For many bright students, making a mistake on a homework assignment or test is one of the worst feelings in the world. But could mistakes play a vital role in the learning process? According to a growing number of educators, the answer is yes.

In his book Teach Like a Champion, Doug Lemov says, "Error followed by correction and instruction is the fundamental process of schooling...teachers should normalize error and respond to both parts of this sequence as if they were totally and completely normal. After all, they are." Lemov isn't the only one who sees failure as an integral part of education. In their excerpt of his book, website Delancey Place notes that his philosophy reminds them of the motto of the engineering department at "the Franklin Institute's nationally recognized Science Leadership Academy": Fail early, fail often.

Even the New York Times is reconsidering the value of failure. In an article on the effectiveness of homework, author Annie Murphy Paul writes, "When we work hard to understand information, we recall it better; the extra effort signals the brain that this knowledge is worth keeping. This phenomenon, known as cognitive disfluency, promotes learning so effectively that psychologists have devised all manner of 'desirable difficulties' to introduce into the learning process: for example, sprinkling a passage with punctuation mistakes..." Making and then correcting errors may be akin to those "desirable difficulties," the struggle ultimately heightening students' ability to grasp a given concept.

We at Thinkwell are no strangers to this approach to learning. In fact, Professor Edward Burger, the star of Thinkwell's math lectures, considers it one of the most valuable parts of the educational process. "In all my courses," he says, "I emphasize the power of failure: learning from failed attempts and taking risks." In The Heart of Mathematics, the textbook Prof. Burger coauthored, the second item in his list of the top ten mathematical ways of thinking is "Make mistakes and fail but never give up."

How have you and your students approached errors and failure? Do you view them as obstacles to success or as stepping-stones on your way to mastering a subject? Have you found any techniques particularly helpful in learning to view mistakes as a good thing?

Halloween is just around the corner, and while most of us will have fun with candy and costumes, there are also plenty of ways to make the holiday a learning opportunity for your homeschool students.

For example, once you get back from trick-or-treating with a bucket full of loot, you can use the candy for math lessons. For younger kids, Surf Net Parents has some sorting and estimating activities, while older students can learn about calorie content and graphing with these exercises from Education.com.

If it's science lessons you're looking for, check out these awesome candy experiments--you can use your Halloween spoils to learn about density, pH, and more! Red, White & Grew also has some anatomy activities that go perfectly with all the spooky skeletons and fake blood on October 31.

What better way to study literature on Halloween than to read Washington Irving's classic short story "The Legend of Sleepy Hollow"? This Red, White & Grew post links to the original version, a children's adaptation, and Disney's take on it (something for all ages!), while Connect the Thoughts has a Washington Irving literary guide on sale for the Halloween season.

The holiday also offers a great opportunity to learn about other cultures. This infographic details spooky superstitions from around the world, while this Soul Travelers 3 post discusses Halloween celebrations in different countries. And of course, the day after Halloween is Día de los Muertos, or Day of the Dead, which you can learn more about here.

And a good old-fashioned Halloween lapbook ties it all together.

How do you plan to homeschool this Halloween? What are your favorite homeschooling ideas from Halloweens past? Let us know in the comments!

Last Friday, Thinkwell got the opportunity to give back to our community through the Entrepreneur's Foundation's annual Fall Service Day. This year, over 500 volunteers from almost two dozen companies came to Austin's Zilker Park to build park benches, weed out invasive plant species, plant wildflowers, and much more.

Thinkwell volunteers were assigned to the Rattlesnake Team, which meant that we got to help clean up the Umlauf Sculpture Garden. The Sculpture Garden is apart from the rest of Zilker in a shady grove full of compelling artwork and verdant plant life--in short, you'd be hard-pressed to think of a more beautiful place to spend the day raking leaves and clearing debris from pathways to make them more wheelchair-accessible.

At lunch, a series of speakers from Austin's Parks and Recreation department taught us about the park's importance. From hosting festivals like Austin City Limits to helping preserve the habitat of the Barton Springs salamander, Zilker plays many roles in the community--plus the playgrounds and swimming areas make it a fantastic place for kids! Thinkwell is proud to have played our part in keeping Zilker Park and the city of Austin beautiful.

