Association of Teachers of Mathematics in Maine
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Week 3: Discussion question (Option 1)

16 Nov 2017 9:07 PM | Anonymous member (Administrator)

Have you ever complained that your students won't try (page 32)? What patterns have you noticed? What strategies have you tried? Do you have any new ideas to try after reading this chapter?

Comments

  • 19 Nov 2017 4:24 PM | Anonymous member
    Mathematicians Take Risks

    The James Tanton quote on page 32, "Math is being able to engage in joyful intellectual play- and being willing to flail (even fail!)" speaks to the issue of risk taking. We need to teach students how to take risks in math class in the same way that we teach other classroom routines.

    For the past two years, I have used Jo Boaler's Week of Inspirational Math. The videos stress brain research, the power of mistakes, and that anyone can achieve high levels of mathematics. The daily problems are open with multiple entry points, a range of solutions and strategies, and more than one answer. The problems are motivating and engaging, and students are willing to take social as well as mathematical risks in this setting. Unfortunately, this doesn't always transfer to daily mathematics class.

    My goal is to incorporate more open math problems into my math curriculum. I also plan to continue to praise effort and risk taking. I like the notion go 'math attack' skills in the classroom.
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    • 23 Nov 2017 10:36 AM | Anonymous member
      Thank you for sharing Jo Boaler's Week of Inspirational Math. I checked it out. Do you show the videos to your students in addition to the activity?
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      • 25 Nov 2017 10:33 AM | Anonymous member
        I was wondering what ages could be used for inspirational math? Can this be used in tangent with the Everyday math curriculum for elementary students?
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        • 25 Nov 2017 2:51 PM | Anonymous member (Administrator)
          We use the week of inspirational math during the first week of school and then start Everyday Math. Throughout the year we use inspirational videos and build the ideas of Jo Boaler's into the math classroom.
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  • 19 Nov 2017 4:24 PM | Anonymous member
    I usually have 1 or 2 students each year in Math class that really have a hard time to take a chance and try. The patterns I have noticed is that they seem to be afraid to try because they will get it wrong. They want me to help them so they can be sure they are doing it right. They are afraid to take a risk.
    I usually start Math class with a warm up Math problem or problems. I tell the class that they do not count for a grade. I expect an answer whether it is right or wrong. The point is to try! After a few times of students doing this, I start to see them trying problems and losing that fear. I did have some new ideas as I read the chapter. I liked the idea of giving more open problems. I try to do this once or twice a chapter, but I should really be doing it more often. I also like the way the teachers worded their class discussions to encourage their students to inspire each other.
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    • 22 Nov 2017 9:30 PM | Anonymous member
      Hi Julie,

      I like the idea of using warm-up problems to get students "trying" right at the beginning of the class period. I use this strategy when I want to find out prior knowledge that is required for a new unit. Fraction skills, working with negative numbers, solving equations, or graphing ordered pairs often fall in this category. It is more of a feedback tool for me, than a "graded task". I really have to emphasize that in order to get full participation.

      Open ended questions do take some planning (and class time). Letting students work in pairs gives students an opportunity to discuss the possibilities without feeling like they have to come up with a potential solution on their own.

      Pam
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  • 19 Nov 2017 9:26 PM | Anonymous member
    I found a lot of great ideas in this chapter. I will definitely be more aware of the type of problems I am asking students to explore. There are also lots of great tips on encouraging risk taking through the language used by the teacher. I loved the way Heidi encouraged Alvin by using his work as an example in class. I will encourage hard work and effort, and be careful not to tell students that something is easy or they did something well because they are smart.
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  • 21 Nov 2017 12:28 PM | Anonymous member
    Sure, I have complained that my students won’t try! My complaints are really frustrations at myself because I recognize I have not yet managed to create an environment in which they feel safe enough to take a risk. The students who won’t try are often members of the “this is easy” crowd and they are facing a task without one clear answer, which is unsafe to them. Other students who don’t try are those who have not been successful in math in the past.

    I have found the sharing and analyzing of student work, and the honoring of different approaches with careful monitoring of the language of my responses to student questions and ideas to be helpful in overcoming the “I cant’s.”

