Teaching developmental math

Comment by Danny Clarke on August 7, 2009 at 9:17pm

My idea for teaching math is to start with a problem and teach just the math necessary to solve the problem. I have experienced this type of instruction at Michigan University in the past and it was very effective. I don't think this approach has a name, but it gives the motivation to learn math for a reason, so I call it "math for a reason".

Comment by Rashmi Kathuria on August 7, 2009 at 11:54pm

I liked this approch...Math for a reason. When I start my new Math lesson then my first priority is to explain the need of the concept to be learnt. It is said necessity is the mother of invention. When students realise the importance of concept they learn faster . Also it helps in creating interest in the topic. We say we are learning it because we need to learn it.
This is one approach.
I believe there is no fixed method of teaching learning Mathematics. One needs to be flexible in selecting a suitable strategy according to the need. The need is based on two factors viz. type of learner and concept to be taught.

Comment by Danny Clarke on August 8, 2009 at 12:31am

Here's an example of the "math for a reason" approach to teaching. The example is based on the problem posted earlier about finding the largest semicircle that can be inscribed in the unit square.

"I have been asked to help paint a mural on the outside wall of a grocery store in my neighborhood. My task is to create the background for the mural. The instructions are to create the largest possible semicircle on the wall, with the semicircle touching all 4 sides. The wall is square with 10 feet on each side. I need to find out how to position the semicircle to satisfy the instructions. I also need to know the radius and center of the semicircle. How can I figure this out with the basic math that I know?"

With this as a lead-in, we can discuss topics like finding the area of a semicircle and how to find the radius and center of the circle using geometry and the Pythagorean Theorem. Visual aids might include a representation of the grocery store wall with cut-outs of various semicircles and how they can be positioned within the square.

The goals of the lesson are:
1. Understanding and following instructions precisely.
2. Learning some basic math.
3. Leaving with a feeling of accomplishment.

Comment by Danny Clarke on August 8, 2009 at 12:41am

Oh, yes -- in the previous example, notice that I never used terms like "inscribed" or "unit square" (although I could not figure out how to avoid terms like semicircle or radius.)

I believe there 3 major components to teaching math to a developmental class:
1. The subject matter
2. The modes of learning for individual students
3. The psychology of learning math

The subject matter is well known; the modes of learning include both visual and symbolic as well as verbal, reading and interaction with a group.

The psychology involves developing a feeling of confidence by the student in being able to solve problems using math. This is probably the most important, because without this feeling, problems can seem overwhelming.

My belief is that those who "get" math early on form an elite club that tends to shut others out with invented language and a superior attitude. A poem that illustrates this by John Markham:

He drew a circle that left me out
Heretic, stranger, a thing to flout
But love and I, with a will to win
Drew a circle that took him in

Comment by MariaD on August 8, 2009 at 3:19am

Danny, I love this approach, and students do too. When my daughter was nine, having no formal prior math instruction (we unschool), we started using SATs books with problems. Just grab a problem and figure out what it takes to solve it, investigating, experimenting and discovering along the way, of course. It's a lot of fun.

Comment by Michael Friedberg on August 8, 2009 at 3:58am

One can get an impression that there is nothing to discuss - everything looks successful and friendly to students. Are there any undecided problems?

Comment by MariaD on August 8, 2009 at 11:21am

Michael: this approach requires high flexibility and high math knowledge from teachers, and a lot of trust from students, because sometimes it takes hours to solve a problem, even days.

In other words, it's a huge attitude and skills shift for the majority of people. It's the right kind of shift, but in practice, it's rather revolutionary, with all the problems that brings.

Comment by Michael Friedberg on August 8, 2009 at 12:13pm

Maria, thanks! These are the words!
If so, the main burden is on the teacher`s shoulders, and the student is guided. I was in similar situations and students asked me: we understand that You guide us right, but how do You know the successful additional construction? I answered to them something like that "It is useful to draw radii to the tangent points in order to get right - angle triangles". And finally said them exactly by Your words, that the way is to try and check, and may be long.
The question is - may we build the public school on such a business?
I think - not/ Not by 100% - at least
Respect

Comment by MariaD on August 8, 2009 at 6:32pm

Public education will have to change. May we build a typical industrial age, assembly-line public school using Danny's "Math for a Reason"? Probably not, but then, do we WANT to build any more such schools?! Speaking for myself, I'd rather apply my efforts elsewhere.

This week, we had a Math 2.0 interest group webinar/discussion on LearnCentral, with John Rosasco hosting, about accelerating gifted math students. The conversation turned toward providing depth and connectivity, and individual approaches, problem solving, powerful mentors... People also said social web tools and technologies can help in many ways. The ways they described, again, assumed structures different from industrial-age schooling. We have not figured out then and there what web tools would support the Socratic method (try and check and be questioned by a master) - but people agreed it to be extremely powerful.

Comment by J Edward Ladenburger on August 8, 2009 at 11:26pm

interesting discussion going. MariaD -- I agree that it is unlikely that any rigid, "written in stone" approach to education is unlikely to succeed, even when it contains seeds of brilliance. Education is an art of reaching and probing and facilitating individuals, each of whom have varying backgrounds and natural strengths which they employ toward learning and growing.

Giving "a reason to learn" is certainly important and motivating, and yet I have heard this pedagogical idea abused by math teachers who will proudly proclaim that they do not teach anything in their classrooms with does not have a direct and demonstrable application to "real world problems". I have also seen the Socratic method attempted in large classrooms such that very little progress was visible at the end of the year. The use of different structures and social networking and technologies is also promising and may yet prove to be a vehicle which connects students to mentors in beautiful and productive manner.

I do not have time to comment further right now... sorry (called to other tasks) ... but leave with a question I would like to see added to this discussion: What do you (all) see as ultimate goals for this developmental math crowd? Removing the negative past experience and math "baggage" ? Developing some sense of math fluency? or basic math life skills? ...

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