By Teachers for Teachers

Planning a math class always takes us longer than we expect. However, over the course of more than thirty years, we have learned what is most important to consider, what questions will help us to prepare a lesson that helps students develop important skills and master important content. We have also learned that no method, no lesson works for every student or every group of students, but we know that our lessons are more successful when we consider WHO is in the room with us, what they enjoy, what they know, what they are able to do, etc. as we are planning our classes.

 

The following questions and thoughts tend to guide our planning:

 

1. Why is what I am about to teach important? Where does this concept lead? How might it be applied? If it is not important, does it deserve much time?  A concept is not important simply because it is in a textbook. Use your own mathematical experience in college to guide you. Did you use this?

 

For example, if I am beginning a unit on functions, I know that understanding how common functions behave (linear, quadratic, exponential, logarithmic, rational, etc.) and how changing various parameters changes that behavior is fundamental to understanding much of what happens in most traditional mathematics classes. Understanding these ideas can also allow students to make sense of data and to predict how various events might unfold given the data that exists at the moment. So I tend to spend a lot of time developing an understanding of function behavior and on using our knowledge of functions to solve problems.

 

2. How can I best give students a sense of why this matters or interest them in discovering this for themselves? What kind of a situation can I construct that will help them/allow them to discover some of this for themselves?  

 

For example, when I begin my unit on exponential functions, I always begin with the story of the chessboard, and I ask students to determine, how much rice they will need to fill all the squares on the chessboard if each square gets twice as many grains of rice as the square before it. We chart the results (with x as the number square and y as the number of grains of rice on a given square) and consider how the y values change. We plot the values and sketch a graph and then find the equation using exponential regression. Then we move on to do an M&M lab (which models exponential decay) and compare the two functions - one growing and one decaying. How do the equations differ? What do the functions have in common? Why? 

 

3. What do I need students to be able to do at the end of this class or unit? What are they already able to do - and how can I help them connect the two?

 

When I teach exponential functions, I want my students at the end of the unit to be able to recognize exponential growth and decay when they see a graph, a set of data or an equation. I want them to be able to find an equation to describe a function given data, graphs or word problems. I want them to know from an equation if the function is increasing or decreasing, determine its initial value and its end behavior.  Thus, I need to present them with problems that require them to do these things.  I further need to set up my assignments. worksheets, etc. so that students have to practice determining which method to use and which process to apply.   

 

4. When learning this concept for the first time, what are the common mistakes and misunderstandings students experience?

 

Experienced teachers within your department should be able to answer this question.  If not, send your questions our way.

 

5. What are the best strategies to engage their interest and develop the skills and understandings I feel are essential to develop?

 

I almost always begin each unit with a problem for students to solve - either as a large group or in several smaller groups.  I find that this forces them to access their natural problem solving skills and to connect what we are about to learn with things they already know and are able to do.  Often it helps them formulate an algorithm in their own language - which helps them to remember it and to make use of it down the road.

 

6. How will I know when they have mastered the material or the skill? What evidence do I need to collect or see? In terms of evidence "collecting," what are my options - test, quiz, problem set, group problem set/quiz, project, etc.?

 

My students do math almost every day in my class.  I am fortunate to have two walls of boards - and I like to have them work at the boards because I can easily see their work and their thinking.  They can also see one another's work which provokes questions of one another and of me.  I usually walk around and coach those who need help or ask a peer to help another peer.  Given the evidence I collect, I sometimes quietly nudge students to see me for extra help in a free block. Mini whiteboards are another awesome device to see what students know and are able to do - as is polling software or the TI-Nspire calculator.  All of these are great forms of formative assessment - ungraded assessments that help me know what students are able to do at the moment.

 

8. What about homework?

 

My friend Martha always reminds young teachers to work the homework problems before assigning them. This helps to ensure that students leave class equipped with the information and skills they need to do their homework.  It reduces their frustration.  For difficult concepts, I try also to include some partially worked problems for those students who need them. I'm also careful to post my class notes every day and to create videos reviewing the concepts or pencasts of solutions to help those students who need the help.

 

7. How do I help them remember the important skills and ideas for the test, the final exam, their next course, etc.?

 

Thanks to continuing research into how the brain learns, we know that interleaving practice is key to helping students remember key information and skills.  Interleaving practice means weaving "learned" material into future lessons so that students are always going back to what they have already learned and using that material over and over again. Research tells us that this is one of the best ways to help students remember key information for the long term. In my class, I usually begin each day with warm-up problems taken from all the chapters we have covered to date.  Obviously, we review and return to key concepts - not minor details.  I also deliberately create worksheets that move students back and forth between the concepts covered in class - working with sine in one problem and cosine in another or using the Law of Sines for one problem and the Law of Cosines for the next. This post on Class Teaching, by the authors of Making Every Lesson Count, is helpful in explaining the practice of interleaving more completely.