A Mathematics for Teaching Reflection

Mediated Field Experience is the term my colleague Esther Billings is using for our preservice elementary math education course that takes place in the schools. The schools have made a classroom open to us, and we teach weekly. What makes it mediated is they use and modify lessons the profs provide, and we observe them and give feedback. When it works out, we also have a student assistant who observes and gives format using a short reflection heuristic. This is a bit of a contrast to many of their school experiences as preservice teachers before their internship, where their time with students is unstructured or less supported. 

This semester both of my sections taught two 3rd grade classes, usually repeating the lesson with their other group. Pairs of college teachers worked with 2 to 4 kids, grouped by the teachers, for 45 min. Before they taught, I did a number talk with the whole class for 10 minutes. (Problem strings, quick images, numberless problems, story problems, etc.) Here's our lessons from this semester.

There is a flipgrid of many of these students talking about their experience, and why they think other students should choose this over the regular college course (which is still an option).

This semester, despite the interruption, I hit on a simple end of semester reflection that surprised me with the connections that these teachers made. I've had trouble giving the principles for effective math teaching much life in an academic setting (or the even harder to communicate high leverage practices). So I thought I would share a representative sample.

I asked: "Read these 8 principles for effective teaching of mathematics. For each, give an example of when you as a teacher did, we as a class did, or read about a teacher doing that principle. (list)" Everything in the bullets is a direct quote from a preservice teacher.

Establish mathematics goals to focus learning

• As a class, we accomplished this task through the content journals that were assigned throughout the semester. The content journals were focused on content standards which stated clearly the goals of each day we spent together in the class. By explicitly stating the standards in text, the instructional goals were shared with each of us. This not only allowed us to have a clear understanding of what we were going to be learning, but also allowed us to understand the content in a more concise way. We then put this principle to work by using the content journals to determine our understanding of each of the goals. For each of the content journals, we were given the standard that pertained to the classwork and classroom learning we had done and asked to show our understanding of that standard.
• Everytime my teaching partner and I meant with the students we were working with before we did our warm up activity we would tell them the goals for the day. We never would say by the end of the activity that they should have full understanding or grasp on the topic, but rather we would say such things as, our goal for today’s activity is to think of addition/subtraction, place value, etc. strategies.

Implement tasks that promote reasoning and problem solving

• As a teacher I have done this multiple times with my lessons. When working with groups of students, not everyone has the same way of getting an answer so I make sure that everyone has a chance to say how he/she solved the problem. If they all have the say way of getting to it I would ask them if they knew another way to solve it or how else they could get the same answer (I would ask this even if they didn’t all take the same path of solving)
• An example of this is when we created the wordless story problems. We also read about the teacher in Chapter six [Tracy Zager's Becoming the Math Teacher You Wish You'd Had, our text] who gave problems with multiple entry points
• We did a lot of story problems with one of our groups because we could see their thinking better. A lot of the times they would disagree on what the correct answer was so they would have a discussion with each other for how they got their answer. This helped them show their solution and problem solving through their work.

Use and connect mathematical representations

• A time I used this as a teacher was when my students and I were doing a number line on a white board, we started at a certain number and I had them added the same number to the prior number, taking turns in a circle. After going around a few times I gave them a number to get to and they had to figure out different ways to solve it. They subtracted the number they ended with from the number I gave them, they added by 10s or 20s until they got close to the given number and figured out the remainder. One of my groups of students even made the connection that adding the number 7 three times is also multiplying 7 by 3. 
• I saw this principle most while we were learning about teaching fractions as a class. We learned about teaching fractions by making sense instead of just teaching algorithms which don’t strengthen student understanding. We learned to make connections for solving to find if one fraction is smaller than another by seeing how close each fraction is to easier fractions such as ½. There is also an activity where each part of a fraction is a cut up circle, so ⅛ would be shown as a circle cut into 8 equal slices. Then students are able to layer the fractions on top of each other to compare and find out questions such as which is larger. This also helps them see that the larger the denominator, the smaller the individual piece / the smaller the fraction will be.
• I am brought back to the video that provided an example of representing fractions through a context. Each group of kids were able to come up with their own method of describing how the subs were split up. I remember commenting that there are so many different ways to understand and compute fractions. I find myself struggling to find the best way to solve a problem. I understand the importance of illustrating different ways to solve a problem as a teacher. This allows for students to freely represent their thinking in a way that works with their mindset. And that is what occurred with this particular example.
• In lesson 8, we focused on practicing multiplication using representations and connections. We played a game where the students drew numbers and made rectangles with the dimensions of those numbers, and multiplied them together to find the area of these. During the game, Joaquin discovered that “multiplication is just like addition” and so I had him explain to the other students why he thought that, using the rectangles as representations. 

