Different Ways to Solve the Problem of Algebra

Professor Jon Star of the Harvard Graduate School of Education, a former math teacher, has been working to make algebra instruction more effective. We know that algebra is a tough subject for many students and have written in the past about ways to make algebra more readily understood by middle and high school math students.

Professor Star’s project, as described in Ed. magazine and online in the Graduate School Newsletter, has two aspects: first, he is working with teacher volunteers to show them that providing more than one way to solve a problem is more effective than insisting that problems be solved in one particular way. Second, he is having math teachers include more discussion in their classrooms. The idea is not to just focus on getting the correct answer, but to discuss how that answer was found or why someone took a particular approach to solving a problem. 

Working with colleagues, Star has created a set of curriculum materials designed for middle and high school students, which incorporates these approaches to teaching algebra. The materials are available at no cost for teachers to use in addition to their regular modes of instruction. And the U.S. Department of Education’s Institute of Education Sciences has included this approach in two new publications, a problem-solving guide for grades 4-8 and an algebra practice guide for middle and high school students.

Algebra in the Real World

A couple of years ago, Queens College Professor Andrew Hacker wrote a provocative piece in The New York Times entitled, "Is Algebra Necessary?". In it, he notes that completing required high school or college algebra courses is a significant bar to graduation for many students. He explains that it is not the useful aspects of mathematics that are the problem; instead, it is the way algebra is taught and the tenuous connection between classroom formulas and real life applications that raise questions about the usefulness and necessity of algebra courses.

We have seen a number of students, with and without math disabilities, struggle with this subject matter and are not surprised by Dr. Hacker's statement that in the City University system in New York, where he taught since the early 1970's "...57 percent of its students didn’t pass its mandated algebra course. The depressing conclusion of a faculty report: “failing math at all levels affects retention more than any other academic factor." "

There are many excellent tools, books,  and games that help students build algebra skills and support the importance of mathematics and algebra as part of school curricula and as a life skill. Take a look at the "tags" along the right hand side of this blog page for the subject "algebra" to see our relevant posts.

Following up on Dr. Hacker's article, the Times came up with suggestions for using algebra within the context of the subjects covered by the newspaper. In a post on its Learning Network titled, "N Ways to Apply Algebra With The New York Times", Patrick Honner lists numerous ways to apply algebraic formulas. These include:

• Exploring the housing market, including the impact of changing mortgage rates and whether it is less expensive to buy or rent an apartment
• Reviewing the numbers used in college rankings and creating an individualized formula for ranking colleges
• Looking at the costs of owning a car and exploring depreciation rates
• Determining whether it pays to buy a seven day, 30 day, or single ride MetroCard

Whether you and your students agree or disagree with Dr. Hacker, we can all agree that algebra is not disappearing from high schools and colleges any time soon. If you love it, or just want to get it over with, spending some time applying algebra in the real world may be a useful way to use your brain this summer. 

Math Games

For your gaming pleasure, here’s our latest list of recommended math games! All of the games below are web-based, so no downloading is necessary, though you will need an internet connection. Have fun!

Cool Math Games  - grades 2 - 7

Students can practice addition, subtraction, multiplication, division, and work with negative numbers and decimals, with this huge selection of games. One we particularly enjoyed, Crazy Taxi, let us drive a cab into cars that had specific multiples on them while avoiding cars that didn’t fit the pattern.

Caution: Parents, be aware that this site is host to plenty of games that have little to no educational value, too. Keep an eye on screens to be sure kids choose games that have merit, or make a list of games your child is allowed to play before turning them loose.

Funbrain  – grades 1 - 8

The math arcade is the best part of this large site. Students will start on the first square of this digital “gameboard” and progress forward as they pass challenges. (The game will save kids’ place if playing is interrupted.) This format gives students less control over which games they play, but provides great incentive to keep trying. The challenges are refreshingly inventive.

Caution: Same as above. Monitor screens to be sure kids don’t stray into playing other games that are simply for fun.

Decimal Squares  – any student struggling with decimals

This site links to eight very high quality games all about decimals. Decimal Squares gives kids a fun way to work on traditionally tricky concepts like place value and estimation through fun games like blackjack and darts. There are also some two-player games here, which provide motivation and discourage mindless clicking.

