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Resources for Math Teachers
Ellen Browne, TI - National Instructor and math teacher at Pomfret School, and Pam Mulcahy, math teacher at Holderness School, happily share their resources and thoughts on math education with other teachers. Asked about math education, Browne and Mulcahy both shared:
We spend much of our time working to make math more accessible to a wider range of students, changing our teaching practices to emphasize thinking over rote learning and depth of content over breadth. At the moment, we have arrived at the following goals for ourselves. To develop in students:
A sense of numbers - and particularly a sense of very large and very small numbers and of how numbers grow.
An ability to understand various representations of data - tables, charts, graphs, equations, scale, etc.
An ability to infer such things as whether something is increasing or decreasing, changing rapidly or changing slowly, remaining constant, etc. from a table, graph, chart or equation.
An ability to encounter a problem/challenge, understand what is being asked or what is needed, access and apply relevant knowledge, and develop a path to a solution.
An abillity to recognize patterns and represent them symbolically.
An understanding of basic statistics.
An understanding of basic probability.
Finally, both of us agree wholeheartedly that we want our students to embrace entrepreneur Adam Osborne's words: "The most valuable thing you can make is a mistake - you can't learn anything from being perfect."
Teach Statistics Before Calculus - Arthur Benjamin
Why Math Instruction Is Unnecessary - John Bennett
Visualizing Ourselves ... with Crowd-Sourced Data - Aaron Koblin
The Surprising Beauty of Mathematics - John Matte
The Beauty of Data Visualization - David McCandless
Math Class Needs a Makeover - Dan Meyer
The Best Stats You've Ever Seen - Hans Rosling
Teaching Kids Real Math with Computers - Conrad Wolfram
We Use Math - Brigham Young University Mathematics Department
Lesson Planning Strategies
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.
Explanations and Lesson Ideas
calculus-help.com - This site has some simple but clear flash tutorials on limits and derivatives.
Dan Meyer - This is one of our favorite sites. It has great suggestions (and actual lessons) for ways to develop problems for Algebra and Geometry
Easing the Hurry Syndrome - Jennifer Wilson - This site has lots of ideas and projects for Geometry and Calculus
Global Math Department - This is a website developed by math teachers to share ideas, teaching tools, and conversation. They offer weekly webinars, newsletter, etc.
Great Math Teaching Ideas - mostly algebra focused
Khan Academy - Khan Academy offers basic but clear videos with lessons and explanations for most high school math topics. Great reinforcement/review for students needing it.
Purplemath - a great website with clear and simple explanations of most algebra topics along with three or four practice problems. The site has existed for a number of years, so the explanations are very thoughtful and complete.
Patrick: JMT - a great collection of videos covering most high school (and some college) math topics. Also great for reinforcement and review.
Slam Dunk Math - great ideas for activity based math classes
Texas Instruments - Math Nspired - lesson plans and other resources for teaching with the TI - Nspire
Texas Instruments - TI-84 Resources - lesson plans and other resources for teaching with the TI-84
Thinking Mathematics - lesson ideas, videos, and other resources for most levels of high school mathematics
youcubed - lesson plans and other resources for teaching math
People to Follow on Twitter
Dan Meyers - @ddmeyer
Jo Boaler - @joboaler
Mathalicious - @Mathalicious
Word Problems and Mathematical Enrichment Resources
Art of Problem Solving - links to challenging problems and advice for students and teachers interested in mathematical competitions.
Cut-the-Knot - interesting problems, writings on mathematics, and otehr resources for students and teachers.
Futures Channel - movies about math in real life.
Mathematics Association of America - links to provocative problems and ideas for exploring mathematics with students.
Mathematics Assessment Project - some provocative questions to structure classes around
Phillips Exeter Academy Mathematics Department - contains all the problems assigned to Exeter students in each class taught at the Academy. Great source for excellent problems.
Robert Kaplinsky - excellent, real-world math problems
Worksheets and Practice Problems
algebra.help - has a bank of worksheets that are quite suitable for most Algebra I classes and perhaps some second year algebra classes.
Exercises in Math Readiness - offers explanations and worked problems for Algebra I, Alebra II and Geometry.
IXL Learning - offers students ten free practice problems per day on the topic of their choice. When a student responds incorrectly, the site provides an explanation of the correct way to think through the problem.
KUTA Software - contains an impressive bank of free worksheets in pdf format for Algebra I through Calculus.
ProveIt! - a great resource for Geometry students as it offers them a series of proofs to work AND provides them with feedback on their work.
Useful Stuff to Have
When I first began teaching, I envied the equipment my science teaching colleagues had to help them make abstract concepts more "visible" or more concrete. Eventually, I realized that I could also equip my classroom so that students could learn mathematics as early people did - by observing, hypothesizing, wondering, tinkering, etc.
Calculators - I have used both the TI-84 and the TI-Nspire - and I like both. Both the TI website (see website section above) and YouTube have helpful videos if you need help learning how to perform a particular operation.
Rulers, protractors, compasses, dice, measuring tapes, colored pens/pencils, paper, markers, etc. should be standard issue in math classrooms.
Elementary/middle school classrooms also benefit from other manipulatives like cuisenaire rods, fraction tiles, cubes, geometric shapes (solid and foldable), algebra tiles, counters, play money, etc. Personally, I find these helpful in the high school classroom as well.
Less Traditional (but still useful) Gear
Coins - for probability
Counters - for combinatorics
Dice - for probability
Mini white boards (and markers and erasers) - allows students to work problems at their seats and then display to share their work with class. You can make your own by cutting a sheet of white board into smaller pieces or by ordering them online.
Play dough - to model conic sections
Playing cards - for probability
Sextant (plastic) - for trigonometry
Slinky - to model harmonic motion
Tinkertoys (or other building toys like K'NEX) - to make simple machines like ferris wheels, rotors, etc. to model harmonic motion, angular speed, etc.
Desmos - is a free, online graphing program that allows students and teachers to create and explore graphs. Comes complete with sliders for functions and so much more to facilitate exploration without fancy software or lots of time-consuming calculator work.
Geogebra - is a free product that allows students and teachers to create geometric shapes and graphs of functions and explore the behavior of the shapes and the functions.
GraphCalc - a free download for Windows machines that enables you to draw and analyze graphs. It also allows you to use colors which is helpful for creating PowerPoints and Smartboard lessons as well as helping students to see the impact of various transformation on graphs.