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.

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.

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 one’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 students’ mathematical thinking*.

This shift has required some further reflection. If we consider the definition of a teacher, according to Merriam-Webster (2018), as “one whose occupation is to instruct[]”and 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 doesn’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 “to gain knowledge or understanding of or skill in by study, instruction, or experience.” 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 students’ 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 students’ mathematical thinking. How will I learn what Maria’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. “learn,” accessed August 13, 2018, https://www.merriam-webster.com/dictionary/learn

*Merriam-Webster*, s. v. “teach,” accessed August 13, 2018, https://www.merriam-webster.com/dictionary/teach

*Merriam-Webster*, s. v. “teacher,” accessed August 13, 2018, https://www.merriam-webster.com/dictionary/teacher

Photo by Matteo Kutufa on 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. 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!

]]>The beginning of the school year is the perfect time for a “pretest” or initial reading of students’ affection for mathematics. I typically have my students share a picture that embodies their feelings toward mathematics and ask them to explain why they chose the picture. Below are some samples that my students have chosen in the past. I also share my own picture (top left below) and explain that mathematics is creative and fun and joyful for me (so I choose a picture to illustrate that feeling).

You might instead ask students to share an emoji that fits their attitude toward mathematics. Throughout the year, you might check in to see how their feelings shift. As students feel better about mathematics, they tend to do better and are more likely to continue on to higher level mathematics. You may find a number of your students start off with bad feelings or a lack of affection for mathematics. In that case, you know you have some work to do this year! Bring in their interests to the classroom. Help motivate why they might learn something by explaining how it is used in this course and beyond (it doesn’t have to be real world connections because not everything has clear real world implications and that is okay!). The important thing is that if you care about how students feel about mathematics, you should try to understand where your students are and make it a goal to improve their relationship to mathematics. As a bonus, you can share this data with your administrators and fellow teachers as another measure that everyone should care about in addition to content scores.

So who’s with me? This year let’s make it a point to care about and foster our students’ feelings about mathematics by measuring their affection for mathematics.

]]>As a student on my first days of school, I used to play the, *how bad will my name be butchered* game. Zandra de Araujo is not a common name in the United States and even though my last name has been Americanized from the original Portuguese pronunciation, my friends and colleagues still butcher it from time to time. My first name can also be said multiple ways. People tend to avoid saying my last name or just say I know I’m going to get it wrong as a blanket disclaimer. Why not try a bit harder this year with your students? I have an easy idea of how to go about this.

Early in the year, ask students to record a short video of themselves saying their full name and answering a few questions. Students tend to have handheld video recorders on them all the time (cell phones) and if they don’t, you can let them use your iPad to record a short video really quick at some point early on. Who better to learn how to same someone’s name than from that person? It is important that you do this with all students (even the John Smiths) because you want to give each student the opportunity to share with you.

In addition to their names, you might ask the students to tell you what you think you should know about them as learners, tell you about their favorite math teacher, ask them about their hobbies, ask them what is something people think about them that isn’t true, etc. This will give you better insight to your students as people and opportunities to draw on their resources in the classroom. Here’s an example of what it might look like:

You can have students share their videos via Google or put them in a shared Dropbox folder or just send you the link via whatever your classroom management system is. It will also help you put names to faces quicker. I also encourage you to make a video answering the same questions for your students. They will appreciate it and maybe you will get called Dr. de Araujo instead of Dr. de all year (not that I mind either way).

Give it a whirl and let me know how it goes!

Other related resources that I enjoy:

- Pronouncing Students’ Names Correctly Should be a Big Deal (EdWeek)
- How we Pronounce Student Names and Why it Matters (Cult of Pedagogy)
- The Lasting Impact of Mispronouncing Students’ Names (NEAToday)
- Facundo the Great (NPR Story Corps)

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I go through this process in every class I teach and even during workshops I give. My preferred approach is to start with the group/class to brainstorm norms together. You can ask them questions to get them started such as,

- What do your fellow classmates do in class that help support your learning? What do they do that distracts you from learning?
- What do your teachers do that is helpful in supporting your learning? What do they do that distract from your learning?
- What are some things that you want your group members to keep in mind as you work together?

These norms are mainly about behaviors, and that is okay. We need to talk about what helps create a supportive learning environment for our students. You can write down the students’ ideas and then have a discussion with the whole class about them to see if everyone thinks they are appropriate or if they should be modified in some way. This allows them to take some ownership over the classroom norms. As the teacher, you should also have some things that help you be the best teacher you can be on the back burner in case they don’t come up. Here are some of my favorites.

**Step in and step out**: Talk*and*listen. Monitor how much you talk and try not to dominate conversations and encourage others to participate as well.**Amplify one another**: Quiet students and traditionally underserved students may not get heard as much in class, particularly during whole group students. Students who are not confident may speak up in small groups but not in whole group settings. Encourage students to amplify one another. Share others’ ideas and give them credit. This builds everyone up and helps give voice to those that aren’t always heard.**Assume goodwill**: Sometimes people say things that don’t sound great. I encourage you to always assume the best motives behind what was said and go from there. This doesn’t mean you don’t address problematic behavior, it just means you don’t leap to the worst possibility. For example, if a student is on his or her iPad, I tend to assume they are taking notes. I might say something like, “you actually don’t need to take notes right now because I am going to put all this online later.” I could just as easily have said stay on task and put away your iPad, however, what if they were taking notes? Even if they were messaging, it is still better to assume the best because assuming the worst and being wrong is far worse for your relationship with students than assuming the best and being wrong (generally speaking).**Don’t freeze others in time:**The whole point of school is to learn and grow. Therefore we need to allow others to grow. Allow people to move past ideas as they learn. It would be unbelievable unfair to freeze me in my beliefs and ideas from when I was 20. Allow people to create new beliefs and perspectives without always bringing up the past.**Assume it’s possible and think big**: Sometimes we might say our kids can’t do something. We say it for many reasons, but it is never an okay thing to say. I encourage you to hold others accountable (and yourself) by adding a yet after statements such as this. My kids can’t do that YET. It is our job to find new ways to help get them there. We also need to make sure our kids have positive self dialogues and say they can’t do that yet, but they are learning how. We need to take the perspective that we can all learn how to do something, sometimes we just need some more time or help.**Remember why we are here**: We as teachers are here to teach children. Our students (and us really) are here to learn and grow. Keep our learning and personal development and care at the forefront of what we do every day.

