Week 5: Discussion Question (Option 1)

07 Dec 2017 5:40 PM | Anonymous member (Administrator)

Consider the list of related but distinct ideas around precision on pages 80-81. Which aspects do your students currently have? Which ones do your students need to work on most? What are some ideas you have to help them work on these?


  • 08 Dec 2017 9:24 AM | Anonymous member
    My 6th graders need work on all of these. In order to help them work on precision, I try to weave it into our daily lessons. Every day students present their ideas and work on the board for their peers. I tell them they are taking on the "teacher's" role. They work on explaining their thinking as they work through their task, using or being reminded to use math vocabulary, logically sequencing their work and organizing their work on the board so their peers can follow it. I stress the power of a visual, the clarity of using the language of math (you wouldn't use English when practicing/learning a World Language. It makes me a better math teacher because I am always modeling for my students. My 6th graders have a "Do Now" at the start of most classes and an "Exit Ticket" at the close of a lesson. Both of these require written support for their work. I find myself saying, " I don't fully understand. Can you clarify? Can you give an example? Can you convince a skeptic?
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    • 13 Dec 2017 9:49 PM | Anonymous member
      I haven't heard, "Can you convince a skeptic?" before and love it!!!! It sounds fun!
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      • 28 Dec 2017 7:31 PM | Anonymous member (Administrator)
        I've seen students get really good with the "skeptic" and "convincer" roles in math class. The teacher did a lot of work in the beginning by having partners practice being a skeptic and then being a convincer. The work came from some Jo Boaler resources that the teacher was using.
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  • 08 Dec 2017 10:10 AM | Anonymous member
    My students started the year viewing “precision” as “accuracy”. Some of them valued clarity, but often only if it made their work more accurate. Sadly, appropriateness, rigor and thoroughness were sorely lacking. They wanted to get an answer and be done. One student would even defend his answer, in a subtle, but physically menacing manner with no attention to thoroughness or appropriateness. Interestingly, he was a popular boy on the football team so other students never tried to intercept his assertions. To develop the kid’s appropriateness, rigor and thoroughness I explicitly taught them how to conduct Math Talks, which gave all students greater voice and encouraged deeper thinking. A strategy I learned from my brother in law called “Too high, Too low,” where students estimate too high and too low answers before beginning a problem combined with analyzing peers work has helped to overcome the lack of appropriateness, rigor and thoroughness. I am really looking forward to implementing “I think ______________is unreasonable because ______________” as the next step. I love how open ended the structure is, how it honors all different types of thinking, and how it “forces” the kids to think deeply about a problem before just jumping in to get an answer. I intend to use the sentence stems suggested and explicitly teach this strategy
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  • 09 Dec 2017 7:03 PM | Anonymous member
    My fifth graders would need work on all of these. The ones they are best at are accuracy and appropriateness. As 10 year olds, they are mainly interested in finding the answer and getting my approval. They like to do their math and then ask, me if they did it right. They think little about if their answer makes sense or if they were careful or focused in their work. They are interested in getting the math done and then moving onto the next problem. The big message I feel myself teaching to them each year are specificity and rigor. Everyday I hear myself asking the class and each student when they have a question. Does your answer make sense? If it doesn't make sense, then you need to go back and look at it again. I plan on working on this by getting kids to ingrain this into their math thinking. Looking at their work and seeing what would make sense to do and then after they get an answer does my answer make sense. Teaching them appropriateness, rigor and thoughtfulness will help with this. Teaching them to think of how can you prove to someone else that you have the right answer. Talking through Math problems and answers is a great way to show you really got it.
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  • 10 Dec 2017 6:08 PM | Anonymous member
    Two of the classes I teach are fourth and fifth grade accelerated math. At this point, appropriateness related to estimation, specificity related to units, and accuracy in terms of calculating the “correct” answer are the daily focus in these two classes. Both the fourth and fifth grade students are asked to estimate their answer and then calculate the answer. The students resist the estimation step. Often the students understand the math procedure so they want to find an answer and move on as quickly as possible. They do enjoy verbally explaining their reasoning and comparing approaches to problems. I am working on getting the students to put more effort into written explanations and helping them develop confidence and ease using appropriate math vocabulary.
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  • 10 Dec 2017 10:29 PM | Anonymous member
    My students have been working on appropriateness this year and they are getting better at it. When they started the year they gave me no fraction answers. Everything was in decimal form. When I asked for them to use their fractions instead of rounding, they gave me decimal answers with decimals brought out to 6 places. Then one day they gave me money answers in fraction form instead of decimals to two places. It was difficult for them to think about what type of answers were appropriate for each situation. They just wanted a rule to follow where we always used one form or another. I think I really frustrated them through this process.

