As a teacher for 11 years and middle-school math teaching consultant, I've seen a wide array of different math programs and classes. I'm sharing here the 10 best teaching tips I've compiled over the years.

1. Provide compelling content to study.

Years ago, a colleague I was working with said, "Maybe class can be fun, but I can't make class compelling. I have to teach math!" It's an assumption worth exploring.

Take Ron Berger's middle-school math project to study levels radon in their own homes. Studying radon is boring. But Berger's class project has got to be one of the most compelling projects in math class history. What if his students discovered dangerous levels of radon in the homes of one geographic area and published the results as they had intended? What would happen to real estate values in that area? What he found is that students were highly engaged in mapping, taking averages, looking at standard deviations- students that heretofore didn't care one bit about radon or the other concepts.

So what's the trick? The trick is that there isn't one. You can't trick students into finding something compelling if it isn't. Take a little bit of time to develop a few topics of study throughout the year that you find compelling- the Economy, the Presidential Campaigns, the Human Body, etc. Find an authentic way to present your result- the paper, the web, a magazine. Keep the project small, authentic and do-able.

Students of teachers that do take this kind of time have better outcomes on state tests than students of teachers who only stick to the text. Almost any social studies context provides a backdrop for learning that adds depth.

Even teachers who hold a math "topics" class only once a month see real benefits, so you don't have to abandon your regular class. And, you'll find that students are more engaged when regular class is held.

If you want to go really deep and have solid administrator support, look into the school reform movement of Expeditionary Learning Schools who have an excellent approach to thematic teaching.

2. Don't use extraneous rewards such as candy, purchase points, stickers, etc.

There is nothing more certain than seeing the culture of a math class decline over a period of years when a teacher bribes them. The intent of the teacher, of course, is good. A teacher cares about his or her students and wants the very best for them. "I don't care how they learn math," one teacher said to me. "I just want them to learn it so that they are prepared." The teacher cared enough to purchase candy out of her own pocket, but the real message to students is this: the "positive reinforcement" of candy means "math isn't worth doing on its own." The research is clear on the matter too, and shows us that extrinsic, non-relevant rewards hurt learning.

Even if the effects aren't immediate, over time so called "positive reinforcements" like these mentioned above erode an otherwise high-quality math program. As a teacher, you are much better off trying to create inherently compelling curriculum than buying candy.

3. Build a culture where students teach each other.

For many teachers, one student helping another is called cheating. But I actually found that the better middle-school math programs all encouraged students to team together at certain times throughout the week. The activities were usually graded as complete or not-complete, and when tied to meaningful tasks, such as building a survey together and collecting original data, student comprehension was greater than on individual tasks.

Building the kind of culture that works for student pairs or groups takes years and lots of practice. But before you give up and decide it doesn't work, determine if you are following tips #1 and #2 first.

4. Give less, but more meaningful work, including homework.

The Trends in International Mathematics and Science Study labels the curriculum in the United States as "a mile wide and an inch deep." Their review of math texts in middle-school found that some were almost 700 pages long. With heavy pressure to teach to the standards, as a teacher you might be tempted to skip and jump to many topics throughout the text. Don't. It achieves little learning.

Choose the most important pieces before the beginning of the year, and keep it simple. Teach the concepts you do teach with depth.

The national advisory counsel formed from the study recommended "put first things first" and suggested that indeed, less is more. Take the time to cull the curriculum to a manageable size for your students, and present them with only that. If you have to "cover" standards, find out what standards and document when you indeed teach them in class. You'll find that teaching with depth often reaches to a broad array of standards.

It's helpful to know what's driving the breadth. As the national study panel concurs, publishers are trying to meet demands of hundreds of different districts by including everything that any school might want. And while publishers have been attempting custom publishing, it is just as difficult to create a math curriculum for a small district as a large one. Thus, the challenges of book publishing lead to a single, uniformly created overarching textbook. Often this is a very large text or an entire series.

In the classroom, teachers and students become overwhelmed and unable to handle the scope or breadth of learning in this form. As teachers, we have to recognize that predominantly negative emotions surround math in middle-school, and that anything we can reduce those emotions will go a long way toward gains in learning learning. Placing a 500 page text in front of a 7th grade student is unlikely to help, so use it sparingly and build little, home-made notebooks for daily use.

5. Model thinking, not solutions or answers.

Don't show a student how to solve something. Instead "think aloud". For example, you might have a whiteboard with a problem up, and start by saying, "o.k., I notice that the 4 numbers I am to sum are all in the thousands category, and that the first is near 3,000, the second near 5,000, and the third... I am confused about..." Model exactly what you thinking including confusion, emotions, skills, strategies and more.

