Here’s an activity that my friend Nic Vitale and I wrote for MTMS‘ Math for Real and then found out that they had published something similar (Bu & Marjanovich, 2017) so they decided not to publish our article. It’s funny that Nic contacted me about this situation and we wrote it up on a whim and apparently this also happened to others. I think ours is a bit different from the one published in MTMS (check that one out here) so I thought I’d share it with you all.

Understanding percentages and their applications is an important topic in middle school mathematics. One aspect of this topic that is particularly important is that percentages, like fractions, refer to a particular whole. In instances in which the whole is not clearly stated, it can be difficult to make sense of situations. For example, if I said that I got an 80% on quiz A and 75% on quiz B, one may think it possible to conclude that I answered more questions correctly on quiz A. This is not necessarily the case; quiz A may have had 5 questions and quiz B could have had 100 questions.

The importance of understanding percentages and their related wholes came to light in the real world during a conversation one morning between Nic and his son Levi. Nic and Levi were drinking milk when the following conversation happened:

Levi: â€śDaddy, whole milk has forty percent fat!â€ť

Nic: â€śWhat do you mean?â€ť

Levi: â€śLook!â€ť (pointing to label on milk container, *Figure 1*)

1.Maria told her mother that she got an 80% on her first quiz and 75% on her second quiz. Her mother concluded that this meant Maria got more questions correct on her first quiz than on her second quiz. Is Mariaâ€™s mother correct? Explain your thinking.

2. How do you think Levi arrived at his conclusion? Explain any interpretations or assumptions he might have made in his reasoning.

3. Decide whether Leviâ€™s thinking is correct. Justify your answer.

4. Based on the label (*Figure 1*), what is the percentage of milkfat in whole milk? Justify your answer and include a math drawing.

5. If whole milk did have 40% milkfat as Levi concluded and reduced fat milk still had 2% milkfat as shown, what percentage should replace the 38% on the label? Justify your answer and include a math drawing.

6. Redesign the label so the information is clearer to the buyer. Explain your reasons for your redesign.

1.No, Mariaâ€™s mother is not necessarily correct. The second quiz may have had more questions than the first and so, although the percentage correct on the first quiz may be greater, the number of questions correct may be greater on the second quiz.

2. Answers may vary. A probable answer is that Levi simply added the 38% and the 2%, interpreting â€ś38% less thanâ€ť as an additive comparison between two fixed quantities of the same unit (similar to: Wendy had $2, which was $38 less than what Mary had, therefore Mary must have had $40).

3. No, Levi is not correct. The 2% refers to the percent of reduced fat milk that is milkfat. The 38% describes how the 2% compares to the percentage of milkfat in whole milk. Because they refer to different wholes, you cannot add those percentages together to get a meaningful answer.

4. Based on the given information, whole milk is about 3.2% milkfat. Students may arrive at this in different ways. The 2% milkfat on the label is 62% of the milkfat in whole milk (100%-38%). Thus, students can set up a proportion to find how much milkfat is in 100% of whole milk as follows .

5. The label would then have to read 95% less fat than whole milk. 2% is 1/20^{th} or 5% of 40% so 2% is 95% less than 40%.

6. Answers will vary.

]]>There are three points and two lines on a plane. All we know is that one or both lines go through each point. Discuss the possible ways this might occur.

I think this would be an interesting exploration. I would encourage students to use a tool like Desmos or paper and pencil as they think it through in groups/pairs or individually. The students could ask all kinds of questions such as (with some possible responses in parentheses):

- What are the points? (
*Does it matter? Can we try different points to see if it matters? Maybe this is a good thing to discuss in your response*) - Where are the lines? (
*That’s what we are trying to think about.*) - Can I graph this? (
*Sure, how do you do that if the points are unnamed?*) - Are the points collinear? (
*Good question, no one knows, so maybe we should account for different possibilities*.)

Solving this question would encourage students to graph and think about points and lines as they try to generalize relationships. Though there are particular answers, there are many ways to explore and students could easily work together. I like that students would need to consider multiple situations and have to articulate them in their response. I also think it would also provide a great basis for discussion to have as a whole class as you compare strategies and discuss whether everyone has considered all of the possibilities.

Thoughts on this task? Add them to the comments.

]]>- Part 1: Notes, but why though?
- Part 2: The Purpose of Notes
- Part 3: Aligning the Purpose and Process of Notes

If you recall, I left off my prior post with the following vision for the purpose of note-taking

“I still want students to take notes so that they have **an artifact of their learning**. I want them to have **a reference for when they study that is of their own creation.** In addition, I want notes to provide a means for my students to think about their thinking: **a metacognitive tool.** I don’t want the notes to be a restatement of my thoughts, rather, I want them to be a record of their interpretation and sense-making of the ideas we discuss.”

