier for initial understanding and "entry into the problem (p. 4)." In some cases where kids are below grade level in math, they are also below grade level in reading too. Math instruction becomes multi-content instruction as one becomes a teacher of reading and math as you bring relevance to daily life.
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    • 01/14/2019 10:31 AM | Sara Kaminski
      I also highlighted this sentence...and I shared the chapter with our Special Ed department. It is so critical that problem solving and opportunities to grapple be for ALL!
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    • 01/24/2019 8:13 AM | Todd Butterworth
      This is a favorite idea of mine too. One of my former colleagues liked to talk about the idea of unitaskers, by which he meant tricks that will help in one situation, but not in others (things like FOIL, thinking about the area of a square as s^2, distance formula, that sort of thing). One of my goals for geometry this year in terms of how I'm teaching it is to make things all about triangles. When I'm teaching area, I'm going to try to get them to see how every figure can be broken into triangles. I'll show them formulas, but only after we've really seen how to make everything a triangle so they can see where formulas come from (formulas like the area of a trapezoid are a bit silly and definitely unitaskers, it's just two triangles, why make things scary?).
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  • 01/09/2019 3:19 PM | Jennifer Rianhard
    On page 15, in the Language Rich Environment section, the line that sticks out in my mind the most when I read this chapter was, " Students begin making sense making by verbalizing their thoughts in their own language." Having students think, pair, share or turning and talking benefits the students sense of their deeper knowledge of their math thinking. I have observed many math lessons at my school and I love to listen to the students turn and talk to their partner about a specific topic the teacher gave them. The students do such a nice job with their discussions and they use math vocabulary when communicating; it makes my heart happy. Students get a chance to share their ideas, but they could learn something new as well.
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    • 01/09/2019 8:34 PM | Jami Packer
      A quick personal tangent on your comment here. I frequently use think/pair/share and turn/talk in my classroom, but in all honestly, I am having a difficult time listening in. I don't think I have a hearing issue, but when there are conversations occurring all around me...I struggle to truly tune in to their individual conversations. I will catch a phrase or two that I can pull from when we regroup, but I was actually just thinking this week about how I could get better at hearing, and subsequently highlighting, some of those conversations (without resorting to a tape recorder and hours of at home review).
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      • 01/14/2019 2:25 PM | David Wees
        I have the same issue Jami, my hearing is just not all that good. I do three things to try to mitigate this issue.

        1. I don't try to listen in on all groups. If I want to make sure I get a chance to listen in on more students, I just try to listen in on conversations from different students on another day. That way I can lean in more and actively listen.

        2. Anticipation of what students may say is key! It helps make those moments when I do manage to overhear part of a conversation more productive.

        3. If I do need to ask for a volunteer to share, I sometimes will ask a question like, "Does anyone have a strategy where they ... ?" in order to get a specific strategy up as a support for students.
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      • 01/19/2019 3:22 PM | Amy Lucenta
        Hi Jami,
        Listening to complex ideas by many students at once is always a challenge, despite the integral role it plays in selecting and sequencing. Of course planning and anticipating is essential, but not sufficient. The barrage of student thinking can be overwhelming. The routine nature of instructional routines provides support to us as we work on listening in-the-moment because we repeat the routine, and can continue to work on this practice over time. And, of course, we develop strategies to support ourselves also.
        Thanks for your honesty!
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        • 01/21/2019 4:33 PM | Jennifer Rianhard
          Jami, I hear what you are saying. It is so hard to get to each group and have a listen to what they are discussing. I need to do a better job with it myself, especially when they are turning and talking about their own work. I have been giving word problems every other week and when they are discussing their strategies I am trying to listen to their thinking and I often forget to look at how they solved the problem so when it's time to share on the document reader I have a few students chosen already to share different work. I'm not sure if I should have a piece of paper to help me remember to do this or if anyone else has any ideas, but I do enjoy their math thinking and talk when I walk around in their small groups. Their math language has definitely strengthened due to using the same program for a few years.
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  • 01/09/2019 8:21 PM | Jami Packer
    The comment that struck me was in the section discussing Language-Rich Environments. In discussing when to push for precision and when to allow students space to formulate their math ideas without shutting down the process, the authors state "in other words, be particularly aware of the balance between the demands of mathematical thinking and reasoning, and the stresses of processing and producing the language." While I agree that this is particularly true of SWLDs and ELLs as the authors note, I've noticed that tension with other learners in my class as well and am constantly working to figure out that balance. More specifically, I am trying to figure out a bit how to more overtly articulate that distinction to students for times when they are giving feedback to one another. Often I will see peers pushing for precision in language at a time when a peer needs some space to express their thoughts. I believe that learning how to articulate that distinction in child-friendly language so that peers can give feedback that is both useful and respectful of each learner's journey would pay dividends in class discourse.
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  • 01/10/2019 8:06 PM | Anonymous
    This sentence from page 13 made me say WOW! “The math practices live not in the final answer a student gets for a math problem, but in the thinking and reasoning a student uses to arrive at a solution…”

    This really highlights how a focus on answer-getting (procedures, computational accuracy, & speed) alone is not enough as a math teaching goal. I have always required students to “show your work” for written problems, but I’ve made a shift in my wording in recent years to “show your mathematical thinking”. I’m trying to make that focus more prominent in my teaching and also give value to it in my grading (what gets graded is what gets valued).

    I’m hoping that the upcoming routines from the book will help me make mathematical thinking more central to my classroom practice than just what students write out on their test papers.
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    • 01/10/2019 9:11 PM | Luke
      I like this one too. I think kids lose something in between getting started and recording their answer. So often once they are able to get started as long as they get the answer all is good. I had a professor that used to say, "if you can't explain it, you don't understand it." It is so true. I sometimes lose sight of slowing down enough to really explore and highlight the underlying thinking and reasoning mentioned above. It is so critical for kids to communicate their thoughts well enough for them to refer to in future problem-solving.
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  • 01/10/2019 9:03 PM | Luke
    On page 4 the authors refer to the three avenues of thinking as "providing entry into and through all kinds of math problems." The entry part really spoke to me and stayed with me while reading chapter 1. Starting, or entering, a problem is an initial hurdle for kids to conquer and they may often need a push to get started otherwise they shut down and math is no longer the focus. Making sense of the problem will help as it provides the time learners need to brainstorm and have a clear understanding of the expectations of the problem. Hopefully, after this, if the math path is clear enough, the "entry" will be easier for the kids to demonstrate their understanding and communicate their understanding of math as it applies to the problem.
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  • 01/11/2019 11:14 AM | Anonymous
    I love the line in the very beginning: "The first is that the eight standards for mathematical practice articulate in the CCSS for Mathematics beautifully summarize the way mathematicians work."

    I think sometimes we forget that no matter what job we obtain in life, some sort of math is going to be required. I hate to hear from students, parents, or teachers say "I'm just not good at math". These math practices show that everyone can do these 8 tasks and can reason mathematics in an ever changing, puzzling mathematical society.
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    • 01/20/2019 2:49 PM | Sarah Giaquinta
      I often have students (and parents!!) tell me they aren't good at math. There is such a fear of being wrong, or thinking in different ways to get to the same idea. I'm hoping that with implementing these new routines, it becomes second nature for my students to persevere and not get scared when they are handed a challenge.
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  • 01/13/2019 9:35 AM | Allison Day
    One passage I found very meaningful what the paragraphs on Building on Learning Strengths. In the first line, it says, "Particularly, in mathematics, where the emphasis can so often be on correct answers, it is easy to focus exclusively on students' learning weaknesses, on correcting their deficits." We often try to develop what students don't know and don't focus on what they do. It's important to think about why they might not understand a concept and what is impacting this understanding. What are the skills needed before? What are the big ideas and understandings in the math we are trying to teach them? Taking the time to figure out what students know, helps us as teachers to uncover students' misconceptions and why students might think that way. Highlighting what students do know develops their confidence and encourages them to take more risks. Having taught two of the routines last year, students were identifying their own strengths. When looking at problems, students recognized how they were solving problems in similar ways each time and challenging themselves the next time to use a different approach or strategy. The reflection part of the routines is vital to understanding students' strengths.
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  • 01/14/2019 10:29 AM | Sara Kaminski
    I found SO much from p. 11-17 related to the high leverage strategies we are using in my school. A big focus this year is on meeting the needs of ALL students, and allowing for ALL students to be engaged in complex tasks, not just those who are finished early or need extension. The information on how the practices and problem solving strategies support ELL's and SWLD's validated that for me. I liked the idea of the teacher being the COMPASS, rather than the GPS... rather than providing step by step, small discrete skills support to struggling students, the teacher can highlight different modalities (CRA) to provide an entry point into the situation.
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  • 01/15/2019 4:15 PM | Anonymous
    On page 8, I found the following passage thought provoking: "However, even though one avenue may feel more comfortable than another, it is crucial that students develop facility with all three of these types of thinking if ther are to develop the tenacity it takes to persevere through complex math problems where problem-solving dead ends or computational road blocks require problem solvers to try another avenue." I agree with the authors that while students are more comfortable with certain strategies, it is important that they have the tools to be successful. I am strong believer in showing students various ways and techniques to solve problems. The more strategies that students have to solve a problem the better chance they have at being successful.
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  • 01/19/2019 1:02 PM | Michele DeMaino
    On page 17, "Instruction in which the teacher does the thinking by creating and choosing the problem-solving tools and in which the student does the calculating can result in students becoming dependent on the teacher and suggest the right tool for the job, putting the teacher in the role of the GPS, telling the student what to do at every turn. Instead, teachers who are in the role of the compass, pointing the student in a fruitful problem-solving direction and reorienting as necessary, can begin to foster the math practices in their students."