When you do this, also let your students know how mathematicians think. One piece of research that is helpful to know is that mathematicians spend a long time thinking about how to set up a problem, a little bit of time doing the problem, and a long time "looking back" by asking the question, "Does this make sense?' Model that for your students, by putting up a complex problem on the board and spending time not just jumping into a solution, but just talking about what strategies you might use to solve the problem.

6. Provide feedback that is immediate, relevant to the task, non-comparative, and leads the way to next steps.

Many teachers believe that grading is a form of feedback. It isn't. Grading, when done well, can be a form of assessment of learning, but the distinction should be clear. Grades are not an effective tool as assessment for learning. Grades are the end of the road, when you assess what has been learned, but they should not be intended to inform a student where to go next.

Take, for example, three groups of students who received different kinds of "feedback" on math papers they had "turned in." The first group received only narrative feedback (no score) informing them where and how they made mistakes. The second group received a grade (or score) and narrative feedback. The third group received just a grade. Not surprisingly, the students who received narrative feedback improved when re-tested. Those who had received only a grade did not have the information to improve, and performed the same when re-tested. But here is the surprising part. There was no difference between "grade-only" group and the group that received the grade and narrative feedback. Why? The students who received both a grade and narrative feedback completely ignored the written suggestions and only looked at the score. "I got a blah, blah, blah... what did you get?"

Because we live in a world where grades and formalized assessments are so important, work with the system by differentiating assessment for learning and assessment of learning.

When you are grading, one guide is to reference Rick Stiggins strategies of assessment for learning. That way, when you are conducting an assessment of learning (i.e. grading), you'll notice that you are momentarily stepping out of the role of improving a student's learning and won't have the conflict of trying to do two things at once.

7. Change mimeographed sheets to problems you and your students personally develop.

A pervasive aspect of our culture is to give out page after page of information. In faculty meetings, business meetings and conferences, hundreds of pages of documents are handed out. It makes us look organized and prepared. It's also a way to "cover" content. But for a middle-school math student, it also makes it hard to determine what is important. Was it the fractions part? Was it the decimals section? Was it the number line? Was it the triangle puzzle problem? Was it the cartoon?

Instead of another mimeographed page, have your student write their own story problems. Tell them to add artwork for comprehension. Give them the latitude to make them fun. Celebrate them by posting them in class. Give them 5 home-made story problems they create for homework instead of a mimeographed sheet with 30 problems, and really dive into improving them through revision.

8. Use story to teach math.

Write a story, a real story with characters and plot, and add the math problem set. Write about wizards that need to use angles for their sorcery. Write about spice trading ships on the deep seas. Write a story that lasts a whole page before even getting to the math portion. You've engaged the right-side, or less analytical, part of the brain and you'll see a powerful effect of enhanced engagement.

9. Get math tutor volunteers once a week for two-months before state testing.

As a teacher or administrator, spend time during the fall months by planning for and scheduling a single day each week during the months of February and March (right before testing) to have volunteers come in to teach math in small groups. But what's nice is that if developed correctly, these volunteers don't need to have any special training in math.

Start with a simple plan. Each student has 10 skills they have chosen to work on during the whole class tutoring session and have written down their practice problems in class. The phone calls are made, the specific planning with an administrator is done, and volunteers come in and help the students answer the 10 questions during class with support. Schedule tutoring once every week for two months before testing and see your scores greatly improve.

10. Work with the emotions your students have for math.

10a. Ask your students how they feel about math. Use a bit of class time periodically to gain a better sense of where they are. And, just let them feel how they feel. If they like math, they like it. If they are bored, empathize. If your students can't stand math, you will gain far more ground by seeing their perspective than trying to prove they are wrong. As a teacher this is hard because we are so accustomed to trying to "fix" the situation, and of course, our ego is tied to student emotion. If our students are bored, we feel like we aren't doing the right thing. But the larger truth is that there is an ebb and flow in all of us for the topics we are learning. When the boredom, frustration and negativity does emerge, try understanding it. Perhaps class does feel a little boring. That's o.k. Sometimes it will. And then slowly, over a period of years, build those compelling pieces into your classes so that you punctuate boring times with excitement and joy.

10b. Go slowly. Changing the direction of your math class is like trying to change the direction of a large ship, especially when dealing with emotions. Even once everything is place for the changes to occur, you will notice the "ship's" momentum going in the same old direction before you sense any real shifts. This is part of the process. It took me three years to develop a coherent math program at my middle-school and even then, we occasionally slipped in to old patterns. Good luck!

Scott Laidlaw, Ed.D. has been teaching for 11 years and served as consultant for best practices in mathematics under a federal grant. The company Imagine Education has been building story-based games to teach mathematics for the past 7 years. This February, Ko's Journey will be its first on-line nationwide release. Built with a design called story-playing, students learn math in context authentic to the story-line. Find out more at http://kosjourney.com

By Scott_Laidlaw

1. Provide compelling content to study.

Years ago, a colleague I was working with said, "Maybe class can be fun, but I can't make class compelling. I have to teach math!" It's an assumption worth exploring.

