Musings on Math and the Art of Problem Solving

For math, as for most topics, there are those students who love it and can't get enough, and there are those who find math as mysterious as a foreign language written in an entirely new alphabet, who are constantly frustrated and annoyed at having to learn something they find so perplexing.

In an ideal world, we could present students with math concepts and ideas from an early age, in a fun and engaging way that gives them the gift of numerical competency and a more comprehensive understanding not just of the how-to's of math, but of the more conceptual whys of math.

Like artists who learn to see colors and shapes in artistic ways, children and young adults can be taught to notice math in everyday activities--apportioning a pizza or pie, building a birdhouse, mixing up a recipe, balancing a checkbook, understanding compound interest on a savings account--exposing it not as a foreign language, but as a vital subject that permeates our daily activities in ways too numerous to, well, count.

(Note: a good resource for the youngest students is "Kitchen Table Math," a book by Chris Wright, Ph.D., and available through the Art of Problem Solving web site.)

It is the conceptual understanding of the whys of math that helps students to see problems in a larger context and, in later academic work, to be able to solve multi-step problems using many strategies that work in concert. Rote memorization of formulae or "to-do" steps may help a student get some answers correct on a specific quiz, but do less for their larger problem-solving abilities. It is also this ability to strategize and logically solve math problems that can spill over into solving problems that crop up in engineering, chemistry, biology, computer science, and other topics that American students seem to find too difficult to bother with.

Gifted math teachers understand that it's the relevance to a student's everyday life experiences that can trigger the sometimes elusive "ah-ha!" moment, which indicates they finally understand how to solve a problem.

In an interview I did on my radio show last week with Nigel Nisbet, a math expert with the Los Angeles Unified School District who is currently working on a math project for HippoCampus, there is a discussion of how students are not necessarily mastering the basic math skills (addition, subtraction, multiplication, division, etc.) taught in the early grades. This lack of mastery tends to first be revealed when students start struggling with algebra. Nigel argues that those standardized tests given in the elementary grades really can help parents, teachers, and schools see how well students are mastering those basic topics. Catching this early leaves enough time left to remediate if a deficit is found. (Listen to the interview here.)

My take from the interview is that if a student manages to bump along with a half-baked understanding of math until they hit algebra, it's going to be much more difficult to go back and fill in the missing information, and by then it's much more likely that the student will be developing an aversion to math that could haunt them throughout high school (and possibly into college).

For the math-loving student who either isn't challenged at school, or is challenged and is looking for like minds with whom to collaborate to learn new strategies, I have a recommendation. Richard Rusczyk, whose own gifts with math are surpassed only by his gift for teaching math to others, has a whole online following that includes many of the brightest young math minds in the country. Richard and a core of other dedicated yet unconventional teachers have taken matters into their own hands, creating an extracurricular community of math-lovers who relish digging into a juicy problem and finding creative and clever ways to break it down and solve it. His company and online community are aptly named, "The Art of Problem Solving." I've done several interviews with Richard on the Countdown to College Radio Show, three of which can be found here.

Note: Guest blogger Beth Pickett is the producer and host of Countdown to College Radio Show, and as such as interviewed numerous teachers and math experts on the current state of math in the U.S. today, and the opinions she has expressed here are those she has formed after listening to those who are on the front lines of teaching math. She is also the primary blogger for the HippoCampus Countdown to College blog.

Elementary Math Education

So far in this blog I have really focused on high school education. I want to talk this week about elementary school math education, especially on how the Internet can help younger students grasp early mathematical skills.

I think the Internet is a great place for kids to practice their skills through the use of games. I searched to find some interesting math sites for younger students. It is important that students are monitored and that the sites are reliable, since we are dealing with a younger audience.

First, I found the website King's List of Math Activities. This is a great list of websites that deal with math activities, many for the younger audience. For example, there is an addition game with frogs from the BBC website that would be very fun for students to try.

APlusMath is another great website. It has a link to a game room that allows students to choose from a list of math games. There are even flashcards and math word puzzle games available.

CoolMath is a site that I have talked about before. This site has many fun math games from younger students up through Algebra.

The Classrooms website also has a great list of online activities for students, especially the younger elementary level math students.

