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Takeaways from Math Methods: How Will You Teach Effectively?

Jennifer Bay-Williams

Mathematics Teacher Educator and Author from University of Louisville.
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In a perfect world, preservice teachers (PSTs) in my mathematics methods course would leave each class session with 8-10 important ideas that I have tried to cleverly squeeze into a 150-minute session. By the end of the semester, then, they might have 120 or more important ideas about teaching mathematics -- barely enough to get started.

Woven in and out of each assignment and field experience is a much smaller list of significant ideas about effective teaching. I try to connect these overarching ideas each week so that PSTs can see what they look like, for example, in a second grade math lesson or on an algebra test. At the end of the semester, I ask my students to tell me three important ideas they want to always remember about teaching (mathematics) effectively.

Here I share three takeaways, grounded in research and aligned with the Common Core State Standards (CCSS), that I hope my PSTs might write.

1. Helping Students Doesn't Mean Showing Them How

Before admitting PSTs, we interview each one and ask, "Why do you want to be a teacher?" The most common response is, "I want to help students," a sentiment that PSTs describe later as "giving good explanations" or "making it simpler" -- notions of helping which are underdeveloped.

A synthesis of research in mathematics education by James Hiebert and Douglas Grouws identified two teacher actions that impact conceptual understanding. One is to engage students in productive struggle. Merely telling students how or making things simpler does not actually help them understand as much as providing challenging tasks and time to "dig in" to a problem.

To help them redefine how to help students learn, I encourage PSTs to embrace their sense of accomplishment when they solve a challenging task, recognize the pride they feel when they share a unique way to solve a problem, and reflect on what such feelings might mean for a student in their own classroom.

2. If You Don't Ask Students to Think, They Won't

As PSTs learn how to write objectives and plan lessons, they should not lose sight of the big idea that their students are actively connecting ideas and making sense of what they are learning. In other words, they are helping their students think.

I like to pose this question to PSTs: "What is it in [this task or lesson] that gets your students to think?" Two ingredients are required:

  1. Higher-level thinking questions
  2. Techniques for ensuring that everyone is thinking of the answers to those questions

Are you wondering what Hiebert and Grouws' second research finding was? Making connections explicit through higher-level thinking questions that prompt students to synthesize, generalize, compare, and so on. It is difficult to "wing" higher-level thinking questions, which are often confused with open-ended questions (e.g., "How did you solve #6?") that are easier to construct in the moment.

I give my PSTs a bookmark with higher-level thinking questions on it. One category includes "making mathematical connections." I offer a second bookmark with the CCSS Mathematical Practices because these practices all focus on thinking. Each lesson plan must include higher-level thinking questions and explicit attention to at least one mathematical practice, as well as how they will make sure every student is thinking of the answer (e.g. a pair-share).

3. Plan with Each Student in Mind

Helping all students learn takes more than desire and patience -- it requires using specific strategies for each learner. Each student builds on his or her prior knowledge and experience, and each student is unique.

PSTs learn instructional strategies for specific learners, yet connecting these ideas to a specific lesson is very difficult. As an example, many PSTs work with English-language learners (ELLs) and can identify a variety of research-based strategies, such as attending to key vocabulary, using visuals and providing opportunities to practice language.

These strategies, however, are not often explicitly incorporated into the lesson plan sequence or the lesson itself. To counteract this oversight when we discuss "attending to vocabulary," for example, I ask:

  • What content and context words have you selected?
  • Where in the lesson might you place the vocabulary support?
  • What will it look like?

Such dialogues in the methods course show PSTs that detailed planning is what effective teachers are doing in order to meet the needs of each of their students.

The Big Question: How Do We Get There?

PSTs need opportunities to connect theory to practice -- not teaching practice in general, but teaching practice connected to specific lessons: "How in this lesson might you . . . "

  • Engage students in productive struggle?
  • Get students to think and make connections?
  • Provide support for each student?

What top takeaways might you like to hear from PSTs? What experiences might we plan so that these are not only on their list the last day of class, but also the first day of teaching?

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Sue J's picture

another one is: include conversations about the concepts they're using *especially* the ones they "already know." This is really part of the "if you don't ask them to think, they won't" -- when concepts are relatively new, students don't naturally work them into their models of understanding. Wearne-Hiebert & Hiebert (1985) tested 6th and 8th graders in nine fraction problems with accompanying drawings, exemplifying concepts of varying difficulty from "find the whole number, given that 1/2 of it is 8" (which 96% of both sixth and eighth graders answered correctly) to "find 11/2 of 12" (which fewer than 15% of students from either group could answer.