Take Ron Berger's middle-school math project to study levels radon in their own homes. Studying radon is boring. But Berger's class project has got to be one of the most compelling projects in math class history. What if his students discovered dangerous levels of radon in the homes of one geographic area and published the results as they had intended? What would happen to real estate values in that area? What he found is that students were highly engaged in mapping, taking averages, looking at standard deviations- students that heretofore didn't care one bit about radon or the other concepts.

So what's the trick? The trick is that there isn't one. You can't trick students into finding something compelling if it isn't. Take a little bit of time to develop a few topics of study throughout the year that you find compelling- the Economy, the Presidential Campaigns, the Human Body, etc. Find an authentic way to present your result- the paper, the web, a magazine. Keep the project small, authentic and do-able.

Students of teachers that do take this kind of time have better outcomes on state tests than students of teachers who only stick to the text. Almost any social studies context provides a backdrop for learning that adds depth.

Even teachers who hold a math "topics" class only once a month see real benefits, so you don't have to abandon your regular class. And, you'll find that students are more engaged when regular class is held.

If you want to go really deep and have solid administrator support, look into the school reform movement of Expeditionary Learning Schools who have an excellent approach to thematic teaching.

2. Don't use extraneous rewards such as candy, purchase points, stickers, etc.

There is nothing more certain than seeing the culture of a math class decline over a period of years when a teacher bribes them. The intent of the teacher, of course, is good. A teacher cares about his or her students and wants the very best for them. "I don't care how they learn math," one teacher said to me. "I just want them to learn it so that they are prepared." The teacher cared enough to purchase candy out of her own pocket, but the real message to students is this: the "positive reinforcement" of candy means "math isn't worth doing on its own." The research is clear on the matter too, and shows us that extrinsic, non-relevant rewards hurt learning.

Even if the effects aren't immediate, over time so called "positive reinforcements" like these mentioned above erode an otherwise high-quality math program. As a teacher, you are much better off trying to create inherently compelling curriculum than buying candy.

3. Build a culture where students teach each other.

For many teachers, one student helping another is called cheating. But I actually found that the better middle-school math programs all encouraged students to team together at certain times throughout the week. The activities were usually graded as complete or not-complete, and when tied to meaningful tasks, such as building a survey together and collecting original data, student comprehension was greater than on individual tasks.

Building the kind of culture that works for student pairs or groups takes years and lots of practice. But before you give up and decide it doesn't work, determine if you are following tips #1 and #2 first.

4. Give less, but more meaningful work, including homework.

The Trends in International Mathematics and Science Study labels the curriculum in the United States as "a mile wide and an inch deep." Their review of math texts in middle-school found that some were almost 700 pages long. With heavy pressure to teach to the standards, as a teacher you might be tempted to skip and jump to many topics throughout the text. Don't. It achieves little learning.

Choose the most important pieces before the beginning of the year, and keep it simple. Teach the concepts you do teach with depth.

The national advisory counsel formed from the study recommended "put first things first" and suggested that indeed, less is more. Take the time to cull the curriculum to a manageable size for your students, and present them with only that. If you have to "cover" standards, find out what standards and document when you indeed teach them in class. You'll find that teaching with depth often reaches to a broad array of standards.

It's helpful to know what's driving the breadth. As the national study panel concurs, publishers are trying to meet demands of hundreds of different districts by including everything that any school might want. And while publishers have been attempting custom publishing, it is just as difficult to create a math curriculum for a small district as a large one. Thus, the challenges of book publishing lead to a single, uniformly created overarching textbook. Often this is a very large text or an entire series.

In the classroom, teachers and students become overwhelmed and unable to handle the scope or breadth of learning in this form. As teachers, we have to recognize that predominantly negative emotions surround math in middle-school, and that anything we can reduce those emotions will go a long way toward gains in learning learning. Placing a 500 page text in front of a 7th grade student is unlikely to help, so use it sparingly and build little, home-made notebooks for daily use.

5. Model thinking, not solutions or answers.

Don't show a student how to solve something. Instead "think aloud". For example, you might have a whiteboard with a problem up, and start by saying, "o.k., I notice that the 4 numbers I am to sum are all in the thousands category, and that the first is near 3,000, the second near 5,000, and the third... I am confused about..." Model exactly what you thinking including confusion, emotions, skills, strategies and more.

