I found this rich presentation on the subject here: Natural Math Blog. I’m really enjoying reading this blog right now. I hope to delve more into the content later and maybe even talk about it here in this blog.
I’ve been reviewing the concept of functions for some work I’m doing now. Much of this post is just me regurgitating my thought process for writing about math.
In my own rough-shot words… a function is:
An equation sometimes written as f(x)=x. “x” is a variable that can have multiple values. f(x) which can also be considered in some cases “y” or the “y” value, is the result of completing the side of the equation with “x.” If you know the value of either y or x you can solve for the unknown (variable).
Here are some other defintions of functions
A function is an equation (this is where most definitions use one of the words given above) if any x that can be plugged into the equation will yield exactly one y out of the equation. (from Paul’s Online Notes)
Here’s wikipedia’s definition:
The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. (from Wikipedia)
I think it’s probably a good idea before you’re teaching any math concept, process or definition to look up how others have defined them or done it. When I took my first Algebra class, I think not understanding the common vocabulary for terms such as variable, value, operation, etc. actually threw me for a loop. Up to this time my math education consisted only of doing ‘problems’ in a book. Nothing was explained about how you did them. You just did. Sad isn’t it, but I suspect that even today many future teachers still aren’t prepared properly for even high-school level mathematics courses.
I feel sometimes that I’m learning about these math concepts very late in the game as an adult. Shame is a sad and unnecessary reaction many of us adults have when we encounter things we don’t know (and feel that we are expected to know). As I get older I’m beginning to think this is one of the more detrimental reactions we have to learning as adults, and It’s really time to chuck it out the window along with that pesky emotion called regret.
I created a very quick simple animation that briefly demonstrates my own exploration of the concept of functions. You can click the image below to view an animated slide show. The Flash/Captivate presentation will open up in a new window.
My favorite advocate for mathematics talks on a Science Friday Podcast.
Take a listen here: http://www.sciencefriday.com/program/archives/200905296
She really advocates point out ‘things about math’ that kids might see in the world around them. I agree a hundred percent demonstrating the power of math or even how to do simple things can help children develop that very important “Number Sense.”
She tells people not to avoid numbers :) and gives an example of a friend who was so phobic of taking math that she eventually gave up her dream to become a doctor. What Danica is doing is so important not just for girls but for everyone including adults who are currently suffering from Mathophobia.
Also, I’m very very excited about her upcoming book on Algebra concepts. Danica has actually inspired me to considering tutoring elementary and middle school students in math. I have actually considered writing a math book for knitters and crocheters because I have run into too many crafters, most of whom are women, who constantly berate their mathematical abilities. It’s terribly frustrating for me. While I can empathize because I experienced some difficulties as a child and teenager, I think that the negative attitudes that many women teachers have had about math negatively impacts how their female students learn and perceive math. Even if they are not teachers their behavior impacts the girls and young women around them. The truth is many of these women are actually doing the math successfully…(((they just don’t realize it))).
Someone on Ravelry.com had this avatar, I had to share it here because it was so funny.
This got me thinking. It might actually be a fun assignment for students to look up cartoons or joke on Math-related humor. A few students share these in the classroom once a week for extra credit, and they have to explain the humor to folks in the class. Or the teacher can find the cartoons, share it with the students as part of a Journal entry prompt. Students have to explain in their own words why the cartoon is funny and the background knowledge behind the joke.
I found a few that made me chuckle:
It’s been years since I had to think about classifying numbers, and to be honest, I really don’t think that I understood what the teachers meant about rational numbers, real numbers… I knew that an integer was a number that could be expressed in either a positive or negative form, but that’s about it. I don’t think I was alone in my ignorance. If you can get maturing generation x’er like myself to understand these concepts after years of misconception; how come it’s so difficult to teach this to middle schoolers and high school kids. Okay, there are numerous reasons, but still… if I can get this, most people can with the right help.
Here’s an image I made to show the relationships of the following types of numbers (are they called types?):
- Real Numbers
- Algebraic Numbers
- Rational Numbers
- Whole Numbers
Rational numbers are algebraic real numbers. Rational numbers include integers. Integers include both whole numbers and their negatives. All integers can be presented or expressed in their rational form. For example:
- 3 can equal 3/1 or 30/10 or 9/3
- 2.5 can equal 3/2,
- -5 can equal -5/1, etc
It makes sense that kids, when then get into Middle School, they have difficulties understanding the other types of numbers outside of “whole numbers.” They spent all of elementary school only focusing on how whole numbers function and are used. Honestly, I think it’s because that’s all that most elementary school teachers have been trained to teach (and feel comfortable with).