    I really like the language Shawn used: “…a room full of not sure…” and my favorite – “We will help.” This is so empowering to all learners and supportive of the risk taker. I want to begin using this language and find some of my own.
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    • 22 Nov 2017 9:39 PM | Anonymous member
      Hi Kathleen,

      Your scenario reminds me of working with an honors class. They are all pretty confident with their math skills and struggle when presented with a question above their skill level (especially word problems or proofs that require them to choose a solving strategy).

      When presented with an assessment, many students that have been successful working on a particular skill set have trouble recognizing when to apply it. As an example, students often assume that two algebraic expressions are equal, when they are supposed to be supplementary or vice versa.

      Pam
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    • 21 Dec 2017 11:56 AM | Anonymous member (Administrator)
      Hi Kathleen!

      As you think about other phrases, I'll admit that I steal my phrases from Lowe's and Home Depot......"let's build something together" , " you can do it, we can help" and others of that sort.

      I really became far more sensitive for my students who struggle by my participation in the Casco Bay Math Teachers' Circle, the last few years. At the math circle, we put ourselves into positions all the time where we don't know the answers to questions and we explore ideas together.

      I understand more about how it feels to not know and the kinds of moves that facilitators make to make it ok not to know. Math circle, for me, has been a place where we support one another on our mathematical learning journey. As I'm getting caught up with the group, I was struck by the responses to the "this is easy" prompt. I've seen that phrase shut people down at our math circle meetings.

      Your first two sentences really struck me as important to think about. I do think my complaints are my own frustrations at having not find just the right entry for students. I agree wholeheartedly that it is important for us to set the classroom culture. It is hard, but important work and we can undo all of our hard effort so easily.
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  • 21 Nov 2017 6:36 PM | Anonymous member
    I am currently coaching in a 5th grade classroom in which there is a student that I know from previous experiences with him that he has the Math skill to be successful in the grade level classroom, but if you observe him during class you would think he was a limited student. He sits and does not engage unless an adult stops to prompt and coax.
    Here's what I notice:
    He is successful :
    1. When working in a small group
    2. The work area around him is quiet and clam
    3. He has time to work, rework and collaborate with others.

    So my wondering as I work in the classroom is how to we facilitate more of these conditions.
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  • 21 Nov 2017 10:41 PM | Anonymous member
    For students with low confidence, “reluctance to try” seems to be a common theme. For the most part though, my students will try at least one or two questions and then ask for confirmation that they are completing the task properly. Frequent check-ins can provide support and encouragement. I usually tell the students that I can’t tell where they might be stuck on a problem, if they write nothing (or they have the wrong answer and no work). Instead of expecting all of my students to complete the same number of problems in the same time period, I have them focus on working at their own pace (slow but sure is better than rushing through an unfamiliar process).

    Connecting a new skill / process to prior learning also seems to help. In a current unit of Algebra 2, solving quadratics can be completed with different methods. Students that are good at factoring find success with this method. Students that did well with graphing prefer use that method. Having options can provide students with an opportunity to stay (at least partially) in their comfort zone. Allowing them to help each other, creates mathematicians within the classroom.

    When working with rubrics and core standards, the focus is on a set of expectations for each unit. Having “high standards” as described in Ms. Towle’s classroom (pg 45) is important. We must focus on the positive, so that all students feel like they can reach the intended goal.

    Pam
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  • 23 Nov 2017 10:30 AM | Anonymous member
    Sometimes it takes just one class... At the time, our high school was on a 2-semester schedule; 70 was passing (read that as 69.5); a math credit = 1 semester of work. Students often came to algebra 2 with low grades, low skills, and low expectations of themselves, so getting them to "try" was often problematic. Students in vocational programs and ROTC were especially prone to this because it was the case that their math class could be (and was!) scheduled in a block where they regularly missed 30 of the 90 minute block. So 6 years ago I tried something new with just one class of juniors and seniors... I asked if it were possible, who in the class would like to earn an A to please stand up. Every student stood, although there were many comments about why it was not likely to happen (one student was taking the course for the third time; most had had minimal success in math). And that was the beginning of the A-team concept. We discussed what "A" students did to be successful, what makes a great team, and most importantly the need to create new mental tapes (replace the "I can't" tape). It was quite gratifying when students gently reminded each other to replace the old tape, and even more so because students were willing to take risks, make mistakes, and left knowing that they could really do math (all earned an A or B, except one who earned a C+; for most it was the first time that they had felt success in math!). Creating that safe environment is so important.
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    • 29 Nov 2017 8:01 PM | Anonymous member
      That sounds like a great decision you made to try that idea. The students in the class were able to reframe their thinking about math class. I would like to give that a try. I will think about how your idea could be used in my situation. Thank you.
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  • 24 Nov 2017 12:29 PM | Anonymous member
    Every year I run across at least a few students who refuse to “try”. I think it can always be traced back to being afraid that they are going to make a mistake, or, in the case of traditionally high achieving students, being discovered that they really DON”T know what they are doing (I was one of those kids for a very long time!). Both scenarios result in embarrassment, and no one likes to feel embarrassed. I suppose Carol Dweck would say it is a result of having a Fixed Mindset. Not trying is safe, but trying…really trying…and failing just confirms to yourself that you must not be very smart.

