Week 3: Reflection

Most of my thinking this week focused on the task I planned for Friday, Task: Math is Everywhere… in Logan.  One thing that I learned from that is that students had many ideas about locations where math can happen and also they had many questions that could be answered by math (e.g., how much water is in the ocean?, how many words are in a book?).

I wonder if a teacher could have a “comment box” for math questions or contexts that students would enjoy? Would that overlap too much with science or social studies? Or would it be too disruptive? It would be important to spend a little more time with some students who might not have connected school math to outside-world-math?

I’m thinking this way because when I teach Elementary Math Methods courses, I ask future teachers to get to know their students and their students neighborhoods in order to bring questions and high-level tasks into the math classroom that allow students to access their cultural funds of knowledge. I think those types of teacher-developed tasks are important. At the same time, I wonder how students can be encouraged to bring their own tasks in?  For older students, Tonya mentioned to potential for students to bring cameras home or bring cameras from home, to take pictures from their daily life that seem mathematical or inspire mathematical ideas or questions.

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Week 2: Class Management and First Task

Observation Focus: Class Management

I saw the teacher use many interesting class management strategies throughout the week. The desks are set up in one region of the classroom in the following arrangement:

classroom-set-up

Student Responsibilities: The teacher expects the students to contribute to the organization of the class. For example, they have timed multiplication or division tests each day. The teacher hands out the answer keys and red pens while the students are working. Students exchange sheets and check each other’s.

Clear Instructions: The teacher gives all students opportunities to contribute by giving tasks to members of each table (made of four desks). For example, after the timings, and after students have had one minute to check each other’s timings, she might say, “Friends closest to the door, please gather the red pens and put them away. Friends closest to my desk, please gather the answer keys and put them away. Friends closest to the flag, please gather the timings and put them in the second math bin. Friends closest to the window, please get markers and erasers for everyone in your group.”

Clear Expectations: After giving clear (but brief) instructions, the teacher also gives clear (but brief) expectations – every time. For example, “While you do this, voices should be off and you should return quickly to your seat. You have 30 seconds.” She uses a variety of expectation words for voices, such as: “Remember, ‘spy talk’ only.” “Boards flat, hands clasped, bodies toward me, eyes on the board.” “Pockets on seats.” “I should hear whispering only.”  She also has posters listing expectations for math centers.

Transitions: The teacher gives clear instructions and expectations before students begin to move, and counts down “Let’s have seats in 10, 9, 8, …” She also uses a variety of phrases to bring students back to attention – for each phrase, she begins and the students respond: [“Focus, focus.” / “Everybody focus.”] [“To infinity!” “And beyond!”] [“One, two, three, eyes on me.” / “One, two, eyes on you.”]

Ensuring Students’ Engagement: The teacher consistently uses many different strategies for ensuring students are engaged and feel like they are part of the work.

She sometimes draws names from a cup; sometimes calls on students; sometimes has students whisper to their elbow partners; sometimes has students whisper in her ear. In the whole-class brief lessons, she often uses manipulative and brings volunteers up to hold them or use the SmartBoard.

She consistently gives wait time, asking students to indicate when they are ready in a variety of ways: “Thumbs up when you think of an answer.” “Share with your elbow partner – let’s have right partner share first.””Finger on your nose when you are ready to share.” “Finger on your ear…” “Silent unicorn…” “Silent mustache…” etc.

Experience Focus: Planning and Implementing a Task

I adapted a task on pages 169-170 of Van de Walle1 to be a 10-15 minute teacher center. I saw that students had good strategies and mostly were engaged.

Two students particularly would not try the task. I asked the teacher about strategies to help them try, and she explained they were both somewhat shy so I might need to let them get used to me. She also explained that one student had good days and bad days, and so it wasn’t always easy to keep her engaged.

Reflecting

I noticed that when I interacted with the students, I spent too much time redirecting their attention to the task. Some students may need to warm up, but I need to practice my teacher moves and group management skills when I interact with the students. Especially when it comes to expectations about using the materials or finding an answer.

I will remind students of expectations with resources briefly before beginning an activity. I will also give more attention to consequences so that students can see the boundaries. I will continue to encourage students to explain their thinking and to look for other strategies.


1  Pages 169-170: Van de Walle, J. A., Karp, K. S., Lovin, L. A. H., & Bay-Williams, J. M. (2014). Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Vol. 2). Pearson Higher Ed.

Task: Ordering Whole Numbers

Planning Task

This week the students were reviewing in chapter 1 of Go Math!. For one day, they focused on “1.3: Compare and Order Numbers.”

I adapted the “Expanded Lesson: Close, Far, and in Between” on pages 169-170 of Van de Walle1 to be a 10-15 minute teacher center.

On individual notecards, I wrote three numbers: 27, 83, and 62. On separate notecards, I wrote the questions:

  1. Order the numbers from smallest to largest. How can you prove it?
  2. Which two numbers are closest to each other? How can you prove it?
  3. Think about three other numbers: 50, 55, and 73. Which of the first three numbers is closest to each of these?

I similarly wrote three larger numbers – 219, 457, and 364 – each on an individual notecard. I wrote similar questions to those above.

As instructions, before I showed the students the first question or the first set of numbers, I told them: You will need to explain why your answers are correct. Use math tools to explain: words, numbers, a number line, pictures, diagrams – whatever makes sense to you.

I planned to ask them questions such as:

  1. Ask one student: What did you decide to do? How do you know? Can you prove it?
  2. Ask other students: Do you understand that strategy? What questions do you have? Who used the same strategy? Who used a different strategy?
  3. After a couple of strategies: Which strategy is most efficient (fastest)? Which strategy is most valid (correct)?

Planning Reflection

Before implementing the task, I wrote a plan for noticing and reflection, based on recommendations in the Van de Walle lesson.

  • Noticing Strategies (e.g., counting by ones, fives, or tens; subtraction; visuals like number line, benchmark, other)
  • Noticing whether students could think of only one strategy or more

Reflection on Task and Implementation

I led the center as one of four centers that students visited. Each group included 4-6 students. I felt generally that the task went well, but we did not have enough time to get past the first group of numbers.

Strategies for “Order the three numbers”: To order numbers, students used a few different strategies.

  • Place value: Most students used place value (we had just gone over using place value to compare numbers in the first part of the math lesson). I felt that I saw some students who used place value procedurally (i.e., simply compared the first digit of each number and could only explain “6 is more than 2 so this number must be bigger”) and others who used it conceptually (i.e., compared the first digit of each number and explained something similar to “6 tens is more than 2 tens”).
  • Counting: Many students counted or used counting as a justification: “I know because I count to 27 first and then 62 and then 83.”
  • Diagram: One student drew the numbers with hash marks between them representing tens: 27”’62’83 (he counted: 27, 30, 40, 50, 62, 70, 83). This strategy was similar to the place value strategy, but a different way to visually show the place values.
  • Number Line: One student drew a number line using time as a measure to determine length between the numbers. She was only able to compare 27 and 62, but explained she knew 62 was larger than 27 because of the number line. This strategy was a visual representation of her counting, because she drew the line with a tiny push forward for each number as she counted from 1 to 27 and then from 27 to 62.

Strategies for “Which of the first three numbers is closest to each of the second set?”: To order the second set of numbers, students used a few different strategies.

  • Order and Connect: Several students ordered the second set of numbers (50, 55, 73) and then connected them directly to the ordering of the first set (27, 62, 83). So, they said 27 was closest to 50, 62 to 55, and 83 to 73. This strategy almost worked except that 50 is closest to 62 instead of 27.
  • Subtraction: Some students subtracted and found that 50 and 55 are both closest to 62. (only 3 or 4 of the 22 students weren’t “tricked”)

Group Management: I noticed that some groups and some students were more engaged than others. I asked the teacher the next day:

  • What are some strategies for keeping students on task and using tools appropriately? (some students were drawing pictures instead of comparing numbers or listening to other students’ strategies)
  • What are some strategies for supporting students in thinking about more than one strategy and trying at least one strategy?
    • I thought one strategy could be to choose the tools carefully and “force” different strategies by giving different students different tools.

1  Pages 169-170: Van de Walle, J. A., Karp, K. S., Lovin, L. A. H., & Bay-Williams, J. M. (2014). Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Vol. 2). Pearson Higher Ed.

Week 1: Welcome to the Classroom

After talking to a mathematics specialist in local elementary schools, I was able to connect with and meet a local fourth-grade teacher and then meet her class within the first couple of weeks of the school year.

The school uses Houghton Mifflin Harcourt’s GO Math!® curriculum1 in conjunction with ThinkCentral2 and ExploreLearning Reflex® 3. The fourth-grade teachers collaboratively plan by meeting one hour during school each week. They discuss scheduling, lessons, assessments, and assessment results.