So, how do I help foster a note-taking process that aligns with these purposes? Here’s what I came up with (though I am always open to new ideas!).

Typically, note-taking occurs concurrently with content delivery. It is difficult to think about what you are writing when you are simultaneously listening. [Note: I don’t mean writing solely in terms or text, pictures and diagrams also constitute writing] In order for note-taking to serve as an artifact of learning, there must indeed be learning. This means, students need to understand an idea or concept and they then need to synthesize their understanding well enough to create a written artifact. Thus, students must be fully present and involved in the discussion of the ideas as they occur. It is often difficult to be present if you are frantically recording information word for word. Instead of note-taking occurring at the same time as the discussion of new ideas, why not provide time/space for note-taking after that discussion?

Here’s an example of what I imagine. Say I am teaching about quadratic functions. Whether it be an activity or direct instruction, students engage in the process fully. They may briefly jot down some questions or note a key term, but they are discussing the ideas that are brought up and asking questions as they arise. Then, I (the teacher) give them some think time to consider the big ideas they think they might want to remember about our discussion. They think it over in their brains, maybe discuss with a partner, and then I ask them to make a record of what they want to recall in their notebook. I don’t have a designated format for this, I ask them to create a record that is meaningful to them. Students might restate things in their own words, draw pictures, create metaphors, note important examples and non-examples, etc. I’m actually not sure what they will come up with, which I find intensely interesting. The goal is for students to create notes that they can make sense of later as they look back through them. Following this process, we can, as a class, discuss some of the ways they noted their understandings.

There are a number of things that appeal to me about this process of note-taking. First, students are present during important discussions . No longer will they be frantically transcribing what I write. Instead, they will be invested in understanding. I also think the time to synthesize is key to learning. Giving students freedom to record their thinking in a way that makes sense to them is exciting. This is not only a skill that will serve them well in many areas of life, but it also serves as a formative assessment. I can see how they are linking ideas or how deep their understandings are. I do think that it may be necessary to discuss different ways you might record your thinking, but I don’t think I need to prescribe particular forms. For example, I might suggest that diagrams or examples could be helpful or that they might consider creating a metaphor or analogy to help them make sense of an idea, but they don’t have to.

I think this process aligns well with my goals for note-taking to serve as a metacognitive tool and an artifact of their learning. However, in order for notes to serve as a reference from which to study, I think we must discuss how one might accomplish this. Not all students know how to study. If we expect students to later reference their notes, we should take time to share strategies for studying with notes. For example, I read through my notes, discuss ideas with friends, do practice problems, etc.

As I think some overarching guidelines for note-taking in my courses, here is where I am at:

**Discuss the purpose and process of note-taking with students.**The note-taking process should not be taken as a given. Let students understand why it is important to note their understandings and think about what they are learning. Furthermore, students may not know how to use their notes as study aids. Discussing the studying process with students is important!**Provide time for students to think about what they intend to write prior to writing.**Don’t expect students to transcribe what is said as it is being said. Multi-tasking doesn’t work well. Also, it is really hard to know what you do or don’t understand if you are worried about recording rather than understanding what you are hearing/discussing. Students need time to think. The writing process is used to further refine their thinking and to record their ideas for future reference.**Don’t force students to take notes in a particular format.**Notes must carry the meaning for our students, not for us. Allow students to experiment and reflect on those experiments. Also, provide space for students to share their note-taking strategies with one another so they can expand their repertoire.

So there it is. I’ve thought through why and how I want students to take notes. I’ve reflected on the past and created a process for the future. I’ve also tried out this process myself as I read articles for work. Instead of taking notes as I read, I read and then pause, think about what I want to take away, and then go back and note some ideas. I link to other ideas that are relevant, note ways I might draw on the ideas for particular projects, etc. I have enjoyed the process and I think it is making me think more deeply about what I am reading. I am looking forward to trying this out with students.

**Professional Learning Opportunity**: Take some time to think about and jot down your own process and rules for note-taking. Remember, these are not set in stone, you should revisit them to see what is and isn’t work. Revise as needed or as you get new ideas!

I’d love to hear from others regarding your purpose and process for note-taking. Feel free to discuss your thoughts and experiences in the comments.

