MathNotations

[There's a wonderful discussion in the comments regarding the challenge problem at the bottom of this post. Read tc's and mathmom's astute explanations that generalize to the combinatorial problem of placing k indistinguishable objects into n containers.]


Just a quiet acknowledgment to my readers, an expression of gratitude for helping a math blogger who was unknown before 1-2-07 to reach the 25,000th visit on October 7th. Thank you...

And for our middle schoolers and on up, here's a simple A.P. (that's arithmetic progression, not advanced placement!) problem that is designed to help students see the variety of problem-solving techniques one can employ before they reach for the calculator or plug into a formula.

Could a 6th or 7th grade student or group find a way to determine the 25,000th positive odd integer? How would the instructor guide the process?
Well, let's see...

1st.....2nd.....3rd.....4th.....5th.....
1.......3.......5.......7.......9.......

If students have tackled similar problems and are accustomed to making and analyzing tables, looking for patterns, making conjectures (forming hypotheses) and testing their ideas, perhaps some would arrive at the result. They may even surprise you with their ingenuity!
I'll share my favorite approach but don't expect students to think the same way:

Position...1st...2nd...3rd...4th...5th...25,000th...nth
Even.......2.....4.....6.....8.....10....50,000.....???
Odd........1.....3.....5.....7......9....?????......???

Now, what is the formula for the nth positive even integer? the nth positive odd integer?

Today's problem may not be sophisticated but the issue of pedagogy is never trivial, is it?


Oh, ok, I know my readers want more of a challenge to sink their teeth into. So, I'm adding the following:

The answer to the above problem is 49,999. The sum of the digits of this 5-digit positive integer is 40. Determine the number of 5-digit positive integers with this property. This combinatorial problem should keep you busy for at least a few nanoseconds!

Posted by Dave Marain at 6:01 AM 16 comments

Labels: arithmetic sequence, combinatorial math, discovery learning, investigations, middle school math, patterns

The following was my comment on Joanne's stimulating discussion on 'Teachers wonder about direct instruction.'

Although my primary focus is currently on WHAT we teach rather than HOW, I must strongly endorse Mr. Strauss’ reasoned and thoughtful comments. Good teachers have always blended successful methods of the past with the best of what is currently known about the different ways that children learn. No single style can possibly meet the needs of our more and more diverse learners we encounter every day. There seems to be considerable confusion about the technical meaning of DI as developed by Mr. Engelmann. One would need to thoroughly study his rationale and approach to make an informed judgment and I suspect many are responding to the ‘label’ rather than its substance just as many react to ‘discovery learning’ as if it is a method to be used all the time. Effective math lessons I’ve observed for the past 10 years included the essential components of instructional/learning theory:
1. Motivated the lesson (a ‘hook’)
2. Articulation of the objectives of the lesson (what students will know and/or be able to do at the end of the lesson) - this must be carefully thought out during planning and conveyed clearly.
2. Connected current learning to prior learning
3. Reviewed the necessary prerequisite skills for success
4. Provided clear explanations both orally and in writing (on board, on handout or in an electronic presentation)
5. Maximized student involvement via questioning, promoting of dialogue or an activity
6. Assessed what was actually learned (e.g.,responses to questions or requiring students to complete a specific task).
When you remove all the labels, Joanne, it comes down to this: How do we know that the objectives of the lesson were achieved? When I am transmitting parcels of information directly to students, I am still engaging their minds by asking many many questions of different taxonomies to check for their understanding as well as checking if they are still conscious! When I propose a challenging problem and give them a few minutes to work on it in small groups, I am still monitoring their progress carefully and asking guiding questions.
If DI includes all of these components and allows children to explore at times and tackle unstructured open-ended questions for which there is no clear blueprint for solution, then I applaud DI and I guess I’ve been using it all along. If ‘Discovery Learning’ includes all of these components, then I guess I’ve been using it all along and I applaud that too.
Again, as Larry so ably expressed it, good teachers FIND A WAY that works for most of their students most of the time. There will always be some in the class who are not able to grasp the material for a myriad of reasons, often having nothing to do with the child’s ability. Rather than continue this general debate, perhaps we should be looking at REAL examples of effective teaching and then we can applaud these efforts and use them as models for the rest of us, rather than debate the category into which the lesson falls. Oh well, this will never happen, because real examples and pictures would obviate all of the rhetoric and we’d have nothing to blog about!

Posted by Dave Marain at 7:44 AM 2 comments

Labels: direct instruction, discovery learning, math instruction, pedagogy