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Chapter 4 – Option 1 (Reflection)

02/03/2019 10:55 PM | Anonymous

Annotation is a powerful instructional strategy.  Reflect on how you have used it in your teaching OR tell us how you plan to use it.

Comments

  • 02/04/2019 11:33 AM | Tofer Carlson
    I was actually going to talk about this on my post for Chapter 3 - but then saw this topic here, and realized this is quite literally what I was going for.

    I come to math teaching from kind of a weird background - I student taught in biology and English before I started teaching math, and my undergraduate degree is in Civil Engineering & Writing and Rhetoric. I didn't realize until about two-thirds of the way through my year of student teaching that I desperately missed the problem solving and puzzling that math entails, and chose to head primarily in that direction. I think that background, in particular, the English side of things frames my use of annotation in math problems.

    A couple of years ago - a colleague (also doing this book study ::waves at Todd::) and I did some work on using annotation in AP Calculus to help students make sense of hard to process multiple choice questions. This involved using direct instruction in annotation strategies, using think-alouds to model the process of annotating, and giving students feedback on the amount of problem digesting they were showing on assessments. This might take the form of something like this:

    The graph of f', the derivative of f, is the line shown in the figure below. If f(0) = 5, then f(1) =
    (a) 0 (b) 3 (c) 6 (d) 8 (e) 11
    Figure showing line labeled y = f'(x) in the first quadrant, x-intercept labeled at (1,0), and y-intercept labeled at (0,6).

    The first layer of annotations that we'd try to get students to make would be based on the problem at hand, and figuring out what we're dealing with for the graph. Marking up (underlining/circling) the phrase "f', the derivative of f(x)", and the label "y=f'(x)" would be examples of this surface level of annotation.

    There's a quote from Chapter 3 (p.71) about students being accustomed to looking for key quantities, which applies here too. Students can do this level of annotation without thinking much, and don't get a heck of a lot out of the process, except for the times where it keeps them from misusing math (thinking this is a graph of f instead of f' because that's how we operate).

    The next layer is really what we were pushing for in terms of annotation - it's not enough just to identify important phrases or quantities, but they need to be contextualized in terms of the body of knowledge our students have. In English terminology, this would be making connections in the vein of text-to-text, text-to-world, text-to-self, et cetera. In calculus, we're looking for the connections between information connecting different levels of the function.

    "f'(x)" is circled, and then annotated with connections like: "this is the graph of the slope of f(x)," or "f(x) means the area under f'(x)." These are the annotations that are useful to students practicing sense making, and are out of the comfort level of what they are accustomed to. It's also much more akin to non-academic (I loathe the phrase real-world) thinking and processing. If we are asked to complete a project or solve a problem, it's not given to us as a word problem. We have to do some research, and make our own connections, and maybe gather a great deal more information. When I'm doing research, or trying to synthesize resources to create a lesson, I do a heck of a lot more than circle important pieces, I have a conversation with myself on paper, which is what annotation ends up being about - with the hope of making sense of what we're doing, and potentially with the ability to come back to the thinking at a later time, and having some hope of understanding what was done.
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    • 02/08/2019 7:45 PM | Karen Campe
      Tofer,
      I love your expression "I have a conversation with myself on paper" because it nicely expresses what we ask students to do when we say "show your mathematical thinking".

      I've also suggested to students that they "narrate" their work; this can be aloud to me if we are working one-on-one, or a written narration through careful annotation, like your conversation.

      Karen
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  • 02/04/2019 9:32 PM | Sarah Giaquinta
    Annotation is a part of the routines that I have found very challenging to execute well. When I was first introduced to this idea in a workshop over the summer, I tried my hand at annotating what anther colleague was explain to the group and I found myself standing next to the problem on the poster not knowing what to write or show, based off of what was being said. I didn't want to annotate anything that he didn't actually say but that I understood just because of my math background. So I was able to fill in the holes and make sense of it, but really he didn't say everything. As teachers, when we model annotation, are we supposed to stick to literally exactly what is being said? And what do we do when the thoughts aren't clear or concrete enough?