]]>**Professional Learning Opportunity**: Take 5 minutes to write down why you **have had (past tense)** your students take notes (or not) in your classroom. If you have never pondered it before, this is a good time to start! Step away from this paper and then reread aloud. Now, edit it (don’t erase, write a new draft if needed) to reflect why you **would like (future tense) **to have students take notes (or not) in your classroom. Keep this handy for the next part in the sequence.

Let’s examine my responses for each of these questions.

As I thought about this question, I had to consider my work as a high school mathematics teacher separate from my work as a univeristy mathematics educator. As a high school teacher, I (as I mentioned last in my last post) did not think about note-taking in depth. I**n retrospect, I thought that having students take notes would allow them to have a reference from which to study and to ensure they were attending to the important ideas. It also, sadly, served as a classroom management strategy at times;** the students were quiet and attentive while taking notes.

To understand how mismatched these purposes were with the process, you have to understand the process of note-taking in my former classroom. Please understand that this process was not unique to me. Since leaving K12 teaching I have observed countless math classes at all levels in a variety of locations and note-taking kind of looks the same everywhere I go. So read this not as a specific case, but as a portrait of traditional note-taking in American schools.

As I started a new lesson, I would write out some important ideas on the overhead project (now mostly done via slides on the Smartboard). I developed these notes after reading through and pulling out key ideas from the textbook. This means that the students could have found the same information in their books in greater detail (mine were summarized and largely comprised of definitions and worked examples). As I wrote, my students wrote. They dutifully copied down all my words and examples without question. I even remember them copying down errors and then groaning as I caught those errors and had them fix their notes.

The speed of note-taking was the speed of my writing and talking. How could they possibly have been making sense of what was happening when they had to keep up with my writing? I now firmly believe that these notes had little meaning to my students. The fact that they copied down errors without pause suggests that they were not comprehending what they were writing. In my dissertation study, I termed this process of note-taking a transcription activity. Indeed, most of my students could have been writing down the notes in Portuguese (a language none of them spoke) or any other language and had the same understanding.

I will note that through the years I changed my note process up a bit. I moved to guided notes in which I printed packets for students that had a lot of the information pre-printed aside from blanks for key terms or worked examples. I thought that these might help them keep up with my writing and speaking (I’m a fast talker). The students did like these notes. They could listen until I came to a key point or example and then they had simply to fill it in. Again, I’d argue that this is a transcription activity and they were not really learning through this process. In both of these processes (traditional and guided notes) I also realized that students were taking notes that made sense of the ideas *to me*, they were not encouraged to think through the ideas and note them in ways that *made sense to them*. This has been a major revelation to me and I revisit this dilemma in the next post.

So, in considering my intent for notes to serve as a reference and to ensure students attended to the important ideas, my process was horribly misaligned. Students would have notes and sure, they could reference them, but what good is a reference that you don’t necessarily understand? Also, why not have them use the book for that process as it has greater detail? In addition, if you don’t understand the important ideas, how can you attend to them meaningfully? These points are all so glaringly obvious now, but they weren’t then.

In terms of my final purpose, classroom management, the process actually did support this. I believe this is the reason that note-taking done in this way continues to persist. However, we must ask ourselves if we are okay with a classroom management strategy that does not support math learning. I am not. Furthermore, I believe that much of the classroom management issues we face are actually classroom engagement issues. Surely we can develop more engaging ways to create artifacts of our learning while actually learning!

As a university educator, I typically use PowerPoint slides to help me keep on track of what I have planned. They don’t have a lot of text, but do have some main ideas. I post these slides and tell students they don’t have to take notes, they can simply reference the slides later so they can focus on the conversation in the moment. However, it still seems as though there is value to taking notes of what is happening and key ideas. Furthermore, there have been studies that state the benefit of hand writing notes versus electronic notes. Most of my students only use devices. Should I rethink my practices in light of this information (spoiler alert: yes!).

So in summary, my purposes and processes for note-taking are out of alignment, save for classroom management, which I can find different ways of addressing. This exercise helped me critically examine my purpose and process and think toward the future.

I still want students to take notes so that they have **an artifact of their learning**. I want them to have **a reference for when they study that is of their own creation.** In addition, I want notes to provide a means for my students to think about their thinking: **a metacognitive tool.** I don’t want the notes to be a restatement of my thoughts, rather, I want them to be a record of their interpretation and sense-making of the ideas we discuss.

In my next post, I will delve further into processes that would help to support these purposes.

**Professional Learning Opportunity**: Take a bit of time to detail your current process for note-taking (if you have one). Then, discuss the ways in which your process and purpose align or not.

