Can I make revisions to the lesson plan parts?

Yes, you are expected to make revisions to the lesson plan as outlined in the feedback provided on the rubric. Making revisions is an important part of the learning process, and continuing to think about the task and goals in the ways described in the feedback while only make it easier for you as you move forward in the lesson planning process.

It is helpful for me if you show revisions in a different color font. Submit any revisions with the next part of the lesson plan assignment (all in the same document). I will adjust your score according to revisions. You are welcome to email me any revisions before you submit with the next part of the lesson, if you’d like to check that you’re interpreting feedback correctly and moving in the right direction.

General

How can we better apply the 5 practices to lessons/units? I need more practice on applying the 5 practices to my lessons.

First, the 5 practices are designed to serve as a guide for a single lesson, specifically, one that is centered around a single mathematics task/problem and that focuses on achieving learning goals based on how students think about that task. Enacting problem-based, student-centered lessons like this is quite sophisticated. The 5 practices help to make this kind of math teaching more accessible to you as emerging teachers. However, I do not expect that you will master the 5 practices in this course! This course is designed to introduce you to practices that can help you develop a sophisticated and effective approach to mathematics teaching, and by using these practices and reflecting on how you can use them better each time, it’ll get easier and be more effective!

What are important things to know if your lesson doesn’t go to plan? How to adapt? First, I would say that if you give students access to high cognitive demand tasks and really put students’ mathematical thinking at the center of your lessons, then there will be very few instances when your lesson is a complete failure. If student thinking is driving the lesson, then students are doing the heavy mathematical lifting and much of the talking and explaining. When they come up with ideas, questions, strategies or solutions that you didn’t expect, you can ask them to explain more about their thinking and you get the chance to learn with your students.

Second, if something comes up that you don’t expect and can’t make sense of right away, know that it is ok (and also important) to sometimes tell your students that you don’t know and that you need time to think more about the mathematics. This sends the message that mathematics is not about getting answer fast, but it is about thinking critically and deeply about ideas. This will give you time to reflect and replan so that you can revisit mathematical ideas the next day.

Finally, for times when you do need to stop a lesson early to do some reflection and replanning for the next day, it can be nice to have some routines to fall back on. For example, you could transition to a number talk that is related to the mathematics students are learning (but is something they’re comfortable enough with to do mentally and flexibly). See: https://elementarynumbertalks.wordpress.com/

How much time do you spend on average preparing a lesson?

In this course (and in your teacher preparation in general), you should expect lesson planning to take a considerable about of time and effort. Remember that you are learning how to do lesson planning, and learning takes time. Most important, you are learning about the thinking that goes into planning a lesson, and you are making that thinking explicit, to yourself and me. On average, I’d suggest you budget 1-2 hours to for lesson planning each week in our course.

Once you are teaching daily, you won’t spend as much time planning. This is partly because you’ve already learned how to go through the lesson planning process, so it’ll be more efficient. But it is also because you won’t need to make all your thinking as explicit - you’ll still engage in thinking about those things, but you won’t need to write it all out.

Can you do some of the practices simultaneously?

Absolutely! You are definitely selecting and sequencing and planning how you'll connect while you are monitoring.

I am wondering how to create a well-ordered lesson plan.

This is the focus of our whole 5 weeks together! So don't feel like you need to fully "get it" right away, or even by the end of the 5 weeks (because learning to effectively lesson plan takes practice). But you can build a solid foundation for lesson planning in math a few ways: (1) look at example lesson plans - such as the one in Appendix B of your text; (2) look at the rubrics for additional details about what is expected of you; and (3) look into research on lessons around the specific mathematics topic or task that you are using. This research will give you examples from actual classrooms about how teachers facilitated effective math lessons around specific topics and tasks with real math learners. One place to start is the journal: Teaching Children Mathematics. You have free access through the UTK library.

Cognitive Demand of Tasks

What's the balance between high cognitive demand and low cognitive demand tasks?

