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## Algorithms or Not

Every time my students encounter a division problem involving fractions, I have to remind them how to do them by hand and beg them not to pick up their calculators.  After a brief lesson, I hear “Oh yeah, you mean Keep Dot Flop.”  While I’m glad my students have a clever phrase to remember math, I worry that they memorize a process and don’t really understand the math involved.  This is evident when they grab a calculator to find 1 – 7/8.  I also have this concern with FOIL and Cross Multiplication.

I think I have come to the conclusion that as teachers we often focus on algorithms and not understanding.  I am one of the worse offenders with this.  I realized this the other day that I told students I had 4 easy steps to solve logarithm equations.  I taught everything logarithm in 3 days and I’m not proud of it.  As I reflect on why I did this, I think I’ve come up with a few reasons.

1.  I have a state test to get ready for and my deadline is approaching.  I have a wonderful Principal who encourages me to teach the students and not worry about the End of Course.  He constantly says that if I teach the students then the test will take care of itself.  I have great support, but I still feel the pressure.  I have a deadline and I’m not sure if I’m going to make it so I cram in material and teach using algorithms instead of allowing students to discover math on their own.

2.  I also believe that teachers teach using algorithms because they do not understand higher level math or the applications.  While I have studied higher level math, I am struggling with my Pre Calculus class.  I taught vectors the other day and my students can find the dot product and do all the great operations that go with vectors.  However, when it came to vector applications I froze.  I took physics 20 years ago.  This week I decided that I need to take a physics class at the community college this summer.  Not every teacher has a desire to do this though and not every teacher is trained in their subject area.

As we look at improving math scores in our district and across the nation, I wonder if we aren’t focusing on the wrong areas.  I really believe that we need to start early with teaching number sense and not just algorithms.  I don’t want to point fingers and I definitely put myself in this category.  It needs to start early in elementary school with multiplication, long division, fractions and continue through logarithms and vectors.  Now, how to convince all teachers to encourage discovery of math with true learning and not just have students follow a set of rules?

Categories: Uncategorized
1. November 15, 2010 at 2:12 pm

Reading this post had me nodding like a bobblehead. We hear these concerns from our teachers all the time. I’d also add a third– teachers believe they should be doing “more” than teaching algorithmically but aren’t sure what that “more” is or looks like, especially since they weren’t taught that way themselves.

Personally, I’d learned math very algorithmically (and I was a super good memorizer) and it all fell apart when I got to multivariable calculus and I realized I needed some type of understanding to get through. Failed my first midterm, spent a million hours reteaching myself algebra and precalc until I really understood the whys behind the algorithms, and promised myself never to let my students suffer in the same way.

Given the pressures you’ve described here, my current hypothesis is that teachers need to see the alternative to teaching algorithmically and to see that the alternative isn’t unrealistically difficult to plan for and to execute. If anyone knows how to do that, let me know 🙂

2. November 16, 2010 at 2:41 am

I, too, would like to get away from more rote/algorithm learning, but like grace, I’m not sure what this sort of class would look like. With our curriculum being a mile wide and an inch deep, I am required to teach so many concepts that I just don’t see how I can devote the time more open-ended learning would take.

I really dan meyer’s WCYDWT ideas, but like you, I just don’t feel confident in my abilities to work with true application data. Unfortunately, my physics skills are as rusty as my integration skills. 😦

3. November 16, 2010 at 3:38 am

This is something I’ve been struggling with a lot personally. A freakish percentage of my student body opts into taking calculus (nearly 40% take it by the end of their senior year). While this sounds awesome, it has a very dark side. Many of these students have gotten to calculus by memorizing algorithms, repeating, and then passing quick assessments. They’re REALLY good at that, I mean scary good.

Now that I’m using SBG, I’m really beginning to amplify this procedural shallowness. When I ask for a moderately advanced recollection of a skill studied more than a month ago, I often have to move a majority my my students’ grades back to below proficient. This is beyond frustrating. It has gotten a lot better under the SBG system, and they’ve started to take more stock in the motivations for things, but we’re fighting years of memorize, regurgitate, repeat.

This also shows in the almost total lack of algebraic proficiency during calculus. Things like common denominators and rationalizing roots are key to calculus, and students learn all these techniques in their previous classes, but they can rarely identify when to use them later. Perhaps more time should be spent on predicting the algebraic future of expression than just following patterns like “completing the square.”

So, what do we do? I’ve found that teaching towards motivation and higher level thinking often leaves students floundering or in such a strange place mentally that they don’t trust me that what we’re doing matters. All of my WCYDWT lessons have been met with excitement and then immediate deflation when they realize that thinking will need to be done after the initial question generation. They have no math courage. As teachers, we have to demand that students actual understand the When and Why, and we have to stop being so obsessed with the How. They do need to know how to do things, but none of that matters, if we don’t take time to generate context or motivation.

Thanks for the post, Amber.

4. November 16, 2010 at 8:31 am

Rather than physics, I would recommend taking an electronic circuits class, one that uses MATLAB to solve the matrix equations for nodal analysis.

• November 16, 2010 at 12:13 pm

Thanks for the recommendation. I will definitly look into it.

5. November 18, 2010 at 8:13 pm

Everybody agrees that we should probably do things differently, but NOBODY has trained me HOW ! As for memorizing and rules, I personally don’t think they are all bad, if you explain the reasoning behind the rules. I like rules, patterns, logic, etc.! Isn’t that why we are math teachers? I do agree that students should spend more time discovering math instead of being “spoon fed” math but there is just not enough time. Whenever I do the “hands on” projects, it takes up so much time that I get behind. I really want to teach my students in the BEST way possible, but I am really not sure that there is a best way, because every student learns differently! So if any of you find the magic answer to this dilema……..please let me know!

• November 18, 2010 at 10:24 pm

I agree with you. I love the projects, but with State Testing and being on a block schedule, I just don’t have time. I do think some rules need to be memorized. I want my students to memorize math facts and so on, but I hope it is built on understanding and not just rote memorization. I also want to send up an amen that I need training on how to teach concepts and understanding not just rules. I love the rules and I’m sure that is why I’m a math teacher. I want to be able to reach my students who don’t want to or can’t just follow rules. Maybe we can get training on that. I think that would be a great use or TN Race to the Top Funds. 🙂 Thanks for reading my blog and your great comments.