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Doing Math vs. Understanding the Math

I found this article, "Doing Math vs. Understanding the Math," and after reading it I began to wonder - What am I doing to help my learners understand the math I am teaching them?  So, I thought I would ask the community - How can we help learners understand the math and not just do the math?

Thoughts, ideas, comments - all are welcome, I am needing a fresh perspective.

Thank you all,

Brooke
istasb@cowley.edu

Comments

ms.jenniferway's picture
First

This is a topic I struggle with. Without that understanding, I'll be teaching the same students the same concepts if they leave for a few weeks or months, and I do not like to duplicate effort. I had one student that had to leave for about a month. When he returned, he had lost all of the progress we made, because he forgot the procedures and never learned the conceptual understanding. 

BrookeIstas's picture
One hundred

Jennifer,

I can relate! It seems like all I teach is fractions, decimals, and percents....over and over and over again.  I did find some curriculum that helps develop the conceptual understanding online.  Here is the link to it: http://tian.terc.edu/TIAN_teacher_resources.html

I hope that this might be of interest to you.

Brooke

misschristine's picture
Ten

I have used materials from them before and I like them a lot.

ms.jenniferway's picture
First

Thank you for the link, I'll check them out!

Carol Bryant's picture
First

I would also like to thank you for the link.  I glanced at the TIAN Bundle 1 and noticed there were articles from Marilyn Burns and Donna Curry.  I really value how they both think math should be taught and their reference materials.

Rachel Baron's picture
One hundred

Teaching concepts is messier than teaching procedures, but I agree that it's really important--especially for retention. It's also critical for the kind of word problems that give you more (or less) than two numbers and a key word (There are 3 gizmoes on the shelf and 7 gizmoes on the floor, how many gizmoes are there altogether?).

The thing that I like here, though, is the dual focus. It's not only important to understand the idea, it's also important that once you understand it, you learn how to use the idea efficiently. If you understand the idea, you can make a good stab at figuring out a similar problem later, but you don't want to have to reinvent the wheel every single time.

The problem then is finding the balance. How much time do I devote to understanding, and how much to practice? I suspect that a few practice problems thoughtfully done are worth much more than a lot of problems done mindlessly. Research on memory indicates that new kinds of problems should be mixed in with familiar ones, and spread out over several days (or weeks), which makes sense if you're trying to build retention, fluency, and understanding. Unfortunately, I suspect that to do this properly, you have to pretty much make your own materials or infringe on copyright laws...

misschristine's picture
Ten

I tend to lean heavily on free online resources to fill the gap and when I need to I create the work I need but this is very time consuming, but eventually I'll be at a point where I have the materials I need for what I'm doing at the time.

Alfons Prince's picture
Ten

I am glad that this has come up because it is something that I struggle with daily. I lean towards building confidence in doing the problem then I infuse lessons that promote understanding. It is often tedious because students want to have mastery of a topic after only one lesson and without much practice. I do give a lot of practice problems to strengthen the foundation but I vary the type of problems and make sure to include word problems.

I also agree that without understanding, students are more prone to forgetting what they have learned but with all of the outside factors that my students deal with, its just something I have to work around.

Brooke, thank you for the link, I will be using this site.

 

- Alfons Prince

BrookeIstas's picture
One hundred

Welcome, Alfons!  I know how instructors have to help out each other!  So if others in the Community have more resources please post them.

Brian Kravitz's picture
First

Back when I was a middle school classroom teacher, I worked almost exclusively with students with disabilities. I found that teaching fractions was incredibly hard, and did quite a bit of research on the subject. In doing so, I found the "big picture" model, which I have very successfully used.

The basis of the model, the big picture, is that fractions, division, decimals and percents are all the same (which helped me understand why students tend to struggle with all 4). For instance:

Fraction: 1/2

Division: 1 divided by 2

Decimal: .5

Percent: .5 = 50%

I have found that, without exception, when students understand this "big picture," it opens the way for understanding.

I also use pattern blocks to teach for understanding. The best resource I have found is here: http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years7_10/teaching/frac.htm

S Jones's picture
One hundred

I've been thinking that way, too -- that it would behoove us to spend  more time with halves and quarters and connecting all those pieces instead of diving into figuring out 7/11 of something... 

ms.jenniferway's picture
First

I found that allowing students to connect and estimate to benchmark fractions, or as I call them in my class, the easy ones: 0, 1/2, and 1. Once students can thoroughly understand this concept, I add in 1/4, 3/4, and any version of 3/3, 6/6, 12/12 and other whole representations. When students can estimate which is bigger or smaller than those core fractions, it is easy enough see where 1/3, 2/3, and others go. 

Rachel Baron's picture
One hundred

I just got a chance to check out the link to the pattern block exercises, and they seem very helpful. Thanks!