In terms of when to use high cognitive demand tasks versus low cognitive demand tasks, think more about the overall unit rather than each day. The state standards (and research in mathematics education) support the idea that units, overall, should have a stronger emphasis on high cognitive demand tasks. In other words, it is important for students' future math learning to use procedures that they understand the meaning of and to learn how think critically and problem solve in mathematics. Very little of your unit should be devoted to memorizing or using procedures or algorithms rotely (i.e., without any connection to meaning).

What you include in a single lesson will be determined by your learning goals for the day. There may be days when your lessons only involve high cognitive tasks - when the goals of your lesson focus on understanding the big mathematical ideas and concepts. And other days, you may only engage students with low cognitive demand tasks - when they are learning about arbitrary things (e.g., learning names of shapes; learning how to tell time; learning how to draw a graph) or when you want to focus on developing procedural fluency with procedures that have already been understood and formalized.

I'm still having trouble differentiating between high and low cognitive demand tasks. What should I do?Anytime you're introduced to a new idea, it can take some time before you feel really confident applying that idea. The Task Analysis Guide in your textbook is a great tool to help you continue to develop and apply your understanding of cognitive demand as you analyze tasks. Remember that the cognitive demand can depend somewhat on context (e.g., what the students have already learned to do), so you might find that it is easier to distinguish cognitive demand once you're in an actual classroom.

What’s the best way to facilitate critical thinking without being too high in cognitive demand?

Remember that cognitive demand is not the same thing as difficulty. In other words, high cognitive demand does not necessarily mean hard. Tasks can have a high cognitive demand while still be accessible to students. Cognitive demand is about the type of thinking that students are asked to engage with - they’re asked to make sense of mathematics ideas or make connections, etc. This can feel hard, especially at first, because in many classes students aren’t asked to think about mathematics in these ways. As students are learning to think more deeply about mathematics, you can help by modeling for them what that looks like and by asking questions that help scaffold new mathematical practices like explaining, connecting, etc. We’ll explore more questioning strategies in MEDU 530, but in the meantime, here’s an introduction: https://www.weareteachers.com/8-ways-to-pose-better-questions-in-math-class/

How can we engage Special Education students in high cognitive demand tasks?

Again, remember that cognitive demand is not the same as difficulty. So it is possible to make high cognitive demand tasks accessible to every student, including special education students. Like all other students, special education students might be new to explaining their thinking or approaching problems using their own strategies. So you’ll want to make sure that you provide adequate supports to help students learn what it means to explore mathematics ideas more deeply. Tools, such as snap cubes, relationship rods, counters, base ten blocks, etc., are great for all students, and especially special education students because these tools can allows students to create concrete models for mathematical problems or to make sense of bigger, more abstract, mathematical ideas.

Learning goals

How can we better explicitly state our learning objectives?

When trying to decide if you learning goal captures the conceptual understanding that you’re aiming for through a problem-based, student-centered lesson, ask yourself these questions. Does your goal clearly state…

• What the mathematics topic is?
  • For example: Students will be able to multiply numbers from 1-9...
• What students will do? And how they will do it?
  • ...by arranging objects into equal groups, determining the total number of objects, and writing a multiplication equation to represent the product.
• How they will understand it?
  • They will recognize the product as the total number of objects in m equal groups of n objects. (i.e., They will understanding multiplication as the process of determining the total number of objects in a number of equal groups.)

If you read through the state standards for the appropriate grade level, the standards often give you insight into the conceptual understanding students are expected to develop. Look for words like “recognize” or “identify”.

Where can I find math standards?

The TN state standards for mathematics are here: https://www.tn.gov/education/instruction/academic-standards/mathematics-standards.html

Common Core State Standards are here: http://www.corestandards.org/Math/

Why is it that all my teachers in this ELED course all have different versions for writing objectives? Is there a different way to write them for different content areas?