    We have also been working on speaking and writing clearly. Since their writing skills are not quite developed, I have been working on this during class discussions too so they can hear what I am looking for. I am still hearing "you know what I mean" but it is less frequent.
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  • 11 Dec 2017 12:04 PM | Deleted user
    Accuracy, Appropriateness, Clarity, Rigor, Thoroughness. These are very similar to the categories on the Math Forum Problem Solving rubric that I use with my students: Problem Solving (interpretation, strategy, accuracy) and Communication (completeness, clarity, reflection). I stress all of these areas with my students.

    I have one student whose work lacks rigor; he has a lot of ability, and learns easily, but communication is more difficult for him, and in both oral and written responses, he leaves out a lot of details and makes misstatements, that he doesn't make an effort to catch.

    Case in point, today I got a very confusing problem response back from him. Although he had a correct final answer, he had inaccurate labels on triangles, unsupported triangle congruencies, and a mix-up in the use of "to the power of" caret symbol. No one else would be able to follow his reasoning.

    I'm not sure how to help him learn how to proof-read his work; if I had a larger class, I could have "math buddies" to look over each other's work but with a class size of two, I don't see this as an option. This time, I pointed out to him that he had a lot of solid reasoning in his work, but the errors made it hard to follow. I asked him to relook at his work. But when it is handed back to me the next time, I will to ask him to lead me through his reasoning orally, so perhaps HE can find the errors rather than ME.
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  • 11 Dec 2017 1:19 PM | Deleted user
    As a 5th grade teacher, I have been focusing on Rigor, Specificity, and Accuracy with success since I hear my students use my language in their conversations. However, I struggled with getting my higher-level students to slow down and consider their Thoroughness, Appropriateness and Clarity. It seems like the students that should be thinking it through are the ones that just want to do "the math" and rush to find one answer, only. I really liked the examples Tracy gave in this chapter; it highlighted each of these important dispositions.
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  • 11 Dec 2017 1:45 PM | Anonymous member
    I would say my middle school kids need to improve in all aspects especially thoroughness. AS we learn new concepts in students are always striving to get the correct answer. They may complete a problem or a few then ask me, "Is this the right answer?" Invariably I always ask them a responding question, " Can you tell me how you arrived at that answer?, Can you support your answer by giving an example? , or Did you reread the question or directions to make sure you answered the right question?" I usually get a little eye roll or huff in response.
    I am really trying to get my kids to think about their answers and how they arrived at it. I want them to question themselves and others. This is definitely a struggle but a worthwhile fight.
    Specificity is another area kids struggle with. Often kids will just go through the motions to solve a problems and not really pay attention to WHAT they are tying to find other than an answer. What does the answer mean?, is a question many kids cringe at.
    I try to incorporate written questions like, "What does _____ mean?" and " How do you know that _______ is correct? in some of my lessons to get the students to reflect on process, answers, and meaning to help increade specificity and thoroughness.
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  • 12 Dec 2017 11:13 AM | Anonymous member
    When I read this chapter I wrote beside this list "have on wall in classroom" meaning I should put it on my wall! Then, beside the words rigor and thoroughness, I wrote "are my students checking their work?" And this was before looking at the prompt for this chapter! I think I need to work with my students more on these - all of these. I think next time I will ask my students - how do you know you have the right answer for example - how can you check? The students are pretty good about using the correct terms - and about estimating. Our focus will be on specificity, rigor, and thoroughness. It was interesting as we were checking our homework today - and they had to show me their math. I think having more conversations, and feedback, will definitely help them.
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  • 12 Dec 2017 1:01 PM | Anonymous member
    I feel like my students need to work on all of these aspects. I ask questions and try to point out the need for accuracy, appropriateness, specificity, clarity and thoroughness. Rigor would be the next step for my students. Word problems are one way I can focus on these aspects. There are different ways that students can show their thinking with a visual or chart. Accuracy and/or appropriateness depend on the solution to the problem. Did they label their work? Write their answer in the correct term? Sharing solutions to word problems gives the students a chance to express their thinking, use the vocabulary.
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  • 12 Dec 2017 2:22 PM | Anonymous member
    I feel that my students focus heavily on accuracy. They want to get the right answer but for me I want them to understand and be able to show me how they got the answer. Too often they just focus on being right. To help my students to truly understand what it means to be accurate I feel that they need to work on thoroughness and rigor to help them understand the process of getting an accurate answer. I really enjoyed reading the ideas in this chapter such as using buddy checks. I could see how implementing this activity would help engage my students in checking work as well as learning from how others solve. I have tried in the past having students solve a problem, crumple up the paper and throw it. Each student picks up a paper and examines how another child solved the problem. This activity help my students to examine the process someone went through to get the answer and help them to become more thorough. I can see how using buddy checks could be another way to get children working together on math and help students to learn to check their own work. I also enjoyed reading about My Favorite No. This strategy gets children to realize that even if their answer isn't correct they might have done some great math work that should be celebrated. It helps students to shift their thinking from just getting the right answer to learning from their mistakes. I really enjoyed this two examples and will start using them in my classroom.
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    • 13 Dec 2017 10:42 PM | Deleted user
      Hi Amanda,