    A few years ago I was lucky enough to be working with a 4th grade teacher who was striving to present students with rigorous problems on a regular basis. This particular class was comprised of the average, below average and very below average performers according to NWEA scores, which is why I happened to be assigned to be in that class as an Interventionist. Nearly EVERY day we struggled to end the class on time and get the kids out the door…because they were so engaged!! It didn’t happen overnight, but it happened! It was so inspiring to see these kids wanting to keep working on math. The other students would filter in for their break time and many of them, from the above average and high achieving math class, often looked at what our class was working on and some of them would want to give the problems a try, but at least a few of them felt the need to tell our students that they’d already done problems like that "a long time ago". Ironically, I can’t remember why, but the math groups remained in homeroom groupings one week. Our approach didn’t change. We presented the same type of rigorous problems from Math Forum, followed it up with our usual “Notice and Wonder”, and proceeded to watch. What unfolded was very interesting. All of a sudden some of the presumed above average and higher achieving math students were at a bit of a loss. They had very little practice at what to do when presented with a problem that they didn’t quickly see a way to solve. The students who were in our class regularly had developed strategies, and patience, persistence, and a tolerance for frustration that the others seemingly had not. I dare say it was a very uncomfortable week for those students who essentially shut down at something they considered themselves to excel at. It drove home the importance of making sure our students go beyond just being able to “get the right answer” so that when we scratch the surface of their knowledge there is substance behind their understandings.

    I think my colleague and I had managed to create a classroom that helped foster the same things that Heidi, Cindy and Shawn did by truly valuing student questioning, wonderings, innovations and attempts.
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  • 25 Nov 2017 4:40 PM | Anonymous member
    In a special ed classroom I don’t get as many questions as I would like. Many times there is silence or a nod of the head. I have joked that they are nodding their heads because they want me to move on or they think that is what I want to see. I tell my students that I cannot read minds and need some help from them.

    Strategies – I use marker boards and do one part of a problem at a time.
    Using the document camera I show students work. We look at how they set up the problem; did someone have a different way? Are they both correct? How can they both be correct? Is one way better than the other? (We generally will discuss that solving the problem that makes sense to the student is the right way, there are no wrong ways just different ways.)

    One of my goals this year is writing our responses to word problems. When taking the MEAs, I am able to read to students as part of their testing accommodations. I have watched students write just an answer, no explanation, no work. We have practiced the release items but many of the students do not seem to know where to begin. I modified a worksheet and divided it into 3 main sections, the problem, work space and the solution. The work space has 2 different spaces one with lines for writing their steps taken to solve the problem and another blank space for whatever work, tables, graphs, drawings they want to create. I read the problem and each student works on their response. Using the document camera we share everyone’s response (largest group has 5 students and it is a safe environment as we are all learning) Students explain how they started, what they knew, what clues they used etc. As a group, we write a class response. We talk about how we can show our work, use different visuals. We need to come to a consensus that it works for the group. I also model how it would be if someone chose to write it all out. We talk about the need to show what we are thinking.