The teacher I am observing structures her 1.5 hour mathematics lesson as follows:

  1. Students return from lunch at noon, and find their appropriate timing sheet for multiplication or division. (The teacher has a record on the wall showing which timings they have passed off.)
  2. By 12:05, most students have returned from lunch and are ready for the timing. The teacher twists the face of the timer hanging on the whiteboard at the front of the class for 2 minutes. As students complete the timing, they pull a book from their desks and begin to read.
  3. As students work, she passes out red pens and laminated sheets with the answers to all multiplication or division timings on each sheet. Students exchange timings with an elbow partner, check them, and mark them with a check or an “x” to indicate if their classmate has passed or not.
  4. The teacher instructs students to return the materials. For example, “Friends nearest the window, return the red pens. Friends closest to my desk, return the score sheets. Friends closest to the door, collect the timings and put in the math basket – third one down.” She sets the timer for another minute and students scatter.
  5. After timings, a brief 20-minute or so lesson on the day’s topic
  6. After the lesson, students break into four groups for centers: Tech Time (Reflect Math), Hands-On (game on the carpet), Teacher Time, and Seat Time. Each center lasts approximately 15 minutes and the math lesson ends at 1:30.

 


1 http://www.hmhco.com/shop/education-curriculum/math/elementary-mathematics/go-math-k-8
2 http://www.hmhco.com/classroom/classroom-solutions/digital-and-mobile-learning/personal-math-trainer/whats-new-on-thinkcentral
3 https://www.reflexmath.com/about

Program in Mathematics Education – Featured Graduate Student Post

Grad Post.jpgI’m in the spotlight on my program website this week:
https://prime.natsci.msu.edu/

The page address is: https://prime.natsci.msu.edu/news-and-events/news/graduate-student-post-eryn-stehr/

Eryn Stehr arrived at MSU to pursue a doctoral degree in Mathematics Education starting with the Fall Semester of 2011. Her research interests focus primarily on “supporting teachers’ professional decision-making about their use of technology to support their mathematics teaching and learning.”

Eryn began her journey in higher education by attending Utah State University, where she received a bachelor’s degree in mathematics. Upon her graduation from Utah State University, Eryn continued her studies at Minnesota State University, Mankato receiving her master’s degree in mathematics.

Eryn Stehr With StatueWith two degrees under her belt, Eryn applied for – and received – a fixed-term position as a mathematics instructor at her alma mater, Minnesota State University, Mankato. In addition to serving as a mathematics instructor, Eryn also volunteered as a math coach at an elementary school.

Now, as a member of the PRIME program, Eryn spends nearly all of her time concentrating on her studies and working on her dissertation, which focuses on technology in the mathematics classroom. Specifically, Eryn will cover multiple dimensions of technology in the mathematics classroom, including:

– “Teachers’ conceptions about the nature of mathematics, the nature of mathematics teaching and learning, and the role of technology in supporting mathematics teaching and learning”

– “Teachers’ attention to features of mathematics-focused digital tools and resources”

– “Potential relationships between teachers’ conceptions and what teachers notice about digital tools and resources.”

Eryn Stehr Apple OrchardWhen Eryn allows herself some down time from studying and working on her dissertation, she indulges in walking, watching Netflix, and reading. In addition to these hobbies, Eryn is earnestly working on developing new hobbies such as playing the piano, cross-stitching, and genealogy. Accompanying her in many of her hobbies is her 15-year-old cat, Pell.

Upon graduation from the PRIME program, Eryn is hoping to work at a university close to her parents and family so she can balance her personal and professional lives.

Written by JJ Thomas

When they take the post down some day, here is the PDF version of the post: Graduate Student Post_ Eryn Stehr – Program in Mathematics Education

“Technology and Noticing” – Psychology of Mathematics Education – North American Chapter

PMENA 2015 - Noticing and Technology - Final

I had a great time at PMENA-37 this past weekend! What a supportive, engaging, and productive space for learning about mathematics education – rather, immersing oneself into the mathematics education community.

Friday afternoon, I presented my poster, “Digital resources in mathematics: Teachersconceptions and noticing” and I learned a lot through conversations I had with visitors to my poster. You can read the overview of this poster in PMENA-37 Proceedings (p. 1256).

This poster represents a moment in time in my dissertation study. I am beginning to look at what teachers notice (or paid attention to, or found important) in online tools and resources by first considering how they responded in our pre- and post-evaluations in the online master’s-level course “Learning Mathematics with Technology.” We (the instructor and I, as a teaching assistant) gave students a broad prompt:

Assume you are considering using these tech tools in your teaching. Evaluate the three tools to decide whether you would recommend using them. Write an evaluation of the three tech tools for an audience of other teachers who might be considering using them.  Make your recommendation clear: which (if any) would you recommend? Argue your position. You will need to decide which elements, features, or characteristics of each tech tool to use in supporting your argument.