JohnK Wright V's picture
JohnK Wright V
Certified Math Teacher 8-12 and Technology Applications EC-12, Plano,TX

I would also include an objective for each lesson, on why the topic is important. Will it help them with an important business decision in life? Understand what happen when we shoot a rocket into space? How it is important to music,science,etc.?

How can you relate this to life?

I liked the idea of teaching to all students. Students tend to learn by visual, auditory,kinesthetic (or combination). Get the students engaged. I once took my 11th grade Algebra II students outside and we used kite string to show the coordinate plane. I had students go to certain points (Coordinates).

All students really mean all students. Think about not just the general ed kids in your class, but what about those who are performing at a high rate (i.e. your gifted/honors/ap type kids) along with those performing at a lower rate? What about your special education, English Language Learners, 504,at risk population kids?

We need to teach students to be problem solvers. Today they will be learning skills for jobs that have not been invented yet.

Sue J's picture

Were you doing 3-D coordinate plane? That's kewl :) I don't get that far but do explain that yes, it's a brain leap to go from "regular" numbers to having two numbers that go to one place (often they'll plot two points on one of the axes when we're starting).

Risa Weiss ETA hand2mind's picture
Risa Weiss ETA hand2mind
University and College Educational Consultant

In your post you mentioned the challenge of preparing the preservice student with enough ideas/strategies to be ready to teach when they are done with their university courses. I hear this from many other math methods professors. Wouldn't it be nice if we could provide all preservice students with a "toolkit" that they can utilize to be "ready to teach". When I went through my teacher preparation program several decades ago I was taught many concepts and strategies, but none that I utilize today. What are some of the tools you have provided to your students so that have a bag of tricks they can use when they enter the classroom on day one?

Carisa Koch's picture
Carisa Koch
Ed Tech Savvy Math Teacher

I wish my program incorporated a mathematics methods course. I took one when I was first looking in to teaching and it gave me some great ideas. I totally agree that PSTs need opportunities to connect methods to specific lessons. Each lesson has its own challenges, and the best ways to address these challenges will vary from topic to topic. It would be great to have many years of teaching under my belt so that I can know in advance what students will struggle with in a given lesson. Having this knowledge ahead of time will shape our lesson design and delivery.

Norah's picture
Early childhood teacher, writer, life-long learner

I love that you are using problem solving and higher order thinking skills in each lessons. Ensuring PSTs have these objectives in mind before they even start teaching will improve the quality of mathematical teaching in the classroom. Great strategies!

Deshaun's picture

Hello, my name is Deshaun Harris and I am currently a pre-service mathematics teacher and math tutor. I would like to start by saying that this article truly transformed the way I teach my lessons when I am teaching my students. The subtopic that I engaged in the most was planning with each student in mind. During my field observations, I noticed that each child has a different way of learning and they way in which the objective is delivered matters. As I plan for instruction, I think about three possible ways in which I can teach a specific objective to make sure that each one of my students are successful. Also, I enjoyed the subtopic "If You Don't Ask Students to Think, They Won't." I believe that all students should be required to think and challenged, no matter if they are struggling or advancing. As a pre-service teacher, what is one piece of advice you would give me when it comes to teaching mathematics?

Jennifer Bay-Williams's picture
Jennifer Bay-Williams
Mathematics Teacher Educator and Author from University of Louisville.

A toolkit is a great idea - I feel like a good methods course is trying to do that, but in all the many content-specific ideas only so much can get remembered. I try to help PST be resourceful - I share favorite print and web resources, as well as organizations that can continue to support lifelong learning...

Jennifer Bay-Williams's picture
Jennifer Bay-Williams
Mathematics Teacher Educator and Author from University of Louisville.

Thank you for the post! You end by asking for advise -- beyond the three things that I posted in the blog, it is to just be stubborn and persistent about implementing practices that best meet the needs of all students - find places to get energized and stay with the vision. I just returned from the National Council of Teachers of Mathematics (NCTM) Annual meeting - heard many great ideas there to support these big ideas and met many others with the same commitment. Hope this helps!

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