Subtangent  -3rd grade and up

There are quite a few fun, unusual math games to choose from on this site. We found Broken Calculator to be especially enjoyable. The player is shown a calculator with missing keys and challenged to make certain numbers using only the few number and sign keys left. We also liked Matching Game which challenges students play a version of memory by turning over digital cards to look for matches like equivalent fractions, multiplication facts and their answers, shapes and their names, etc.

Remember, you can always visit our Resources page for links to our favorite games and resources for math, reading, writing, studying, and more!

NCTM’s Illuminations is a Fantastic Resource for Math Instruction

After sifting through a seemingly endless list of math apps and online math games recently, we were pleasantly surprised and energized to discover Illuminations. While much of the content we viewed on other websites was more of the same, Illuminations was, well, illuminating. A resource developed by the National Council of Teachers of Mathematics is bound to be impressive and insightful, and Illuminations, we discovered, is both.

Illuminations offers 108 online activities that teach a multitude of skills, from counting to calculus. With material for students in, and teachers of, kindergarten through 12th grade, there is almost certainly something here for everyone. Illuminations’s resources cleverly meld the “why” and the “how” of math so that students will not only understand what to do but understand the reasons they’re doing it.

Some activities are very simple, geared toward helping students to internalize a single concept. For example, Geometric Solids invites students to explore the relationship between the number of a solid figure’s faces, edges, and vertices by tallying up the numbers of those features on different three-dimensional shapes (all of which can be rotated by dragging the mouse and made to appear either opaque or transparent). Half Angle provides a simple, visual explanation for why bisecting the angles of a triangle can reveal the location of the center of a circle inscribed within that triangle. For the most part, though, the tools focus more on the exploration of broader concepts and skill building. Here are a few more examples of activities found on Illuminations:


Coin Box

Young students mastering money can move coins around the screen as they count them, practice trading coins for other coins of equivalent value (for example, five pennies can be exchanged for a nickel), and drag coins onto a grid with 100 squares to help with making change from a dollar. The coins even make a satisfying clinking noise as they are moved!


Factorize

To teach why a number can have multiple pairs of factors, Factorization provides a 100 square grid and invites students to draw rectangles. The number of squares within the rectangle (i.e. its area) is displayed as the rectangle grows or shrinks, and when students get the number they want they can count the squares on each side of the rectangle to find the factors of the number. The activity indicates how many factorizations there are for each number so students will know when to keep drawing for more solutions. For example, if the number to be factored is 18, the game will prompt the student to draw three different rectangles as they work to factor it (i.e. 18x1, 9x2, 6x3)


Algebra Tiles

This insightful activity allows students to represent algebraic expressions visually using colored squares and rectangles for terms. Once students have set up the expression (Illuminations will check to be sure they’re right), they can move terms from one side of equation to the other and change colors to indicate whether terms are negative or positive. The “cancel out zero pairs” button makes instructions like “subtract from both sides” concrete and visible.


One big caveat applies to this wonderful site, however. The activities are fantastic teaching tools, but most students will have trouble using them independently. This is the kind of site that a frustrated parent or teacher could turn to when the umpteenth explanation of a concept doesn't sink in, but they’ll likely need to sit next to the student and explain how to use the activities and the relationship between the activities on the screen and the paper math work the student must complete. But please, don’t be deterred. We are enormously impressed with the astute instruction provided by Illuminations and recommend it without reservation for both instruction and practice.

TV's Danica McKellar Writes on Girls and Math

Most readers of a certain age will remember Winnie Cooper, the object of Kevin Arnold’s (and many viewers’) long-standing crush on the TV series The Wonder Years. After the show concluded, Danica McKellar, the actress who played Winnie, continued to be crush-worthy off-camera, graduating summa cum laude from UCLA with a degree in math and helping to pen a groundbreaking mathematical physics theorem. She’s devoted to promoting math education for girls and by all accounts is a whiz with numbers, though apparently in middle school she was terrified by math. But then she began to make up “tons of cool tricks and ways of remembering things,” and math began to fall into place. And in her book, Math Doesn’t Suck, she shares those tips and tricks with middle school girls.