I think it's important that we make math education as fun as possible for students from the very beginning. We all know that math can be a difficult and scary subject, so why not start the process on a good note. Students learn when they are playing and having fun. The Internet can be a great place to start that process.

Online vs. Brick and Mortar Schooling

Lately I have had a lot of questions about why students and parents would choose online over traditional brick and mortar schooling. I am no expert, but I have worked in both arenas and have my own opinions about the growing trend of online schooling. It is no secret that online schools are popping up in every state in great numbers. I project manage math curriculum for a company that writes online courses. I've also taught in both a traditional high school and a brick and mortar high school.

So why would a student choose an online program? I think its important to understand that there are different types of online schools. Some students choose a full-time program where they take all their classes online. Some students are involved in supplementary programs where they might take one or two classes that are not offered in their traditional high school. This is often the case in smaller, rural schools that can't staff teachers for every subject. There are also hybrid schools that are being developed where students take a portion of their courses online and a portion of their courses in a classroom. I believe it is this hybrid classroom that will become the most popular. This combination of convenience mixed in with the social aspect will be very appealing to students and parents.

From my experience in the online schools in Colorado, we have a reverse bell curve. We have a lot of very high achieving students who like working on their own, like to be able to set their own schedule, and like working at their own pace. They are able to take classes that may be at a higher grade level then what they are used to and they feel comfortable working in a home environment. We also have the low achieving students, who may need more time to grasp subjects, and feel that working at home is the best alternative. We don't have as many of the students that fall in the middle of the bell curve. It is very important to note that this is the not the case in all states. This just seems to be the demographic in the state of Colorado.

Parents have told me in the past that they chose online education because if fit the personality of their children. That is an important aspect in this whole process. Online education is not for everyone, but the students that thrive, really enjoy the opportunity to work from home, in a safe and encouraging environment.

I hope that answers some of the questions. Please let me know your thoughts.

HippoCampus

Throughout the time I have been writing this blog, I have referred back to the HippoCampus website. I want to do this week's blog on what is available on this website. I feel like this website has a lot to offer students, especially in the discipline of mathematics.

Entire classes are available on the website. A very easy to follow menu is presented for each class that divides up the course based on the concepts. A student clicks on a link and is taken to an interactive lesson.

For example the following is the lesson on number properties in Algebra 1.
Another example is a lesson on limits from the Calculus class. The Calculus class is even subdivided into AP and non AP, AB and BC, General and Introductory. Each level is catered to the level of the learner.

I have used these classes in my own course writing, directing students to these pages for more information, more practice, and remedial work. Students have been happy with the lessons and feel like this is a great addition to their curriculum. Our overall goal should always be to give our students as much information as we possibly can, directing them to helpful websites, good examples, and the discovery of learning.

The HippoCampus website has Algebra, Algebra (Spanish), Calculus, Calculus (Spanish), and Statistics.

Rubrics in Mathematics

As I write more for the online curriculum world, I find it is extremely important to include rubrics with any assignments that are created. If you think about the environment that online students work in, they are home, assignments are done on their own, and it may take teachers a while to return phone calls and emails, students really need to know how they are being graded. In a brick and mortar environment, I hardly ever used rubrics. Students did homework from the book, they completed short answer assignments, and occasionally they would complete a project that required a rubric. The difference was that I was always there to answer questions and clarify any grading issues. That is not the case in the online environment.

I don't know about you, but I have a very hard time writing good rubrics. I either don't explain enough or I go into so much detail that my students get lost in all the wording. I found some helpful websites that can be used to write rubrics. I wanted to share these with you so that if you are rubric challenged like myself, you know there is a little help out there.

Rubrician.com gives you a list of websites that offer help and templates for rubrics. I found that this site is a great place to start if you are writing a rubric.

Rubrics4teachers.com has a great list of sample rubrics for all math levels and math classes. This is a great place to get ideas.

Mathforum. com also has a teacher to teacher post that has a list of sample rubrics. This is another great place to get some ideas.

Finally, I found this great website from missioncollege.org that gives you steps to creating a rubric, examples of different kinds of rubrics, and shows examples for future use.

Do you have any good websites that can help us with writing rubrics?