    The first problem was interpreting a pie graph and the second problem was on mean, median and mode.(skills we had been working on)
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  • 26 Nov 2017 8:21 AM | Anonymous member
    I'm certain I had a few math teachers who taught by the method of..."because that's the rule!" I'm grateful that this is not so much our thinking today! As a teacher, I truly enjoy seeing/hearing students step outside of the box to problem solve and feel that I foster them to take risks. I really enjoyed the samples of language that encourage risk taking and will add them to my toolbox. Highlighting student efforts and giving necessary supports are valuable tools, too. Lots of good things to consider!
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  • 26 Nov 2017 8:40 AM | Anonymous member
    Well, I did my low floor/high ceiling activity called "Circle Fever" and built in 4 minutes for the students to read and start the problem by themselves (before they started working on it in their groups). One of my students didn't even start the next diagrams. I was carrying around an observation form for my Illustrative Activity for a follow-up workshop in 2 weeks, so I just let him be. I did continue to watch him though and he never participated - he even was looking at his fingernails. So, I've read the comments here, and I always do a warmup at the start of class. I am going to start watching him more closely so that I can look for opportunities to praise him on his thinking. I am also going to have a third conversation in my classroom about what to do if nothing strikes you at all when you have a problem to do - and see what the kids come up with. I am also going to go back and show some of the Jo Boaler videos about making mistakes, and continue to look for low floor/high ceiling activities.
    So, I can this that this is going to be an area that is going to require some thinking and doing on my part.
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  • 26 Nov 2017 3:40 PM | Anonymous member
    I have had students give up too easily on their math work, even when they know the procedures. One way I have tried to engage them is to include their names and some local event in the problems we are solving. I change the examples from the text to reflect ourselves in our community. That goes a little way....
    I have also tried having student partnerships where students can help one another with their work. However, the students had little experience with this idea, so they would check out while a more able student did all the work, and the others would copy down what they wrote.
    Some new ideas that I would like to try out include starting every class with sharing how someone needed or used math to solve an everyday problem, and I would like to have students make more personal connections to math in their world.
    I have found that just going by the book and doing the math program is not necessarily causing students to learn a whole lot about math, nor is it necessarily improving their mathematics skills.
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  • 26 Nov 2017 6:44 PM | Anonymous member
    It is frustrating when students won't try. I find these students do not engage and wait for other students to answer. Some new ideas that I will try is to teach students what is expected and how to have a discussion. I will also allow students to have time to think before responding. Students will have time to explore with math materials and explore math problems. Some new ideas I will try are to try a mathematics problem each week and try grouping students in different ways.
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  • 27 Nov 2017 4:13 AM | Anonymous member
    Of course I have complained that students won't try and have noticed that they need encouragement to take a risk especially in front of their peers. In a small group they seem to feel less anxious about their process or answers. One other habit I am trying to break is students who erase how they work out their problem or the "show your work" part. However, we are definitely making progress there. Math has become my favorite part of the day since adding three act tasks and open math tasks. It's great to get out of the curriculum book and worksheets to just discover, discuss, and share about math together.
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  • 29 Nov 2017 6:57 AM | Anonymous member
    As I read through this chapter I tried to take a step back in my classroom to see how I was presenting new material plus reinforcing prior skills, along with encouraging my class to take the risks needed to become more confident, and successful, in math. I then asked the students what skills I have been giving them to help them become more confident in math, trying new concepts and feeling safe in whole or small group work. I also encouraged them to give me feedback on strategies I could improve on. The improvement area gave rise to many suggestions ranging from less work each day to no more homework, with only a few asking that I extend math period. Not many constructive comments in this area. For positives i received some great feedback. One piece of feedback that was repeated throughout our discussion was the fact that I make them feel safe by responding positively to incorrect answers because, as they said, I always complement them for thinking and trying even though their answer might be wrong because this may lead us to the right answer. This is a group that does not have a strong foundation in number sense so we are constantly reinforcing this as we move through each lesson. We start each class with a Mad Minute geared toward the operation we are working on that day. Then, we solve 1 word problem dealing with that same operation. Next, we move on to a short lesson followed by independent or small group work as I work with struggling students. At the end of the period we play, if time, a quick game to reinforce the lesson.
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    • 29 Nov 2017 8:04 PM | Anonymous member
      It sounds like you have made good working routines a reality in your classroom. You are set up to accomplish a lot of math in one class meeting.
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  • 02 Dec 2017 4:22 PM | Anonymous member
    I am absolutely guilty of complaining that students won't try. I think many times we have conditioned our students that if we pose a question and they don't immediately jump in, we will answer it for them. I've made a conscious effort recently to just stay quiet. After we hit the 10-second mark, I'll offer, "Come on, I know you're thinking about SOMETHING!" but I really, really love Shawn's quote on page 47. ("Is there a volunteer to try? We'll help.") I love the idea of asking for a volunteer, saying that we know they are TRYING, not that they have to be right, and that *we,* not the teacher, will help. Likewise, I love the "you've got a whole room of 'not sure.'" These are so encouraging!