    At this point, when implementing routines, I have been the one to do the annotations. I would love to get to the point where they can annotate for each other. I noticed that Mr. Smith started out annotating, then invited two students to come up and finish the annotation. It sounds like this is the first time students have done it and they have practiced this routine a few times already. I'm sure it comes much more easily with practice, and I'm hoping students become more thoughtful and clear when explaining so the annotating can be a bit more productive of a piece of the routine. Just the other day, when one of my students was finishing his reflection prompt, he said "when you are trying to convince a skeptic, it is important to see things from their perspective". I think he hit the nail on the head here! When annotating, you are getting into the head of another person and trying to make a visual for their thoughts. This is a powerful skill to have and one that takes some serious time and effort to develop. I find that many students are getting frustrated when others aren't understanding what they are saying, so hopefully, with more practice using annotation in these routines, both my students and I can get better at this part!
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    • 02/05/2019 10:15 AM | Todd Butterworth
      As Tofer talked about, we've worked on annotation in calc a decent amount and it's so much more than just circling stuff :)

      But in terms of your question, Sarah, when I annotate what kids are saying, I try to only stick to what they are saying. So if the student isn't saying something clear, I can't annotate anything, but I can listen and then ask for clarification, either from that student or another student. Sometimes I intentionally look confused when kids say something that I, as a teacher, understand, but that is not generally understood, to encourage students to clarify their language. I try to be a bit confused when annotating if kids aren't clear. Sometimes I'll intentionally annotate something incorrectly so that the student or another student has to correct me.

      Annotation and explanation is definitely something that students need to practice (and it's super important to try to work with kids and make sure they realize that you are focusing on what they are saying and how they are saying it, you're not just "picking on them" because some kids can take it that way). How far I push kids in terms of their explanation is really a kid by kid basis. I have a kid in calc who has a very expressive face, which is really helpful for me because if I'm explaining something or someone else is I can look over at him and see right away if what is being said makes sense or not to him. I use him as a way to gauge understanding and then ask him questions about where his confusion is so I can make sure we clear things up (often that means we do further annotation of something or a different method of annotation, depending on the confusion).
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      • 02/08/2019 7:52 PM | Karen Campe
        Sarah & Todd,
        You raise an important question about whether annotation should just mirror what the student says or extend it to "fill in the holes" based on our mathematical knowledge as teachers.

        On the one hand, I think the initial annotation needs to be grounded in what the student says and means, and therefore the teacher might ask for clarification (or urge other students to ask for it) like Todd says.

        On the other hand, I do think there is a time within a lesson arc where the teacher will come in to fill in those mathematical holes for the large group, to draw the connections that a student might be hinting at but not articulating well. I think the Four R's strategy can be helpful here to build up a more detailed mathematical concept: Repeat (by another student), Rephrase (by another student), Reword (by the teacher, substituting mathematical language/notation as appropriate), Record (document the concept for the class).
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  • 02/05/2019 11:59 AM | Hala Sahlman
    I really enjoy the annotation piece to this routine and have tried to use it in other areas of my teaching as well. I agree with other teachers' comments that it can be very difficult to show a student's thinking on paper. One part to annotating that I really enjoy is that it forces students to use specific language when explaining their thinking. They can't just point to an area, or mark it up themselves, they have to clearly tell the person annotating what they are visualizing in their mind. Many students have gotten frustrated trying to put their thoughts into words, but I think the challenge has helped them to be more thoughtful in their explanations.

    I also have found the "retelling" portion of this routine helpful for annotating. Waiting for another student to repeat what the first student saw can sometimes give me time to think about the annotation and can help to bring out some other math concepts that maybe the first student didn't explicitly say, but the second was able to understand and put into more concrete wording.
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  • 02/07/2019 1:21 PM | Jill D'Amico
    I agree that annotation is a powerful instructional strategy. I have been using annotation with my K-5 teachers through the use of Number Talks. When first beginning with Number Talks it was very difficult to not annotate a students's thinking because as the teacher you "knew" what the student was saying/thinking. The use of annotation provides an opportunity for students to be very precise and accurate when explaining their thinking and encourages math vocabulary! Our ultimate goal is that students then begin to use this math vocabulary in ALL of their math work.
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  • 02/08/2019 7:24 AM | Leah Frazee
    I think annotation is very important for my students when learning about solving systems of equations that describe a physical situation, such as how much fencing is needed to enclose a rectangular space with a divider. I see students use perimeter equations without considering the structure of the object to see that the amount of fencing needed for a rectangular space with a divider will not be the same as the perimeter. The way I would use annotation is by helping them develop a "fencing" equation whose mathematical structure could be illustrated on the diagram of the rectangular space. I would color-code each segment of fencing and show how each of those segment lengths are included in the new fencing equation--I think this may help students connect the physical representation to the mathematics in a similar way as Chapter 4 describes thinking structurally about the area of the net.
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  • 02/08/2019 7:43 PM | Karen Campe
    I work with a low-level Algebra 1 high school student one-on-one, and we've started to do some annotation with one of those two-side teacher pencils with a red point on one end and a blue point on the other.

    My student was learning equations of lines: how to graph them and write equations from given information or to model real-context situations. We marked anything relating to slope in red in the problems and equations, and used blue for points or Y-intercepts.

