Students taking over!

Recently I was introduced to the idea of a flipped classroom. The idea is an expansion of what I have been doing by limiting my lecturing and increasing the amount of time students spend working problems in class. The flipped classroom involves taking the teacher's instruction out of the classroom and placing it online outside of the classroom. Then classroom time is spent with students asking questions, working on problems, and would allow for expanded exploration. [Here is a nice presentation about what a flipped classroom is.]

So, how could this be used in the developmental math classes?

The first thing is to get "buy in" from the students. Some students may be resistant because they are used to the teacher telling them what to do (or what has been called the '"sage on the stage" model of education). Getting students who have signed up for an on campus class may be resistant to having to spend time online. Since I am already assigning online and paper homework I am used to this resistance. One thing that I have started doing is helping students individually getting signed in, and when possible having the class in the computer lab room.

Making sure that students are accessing the material could be difficult. There are always students who will not participate no matter how you 'incentivize' it. The main thing, as I see it, is starting the class with a warm up that comes straight from the online preparation material. Also, it is important to make the material entertaining and informative. Just throwing up worksheets is not going to work. The instructors that I have talked to that use this techniques record lectures, and I will need work on how to do it with online whiteboard software. Also, breaking up the lecture into smaller parts is recommended so that students can go back over the parts that they need to without having to go through the whole lecture.

The really exciting part of this, for me, is being able to expand what happens in the classroom. This has been where I have always struggled. There are so many interesting things that I wanted to do but had never figured out how to fit in. By answering student's questions at the beginning of class, there will be more time for collaborative projects during class. I have shown examples of arithmetic of worksheets in the prior two blogs. Another thing that would be great is having students develop FOIL and from that Special Products in introductory Algebra classes. There are so many resources out there, and being able to take those and expanding them for adult students will present new challenges.

I would love feedback from those who have already started using this methodology, and to discuss any concerns that people have.

Calculators in Developmental Math??!!??

I continue to ponder the issue that I raised in my last post, how do we deal with calculators and other technology in the classroom? My first response is still the one drummed into me from my earliest days teaching developmental class; maybe in later classes but not now. But, again, does this really make sense?

In talking with other instructors it was pointed out that these days we are all carrying around powerful computers in our pockets. By not allowing students to use them we are denying the present reality, and missing an opportunity to expand what we are teaching in the developmental classes.

In the previous post I discussed helping our students develop their mathematical thinking using arithmetic. In the example that I gave in my last post:

*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*

           9 + 1 = _____

           9 + 3 = _____

           __ + __ = ____

           What pattern do you see?

           Can you come up with another equation that fits this pattern?

           Can you justify, that is show, that your equation is the correct answer?

*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*

What is lost by allowing students to use their calculators for the first few problems before asking them to extrapolate and justify their answer? The only change that I would make if using calculators would be to instruct students to do their justification without using the calculator.

Moving past Basic Math Skills, there are usually two or three other developmental classes, Pre-Algebra and a pre-collegiate Algebra course (or courses). Here the ban on calculators makes even less sense. The arithmetic in these courses is incidental to learning the material in these courses.

One argument that I have heard, and often used myself, is that students need to learn how to manipulate rational numbers (fractions) by hand, so that when the get to rational expressions (algebra fractions) they know what to do. The same method of pattern recognition I showed above could be used, while still allowing students access to their calculators.

*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*

Find the Lowest Common Multiple (LCM) of the following groups of fractions. Remember that the LCM is the smallest number that the original numbers divide (go into).
      
          3, 2         LCM =                                    

          3, 9         LCM =                                    

          6, 9         LCM =                        

          12, 18     LCM =                                    
          Explain how you found each of the LCM’s above.
          Do you see any patterns or procedures to do other LCM problems?

Use the above arithmetic problems to guide you in finding the LCM’s for the following groups of algebraic expressions.

          x, x + 1         LCM =                        

          x, 3x         LCM =                        

          (x + 1)(x - 2), (x + 1)(x - 1)         LCM =                        
           x(x - 3), x(x +2)         LCM =                        
       What pattern(s) do you see?
       What is the correlation between the arithmetic problems and the algebra problems  
       Come up three problems using rational expressions explaining how you are using the pattern(s)
       that you found.

*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*_*

Again, allowing the students to use calculators for the arithmetic will detract nothing from the learning experience. In fact it may actually help student comprehension because rather than concentrating on how much they hate fractions, they can concentrate on what we really want them to pay attention to, learning how expand what they already know.

I am sure that, up to now my resistance to using calculators is the age old problem of, “I didn’t learn it that way and I made it through.” This is true, but I also never had to use an abacus or a slide rule like previous generations did. Would making me use those tools have made me a better mathematician? No, the tools of the past have been replaced by the tools we have now, and we need to learn how to adapt.

Just recently I had with Dr. Michael Gigliardo of California Lutheran University. He has started using a 3D printer to be able to have models of the shapes that you are trying to find the volume of in Calculus III. These can be very difficult to visualize when you try to graph them on paper. I recounted to him how my ex-husband would try to carve the shapes out of balsam wood, and how his method was clearly superior. He told me that they were able to take a shape and determine the equations that would generate it.

This ability to expand what is being done while covering the requisite material is what we are depriving our developmental students of. By using the increasing power of technology we can deepen our student’s understanding.