    The table on pages 51-53 had some great phrases to put in my back pocket. I learned a long time ago not to say something is "hard" or "easy," because it loses kids before we do a single thing. But I love the phrasing "this problem is really interesting," and I will start using that immediately. Right now in Geometry we are working on formal proofs. The idea that there are many, many correct possibilities is liberating for some students and intimidating to others. But no matter what, it definitely makes it interesting! Pausing to converse as a class with questions like, "Why did you take this approach?" will be a rich addition to our class. I think answers as simple as "I wanted to put all the given statements first" and getting students to think about what they did and why they did it ("so I wouldn't forget them" would be a perfectly valid answer!) can make this otherwise-dry stuff interesting AND help grow confidence. You didn't do it the way Suzy did on the board, maybe, but you still did it RIGHT.
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  • 03 Dec 2017 9:31 PM | Anonymous member
    Yes, I can't imagine not having a year where there are students who won't try. I have noticed that it often comes from both my higher level students and my lower level ones. Sometimes a higher level student won't understand something, but they will try to play it off by not trying. And then, in the case of lower performing students, not trying has become somewhat of a habit for many of them.
    My main method of attacking these issues comes in the form of discussing life lessons (academic, athletic, etc.) in general with the students and interjecting my own experiences. The key here has been making it a discussion and not just a lecture. I have found students will buy in more when they have the time to tell their own stories and experiences, and then you can refer back to them when needed. We especially bring up some basic psychology and meta-cognition ideas, which they seem to like. Being at the middle school level (6th grade), students are often much more impressionable and willing to partake in such discussions.
    Additionally, I will also just give students time to practice some math concepts without any grade attached to it. Then I stress to them that there really is no reason not to be giving things a try. They literally have nothing to lose, only only something to gain.
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  • 05 Dec 2017 7:10 PM | Anonymous member
    I chose to recognize and celebrate different approaches to solving a problem. During our morning meeting time one student is responsible for taking attendance usually by counting each child in the group. When the student gave his final count of 16 I ask “how do you know?’” he replied that that was the last number he said when he counted. I they had a different way of figuring out how many were in our class today. Several students said they knew we had 16 kids in our class and there were no empty mats on our rug. Sophisticated strategy for a four year old.
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    • 13 Dec 2017 5:59 PM | Anonymous member
      Yes, I like to recognize and celebrate several different ways to get the same answer. Students use the document projector to show their work daily and more often than not there are many different approaches and strategies. Modeling and requiring (strongly encouraging!) explanations using meaningful math language is so important. The language becomes familiar and students feel safer using it in their own explanations.
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  • 08 Dec 2017 12:08 PM | Anonymous member
    I have most definitely complained that my students won't try or take risks in my room. I encourage mistakes and risk taking, but as I was reading, I am not sure I model risk taking and making mistakes often enough. I think part of my frustration is not providing enough opportunities to explore mathematics and share out their thinking. Students don't always like to have to think and really try new things, but why would they if they know you will just show them a process to make it easier. Well, what are they going to learn from that? They will memorize the process for the time they need to and then forget. BUT, if they are made to explore and try, they will get a deeper understanding of the math being taught and hopefully remember it later in the year when it comes up again.

    This chapter was a great reflection for myself to remember they need modeling and repeated support when I want them to take risks in the classroom. I need to do more than TELL them it is okay to make mistakes. I teach 8th grade math and Shawn Towle's example resonated with me. Sometimes it is just all about the language to show students we will all help.
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  • 09 Dec 2017 12:03 PM | Anonymous member
    I have noticed that they won't try anything that is different from what we do in class or anything that requires reading. I frequently tell them that I wouldn't have given it to them if I didn't think that they could do it. I give try to encourage, help them break it down, ask leading questions to help them discover a way to start.

    New strategies, we as a department are giving our kids more open-ended, multiple ways to solve type problems. They have time to think individually on the problem and then they work in groups on the problem. I am seeing students that enjoy doing math more and are more willing to take risks.