As you think about the thing you learned, consider carefully your learning process. Did you take notes? Did you try over and over? Would you process differ if what you were learning was a concept or a skill or was for school or for a hobby? I imagine it might. When I think about my hair braiding example, I found the best way for me to learn was to practice along with the video. I paused and rewound again and again as I tried to braid and re-braid my hair.Â I didn’t write anything down, I had a tutorial I could go back to again and again as needed.

If I instead consider trying to understand the ideas presented in an academic paper, my process would differ. Clearly one difference is the format, I’m now reading rather than watching. However, there are other differences too. For one, reading academic papers is something that is important for my job, not just a new way to do my hair for the day. For another thing, the ideas in the article are things that I want to understand, not just do. I also want some record so that I can come back to it later. In this instance I will read the article (it exists digitally on my computer) and I will highlight the PDF and stop periodically to take notes in my journal as I read. This affords me the opportunity to interpret what I am reading in my own words and also to keep track of important aspects that I might later want to return to (via highlighting).

In my two examples what I was learning and why I was learning differed. When I set out to learn something new I often draw upon my method without much thought. Lately, I’ve been trying to be more mindful of my methods which again led me to note taking in the mathematics classroom. As I sit in math classes and think about the purpose of notes it occurs to me that the students and the teachers in the classes may not have done the same.

I don’t remember teachers telling me about the purpose of notes beyond, “so you remember this later,” but if that was the case, most of what was written had already been written in the textbooks prior to my arrival in the classroom. In addition, copying down what was happening as it happening wouldn’t help me remember unless I went back to them at some point and if they make sense to me, right? As a teacher, I made my students take notes because it is what one does in math class. I had evidence of this; as a math major I graduated with notebook upon notebook of notes from all sorts of math classes. (for some reason I moved these with me from place to place, even though I never referenced them, until I finally got rid of all of them a few years ago). However, in retrospect there was another purpose for making my students take notes, and this revelation is a bit embarrassing in retrospect. The students were so much easier to manage when they quietly sat taking notes.

I readily admit that as a new teacher I was lacking in classroom engagement (a term I prefer to classroom management) strategies. Note-taking-time was this magical portion of class during which I felt in control and knew that the students must clearly be learning since they were so quiet and attentive (*snort*). What I failed to realize was that they could have been taking notes in a Klingon with the same degree of success and learning that resulted from the process. So now, I am trying to reevaluate my own purpose for having students take notes (or not) and I hope you join me.

Below, I have an exercise that can serve as a professional learning opportunity for you prior to my next post. I hope you try it out (I will post what I come up with next time) and join me for the next post early in the new year!

**Professional Learning Opportunity**: Take 5 minutes to write down why you **have had (past tense)**Â your students take notes (or not) in your classroom. If you have never pondered it before, this is a good time to start! Step away from this paper and then reread aloud. Now, edit it (don’t erase, write a new draft if needed) to reflect why you **would like (future tense)Â **to have students take notes (or not) in your classroom. Keep this handy for the next part in the sequence.

As a math student in school (at all levels) and as a math teacher, I too took notes as a given. My teachers would teach and I’d be told the importance of taking notes. As a teacher myself, I’d carry on the mythos of notes proclaiming the importance of this process for learning. Lately, I’ve been questioning this practice for a number of reasons (which I’ll get into in subsequent posts). Thus, I’ve decided to take a deep dive into the process of note taking. This process will involve me questioning the purpose, process, and need (couldn’t think of a third ‘p’) of notes in the mathematics classroom. I hope you join me for this journey over the next few posts, the first of which will be released the beginning of next week. So gather your thoughts, be ready to critically question conventional practices, and join me in finding out an answer to “but why though?!?”

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The first time this phenomenon really perturbed me (apparently I’m in a GRE vocab kind of mood) was a few months ago when I was working with a student in a classroom. In this class, it was common for students to work on individual whiteboards. I get it, students who don’t enjoy writing on paper will sometimes jump into action when they get to write on something novel like a whiteboard. However, the problem was that we were working on problem after problem and I couldn’t draw connections between them because there was no record of our work.Â

This was incredibly problematic. How can we look for patterns and make sense of structure without being able to look at multiple problems? How can my students go back and revise their thinking or strategies if there is no record of where they’ve been? How can we look back at how far we’ve gone without our historical records? How can students study their work to prepare for assessments? How do families discuss mathematical ideas from school if there are no artifacts to anchor these discussions? How can teachers connect one worked example to another without the ability to compare and contrast the images simultaneously? How can teachers really understand student thinking and writing (particularly in large classes) if there are no artifacts to consider? These questions have been troubling me for several months.

In light of my reflection on this (more and more common) approach in school mathematics classrooms, I encourage you to leave behind a trace. Have students use a notebook to document thinking as they work on whiteboards or screens. Embed questions that they should explore as they go. For example, have them stop after 5 minutes on a computer game and discuss the strategies they are using and how their thinking is progressing. Then, have them look back on prior updates to see how their strategies are evolving. If students are on whiteboards, have them note their thinking for problems every so often so you can see how they learn from their practice and they can revisit their thinking.Â

I’m not saying that there should never be activities for which there is no record. I am saying that we should consider what we lose and what we gain from the creation of records of practice so that the process for our and our students’ documentation (or lack thereof) of mathematical representations and/or thinking is purposeful. I’d love to hear your thoughts below!

]]>My favorite way to solicit student feedback came from one of my favorite professors (and math ed legend) Dr. Jeremy Kilpatrick. Dr. Kilpatrick had a simple system for gathering course feedback and he did so in lieu of the university’s largely useless (IMO) evaluation system. Here’s how it works:

- Step 1 – Have each student get a piece of paper and fold it into quadrants.
- Step 2 – Mark the quadrants (in no particular order): Do More, Do Less, Start, and Stop
- Step 3 – Ask students to describe what they wish you’d do more, do less, start doing, and stop doing in the relevant quadrants.

That’s it! I just did this with my students. I got useful feedback to improve my course moving forward. I preface this activity by talking about what is useful feedback to me and what is not. For example, saying stop doing readings is not useful. I don’t know why you want me to stop and what would be more helpful in their place. So, I tell them a more useful comment would be something that explains what you don’t find useful about the readings and a possible alternate to that practice. Giving useful feedback is an important skill that will serve them well for years to come.

As a K12 teacher I seldom did gathered formal feedback on my teaching from students, but I sure wish I had! I’d love to hear other ways you collect feedback from your students, feel free to share in the comments.

]]>For the past 15 years as a teacher, and now teacher educator, I have shifted my internal dialogue about what it is I do as a teacher. These shifts were not entirely intentional and it was only recently that I took the time to reflect on the changes and the ways in which they manifested in greater motivation for and satisfaction with the work I do every day. I still find it shocking that the ways I discuss, with others and internally, what my work entails so clearly reflects my practice and perspective.

When I started teaching, when people asked what I did for a living I would say, *I teach mathematics to students*. This is a common way to describe oneâ€™s work as a high school mathematics teacher and it is also reflective of traditional models of teaching. To say* I teach mathematics to students* is to center me as the teacher and portrays the notion that I am doing something *to* students rather than *with* students. This interpretation of my description reflected well my early work in the classroom.

Early in my career, I tended to create lessons based on the content that I was teaching without much consideration of the students I would be teaching that content to. However, as I taught in this way, I noticed I was not reaching all of my students and so, over time, I began to center my students more and more as I planned and implemented lessons. These shifts in my practice came alongside a shift in my description of what I did. After several years, when people asked what I did I told them that *I teach students mathematics*.

This shift in description was also reflected a change in my teaching practice. As a more experienced teacher, I was centering students; I taught *students* mathematics. I realized the human beings I teach are the most important (and interesting) aspect of my work; content is second. When we think about our students first, we can then better think of how to create access and equity for our students, as we teach mathematics. With this new perspective, I found myself motivated and continually challenged to first know my students and then figure out ways to help them learning interesting and meaningful mathematics. It was always a challenge and it was (almost) always rewarding.

This shift was a good start in reconceptualizing my career, but I have since further reframed my practice, and perhaps consequently or concurrently, my view of what I do as* learning about studentsâ€™ mathematical thinking*.

This shift has required some further reflection. If we consider the definition of a teacher, according to Merriam-Webster (2018), as â€śone whose occupation is to instruct[]â€ťand the definition of what it means to teach including:

- to cause to know something
- to cause to know how
- to accustom to some action or attitude
- to guide the studies of
- to impart the knowledge of (https://www.merriam-webster.com/dictionary/teaches)

We see a portrait of teaching that likely aligns with the majority of those in the United States. Among each of these meanings the implication is the teacher contains the knowledge and he or she is sharing it with others. This just doesnâ€™t seem to be enough to motivate me; the interesting thing about being a teacher is that I get to work with students! Students are pretty much absent from these definitions. I needed to think more about why do I have students in my room with me and what they contribute.