I considered the teachers in my study who chose the addition set of tools, and as I read and re-read their evaluations I created a list of tool features that the teachers paid attention to: Purpose of tool; Classroom use; Understanding (e.g., instrumental, relational); Thinking (e.g., critical, level); Engaging / Aesthetically pleasing / Distractions; Directions and Instructions; Customizable; Progress / Tracking; Preparation; Age / Grade-level; Interaction / Manipulative; Feedback / Responsiveness; Ease of Use / Accessibility; Differentiation / Learning styles / Learning needs; CCSSM / Standards Alignment; Mathematical representation; Teacher Support / Resources; Breadth / Variation. These were too many for the poster, so I narrowed them down to 12, and sorted them according to whether they seemed to be general, pedagogical, or mathematical characteristics (but recognizing the overlap in the categories, similar to the TPACK framework):

  • General: Customizable; Engaging / Aesthetically pleasing; Ease of Use / Accessibility
  • Pedagogical: Differentiation / Learning styles/needs; Progress / Tracking; Feedback / Responsiveness; Teacher Support / Resources
  • Mathematical: Mathematical Purpose; Standards Alignment; Understanding / Thinking; Interaction / Manipulative; Mathematical representation

In the tables, I noted whether teachers mentioned a particular characteristic was included or favorable included in the tool (Y), was not included or included in an unfavorable way (N), and whether the teacher did not mention a characteristic (–). Because of the limited space of the poster, I chose three mathematical features to show teacher comments about: Mathematical purpose, Understanding / Thinking, and Interaction / Manipulative. I included comments for each tool from each teacher in the pre-evaluation (left) and the post-evaluation (right).

Discussion of Results (which are shown in the poster above)

It goes without saying that the online tool itself seemed to impact the features that teachers paid attention to and I had chosen the three tools to be different in (hopefully) interesting ways. One tool is built around a ten-frame representation of addition, and includes a symbolic representation (horizontal addition statements). One tool is a quizzing software that includes a symbolic representation (horizontal statement) of addition only. One tool is built around a base-10 blocks representation of addition, and includes a symbolic representation (vertical addition statement). The tools differ in a number of ways and the teachers noticed many of their differences. There is evidence that teachers moved, not just to noticing more characteristics, but also to more profound noticing by the end of the course.

One interesting result was the teachers’ attention to thinking and understanding in the Math Trainer applet. In the pre-evaluation, none of the four teachers commented on opportunities for students to think or develop understanding. In the post-evaluation, all four teachers commented on Math Trainer’s support of instrumental understanding (we read Skemp (1976) early in the course), and on the lack of opportunities to “explore or gain a deeper understanding,” or for “critical thinking.”

Another interesting result is the change in Nicole’s use of words from the pre-evaluation to post-evaluation in her description of the mathematical purpose of each tool. She did not change words at all in her description of Math Trainer’s purpose “to practice math facts,” but changed from practice to explore for NCTM Illuminations: Ten Frame Addition and NLVM: Base Blocks Addition. Her description of the purpose of NCTM Illuminations: Ten Frame Addition in the pre-evaluation was “to practice using a 10-frame to work with numbers” and in the post-evaluation was “to work with a ten-frame (or five-frame option) to explore addition and subtraction of whole numbers.” For NLVM: Base Blocks Addition, she wrote “to practice addition with base-ten blocks” in her pre-evaluation and “to explore addition with base-ten blocks” in her post-evaluation.

PMENA 2015 – Noticing and Technology – Final (pdf of poster)

PMENA 2015 Proceedings – Noticing and Technology (pdf of proceedings paper)

PMENA – November, 2015

PMENA-banner-1140

I’m excited to participate in PMENA this weekend – very conveniently held at my school, Michigan State University. I have helped out the Local Organizing Committee, but mainly I have enjoyed watching the “behind the scenes” action. Led by Drs. Tonya Bartell and Kristen Bieda, we have worked hard for about a year and a half. I’m excited to see the end result!

I will be participating as a presenter on a brief research report and three posters:

  • Friday, November 6, 9:20 AM – 10:00 AM (Kellogg: Conference 62)  Assessing Teacher Knowledge and Practice
    • Mathematical Knowledge for Teachers: Opportunities to Learn to Teach Algebra in Teacher Education Programs
      Presenters: Eryn M. Stehr, Jeffrey Craig, Hyunyi Jung, Leonardo Medel, Alexia Mintos, Jill Newton
  • Friday, November 6, 4:00 PM – 5:30 PM (Kellogg: Lincoln)  First Poster Session
    • Digital Resources in Mathematics: Teachers’ Conceptions and Noticing
      Presenter: Eryn M. Stehr
    • Building Algebra Connections in Teacher Education 
      Presenters: Hyunyi Jung, Jill Newton, Eryn M. Stehr, Sharon Senk
  • Saturday, November 7, 10:30 AM – 12:00 PM (Kellogg: Lincoln) Second Poster Session
    • Secondary Preservice Teachers’ Opportunities to Learn About Modeling in Algebra
      Presenters: Hyunyi Jung, Eryn M. Stehr, Sharon Senk, Jia He, Leonardo Medel