    This reading really hit home. I have a combination of kids that one-on-one are great kids, but together in a room, are a nightmare. For whatever reason, on Wednesday they were excellent. They worked really had with algebraic proofs and identifying the properties. We had a great class. On Friday I reminded them of how hard they worked and all that we accomplished, hoping for a repeat of the same. They worked fairly well. However, when it came time for them to work on the assessment, I tried to assure them that it should be "easy" for them that they are good at it. I knew when I said it, that I shouldn't have, but it was too late. It was already out there. I wished that I said "You have been working really hard at this and I just want to see what you can do." During the assessment, some of them went along with confidence and did fine. My kids that struggle with math, did a couple and guessed on the rest. They shut down.

    Oh the power of our words. I don't use the words that you are smart and I am trying to be aware of my use of the words easy and hard. I want to use the words "you worked hard" and "Challenge yourself".
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  • 13 Dec 2017 6:06 PM | Anonymous member
    Love the idea of setting the tone early and often about challenging oneself. Safe risk taking environment SO important. Somewhere along the line I learned that students in Japan actually thank a classmate for making a mistake--they acknowledge the value of mistakes in the learning process. In our class (4th grade) when a peer is brave enough to share work and there is an error, we respond with a respectful "Domo!" Then we take a close look at the error and use math language to respectfully describe the error and the correction.
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  • 30 Dec 2017 12:50 PM | Anonymous member
    I enjoyed the description of Shawn Towle’s classroom discussion. The role of teacher as protector and advocate for students brave enough to share their attempts to solve a challenging problem is so important to engage all of the students in the class. Too often math class can become a sprint to the ‘correct’ solution without providing the less confident students a chance to participate. Seeking multiple strategies for solving a problem allows many students to participate and learn from each other. The discussions become richer.

    The contrast between this and the story opening the chapter of the math teacher who referenced “the rules” rather than explain why a number raised to an exponent of zero equals one is quite striking. Zager felt the teacher himself did not understand why the answer was zero. This seems to be true considering he allowed her to be mocked into silence. I suspect there were others in the class with the same question.
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  • 15 Jan 2018 5:46 PM | Anonymous member
    It is without fail, that every year I have students who won't attempt (not just in math but other subjects as well) or put the work effort into trying something new. For math, some examples where not trying is most evident at first glance such as: solving word problems because it involves reading (some who struggle in math are low readers as well), attacking an equation they are unfamiliar with or it has "bigger" numbers. Most of the time, there are a handful of students in the class who, when asked to practice their math, immediately shut down and say phrases like "I can't" or "I don't know how" or "It's too hard," particularly at the beginning of the year. I have also found that during math lessons or practice, the students who are less confident in their math skills tend to feel the pressure from those students who may get to their answer quicker or be the first ones done and want to share with the teacher. I notice that the lower students will often use "wandering eyes" and copy off their neighbor who gets it just so that they are not the last one to finish or feel embarrassed that their page/whiteboard is blank. Those students are the ones I try to check in on and focus a lot on when doing a math lesson. Unfortunately, and maybe too often we as teachers can tend to let the ones who already know how to solve math "slip through the cracks" with no worries because they are meeting the math curriculum expectations. So a goal that I have been working on is ensuring that doesn't happen and recognizing students who do try, praise them for their effort more, and use them as examples for the weaker students. Heidi's example of using Alvin's work to "inspire" others really resonated with me and I would like to find more time and opportunities to have math discussions around student thinking and understanding because that is important. I do often tell my kids to "challenge your brain!" and will often hear them say they are picking challenging problems or apps when they practice on their own and that is rewarding. Learning to stray from phrases like "you are so smart" with the little ones has been hard as we want to build their confidence in themselves but this chapter really helped with reminding us of the power of our words. I hope to use more open math questions and have discussions at least once a week or perhaps touch upon it a little bit at the beginning of our math lesson. We are trying a new math program this year and both the students and I are getting used to the program's language so we are incorporating taking risks together already! But I plan to use more feedback into the routine to inspire kids to take risks on their own more frequently.
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  • 16 Jan 2018 4:05 PM | Anonymous member
    I often have students who are not willing to try on their own. They had rather come and say " I don't know how to do it". When pressed into identifying what part they don't really know how to do they aren't sure because they haven't even attempted the problem. So I usually say to them "well you start and I'll help you when you get stuck" . More times than not they are able to complete the problem without getting stuck. I like the page of encouraging comments to put on student papers. I think when students see these type of comments in writing it is more meaningful than just telling them because they can see it. I also tell them there is more than one way to come to the answer of a problem. We often have two or three ways that students thought about the problem and solved it up on the board.
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