      I agree, as my students also tend to focus on whether their process or answer is "correct". Refocusing on the parts that they have done correctly, before talking about what might have gone awry is a much better feedback tool. I had a student today working on the midpoint formula on the coordinate plane (for the diagonal of a quadrilateral. She had done the process correctly, but had copied a number incorrectly. Instead of having to redo the whole problem it just took a couple of changes. A "buddy check" would likely given her the same feedback.

      Sometimes, when I give any indication (whether a facial expression or pause in responding) that the problem is not done correctly, students will erase all of their work before I get a chance to explain what they had done correctly. I think that it is a "learned behavior" to assume that everything is wrong, instead of considering that parts of what they have shown is correct.

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      • 14 Dec 2017 7:53 PM | Anonymous member
        Sometimes I find just asking a student to explain their process is enough for that erasing behavior - even when the original answer was correct!
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        • 20 Dec 2017 7:55 PM | Deleted user
          Great strategy. I might have to borrow their pencil first, too.
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  • 12 Dec 2017 7:50 PM | Anonymous member
    Precision: I thought my students had it, they were precise in my elementary definition of the word, but as I dug deeper into 6 different elements of precision, I clearly was able to find multiple aspects that my students needed to work on. Appropriateness and specificity jumped out at me because we are working on interpreting remainders. Students have discovered that sometimes the remainder doesn't matter based on the question posed, while other times we need to include the remainder especially if we are trying to figure out "how many buses a school needs for a field trip", or when we calculating "how many brownies you get if your mom said you can have the left overs". While discovering these differences we also learned that the unit matters. Is the remainder students, or buses?

    I think I assume to often that my students have clarity in understanding vocabulary, or understanding what the question is actually asking them.
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    • 13 Dec 2017 10:00 PM | Anonymous member
      Assumption is a dangerous word. I do it all the time. Thank goodness some students speak up! :)
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  • 12 Dec 2017 8:50 PM | Deleted user
    Even though I teach 2nd grade I feel like we can work on all of these but in a level that would reach my students. I talk about Rigor all the time but don't ask the questions to help the students start thinking more. Thoroughness is also something that if I could teach it more explicitly it would help the students understand their work and their thinking more. Clarity is something that I need to work on in my teaching more, if I am showing them clarity in my thinking and representations with can all work on it together. I truly love all these strategies and want to remember to use them in my teaching!
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    • 17 Dec 2017 2:36 PM | Deleted user
      I teach first grade and find students needing work in all areas listed on pages 80 and 81 in the text. This year I have focused my teaching around the accuracy, appropriateness and clarity. I find that these young learners need repetitive practice in building foundational skills in number sense, place value, and explaining their thinking around how they got the answer. Many of my young learners have been explicitly taught math talks as a way of expressing understanding on a topic. Some of the students just want to be done and show me their math work. I am working hard to explain to my first graders that it is not about working quickly to get the answer but that you have to understand how you got your answer.
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  • 13 Dec 2017 9:59 PM | Anonymous member
    I guess my pre-k students are working on clarity right now. Last week, we had a math night again. My activity this time was an origami gift box. As we folded, shapes were "clarified" with the new folds. As a mother was helping her son fold, the father said, "(Mom) can just come next time and do it. We (father and son) should have stayed and watched the game. The mother then came back with, "He is learning crease, fold, hexagon, half..." Thank goodness the mother realized how much her son was learning, even though he was mostly working on the vocabulary and not able to manipulate the paper by himself well enough. It was a rigorous activity, and it was most important to have families working together.
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  • 13 Dec 2017 10:31 PM | Deleted user
    Distinguishing the elements “accuracy, appropriateness, specificity, clarity, rigor, and thoroughness” is something that I do individually depending on the current topic. I have not really thought about discussing / explaining them as a group.