Photo courtesy of Umlauf Sculpture Garden website.

Labor Day weekend has come and gone, and that means today is the first day of school for lots of students--including homeschool students. How can homeschool parents make sure that first day is exciting and fun for their children? Here are some tips from around the homeschool blogosphere.

The Creative Homeschool recommends borrowing some traditions from the conventional first day of school, like buying new school supplies or a new outfit:

It's fun to have something new to wear, even though we don't invest in a whole wardrobe at one time.

Titus2.com advises marking the day with a special ritual:

...when we began homeschooling, we decided to continue first-day-of-school photos. Without lunch boxes, the children hold their favorite schoolbooks in their hands for the picture-taking session.

Terrie Bitner points out that the first day of homeschool gives you opportunities that the first day of public or private school can't: 

Let your children help plan your curriculum. This is one preparation the public schools can't match. For instance, younger children might be asked to make a list of things they want to learn about in science. You can work together to choose the topics you want to study.

Rubyellen at My Cakies has a similar take on the advantages of the first day of homeschooling:
 

...the "official" [first] day is on Tuesday, but we want Friday off, so we started today. It's kinda nice having that flexibility!

What sort of first-day-of-school traditions does your family have? Tell us in comments!

Back-to-school season is upon us, but there's still plenty of time to take advantage of summer freedom with a trip to the zoo. Visiting the zoo is the perfect way to combine education with interactivity, and it's fun for both kids and adults--not to mention that it's a perfect activity for homeschool students.

If you're studying biology, reading about animals is great, but it doesn't compare to seeing them in person. A book can tell you about a 14-foot crocodile in words or even photos, but until you stand next to one, it's hard to fully grasp just how huge and powerful and, well, scary it is. Similarly, it's a thrill to get to see less intimidating animals like squirrel monkeys scampering around and playing with each other.

But, entertaining as it is, the zoo isn't just about fun. It's also educational. The signs in front of the exhibits often include interesting facts about an animal's behavior or anatomy. For instance, did you know that flamingos get their pink color from the shrimp they eat? Or that komodo dragons' mouths contain both venom and virulent bacteria? You may even learn about an animal you didn't even know existed! On a recent trip to the zoo, for example, I got to see a South American tree porcupine. I had no idea there was a type of porcupine that had a prehensile tail and lived in the trees.

To get the most out of this learning experience, pair your excursion with lessons from Thinkwell Biology. If you download our Biology iPhone app, you could even watch lectures about topics like species diversity and terrestrial biomes right at the zoo!

Make sure to check your local zoo's website before you head out. You may find special events you'd like to attend or summer discounts you'd like to use. And if you can't make it to the real thing, there's always the Lil' Fingers YouTube Zoo, which features videos of animals classified by region and type. It's made for younger children, but is anyone really ever too old for videos of otters and macaws?

As the Internet becomes more omnipresent, students use more and more academic resources online. Conducting research online can be daunting, whether you grew up hunting and pecking on a typewriter or had your own website by age nine. How do you tell which sources are reliable? And how do you incorporate them into a paper or essay? We're here to help! We previously blogged about how to use Wikipedia for research, and this post will cover how to cite Thinkwell and other online sources.

First, it's important to know that although certain information is always important, there's no universally correct way to cite any source. There are several different style manuals, each of which has its own way of organizing information in citations. Two of the most common styles are American Psychological Association (APA) and Modern Language Association (MLA). APA is usually used in social sciences, whereas MLA is usually used in the humanities, but you should use whichever one your instructor prefers. Here's a guide to APA and MLA citation styles for online sources:

APA

The APA format for citing a website in your "works cited" page is as follows:

Author. (Publication date). Document title. Retrieved from URL

For example, to cite the lecture on chemical digestion from Thinkwell Biology, you might write the following in your "works cited" page (I put extra information in the "document title" section because Thinkwell is set up in such a way that I can't provide a more detailed URL):

Thinkwell. (2000). Biology: Animal systems and homeostasis; The digestive system;

The beginning of chemical digestion. Retrieved from http://www.thinkwell.com

Within the body of a paper, the APA format for parenthetical citation is:

(Author, year, page number if applicable)

The parenthetical citation in your paper would look something like this:

Chemical digestion is an important part of the digestive system (Thinkwell, 2000).