When we consider the definitions of learning we see they include â€śto gain knowledge or understanding of or skill in by study, instruction, or experience.â€ť I like framing my work as the work of a learner because it reflects that I, the teacher, am not just imparting knowledge; I am learning something by engaging in my work. I am even more inspired by the other definitions:

- to come to be able
- to come to realize (https://www.merriam-webster.com/dictionary/learn)

With the description of my work as* learning about studentsâ€™ mathematical thinking,* I now enter my classroom wondering about what I will learn from my students today. What will I learn about their understanding of whatever it is I am teaching? What new ways of solving or talking about mathematics will I hear? How will I learn what they know and then pose new questions to know how deeply their understandings are? These questions guide my work and motivate me as I notice and wonder in my classroom. Instead of planning to teach content, I plan experiences to help me learn about my studentsâ€™ mathematical thinking. How will I learn what Mariaâ€™s understanding of quadratic functions are? How can I devise a task that will help me learn how Dan decides whether to use elimination or substitution to solve a system of linear equations? These questions lead to a more interesting and engaging lesson for me and my students; it never gets old and I find myself excited to know more about my students and find new ways to uncover and develop their understandings.

As I continue to work as a mathematics educator, my practice will continue to evolve. One of the most motivating aspects of teaching is that you can always get better; there is no such thing as a perfect lesson or a perfect teacher. As I continue to learn from students, I will inevitably continue to grow as a teacher. As I grow, I look forward to seeing how my dialogue shifts once again as I come to be able and come to realize what is possible from and with my students.

**References**

*Merriam-Webster*, s. v. â€ślearn,â€ť accessed August 13, 2018, https://www.merriam-webster.com/dictionary/learn

*Merriam-Webster*, s. v. â€śteach,â€ť accessed August 13, 2018, https://www.merriam-webster.com/dictionary/teach

*Merriam-Webster*, s. v. â€śteacher,â€ť accessed August 13, 2018, https://www.merriam-webster.com/dictionary/teacher

Photo byÂ Matteo KutufaÂ onÂ Unsplash

]]>As a math major, I was surprised at just how much writing there was in college mathematics. Where was all the writing in high school math? Why was written communication so much more prevalent in college math? Didn’t I say I was a math major because I didn’t want to write? It may then come as little surprise that as students enter their first “proof course” in college, many students struggle to communicate their thinking. How can we help support students in expanding their comfort and skill in writing about mathematics?

One thing that helps is to explicitly discuss the purpose of writing in math and what a strong mathematical explanation entails. For example, I have told students that we are writing to share our problem solving processes, convince me (mathematically) that that process has resulted in a solution, and that you understand why it is indeed a solution.Â I don’t stop there, I also talk about what a good explanation entails. This might look very different from class to class or grade to grade. For example, in kindergarten (yes we write in kindergarten) we show our thinking with pictures and numerals as best we can. In high school, we might use complete sentences that include well labeled diagrams to further evidence our thinking. I also ask students to avoid pronouns because I often don’t know what it refers to and I want them to be precise in discussing their ideas.

As we develop writing, something that is helpful is to allow students to share their writing and follow along with it to see if you can recreate their strategy and solution. Doing so usually uncovers aspects that are not clear or not logically consistent. For a recent elementary PD session, we made a list of expectations for math writing so that students could begin thinking about what constitutes an adequate explanation as they wrote them. The hope was students would start thinking about their writing and understand what makes a clear explanation. We adapted down to kindergarten and up to fifth grade (see below for the general idea). We didn’t just give the list to the students, we practiced writing and then pulled out things that were clear and less clear and talked about what made them clear (or not). We then talked with the students about each of these in the context of their writing. It takes a lot of time, but we hope that over time if we continue to use it then they get used to communicating their thinking in writing.

Some of my favorite prompts for writing in math involve things such as write a letter to your family to explain how you solved this problem. Pretend you have a pen pal and explain your thinking to them. Somebody said they don’t believe your answer is correct, write a letter to prove you are right. Write about an error you think someone could have made on this task and explain why you think people might make that error. There are lots of ideas for prompts, but you could also simply ask students to explain their thinking.

So, as you kick off the school year and discuss expectations, make sure to develop and discuss your expectations for writing and help students understand them. Also, make sure not to shy away from writing; if your students aren’t good at something that is a pretty good clue you probably need to do more of it, not less. If you have some routines or ideas or resources for addressing writing in math I’d love to hear them!

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