    in Geometry class this fall, we did a unit on angles and then one on triangles. When the students wrote their answers, there are three common imprecise results. In the triangles unit, when students solve for side lengths, they list answers with a degree symbol. Alternately, they use no symbol or unit at all for both angle measures and side lengths. The third result is assuming that algebraic expressions represent angle measures and they should always “add to” or “be set equal to” 180 degrees.

    I think that would be effective to put the list of terms on a poster that says,

    Is your solving process:

    rigorous, thorough, & clearly shown

    Is your answer:

    accurate, appropriate, & specific to the problem

    Under testing situations, students often forget the “appropriateness or clarity” of their answers, but when you ask them after the fact they usually say something like, “What was I thinking.” or “I knew that.”

    Using a “Buddy Check” during practice, which would allow students to ask each other if their solutions meets each of the six “precision” terms, has definite possibilities. This might even work with a “warm-up” problem, where students might be asked (after I collect and choose a “My favorite No” to share with the class) to rate the answer / solution in each of the “precision” categories. I would rather refocus the students on the content of the solution (and not just the correct value of the answer / solution).

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    • 16 Dec 2017 3:11 PM | Anonymous member
      I love that you thought of putting these ideas onto a poster...but I am wondering if they should be split? Sometimes my students do not use an appropriate strategy when solving a problem or they may not use the strategy accurately. hmmm..you have given me something to think about. :)
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      • 20 Dec 2017 7:57 PM | Deleted user
        Let me know what you come up with. I love new ideas / strategies.
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  • 14 Dec 2017 1:54 PM | Anonymous member
    Definitely, my students focus on Accuracy the most! They just want to know they have the correct answer, and move onto the next problem, whether they even did it, or know how....they got the answer. I have asked students more this year, "How did you get that answer?" Can you explain and teach us how you got that answer? We continue to work on our vocabulary, and use it daily to clarify how we did our work. If appropriate to the lesson, this year I have had students show me in more than one way how they could come up with the answer. I think this forces them to think a bit more, other than just the first way that comes to mind, otherwise they will always use this one same method. I liked the Buddy work. I have been having students work together much more often, or independently, but get together to see if they had the same answers(if only one answer to the question). If they have different answers they must work together to find which student, maybe both has the wrong answer, and explain what step they made an error, and maybe for what reason did they make this error. Often they find they are simply not thorough enough, they made some silly error. Maybe by using the Buddy system, they will slow down, read more carefully, and check their answers when complete.
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  • 15 Dec 2017 2:03 PM | Anonymous member
    I love the idea of having student voicing their mathematical thinking. I think they as teachers we need to train the students how to have productive conversations. One of the third grade teachers I work with has taken this on in her classroom. She has provided students with sentence stems to use as they discuss their math work. She used a FishBowl protocol to introduce the strategy where students observed what they noticed this looked like and sound liked. The students then set off to practice in student groups with student observers providing feedback on process and math talk. I am excited to watch this play out in the classroom as she gradually releases the scaffolds.
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    • 28 Dec 2017 7:29 PM | Anonymous member (Administrator)
      I love the fish bowl idea for helping students learn how to "voice their mathematical thinking" and the conversation starters are a way to encourage students to share their thinking. The starters can become a good visual que for students. I look forward to hearing more.
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  • 15 Dec 2017 8:50 PM | Anonymous member
    I feel that my students need a lot of work in all of the areas. I focus on all but rigor. My pet peeve: complementary angles equal ninety degrees rather than they have a a sum of ninety degrees. This becomes a problem when they are asked to find a pair of angles that are complementary. Inevitably, I get two angles that are ninety degrees each.