    This intentional labeling helped her to "see" where the information was and make connections between the verbal, algebraic, and graphical representations.
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  • 02/10/2019 4:54 PM | Karen Rivero
    Although I teach 4th grade, I have been following what many of you middle and high school folks do, hoping that when my students get to that level, they'll have the tools in place to be successful, thoughtful mathematicians!
    I recently scored a fractions/decimal quiz, (coincidentally, a "garden" word problem) which required explanation of their thinking in one question. In their haste to finish, they gave the correct answer, but neglected to explain, therefore losing credit; in the very next question, they were to show equivalent fraction AND decimals. They only gave one, again, needlessly losing points.
    In pairs, we later reviewed similar questions, stopping before the question to NOTICE the relationship of the numbers, and took a minute to label the representation accordingly. What a difference!!!
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  • 02/13/2019 11:41 AM | Laura Larson
    Before reading about the routines, I used to try and get my students to annotate their own work in a more productive way, so I love that many of the routines incorporate it in an integral way.

    As I'm reading and/or giving feedback on student work, I often have to struggle to follow the sequence, and they're not always great about solving a problem from top-to-bottom and/or left-to-right or even knowing whether one is more appropriate for a situation. The students themselves have a hard time keeping track of where they left off, which result gets fed into the next stage of a multi-stage problem, etc.

    When I ask students to add to their work, sometimes I just suggest letting me know "what they're up to" at a particular stage, in the hopes that not only will their work be easier to follow, but it will help them keep it organized as well.

    For example, when asked to find the "better deal" between two prices, it can be hard for students to manage telling the two offers apart unless they label them (as Deal #1 and Deal #2, or something similar). But they also lose track of the quantities themselves.

    So I think of annotating as going beyond just marking something up or color-coding, but actually using little words and phrases to help organize and lend structure to work that may have no structure of its own.

    I especially like the role of annotation (on the part of students) in the "Decide and Defend" routing, but that's outside of the book :).
    --Laura Larson
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  • 02/20/2019 2:45 PM | Terry Onofreo
    As a person who works best when I can visualize a problem, annotation makes so much sense to me. I find myself looking to do that when a student is explaining a strategy or some thought process - asking for the student to label what they mean. This helps in first grade when a student has what looks to be random circles drawn to solve an addition problem but, when labeled, show each part of the story. It helps when a fourth grader is explaining the strategy she's attempting to solve a multiplication of fraction and whole number but is getting herself lost in her explanation. "Label what you've told me so far" is a prompt that often gets their thinking back on track.
    When I read this chapter on annotation, I connected it to this "labeling" I've been prompting students to do. I am looking forward to working through this routine with students and having them annotate someone else's thinking.
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  • 02/24/2019 11:56 AM | Kerin Derosier
    Through our PD with Grace and Amy and reading through the chapters, I have come to realize how important annotation can be for students. I have always loved the discourse that comes with a hard question, but usually have left it at a students explains, a student retells, and if any clarifying needs to be done in the process I guide that. I think based on the limited time we have in class, sometimes my notes or way I write things on the board was extremely messy and confusing. The routines have really made me think about this and even prethink what annotation might look like. I now make sure to use colors thoughtfully and make sure that anything I am writing down for the students is clear and makes sense. It is great that this is a skill that as I am teaching my students I can talk to them about my weaknesses and the growth I have had in it as well!
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    • 02/28/2019 1:01 PM | Michele Hanly
      As educators, we are always aware of the time constraints during our lessons. We are constantly looking for more efficient ways to convey our message to students. However, sometimes we need to take the time necessary to "model" accurate and comprehensive annotation. "Children learn what they live" in school as well as at home. I have found that showing them, in addition to telling them, the correct way to complete this task can be an important investment in student understanding.
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  • 02/24/2019 9:24 PM | Luke
    On page 84 the authors refer to creating a record, or "visual residue"... as the purpose of annotating. I use underlining, arrows, record with different colored dry erase markers, circles, etc. almost on a daily basis. Sometimes I think I do too much and perhaps confuse the kids as a result. Sometimes I think the kids get too wrapped up in the recording of the notes that there is no "residue". I do not provide my students with additional problems for the purpose of reproducing annotations to refer to. It is one of those things that makes sense but doesn't happen.
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  • 02/25/2019 11:02 AM | Jennifer Rianhard
    Annotation has been a routine that I've been trying to implement when I go into the classrooms to teach problem of the week. Students have to dissect the problem, annotate their work on their own or with a neighbor, share with the class and then write about the problem. The two hardest parts is the annotation and having the students write about their math thinking. All of these suggestions to record their thinking: illustrating, chunking, highlighting, dotting, drawing arrows, labeling, etc. are all good strategies for all students to use to solve and understand the problem.

    I'm sorry this is so late. I fell behind. I'm also taking a rigorous Bridges enrichment class online so it's a lot of reading for me.
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