Often the things that a teacher needs to think about in order to set goals and plan a lesson effectively can be very similar across different content areas. However, each instructor likely has different expectations for how much of the goal setting and planning you need to make explicit. These differences are likely based on different expectations for learning to lesson plan in different disciplinary (e.g., math, literacy) traditions. In mathematics education, research supports the idea that when learning how to lesson plan, it is best if you make the goals and plan for the lesson as explicit as possible. The more explicit you can make your plans, the better feedback you'll receive so that you can refine your planning practices. Also, one thing that might be somewhat unique to mathematics is that we are often asking you to teach in a way that you did not learn the math content. So it helps if you think through the goal and the lesson plan in detail. When you're teaching on a daily basis, you won't necessarily write out everything, but ideally, you'll think through the lesson and goals with the same detail.

Anticipating

Is anticipating about what students will struggle with or what they will know?

Anticipating is about both! You want to anticipate potential roadblocks that students will face (i.e., part of the task they might struggle with) so that you can prepared to support them without lowering the cognitive demand of the task. AND you want to anticipate ways they might solve the problem so that you can prepare for your selecting, sequencing, and connecting for the whole class discussion (i.e., the summary phase of the lesson).

How do you come up with strategies that you know your students will do?

As you get more experience teaching, you’ll learn from your students and be more easily able to anticipate strategies they might use. Until then, one of the best ways to anticipate possible strategies is to solve the problem in multiple ways, using different representations, such as direct modeling (drawing pictures, using manipulatives), using equations, etc. There is also a fair amount of research that can tell you common ways that students think about specific mathematics topics. One place to start is the journal: Teaching Children Mathematics. You have free access through the UTK library.

What if I predict my strategies and none of my students work the problem those ways?

This could happen, but it is unlikely if you use what you know about your students or draw on research (see previous response) to help you predict strategies they’ll use. If you do find yourself in that situation, that’s ok! It’s a great opportunity for you to learn more about student thinking. Ask students more questions about how they’re thinking about the problem. If you’re having a hard time making sense of student thinking in relationship to your learning goal, then it is ok to stop the lesson and give yourself more time to plan for selecting, sequencing, and connecting in the whole class discussion. You can always do the discussion the following day.

How do we make sure we stay we keep within our allotted time period if problems arise we may not have thought of in the anticipation?

Student-centered, problem-based lessons can be done within a 45-60 minutes lesson, but not always. Sometimes it might make sense to extend a lesson over 2 or even 3 days...especially if students struggle with the mathematical ideas more than you expected, or if students come up with a lot different ways to solve the problem and want to keep sharing. Textbook lessons often explore the same mathematical ideas over 4-5 days worth of lessons. So you should be able to stay within pacing guides even if students spend multiple days doing high cognitive demand tasks. This is true because high cognitive demand tasks often give students in engage with multiple ideas and multiple strategies, that are often broken up across lots of problems in textbook lessons.

Launching the Task

What are some good ideas for launches?

Chapter 7 in your textbook provides some tips for launching your task, and the sample lesson plan in Appendix B shows you what a launch plan can look like. Additionally, the recommended Jackson et al. (2012) reading for Week 3 goes into more detail about how to launch your task.

To meet the needs of each student in a lesson launch, depending on your class, is it important to meet each requirement or can you focus on just a few of the key parts in a lesson launch (e.g., common language)?

As we talked about in class, the goal of the launch is to make sure that every student is able to get started working on the task. So you need to decide what to include in the launch based on the task itself and based on what you know about your students. The areas to address that we talked about in class and that you read about (e.g., common language) are suggestions for what you should pay attention to so that you make sure every student can get started. You do not always need to address everything. For example, the border task that we did in Week 2 does not require attention to contextual features because there is no context for the problem beyond the picture. Also, the mathematical ideas required to do the task only involve counting. So as long as you're confident that your students all have experience counting, then you don't need to spend time go over any mathematical ideas.

Is it possible to go through a lesson without excluding any of your students that do not have certain previous knowledge?