Facilitate meaningful mathematical discourse

• Reading the Becoming book I really liked the story from chapter 13 the teacher Ann’s story with the Powerball. It was exciting to see all of those kids questioning and looking at a whole new view of a question not even the teacher hadn’t thought about before so it brought a lot of meaningful conversation into their classroom even before class was actually started.
• A specific example of this happened during lesson 7 when we briefly showed our students a picture of a piece of toast with blueberries on it and then they had to tell us how many blueberries were on the toast. We had the students share how many they counted and how they did. They both counted 12, but differently. They talked amongst each other about the ways they had counted, and each acknowledged that both ways were effective, but decided that one's student way was more effective. Count the number of rows and times it by the number of columns.
• This reminded me of when after we met with our third graders for the day we would share stories about what happened that day. I really think this was helpful because if it was a Tuesday and my group had a rough day with the lesson but another group shared how they had a successful day, I could use their ideas for my lesson on Thursday so it would be more successful. 
• This is something my partner and I did a lot to keep the students engaged, to build relationships, and to listen for reasoning. During many of our lessons we asked our students to share outloud how they thought about something or their approach to a problem. We encouraged them to share their thinking and to used it as a way to talk about new approaches and different strategies. There was many times that one student answered a problem a different way than another student. We wanted our students to talk to each and learn from each other. Sometimes we can learn the most from our students or from our peers. 

Pose purposeful questions

• I think our numberless story problems are a great example of this, I had never had experience with them as a student but I love them and when I am a math teacher one day I will definitely use them. The reason I like them so much is because it slowly introduces the idea of the problem and the context of the story before throwing too much at  them by giving them everything all at once. 
• An example of something I did as a teacher to fulfill this principle was asking the question “how do you know that?”. I had listened to Professor Golden ask us this question in class, so I made an effort to use it when working with the students. This question is simple yet it invites the student to consider their thought process and how they arrived at their understanding. This question is purposeful because not only does it allow the teacher to have a greater understanding of the student’s understanding based on their reasoning, but it also gives the student the opportunity to self-assess and consider their own thought processes and reasoning when learning mathematics and interacting with problems. 
• Professor Golden's number talks with the children to begin each session with them was a great way to get a routine down with students that help them be able to recognize procedures and ideas about math. The number talks were good ways to emphasize all students' thinking and model and represent the various strategies and thoughts students have on the problem at hand. It is also an inclusion and safe space to talk about wrong or right answers, modeling that mistakes are okay and that struggle to find multiple strategies will come with time.
• I enjoyed the Notice and Wonder that was encountered in our MTH 222 classroom. From Homework to the discourse we had in class/with our tutees, questions were facilitated in this pattern. It was especially incremental with our Numberless stories! The basic structure of Notice and Wonder follows from ch. 7 of our Becoming Textbook: 
• Give students an image or scenario without a question. If you like, you can use problems from your curriculum and obscure or delete the question.
• Ask them, “What do you notice?” (You might want to use think-pair-shares at each step to increase engagement and thoughtfulness.) Record their noticings so students can see them. 
• Ask students, “What are you wondering?” Record their wonderings.
• Ask students, “Is there anything up here that you are wondering about? Anything you need clarified?” Pursue any follow-up questions.
• Once students have had ample time for noticing and wondering, you can either reveal a question you’d like them to solve or have students come up with a question by asking, “If this story were the beginning of a math problem, what could the math problem be?”
• This reminded me of when after we met with our third graders for the day we would share stories about what happened that day. I really think this was helpful because if it was a Tuesday and my group had a rough day with the lesson but another group shared how they had a successful day, I could use their ideas for my lesson on Thursday so it would be more successful. 

