On Being a REALLY Good Math Teacher

I started a grad school class this week. It's been a while, so on my way out the door, I grabbed an old notebook off my shelf and shoved it in my bag with my new textbooks. As it turns out, this was a fortuitous move, for the first half of this notebook was filled with journal entries from my days as a student teacher and a first year teacher. It's been a fascinating read.

One particular entry contained notes from a lecture I attended by John Benson, one of my highly admired mentors. His topic: "The difference between good teaching and really good teaching." He said that GOOD teachers:

1. Have a clear idea of what students know and can do,
2. Know what is necessary for success on a particular task,
3. Use a variety of instructional methods to reach many learning styles,
4. Are eager to spend extra time outside of class to answer questions,
5. Establish clear guidelines for student success and performance,
6. Hold students to high standards,
7. Account for individual differences,
8. Provide clear explanations of the concepts that students are expected to master, and
9. Continually preview and review.

Competent teachers may accomplish a satisfactory subset of these objectives, but do not provide the care and variety that is evident in the classroom of a good teacher. On the other hand, REALLY GOOD teachers:

1. Have a second set of objectives that go beyond the mastery of today's content,
2. Can seize teachable moments and move towards higher objectives,
3. Are able to recognize when students have lost interest and can seize the opportunity to teach something really interesting, and
4. Believe that these higher objectives are the really important part of their mission.

There's more in my notes, but I'll stop there because these lists still make an impact on me. The 'short' list for really good teachers is appealing to me. I believe in these four objectives and I like working to improve on them. They seem to be about passion and that makes me feel good. But as I read #1 a little more closely, I pause on the word 'second,' and realize that there is no hope for me as a really good teacher unless I can also move towards better mastery of the objectives of a good teacher... and that's a LONG and difficult list. I think John Benson is spot on when he suggests that really good teaching comes from a long list of grueling, difficult, AND passionate objectives. It's a package deal, and it's really hard. But the company is great.

Can You Help Me with an Algebra 1 Sequencing Project?

I created a checklist style review guide for my Algebra 1 students at the midpoint of the course several years ago. Over the years, I have reviewed and revised this study sheet multiple times… tweaking phrasing and sequencing, but also changing my mind again and again about what is (or is not) a basic skill in a beginning algebra class. Recently, I took the leap and created the second half: committing myself to a firm opinion about the essential nature of 55 basic algebra skills. The problem is that every time I pick it up, I find something else that I want to change. It is becoming an albatross for me, and so this is where I could use your help. Take a look and let me know what you think. Look at language, sequencing, design, etc. Did I miss something important or include something they already mastered (or are not ready for)? I really want it to be great, but have become overwhelmed in my solitude.

If you click on the image of the study guide, you will be directed to a site where you can upload it for free*. (*This project has been completed, and the free version is no longer available on the site.) As you read through this study guide, please bear in mind my goals/objectives for this type of algebra review guide:

• The individual skills are meant to highlight the essential tasks that an Algebra 1 student should have mastery of. I firmly believe that good algebra teaching revolves around problem solving and applying multiple skills to illustrate, communicate, generalize and verify solutions to problems. I have purposefully left off any topics that I consider to involve multiple skills and the critical thinking of deciding which methods are appropriate and useful. I lovingly refer to the chosen topics as our ‘bag of tricks,’ or the tools from which we pull from to solve problems.
• I have tried to include only topics that are learned in an Algebra 1 class (and not earlier) although there are a few that I have found to be so essential that they bear highlighting again (like order of operations, graphing points on coordinate plane, and properties of real numbers).
• I have attempted to order the skills according to my best sequence of instruction, but I have found that there are some that tend to bounce all over the place. Ratios, proportions, and cross products, for example, have felt comfortable to me in many different locations in the course. The same is true of datasets and statistical analysis. Bear in mind that this is meant to be a cumulative review and not necessarily a course outline. For example, I thought it best to group box plots with scatter plots on the study guide even though I don’t necessarily teach them at the same time.
• The code after the topic name is my newest attempt to align this sheet with the Common Core Standards. If you are a Common Core expert, I would greatly appreciate fact checking and additional input with this alignment.
• The last two columns are intentionally blank, to provide teachers the flexibility of aligning the guide with their class textbook and supplementary materials. I go back and forth on the usefulness of this.