    This make me think because even when I think I'm giving my students real world thought provoking problems for students to work through, they still need me to introduce the problem, clarify directions and some students become dependent on me as the teacher to guide them during the lesson. By incorporating these routines throughout the year, I want my students to become more independent and be able to use these thinking strategies to think as a mathematician to work through problems.
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  • 01/20/2019 10:37 AM | David Shimchick
    Respondents already mentioned a couple of the sentences which resonated deeply with me in the thread. So I will share I am intrigued by the following sentence because I am eager to see how the routines are structured and implemented in order to achieve what they aim to do: "Each routine is designed to emphasize the interconnectedness of mathematics and to provide a structure that helps students learn how to intentionally reason along a different avenue of mathematical thinking."
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  • 01/20/2019 2:45 PM | Sarah Giaquinta
    One section that I found particularly powerful was about the use of language and the understanding of the mathematics being produced. On page 16 "the goal is to build conceptual understanding and language production" really brings together two huge ideas. Sure, it is one thing for students to be able to do math, reproduce what they have seen performed, but to be able to talk about it in a meaningful way and truly understand what they are doing is a whole other goal. On the first day of school, I explain to my students that class participation is so important, and I always get a few kids who tell me they aren't comfortable saying answers in front of the class. After realizing that this is what many students assume participation is, I now have a rubric that helps explain what it means to me to be an active participant in the learning of our classroom. We have to get the idea that math is right and wrong out of both kids and their parent's heads! Helping to facilitate discussions about what they are learning will enrich their level of understanding so much more! It is one thing to be able to "do" math, but there are so many more levels to learning math than that! They need to learn the language of the content so they can more specifically describe their thought process. They need to be given opportunities to discover their own ideas about material they are learning. As educators, the more opportunities we give them to parse through their thoughts and defend their thinking, the more they will develop their language and gain a deeper understanding of what they are learning. It can be a challenge to come up with meaningful ways to set up problems to do this, so I am looking forward to learning more strategies to help reach this goal!
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  • 01/25/2019 4:14 PM | Walter Pohle
    "The CCSS for mathematics asks us to expand our view of tools beyond calculators and compasses to consider such mathematical tools as number lines, diagrams,graphs, and tables"

    I like the fact that the "one size fits all" approach to solving math problems is a thing of the past. Students are now being taught and using multiple strategies (ie. number lines, diagrams, etc) to solve a math problem. Having a choice of strategies allows students, especially SWLDs, find one strategy that works for them. Students that want to challenge themselves can solve the math problem using multiple strategies. Win Win
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  • 01/28/2019 11:01 AM | Anonymous
    Pg. 17 "these routines are not problem-specific, they are 'thinking specific'; directions to students within the routines help keep the focus on different avenues of thinking."
    This section has helped me think more clearly about implementing MP's to support student thinking in a way that has been struggle. This is helping me to see how I might structure the content within the practices to further student thinking.
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  • 01/28/2019 1:16 PM | cindy noftle
    he following sentence on page 17, "...routines are not problem-specific, they are thinking-specific." resonated with me. It is exciting to think that the routines are helping the students to think! The students will be able to increase their math skills and not look to follow the same process over and over again. No longer will they see a problem and ask for specific instructions to solve that specific problem. Students tend to view each problem as a distinct item that uses its own individual algorithm to solve it. They need to be able to connect what they know to a wide range to problems. Hopefully, the routines will strengthen their ability to reason and then they will feel confident in implementing a process t solve that problem.
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  • 02/10/2019 7:00 AM | Anonymous
    Moved from Welcome section:
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    01/29/2019 11:46 PM | Cesar A. Llontop
    For chapter 1, for the first question, on page 16, under Growth Mind-Set, it states "when our focus shifts from a problem's answer to the various solution strategies and the thinking that leads to those strategies, students begin to see that mathematics not only makes sense but it is learnable." Because I work with a group of adult ed students, I constantly promote performance task questions, which enable the students to think outside the norm. For instance, at one point, I asked them to show me how would they solve a simple multiplication question of 45 times 24. The majority of the students did it the conventional way, basically multiplying in vertical format because this is how they learned back in their elementary school years. When I gave them a context clue to write 45 and 24 by breaking it down in terms of place value numbers, they could not connect that they could use the distributive property to get to the same answer as the conventional vertical form. 45 = 4*10 +5 and 24= 2*10+4. So If we apply the distributive property, we would arrive to the same answer. At first, they were fully overwhelmed because they were used to the conventional norm and they resisted to do it in a different way.
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