Build procedural fluency from conceptual understanding

• By picking apart both addition strategies and subtraction strategies, both in class and in the homework in articles and in the book- we learned about a lot of procedures and created fluency in those procedures, from understanding how different methods work and the underlying meanings of the numbers and their many properties. By understanding place value, and the numbers, and groupings and factors of numbers, we are better at understanding and applying those concepts together to understand both the procedures and strategies of, say, addition or subtraction, and develop fluency from correlational learning.
• This was something I noticed about my own learning. I think I came into the class knowing mathematical procedures for a lot of concepts. I left the class knowing why I do a mathematical procedure here, and how I can use my problem solving skills to build on what I already know. For example, I learned a lot about story problems. I knew how to answer story problems but I didn’t understand how they work and really the differences between them. There are a few different kinds and while they focus on different things they are meant to be used as a mathematical tool. Story problems are designed to make students stop and think about what the question is asking them. They are also used to help students visualize a situation and determine the right approach to it. I used to have the mentality that you should find and pull out the numbers and do something with them like add, subtract, or multiply. I now know to stop, slow down, and think first. 
• One of my students, ____, had a hard time doing addition. We tried a lot of different techniques and ideas with her, but one class she just decided to try a problem using her own technique and she got the answer correct. She was flexible and once she was able to do it her way, she began to understand the ways that we were presenting to her. 

Support productive struggle in learning mathematics

• This we used in class and myself as a teacher. This was helpful with the number circle because we were doing this with difficult numbers to see if my students and myself were able to find easier or different ways to find the next number. We supported answers and time to think and did not shout out and point out if the students added or subtracted incorrectly. Figuring out problems in your head and by yourself is more productive in learning new ideas/concepts.
• This was something I had to learn to do. After Ms. Cordy came and talked to us in class. I got a better understanding on how to do it. A productive struggle is so important in helping a student being able to learn a topic that might be hard for them to grasp, you can't just bail them out with their first answer, or “I don’t know” and move on from the topic, you want to make sure that know what they are talking about, or else you could just move on leaving them behind.
• We gave our students the problem 123/10 to do, and they were discouraged at first due to the big numbers and division, _____ even said “this is too hard, we can’t do this!”. However, we let the students sit and think for a while, trying out many different strategies and giving encouragement when they seemed stuck on an idea. Eventually, they focused on counting up and down by 10s to find the answer, and were debating whether it is 13 or 12. Hannah and I reminded them there can be a remainder and this helped them decide the answer was 12 with a remainder of 3. They all had tried multiple methods of solving, and Hannah and I encouraged them to keep trying until they figured it out, and they seemed to feel very accomplished after doing so. 
• A way we supported productive struggle when learning mathematics was by not just giving them the answer when they were wrong. We would ask how they got the answer they got and we would discuss it as a class, hearing ideas from multiple students. We also let everyone know that it is okay to not always get the answer right, and everyone can learn from the mistakes they make including the teachers. We also make mistakes and are learning as we go on, this helped not put so much pressure on them to just “be right” but to actually understand the content they are learning and feel comfortable asking questions. 

Elicit and use evidence of student thinking

• Throughout the semester I kept a lot of evidence of students thinking through photos as well as notes.  I began writing down student thinking on a white board to help myself understand the student’s thought process.  This proved to be an excellent resource to look back at and analyze.  Oftentimes I learn a lot from the students and they showed me ways of thinking that I would never have thought of.
• Something I think I developed a lot over the semester was to ask better questions to the students. I wanted them to be able to explain their thoughts to me. I would often ask, “how did you get that?” “why?” “is there another way?” I remember constantly saying, “I want to see your thinking.” That’s where I got a lot of feedback from the students. 
• The logs we kept on each meeting and observations we made about each individual student helped us pick activities that challenged them appropriately.
• Hannah and I realized that our students in Mrs. Caterino’s class had a hard time subtracting when the number being subtracted had a larger first digit than the first number (ex 32-18). We decided to change some games from addition to subtraction to help add practice of that, and adjusted them so there were more problems like this so the students could work on doing subtraction problems like that. 
• Since all of our students were on such a variety of different levels we had to develop our lessons to make sure that everyone would be able to try it and be challenged mathematically. So we’d have follow up questions for our student who would fly through problems and made sure our student who was struggling understood what he was doing mathematically and why.
• We would evaluate students' work from past weeks and what they were knowledgeable on, and use this student's thinking to tailor our future lessons and make them accessible to all the students.

I was pleased to see connections to our reading, our content time, the number talks they observed, and some progress in their thinking and use of these ideas over the semester. I think the principles became real for them in a way that I haven't seen in traditional teacher prep.