When you do this, also let your students know how mathematicians think. One piece of research that is helpful to know is that mathematicians spend a long time thinking about how to set up a problem, a little bit of time doing the problem, and a long time "looking back" by asking the question, "Does this make sense?' Model that for your students, by putting up a complex problem on the board and spending time not just jumping into a solution, but just talking about what strategies you might use to solve the problem.

6. Provide feedback that is immediate, relevant to the task, non-comparative, and leads the way to next steps.

Many teachers believe that grading is a form of feedback. It isn't. Grading, when done well, can be a form of assessment of learning, but the distinction should be clear. Grades are not an effective tool as assessment for learning. Grades are the end of the road, when you assess what has been learned, but they should not be intended to inform a student where to go next.

Take, for example, three groups of students who received different kinds of "feedback" on math papers they had "turned in." The first group received only narrative feedback (no score) informing them where and how they made mistakes. The second group received a grade (or score) and narrative feedback. The third group received just a grade. Not surprisingly, the students who received narrative feedback improved when re-tested. Those who had received only a grade did not have the information to improve, and performed the same when re-tested. But here is the surprising part. There was no difference between "grade-only" group and the group that received the grade and narrative feedback. Why? The students who received both a grade and narrative feedback completely ignored the written suggestions and only looked at the score. "I got a blah, blah, blah... what did you get?"

Because we live in a world where grades and formalized assessments are so important, work with the system by differentiating assessment for learning and assessment of learning.

When you are grading, one guide is to reference Rick Stiggins strategies of assessment for learning. That way, when you are conducting an assessment of learning (i.e. grading), you'll notice that you are momentarily stepping out of the role of improving a student's learning and won't have the conflict of trying to do two things at once.

7. Change mimeographed sheets to problems you and your students personally develop.

A pervasive aspect of our culture is to give out page after page of information. In faculty meetings, business meetings and conferences, hundreds of pages of documents are handed out. It makes us look organized and prepared. It's also a way to "cover" content. But for a middle-school math student, it also makes it hard to determine what is important. Was it the fractions part? Was it the decimals section? Was it the number line? Was it the triangle puzzle problem? Was it the cartoon?

Instead of another mimeographed page, have your student write their own story problems. Tell them to add artwork for comprehension. Give them the latitude to make them fun. Celebrate them by posting them in class. Give them 5 home-made story problems they create for homework instead of a mimeographed sheet with 30 problems, and really dive into improving them through revision.

8. Use story to teach math.

Write a story, a real story with characters and plot, and add the math problem set. Write about wizards that need to use angles for their sorcery. Write about spice trading ships on the deep seas. Write a story that lasts a whole page before even getting to the math portion. You've engaged the right-side, or less analytical, part of the brain and you'll see a powerful effect of enhanced engagement.

9. Get math tutor volunteers once a week for two-months before state testing.

As a teacher or administrator, spend time during the fall months by planning for and scheduling a single day each week during the months of February and March (right before testing) to have volunteers come in to teach math in small groups. But what's nice is that if developed correctly, these volunteers don't need to have any special training in math.

Start with a simple plan. Each student has 10 skills they have chosen to work on during the whole class tutoring session and have written down their practice problems in class. The phone calls are made, the specific planning with an administrator is done, and volunteers come in and help the students answer the 10 questions during class with support. Schedule tutoring once every week for two months before testing and see your scores greatly improve.

10. Work with the emotions your students have for math.

10a. Ask your students how they feel about math. Use a bit of class time periodically to gain a better sense of where they are. And, just let them feel how they feel. If they like math, they like it. If they are bored, empathize. If your students can't stand math, you will gain far more ground by seeing their perspective than trying to prove they are wrong. As a teacher this is hard because we are so accustomed to trying to "fix" the situation, and of course, our ego is tied to student emotion. If our students are bored, we feel like we aren't doing the right thing. But the larger truth is that there is an ebb and flow in all of us for the topics we are learning. When the boredom, frustration and negativity does emerge, try understanding it. Perhaps class does feel a little boring. That's o.k. Sometimes it will. And then slowly, over a period of years, build those compelling pieces into your classes so that you punctuate boring times with excitement and joy.

10b. Go slowly. Changing the direction of your math class is like trying to change the direction of a large ship, especially when dealing with emotions. Even once everything is place for the changes to occur, you will notice the "ship's" momentum going in the same old direction before you sense any real shifts. This is part of the process. It took me three years to develop a coherent math program at my middle-school and even then, we occasionally slipped in to old patterns. Good luck!

Scott Laidlaw, Ed.D. has been teaching for 11 years and served as consultant for best practices in mathematics under a federal grant. The company Imagine Education has been building story-based games to teach mathematics for the past 7 years. This February, Ko's Journey will be its first on-line nationwide release. Built with a design called story-playing, students learn math in context authentic to the story-line. Find out more at http://kosjourney.com

By Scott_Laidlaw