    My students need the most work on thoroughness. I am thinking about having my students keep a math journal. Having to describe a procedure to someone not in the class, will help him with the level 4's
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  • 16 Dec 2017 3:04 PM | Anonymous member
    I think that my students are pretty strong at with specificity and clarity. The idea that I have been struggling with is thoroughness. I have a few students that just like to get it "done." I have been stressing to them that being the first done is not always a good thing. I encourage them to go back and try to ask questions of them to push them to be more thorough but I am still finding that they are rushing.

    I liked reading about the example of the eighth grade teacher who added an extra box to a recording sheet in order to get his students to not just stop once the sheet was full. I think that I will try this and see how my third graders respond. This strategy is powerful because it causes students to not only gain a deeper understanding of the work they are doing but it also encourages them to be precise in proving their answers and thinking to themselves.
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    • 28 Dec 2017 7:27 PM | Anonymous member (Administrator)
      I agree with Jennifer and the value of having more spaces than needed for a task. Let's say there are 8 possibilities for a problem, providing 10 spaces can encourage students to push to be thorough. If there are the exact amount of spaces students think they are done because they filled in all the spaces, by having extra spaces students need to be sure that they are confident that they have all the answers - encourages rigor and thoroughness.
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  • 17 Dec 2017 6:59 PM | Anonymous member
    My students (Pre-K) need to work on all of these aspects but I think clarity is the most important for these young learners. Accurate knowledge and use of appropriate vocabulary is essential for students to build a strong foundation for learning in any subject. When thinking about clarity/vocabulary and where it really matters to young children I think of the very popular block area found in most Pre-k classroom. These 3-D objects are often the first experience children have with manipulating shapes yet we refer to them in 2-D terms. I will try harder to refer to the many different blocks using the proper term for each shape.
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  • 22 Dec 2017 10:11 AM | Anonymous member
    My 8th graders need to work on all of these. There are times that I'm impressed with their thinking, like when I hear things like, "That answer doesn't make sense," or when I see correct labels and math vocabulary that supports an explanation. On the other hand, I when I ask the question, "How did you get the answer?" I often hear, "I used a calculator." So I did deeper about what the student did on the calculator. It takes a lot of training and time to work on all of these aspects of precision. My students and I are all working on it, and I occasionally use the "My Favorite No" strategy. I love the idea of whole class critique for student work. It speaks to the class culture that mistakes are ok, and we can help each other learn from them. I will keep this idea in mind as I have students work on precision in their daily work and projects.
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  • 28 Dec 2017 7:22 PM | Anonymous member (Administrator)
    My students struggle the most with clarity, rigor and thoroughness. Almost every class includes a discussion about student work and through the discussion, students need to be clear with their language and how they are describing their thought process. I encourage students to ask questions if they don't understand one of their classmate's explanations and if they don't, I pose a question that pushes at the need for clarity. I might ask, "Why did you do...?" or "Can someone else explain ..., I don't understand." The example of Cindy Gano's class (p.84) is something I'd like to try with my students (all of whom struggle in math). I like the idea of pushing students to dig deeper with a task. I remember this fall, a student was working on a task and said that it was the hardest task we'd done this class. He'd been working on it several class periods and had a solution but I pushed him to think about how he could represent his solution by applying the algebra we were learning in class. He had an answer and could explain it well. The push was for him to keep thinking about the problem and find another way he could solve it - since it was an algebra class I was pushing for him to apply what we were discussing in class to this particular problem. He did eventually "get it" and the feeling of success was evident. Providing students the appropriate push can be the challenge, it has to be enough that they keep working but not so much that they give up.
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  • 29 Dec 2017 12:50 PM | Anonymous member
    I really enjoyed this chapter stressing the importance of slowing down the expected math work to focus on the ideas that represent precision. I have already taken one day a week to focus on one multi-step problem for partners to discuss and solve. However now, I have more strategies and activities to add to this day such as the Warm Up question that can lead to more in depth discussions using the My Favorite No activity. More analysis of mathematical thinking is needed for children to be able to better understand the processes and capture them in writing. My students have enjoyed working on difficult problems in partners. They look forward to the challenge. Now I will build on this activity to make math more precise.
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  • 29 Dec 2017 5:50 PM | Anonymous member
    My students in grades 3 and 4 need to continue working on all of the areas mentioned on pages 80-81, but I have seen changes in their thinking and mathematical ability and feel optimistic about continued growth in these areas with the Math program (Eureka) that we have been using in our district, currently in it's second full year of implementation K-5. However, with this being said, it is important to stress the importance for all of us (teachers/support staff/administration in our district) to understand the significance of teaching and implementing the mathematics program with fidelity and high expectations across the board if we are to expect precision and the other Mathematical Practices to be developed in all of our students regardless of the classroom they are assigned to.
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  • 30 Dec 2017 11:36 AM | Anonymous member
    I work with 5th and 6th grade students and I would honestly have to say that we need to work on all of the listed ideas around precision. The one area I have worked most on is rigor. Too often my students want to quit if the problem is too hard or will take a long time. I have been trying to help increase rigor by asking students who finish early to find ways to prove they are correct or to use a different strategy to solve. I really want to increase clarity in my math teaching. I agree having students hear and use the correct math vocabulary will increase their understanding and help math make sense.
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  • 01 Jan 2018 12:45 PM | Deleted user
    Being a teacher of 5 year olds, they aim to please so accurracy is definitely their focus. Since Kindergarten is the beginning of their educational career, most want to do everything right. They look for me to approve or disapprove so they can proceed with confidence or wonder if answers are right or not. I strongly believe that effort, trial and error, and participation are what my students are encouraged to believe is what they need to be successful in Kindergarten.. I can confidently say that my students all feel comfortable with putting their thoughts out there whether it be paper and pencil or oral and whether it be right or wrong. We do a lot of demonstrating and oral explanations in math, teacher as well as students, on how we solved a problem. Students want to get the right answer but even if it is not we talk through the process of their thinking and so I hope to switch their learning from accuracy to rigor, and appropriateness. I will be encouraging my students to explain their thoughts and then questioning them to bring their thinking to a new level for the rigor by asking questions like, " How else could we think about this problem?", "Can you give me evidence of why this strategy works?", "Can you show me other ways to solve this problem?" and "Does this make sense and how do you know?".
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    • 07 Jan 2018 8:00 PM | Anonymous member

      I think it is great that you are asking so many open-ended and problem-solving questions of students at such a young age... you will be setting them up for an easier life of solving math problems as they go through school! (Which will make my job easier in the long run... Haha!) So much of "getting" math is understanding the process! I try to explain that to the middle school students... they need to not only look for the "right" answer, but also the "logical" answer... and how they got there!

      Thanks for sharing!
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  • 03 Jan 2018 7:10 PM | Anonymous member
    This varies greatly across my classes, although thoroughness would be one that I'd say students need to work on most - both in terms of the numerical answer and supporting statements.

    Today was the first day back from vacation (+ an extra "snow day"), so I used the pattern game with "84" in the middle (p. 84) with some modifications as a warm-up for all my classes. (One modification is that each succeeding class had to try to come up with patterns that they thought hadn't been added to any previous class's list.) One added benefit was increased energy level and focus for the lesson of the day.
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  • 04 Jan 2018 11:26 AM | Deleted user
    Precision is an important skill that is developing in Kindergarten. We look for accuracy in doing number facts. For some students that means using your fingers or manipulatives for number sentences and being accurate with 1 to 1 correspondence in counting. Appropriateness, we use when making estimates of items in a jar. Is an estimate of 10 appropriate for a jar with 100 items? We count out 10 items and compare and discuss if that would be an appropriate estimate. We are using numbers all the time in the classroom. We often check the temperature and tell the class what the temperature is. I need to remember to make sure we are specific with these numbers and say 20 degrees F. Recently we are learning about 2-D and 3-D shapes. Students are learning how to describe the shapes with clarity. Rigor is developed in Kindergarten by having them listening to other ways of solving problems. We are always working on thoroughness by reminding students to be focused and careful in their work.
    I gave examples of activities that highlight each term for precision. However I think that I can be better at using these terms with my students. I like the idea of putting the terms on a poster in the classroom as reminder to me. In explaining these terms to the students they will have a better understanding of why I am asking them to be more precise and will be able to work toward that goal.
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  • 04 Jan 2018 2:18 PM | Anonymous member
    My 5th graders primarily focus on accuracy. I constantly work on appropriateness (does an answer make sense) and if not appropriate, try again (rigor). Thoroughness also is lacking as students just look at numbers and instantly do something with them rather than taking their time to think through a problem. I tell my students there are 3 parts to a math problem - getting an answer, understanding/explaining the process, and knowing what they are doing/solving. I tried Cindy's example on page 84 and the results were interesting. Students had a very difficult time with the problem. As they started to work, they kept asking, "is this right?". I like the idea of continuing the activity to see if students can/will go deeper.
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  • 05 Jan 2018 4:23 PM | Deleted user
    I teach 6th grade and my students need work on all of these. I am working on accuracy and always asking if their answer makes sense and continue to work on their estimations skills. We talk about appropriateness as well when we get an answer that may not make sense, I have students discuss with each other and try to convivce whether the answer is correct. I continue to stress how important being specific and clear with work in math is so important. I stres with my students if your work is clear then students that missed a class will be able to follow the work and learn from it. We continue to work on rigor and thoroughness as well. It is always a work in progress and having students rise to the occasion
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  • 07 Jan 2018 7:56 PM | Anonymous member
    Sorry this is coming in so late... I misplaced my book for a couple of weeks (switching between two classrooms this year is throwing me for a loop!)! I apologize-- I am now playing catch-up!