You should revisit these norms from time to time, particularly early in the school year. As you move to new groupings and settings, you may find you need to modify or adapt your norms. I imagine my norms as the contract our community has developed together to help care for one another and to foster meaningful learning.

In addition to these more behavior-specific norms, mathematics teachers should also talk about norms for mathematical work in their courses. For example, what does a good mathematical explanation look like in your class? Should it include complete sentences? Math drawings? Something else? It might be quite different in your classroom from another teacher’s classroom. We need to be explicit with students in discussing our expectations. This goes for how we share mathematical ideas, how we present work, etc. It also goes for other small things like how we sharpen pencils, get late work, use electronics, etc. Students aren’t mind readers; share your vision of what it means to be a good community member and develop an effective learning environment with your students early in the school year.

Feel free to share your ideas and practices regarding norms in the comments below!

]]>The first days of school are critical to paving an easy and productive way forward for the rest of the year. For the remainder of summer I will devote a few of my 180 Ideas to discuss strategies that you can use to kick off the school year right. So be on the look out, ~~winter’s coming~~ school’s almost back in session!

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You arrive to school early because you are so excited to give the lesson a try. The students come in, you launch the lesson, and then it all goes wrong. Maybe it goes wrong because the students aren’t making connections to the prior knowledge required to solve the task. Maybe your directions were unclear. Maybe it’s a bit too complicated in terms of language or content. Maybe the students have a pep rally next period and they can’t focus or they just interrupted class to announce the homecoming course. Whatever the reason, this lesson is going south fast.

You think about all of the time and effort you’ve put into creating the lesson. You play the video of how you imagined the lesson going over and over in your mind. You lament what could have been. You can’t bear to think of abandoning your beautiful lesson.

At this point you need tostop, collaborate, and listen! (sorry for the Vanilla Ice reference)

Remember the reason you poured your heart and soul into this lesson was to support student learning. If the students aren’t working productively toward the lesson goals you need to change course. What does this mean? Well, you have to **stop** the lesson and check in with the class (**collaborate**). Ask your students what is going on and **listen** to what they have to say. It is quite likely they will tell you what they need or at least what is going on. If the lesson is too complicated or the students are unproductively struggling, have a brief discussion to allow students to discuss aspects that are unclear, approaches they’ve tried (that worked or not), ideas they have for what strategies to try next, etc. This will provide students with ideas for how to keep moving forward. The goal is to keep students engaged with the mathematics, and unfortunately for your lesson, that might mean abandoning some aspects or creating a revised vision for the lesson moving forward. You may also have to help them to connect to some other ideas or ask some questions to get them working again. If they are just too excited about non-mathy happenings (e.g., pep rallies, homecoming courts) then set a timer, give them a couple of minutes to get it out of their systems and then move on. Sometimes you just need to let them be kids for a bit before they can concentrate on mathematics.

So to summarize, we have to be responsive to our students, not our lesson plans. If something isn’t working, you need to **stop**, and then **collaborate** with and **listen** to students so that you can change course. Remember your commitment is to your students and their learning. We must evaluate the impact of our lessons *with* students; you can’t say whether a lesson is great or not absent students. Also, there’s no such thing as the perfect lesson-you can always make a lesson at least a little bit better-that’s what makes teaching both challenging and interesting! I’d love to hear any of your experiences changing up lessons in response to students.

Once students have developed some understanding of different ways to approach a particular type of problem, we need to have them make sense of when to use which of these strategies. My favorite way to do this is to just make this the whole task. For example, instead of having students graph the following lines or lines containing the given points,

I would instead ask them to:

Choose the strategy (calculator, determining two points and connecting them, using the slope and y-intercept, mental math*, etc.) you would use to graph each of the problems and explain your thinking.

*mental math means you can do it in your brain without showing work. Students should still explain what their brain is doing.

Notice, I didn’t ask them to actually graph the lines; they’ve already practiced that. Instead, I asked them to think about *how* they would graph the lines. The difference is subtle, but important. When we say our students aren’t making sense of the math, it’s often because they never had to do so. If we design tasks where this is the entire goal, they will grapple with choosing a strategy and, consequently, get better at it. As a bonus, you can also have students share their choices. If students choose different strategies for the same problem you can have an interesting discussion about the different reasons behind those choices. Having students realize that different strategies just make for different adventures en route to solving a problem is just another great benefit of this idea!

If you want another example of lesson that uses this idea, a few years ago my friend and frequent collaborator Sam Otten and his brother Andrew (who is a math teacher) created a lesson on systems of linear equations that was published in the Mathematics Teacher.

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