Yes! Well launched tasks can be accessible to every student regardless of their previous mathematics experience or their experiences with the contextual features of the task. As much as possible, you should ensure that contexts are accessible to your students, and sometimes it even makes sense to do mathematics problems without a "real world" context. For example, the border task that we did in Week 2 does not have a context beyond the picture, and every students can engage with the task as long as they understand what is being asked.

How do you introduce important language?

There can be lots of different ways to introduce the necessary language that students will need to complete a task. In Week 1, I introduced important language (e.g., caterpillar) through a children's book, and literature can be a great tool for launching lesson. In Week 2, we did a problem (the border task) that wasn't set within a context, so there was less need to introduce language that might be confusing for students. During Week 4, I'll also launch two tasks, and you can see how I highlight specific words that are necessary for students to use to complete the tasks. For some additional ideas, look at Chapter 7 in your text and the recommended Jackson et al. (2012) reading.

Monitoring

How can I ask students questions without making them feel wrong?

Commonly, teachers only ask students questions when the students have made a mistake and the teacher is trying to get the students to correct it. Because this is the experience students have had, they'll often think they did something wrong if you start asking questions like, "How did you get this?" or "Why did you do that?" Most students are not used to explaining their thinking, especially in math and even more so when their answer is right. So you'll have to help students learn that when you ask them to explain their thinking, it is because you are genuinely interested in how they solved the problem and what they were thinking. Over time, students will get used to you asking questions all the time, and they won't feel wrong anymore. But for the first month or so, you might need to keep reminding them that you are asking questions because you really want to know what they were thinking, right or wrong.

Selecting & Sequencing

When is the best time to utilize incorrect answers in sequencing?

I would recommend selecting a student to present an incorrect answer when you think that student and other students will learn by discussing the incorrect answer. This can be the case when a lot of the students have made the same mistake. Or it might be helpful if correcting the mistake together as a whole class will help you emphasize a mathematical point that is relevant for your learning goal. In terms of when to sequence the incorrect answer, that can really depend on what mathematical point you are trying to emphasize by discussing the strategy that produced an incorrect answer with the class. If it was a common mistake, it might make sense to talk about the incorrect answer first so that the correct approaches will be more accessible to students. If it is something less common, it might be better to address it later, at a time when it fits into the "mathematical story" you are telling through the discussion.

Would a formula (I think you mean, equation) have been better for the Text Message plan task, or would you still have faced the same problem?

An equation would not have necessarily been better or worse than tables and graphs for answering the question, “Which is a better plan?” However, an equation would not have been appropriate for addressing 5th grade learning targets. In 5th grade, students are expected to learn how to graph ordered pairs on a coordinate plane, but they are not expected to write formal linear equations (or graph from equations). Even in high school, giving students the option to visually look for patterns on tables and graphs can be an effective way for them to develop an equation, with connection to the meaning. Once students are familiar with writing linear equations to represent certain situations, that can often be more efficient, but the focus on tables and graphs can help while they’re learning to write equations.

How do I eliminate my learned way of doing math so as to not impose it on my students?

You don’t need to eliminate your own understanding of mathematics. You just need to remember that there is more than one way of solving problems and making sense of mathematics, and that your way is just as valid as the way a student might make sense of the mathematics. You can avoid imposing one way of doing mathematics on students by creating a classroom culture where different ways of thinking are valued and important to making sense of mathematics.

I still have questions pertaining to how to demonstrate various means of representation when demonstrating math problems.

The idea with student-centered lessons is that the lesson is driven by the ways that students solve and make sense of problems. So the representations of strategies should be based on what students did to solve the problem. Sometimes, we might introduce specific ways of representing that we want students to learn - like writing equations or making graphs, but we can connect those more formal mathematical representations to the ways that children more informally solve problems and represent their strategies. Research suggests that this is the most effective way to support mathematics learning with understanding (versus memorization).