    Regarding the ideas around precision, my students deal best in "specificity" and "clarity", because I think we have created a culture where we are constantly asking for these traits in solving math problems. I have tried to model these traits in my instruction-- it is important for students to understand that something as simple as not labeling their work will make it difficult for peers to understand their mathematical processing.

    I feel like my students struggle most with "appropriateness", "rigor", and "thoroughness". I talk with my middle school students a lot about using "logic" when solving math problems. For example, could the answer to "100 divided by 3" really be 3??? Does that make logical sense? (That is just an example... hopefully they can figure something like that out!) I think that, many times, my students do not want to participate in "productive struggle", which is an important part of solving math problems. My students are students who typically struggle with math anyway, so I am not sure if they just perceive math to be difficult, and therefore, do not always want to put in the effort that it takes to puzzle out a logical answer to a problem.

    I have tried to create situations where students have to explain their answer, rather than just simply saying "Yes, that is right" or "No, that is not right." I try to ask students to explain how they got their answer (whether it is correct or not), and have them puzzle out where they might have made a mistake. Or, in the case of a multi-step problem, I ask students to walk me through how they solved the problem, or explain it to their peers. It is my hope that, by having them go through the steps, they will become more independent problem-solvers as the year goes by.
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    • 15 Jan 2018 4:46 PM | Anonymous member
      I am playing "catch-up" too.....no apologies, just so busy! My students need the same basic prompts I have for reading......Does it sound right?Does it look right?Does it make sense?......of course this requires them to review their work. I am trying to change the "hurry up and get it done" culture. Before work is handed in, I want students to look over their work and ask themselves about their work, process, and final answer. It would be great if they did this once for every assignment, or during class time discussions. I want them to gain a better feel for what they expect the results to be. An awareness of quantity, reason, and accuracy is necessary here.
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  • 15 Jan 2018 2:44 PM | Anonymous member
    My high schoolers need to work on all of these in different ways and at different levels. My freshmen are still very much convinced that decimals are more accurate than fractions or irrational radicals, and don't understand how "exact form" isn't a decimal. Isn't that more EXACT? I get asked all the time, "What should we round to? How many decimal places should we have?" and I always answer to use your judgment. They have sig figs beaten into their brains in science, but in a math problem, or if we were to order something cut at Home Depot, that wouldn't be the case. Likewise, when is a mixed number appropriate? We wouldn't order 9/4 feet of fence, but would absolutely order 2 1/4 -- or, better yet, 2 feet, 3 inches. In AP Statistics, we spent a lot of time on assumptions but I think that in other classes there is still room for playing with them. Just last week we were exploring the triangle inequality theorem, and I put on the board: Two sides of a triangle are 4 cm and 19 cm. What could the third side be?

    This generated some fantastic conversation, and the longer I stayed quiet the better the conversation got, including this gem: "She never said the 19 has to be the longest side," which naturally led to a great little tidbit about assumptions!
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  • 18 Jan 2018 12:26 PM | Deleted user
    When I looked at the list, I thought of the whole school i work in because I am a math coach not a classroom teacher. I also thought back to when I taught 3rd grade and I feel that the majority of student I have been in contact with need to work of accuracy and thoroughness. Many students read a problem and then begin on the calculation without thinking of the context. Most books show a process, then give word problems that practice the process. This does not require students to think about what the problem is asking and what would be a good strategy to begin. Therefor, students don't stop to think about thoughtful their reasoning is, or wether or not their answers support what is being asked. They very rarely so back and reread the problem to see if their answer makes sense because their focus is on getting the work done, not necessarily correct or to understand the proess.
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