I never really understood or appreciated tangrams. I always saw them as just shapes for building images. As a child, I never explored using them to solve visual puzzles.
I’ve been fiddling around with an electronic tangram template. You can see my first attempt to build and label a shape here.
The video feels a little cheesy, but it is an advertisement for a learning program and pre-packaged software and manipulatives. After seeing really poor quality software being shoved down district throats, I sometimes feel this mild revulsion to some education software companies.
Still, it makes some good points about engaging students in math using media that they are familiar with. Also I like the example of using the “Smartboard” where students circled the common factors.
The scene about the “fractions” actually sent me into a flashback mode. The Reggae fraction song was pretty funny too.
Click on the link to view the video in another window:
Factors and factoring are extremely important concepts to understanding math. I found a pretty cool game/applet on the Shodor.org site that allows students to use virtual manipulatives to explore factors.
The applet/game is called Factorize. Check it out.
You’re given a number and asked to illustrate the factors by selecting the appropriate dimensions on a grid. After you select a dimension (area of squares), you press the “Enter” button.
I have to admit that when you’re working with larger numbers the grid squares become a little difficult to highlight with the mouse, but I think this is a great visual activity for students. It also helps demonstrate the Commutative property, you simply have to de-select, the “Do Not Show Commutative Property” button.
I first heard about Danica McKellar’s book in an NPR Podcast interview with her. I was very impressed and moved by Ms. McKellar’s dedication to promoting math education for young women. Too bad I didn’t have such a role model when I was a child.
I like this book.
There are a few flowery and girly things in the book that I have a little difficulty with, but I believe that it really does capture the main math concepts needed by students before they dive into Algebra: factors, primes, rational numbers/fractions, decimal representation of fractions, word problems, and solving for variables. More, importantly she tackles problems and examples of these concepts from subjects that girls can relate to. Some how, she successfully ties in beading, pizza, shoe shopping, espresso, the schoolgirl’s crush, etc. into the examples and problem sets in the book. More importantly, the book doesn’t talk down to girls.
Also, throughout the book Danica features the stories of several successful women who used math in their careers. I think these sections of the book are key for young women. This may date me, but I was told by a teacher that I didn’t need to know math because “I was a girl.” Infuriating as this was, I had to realize that this woman was still stuck in the Dark Ages. Ms. McKellar aptly states in her book, “… we have to be careful that we don’t fall victim to other people’s low expectations.” Kudos to Ms. McKellar for developing a wonderful book that reaches out to young women who are just starting to engage in the wonderfully empowering subject of Math.
P.S. I think the examples throughout this book can be used by Middle School teachers in addition to any examples they come up with in class.
You can read more about Danica McKellar and her book at her website: http://www.danicamckellar.com/
I’ve been using wikis for different projects lately. If you’re not familiar with how wikis are used you can view the quick video below from common craft that explains how wikis work.
A few months ago, I ran across an excellent wiki developed by a chemistry teacher: Chemistry with Mr. Olson
Chemistry with Mr. Olson has a periodic table on it’s home page. If you click on the elements in the table the links lead you do a profile page on that element. I image that each element was assigned to a student or group of students who then shared properties and descriptions of that element.
Click on the image to view the page on NA/Sodium on the Chemistry with Mr. Olson wiki
I can imagine a similar wiki being developed by a math class as they examine the different types of numbers. Or perhaps they could develop a wiki where students illustrate and explain how to complete some operations such as:
- Order of Operations
- Adding, subtracting, multiplying and dividing rational numbers
- Demonstrating operations on positive and negative numbers
For students and teachers who feel comfortable, they can actually film themselves explaining the concepts using manipulative, diagrams and whiteboards. They can post their filmed explanations right on the wiki. I’ve been reading a great deal about working with students today (also known as Milennials and Net Generation students). It seems that there is a growing argument that working effectively with these students requires a different approach than just lecturing and using workbooks. Engaging students in learning can take place when you have them create content or projects. A wiki would make a nice team project for a class.