Monday, May 12, 2008

Components of the Effective (Math) Lesson Gr 5-12 - Part I

One of the reasons I began this blog was to share the collective wisdom of experienced math teachers as a benefit to the novice. Well, here I am 18 months into MathNotations and I don't believe this has yet been specifically addressed. I expect the comments or follow-up posts to be even more beneficial than what I'm writing below.

Here's what I'm asking my readers --
In this post, I will begin enumerating one or two instructional components which I believe should be an integral part of most (math) lessons. Since I have strong antipathy towards jargon, I will try to avoid technical phrases like 'set', 'hook', although closure is ok.

Note that I put math in (..) to emphasize the point that I regard many of these suggestions as integral to effective lessons in general!

Note: These lesson components should be independent of teacher style, makeup of the class, content, etc.


I do know that newbies often feel overwhelmed by all of the differing expectations coming from their immediate supervisor, colleagues, principal, other administrators, courses of study/syllabi, district technology initiatives, state standards, state standards, NCTM Standards/Curriculum Focal Points, standardized test specs -- just to name a few! I haven't even mentioned what they learned from their methods classes, the influence of their math teachers in their formative years, advice from just about everybody. When all is said and done, it seems that the number one concern on the part of most evaluators in the beginning is classroom management, effective delivery of content being number two. Of course, evidence of content knowledge becomes of greater importance if there is an immediate supervisor who has math certification.

How does one navigate through this morass without losing one's mind? Prioritize! Less really is more! Rather than attempt to build the perfect lesson to please the observer, be guided by what you know will lead to demonstrable evidence of learning. Yes, planning is critical. I will comment on that further.

Here then is just the beginning of what I expect to be an extended discussion and one which I am considering publishing as a pamphlet. Please adhere to the Creative Commons License in the sidebar if reproducing any of this.

I am stating unequivocally that these are my own personal ideas of what makes an effective math lesson. I do not want anyone to say that I am telling anyone how to teach!

Each of you out there will have your own list, although I'd be surprised if there wasn't considerable overlap. The order of course will vary. These are the principles by which I was guided both as a classroom teacher and as a supervisor. At the beginning of the year, I would meet with teachers to discuss what I was looking for in the lesson. For clinical observations, I would also have a preconference to discuss specifics. This was particularly of critical importance before observing the non-tenured teacher.

1) Class Opener - Critical first 5 minutes - Establishment of Routines

a) Allow students to socialize/decompress for a couple of minutes as they enter, but let them know what is expected of them; close door at late bell. Establish iron-clad routines for students to follow if they arrive after that - stick to it!

b) Math Warmup/Problem of the Day already on the board or projected on a screen using the overhead or PowerPoint (or Word) from the computer; the warmup can be used to review prerequisite skills for the upcoming lesson, SAT review, an opportunity for students to practice their communication (e.g., writing) skills in math, etc.

c) Answers to some or all of the homework exercises can be written on the board or projected on a screen from overhead or computer. Virtually every publisher of current texts provides ready-made transparencies both for WarmUps and answers to homework, not to mention PowerPoint presentations for every lesson! Some educators object to displaying answers like this as it invites students to quickly copy these on their paper. You may want to have selected answers displayed rather than all. There is no foolproof method here, so use your own judgment. The important thing is to busily engage students from the outset. While students are working on their warmup problem, the teacher is circulating, checking homework and engaging students. This personal interaction with students means so much (e.g., Lily, I saw you in the play on Thu night -awesome!).

Ok, folks, this is just a beginning...
Please contribute your suggestions!


Anonymous said...

Right on with this one, Dave. The only thing I would add is accountability, particularly for the new teacher. Students like to know that giving it their best shot is 'worth' something. I would suggest a check mark, next to that homework check or attendance check. The important thing for the teacher is not answering student questions while strolling around the room and keeping it to a maximum 5 minutes. The important thing for the students is getting their 'math mind' going. The students can share their answers with a student partner or with the class when time is up. This should be a good lead-in for the day's lesson.


Dave Marain said...

You added precisely the details (5-min limit, share with partner, etc.) that was needed! That's why I need the experienced educators to contribute. Thank you!

I really believe that simply-worded, specific suggestions can be helpful. I fear that most readers will not reply to this post. There often seems to be little interest in posts about actual pedagogy. This is disheartening to me, since I write this blog primarily because I feel that experienced educators have an obligation to support, even inspire, the next generation of our profession. You and so many others can make a difference. Again, thank you...

My plan is to move past the beginning phase of a lesson and onto techniques of introducing a new topic/concept/skill/procedure/whatever.
I am convinced that there is an inverse ratio between the length of verbal information provided and comprehension on the part of the learner. Even at advanced levels, one needs to make sense of the formal mathematics via concrete examples or the apt metaphor. Therefore, the next part of this series may be entitled, "Setting a Good Example!" What do you think?

Anonymous said...

"I am convinced that there is an inverse ratio between the length of verbal information provided and comprehension on the part of the learner."

Actually there's lots of research to back you up here. I'm including the following link. I like that second medical school example.

The pause that refreshes idea here. I'm not much on ed-school jargon, but I do think whatever you call it, you've got to let students assimilate that info you're dishing out. (Particularly at the advanced level.) And yes, yes, yes to the concrete examples.

I agree with you that this may be a two-way dialogue, but there really are so many problems facing new mathematics educators, particularly in schools that have no experienced teachers. I think the hierarchy goes something like this:
[1] Classroom management
[2] Content knowledge
[3] Instructional strategies
[4] Curriculum development

Of course, #3 helps #1. But, it's that #2 issue that will draw the most responses and create the most controversy.


Dave Marain said...

Followed your link, Hypatia -- excellent piece of research. Should be required reading for all prospective teachers and perhaps even us veterans!

I strongly recommend all of our visitors to read this thoroughly.

Here is the introduction:

A consistent theme of faculty at colleges and universities is that they are not aware of the educational research that currently exists or how it applies to their teaching. The purpose of this section is to provide summaries of research articles that can inform classroom practice. Explore one of the following.

* Talk six minutes less and students learn more!
* How dense should information in a lecture be?
* How to make lectures more clear.

More quotes (translates to "Less is more!"):

This study suggests, however, that we would be better off presenting only the basic material necessary to achieve our learning objectives: approximately only 50 percent of the material presented in any lecture should be new. The rest of class time should be devoted to material or activities designed to reinforce the material in students' minds.

This study is significant since one of the chief barriers always presented by faculty to the acceptance of active learning is that "there is simply too much content to cover." Apparently less new content and more time reinforcing the facts and concepts presented [which could include active learning] will lead to greater student learning.

Hypatia, most of what I would have composed in my 'pamphlet' is contained in this excellent summary of research on Active Learning. Thank you!

Research likes this provides reasonable validation for what we learn from the practice of our profession. All of the hypotheses being tested came from the belief systems of the researchers, yes? These are beliefs that come from experience and reflection of one's practice.They certainly reflect my beliefs.

By the way, the last part: "How to make lectures more clear" is phenomenal! I could write a post about each one of the points being made! Even though this research referred to university teaching and did not specifically address mathematics teaching, it is, nevertheless, invaluable. Many of us have experienced the joys of 2 hour lectures in a large university hall, so we can appreciate why this research was done. However the principles embodied in this paper can be applied to K-12 classrooms.

Much more needs to be said...

bill said...

I'm using a spreadsheet to project home/class work. After a brief review I project the spreadsheet letting students know at a glance how they are doing in terms of work turned in. Its not grades, just different colors showing assignments still missing and those turned in.

I got the idea from this post: