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?