Thank you for your assistance with this project. I will happily share the final results with you.

Algebra is Not a Four Letter Word

Despite the overwhelming evidence that a foundation in algebraic thinking is essential to a sound mathematics education, algebra continues to get a bad rap among the populous. Billy Connolly's Algebra rant (foul language warning) is hysterical, but sad because it spotlights the popular opinion that algebra is incomprehensible and useless.

I've been thinking about algebra's reputation a lot lately:

• What do we need to do to make algebra seem approachable and  useFUL for everyone? 
• How can we improve the way we rationalize algebra - so that our arguments are convincing and appealing to even the most jaded among us?
• What can I do in my own classroom, so that my students better understand both HOW to use algebra and WHAT algebra is useful for?

These questions have been driving my thoughts lately, most likely spurred by the heightened student frustration I have experienced as we switch gears from one semester to the next. There is nothing like a cumulative exam to dredge up student anxiety and feelings of hatred for the source of those anxieties!

What I Read

I know some of you have blogrolls as long as my arm. I don't know where you find the time. But, without further ado... my blogroll:

• For their overwhelming enthusiasm, extreme generosity of ideas and resources, and high quality of mathematics education, I like Dan Myer (dy/dan) and Kate Nowak (f(t)). Both blogs are friendly, helpful, and FULL of endless discussions about best practices for teaching math.
• I also like Sam Shah (Continuous Everywhere but Differentiable Nowhere), Vi Hart (Vi Hart - Blog),  and Christopher Danielson (Overthinking My Teaching) mostly because they remind me of the colleagues I've had who are wonderfully smarter than me - with all the quirkiness that comes with overactive synapses.
• Richard Byrne (Free Technology for Teachers) introduces me to something new every single day. I love/hate him for that!

These blogs pretty much satisfy my need for young, enthusiastic, creative, and innovative voices in the math education blogosphere. I'm sure there are more great ones out there, and of course I'd love to hear about them, but these have made me happy so far.

But there is another type of voice that I continue to search for - the voices of experienced, wise, veterans. I wouldn't mind reading a few more blogs in this category, but I currently recommend:

• John Benson (along with a young colleage, PJ Karafiol) who writes Angels of Reflection which is filled with tough lessons and ideas with teeth - all backed with demonstrated success and not theoretical success. This is immensely appealing to me.
• Vicky Rauch (aka Scipi @ Go Figue!) who writes from the sobering perspective of a veteran teacher in the midst of products from a failed secondary mathematics education. Her wisdom in the context of community college mathaphobes makes me consider the cost of too much innovation in math education.

A Versatile Blogger

I was recently awarded the 'Versatile Blogger' award. I know! When I read the announcement I was instantly overcome by a rush of pride and disbelief - "My wit, charm, and unique spin on teaching math are getting acknowledged after just a few short months of blogging! I didn't know I was so awesome!" I read on to the award description and requirements for acceptance:

• Thank the person who gave you this award. Include a link to their blog.
• Next, tell 7 things about yourself.
• Finally, select 15 blogs/bloggers that you’ve recently discovered or follow regularly. Award those 15 bloggers the Versatile Blogger Award.

POP!

According to my most casual research, the 'Versatile Blogger Award' appears to have originated this past September. According to blogpulse.com  there are about 182 million public blogs in existence today. I have three questions:

1. Assuming awardees all accept their award and complete the requirements in the space of one week, how long would it take to award ALL 182 million blogs? 
2. How many people have a blogroll long enough to discriminate 15 award winners from it?
3. Didn't you just award me a blogging award? If you don't already know 7 things about me, I probably should give the award back.

And my bubble has completely popped, leaving me with only one lingering question: "Why did it take so long - after 16 weeks of circling the globe, why was I not recognized sooner?" Mathematics can be seriously depressing sometimes.