MLA

The MLA format for citing a website in your "works cited" page is as follows:

Author. "Document Title." Website Title. Name of organization running the website, date of publication. Web. Date of access.

For example, to cite the lecture on weak acids from Thinkwell Chemistry, you might write the following:

Thinkwell. "Weak Acids." Thinkwell Chemistry. Thinkwell, 2000. Web. 15 Jul 2011.

<http://www.thinkwell.com>.

Within the body of a paper, the MLA format for parenthetical citation is:

(Author)

The parenthetical citation in your paper would look something like this:

Strong acids are more dissociated than weak acids (Thinkwell).

Still confused? Check out the Purdue Online Writing Lab, BibMe, or Citation Machine for more help with bibliographies and citations. Citing Thinkwell in a research paper will be a breeze!

Photo via Wikimedia Commons.

When you think of non-linear mathematics, your mind may not jump immediately to crocheting, but Thinkwell's summer intern Gregory Becker found a really cool way to combine the two. For his high-school senior thesis, he crocheted a Lorenz manifold. What exactly is a Lorenz manifold? I'll let him explain in his own words:

"The Lorenz system is a chaotic system of dimension 2.06 (+ or - 0.01, empirically discovered). The most famous image of the Lorenz system is the Lorenz attractor. You can picture the two 'wings' of the Lorenz attractor as the collection of possible paths that a leaf floating downstream could take around an obstacle.

The less famous image of the system is the Lorenz manifold. Impossible to represent accurately in a stable two-dimensional medium, the Lorenz manifold sits between the wings and represents the collection of paths by which the leaf collides with the obstacle.

Because of the challenge in representing the Lorenz manifold, a small niche of the mathematical community has taken to crocheting it. Crocheting works well for representing the manifold for two reasons: It can warp, and it can represent a lattice."
 

Here's a close-up of Gregory's Lorenz manifold, which has 25,511 stitches (!) and took just over 150 hours (!!) to crochet and mount.


Gregory is interning at Thinkwell as an advanced mathematics adviser for our Trigonometry and Calculus titles. He graduated from the Austin Waldorf School and will be attending Williams College in the fall as a mathematics major. We can't wait to see what his senior thesis at Williams will look like!

Here at Thinkwell, we make multimedia textbooks that our customers are proud to recommend to everyone they know. We love when you spread the word about Thinkwell, and we want to give a little something back to you when you do. That's why we're introducing Thinkwell Referral Rewards.

Thinkwell Referral Rewards allows current Thinkwell subscribers to earn $25 Amazon gift cards by recommending their friends and family. Friends and family receive $10 off all 12-month subscriptions at the time of purchase, and the current subscriber receives a $25 gift card for each subscription.

That's right. All the new subscriber has to do for a $10 discount is type the referrer's email address in at checkout--and the referrer gets a $25 Amazon gift card! So go tell your friends and family (and neighbors and coworkers and dog-groomers) about how much you love Thinkwell. You'll be glad you did.

Did you enjoy yesterday's video on algebraic proofs? Do you want to see if you remember everything from the lectures? Check out today's free Geometry worksheet covering the Algebraic Proofs video. It's two pages of word problems and equations in which you'll need to justify each step you take on your way to the solution. Don't forget the different properties of equality and congruence, as you'll be asked to identify them in a few examples.

In addition to videos and worksheets, students in our courses get interactive activities, notes, transcripts of all the lectures, quizzes, and tests. No stone is left unturned when it comes to ensuring you grasp the fundamentals of the course you take. So click on the worksheet below (PDF) and see what you think!