Math Doesn’t Suck is a breath of fresh air. The style is fun and engaging, and McKellar’s tips really are insightful. Girls will be drawn in by the style and held by the substance; they’ll actually want to read chapters of a math book from beginning to end! McKellar explains tricky concepts like the relationship between decimals and fractions, proportions, and percentages in ways that are easy to understand, and provides plenty of clear, interesting examples to give the concepts context. “Step-by-Step” sections provide handy lists of steps girls can use to get through procedures; these pages would be good to bookmark, though McKellar provides lots of memory tricks so girls may not even need to go back and reference them too often. There are also practice problems. The answer key is in the back of the book, and so is a useful Troubleshooting Guide with helpful advice about where to start when you don’t know what to do and lists of useful websites. One of them is McKellar’s own math website, which is available to everyone, whether or not they buy the book. Girls ready to take it to the next level should check out McKellar’s other offerings: Kiss My Math (pre-algebra), Hot X (algebra) and Girls Get Curves (geometry).

An important consideration to keep in mind is that this book is very “girlie,” at least on the surface. The subtitle is “How to Survive Middle School Math Without Losing Your Mind or Breaking a Nail.” Chapters have titles like “Choosing the Perfect Necklace” and “You Can Never Have Too Many Shoes.” This is the aspect of the book that is either brilliant or off-putting, depending on one’s perspective: McKellar uses all of these stereotypically tween-y topics to hook girls, explain mathematical concepts, and make it all feel relevant. “Choosing the Perfect Necklace” starts with a situation: The perfect necklace to complete your outfit is tangled up, so before you can wear you have to straighten it out. McKellar moves, unexpectedly and smoothly, to an analogy about complex fractions, which also need to be “untangled” before they can be used. Similarly, the shoe chapter is not actually about shoes; instead, it introduces the concept of multiples. If you have one pair of perfect heels, you have two shoes, but if you want to stock up and get two pairs, you’d have four shoes. Buy three pairs and you’ll have six shoes, etc. Four and six are multiples of two. Who knew math could be so stylish?

We urge those who bristle at the idea of over-feminizing adolescent girls to take a deep breath and read on. Yes, the headings are written in a font replete with curlicues, and the Step-by-Step sections are each adorned with an image of flowered, high-heeled shoes. But you can’t deny that the book has style – it looks compelling instead of dry. And McKellar has sprinkled in insightful quotations by adolescent girls about intelligence, and information about inspiring women, some famous (Eleanor Roosevelt) and some not (McKellar’s real-life acquaintances who use math to be successful).

In fact, the overall message of the book, which it is nearly impossible to miss in spite of all the pink, is that being smart is more than OK – it’s wonderful. McKellar and her book are living proof that intelligence and feminine style are not mutually exclusive. She seems to want girls to know that they don’t have to choose between being smart and being glamorous, if that’s they want. They can have it all. And they should. That’s a message that’s hard to object to.

Get the Math Makes Algebra Relevant

Math, particularly algebra, can sometimes seem irrelevant to frustrated middle and high school students. It’s hard to see how solving for a variable or graphing a line could ever be useful. They’d much rather flip through a fashion magazine, shoot hoops, or listen to music. Thanks to Get the Math, however, algebra students can see real-life examples of how the math they’re learning is indispensable to those very industries and try their hand at solving the kinds of problems professionals face every day.


Get the Math, funded by WNET, plays host to a series of videos to set up real world math challenges. Algebra students can watch teams of teenagers travel to restaurants, music studios, video game designing companies, gyms, and other interesting workplaces where they meet young, dynamic professionals. After explaining their jobs, the pros present teams with practical challenges they’ll need algebra to solve. For example, fashion designer Chloe Dao wants a shirt to retail for less than $35, but it currently costs more than $40. So she presents the teams with the list of material and production costs, provides the mark-up, and sets them loose to change the design so the shirt will be more affordable. Video game designer Julia Detar of Arkadium challenges the teams to save a spaceship from colliding with an asteroid in a game she’s designing. The asteroid is traveling on a coordinate grid along a linear path. The teams are given only a few points and must figure out the asteroid’s trajectory, then plot a course to guide the spaceship to safety.

The introductory video ends once the problem has been presented, allowing viewers to tackle the problem themselves. The Challenge link provides access to screens that review the information in the problem, and they can enter and check their answers, too.

Once they’ve got it (or once they’re stumped), the final video in the series captures the team working in partner pairs. They discuss how they will solve the problem and critical information appears in captions at the bottom of the screen to make it easier to follow. Once the teams are happy with their answers, they present their solutions to the featured professional for feedback and reflect on the experience. Most of the time, both partner pairs will arrive at different correct solutions, highlighting the important lesson that real world math isn’t black and white. In redesigning the shirt, for example, one team opted to forgo the beading to bring down the cost due to a bad experience she’d had with beaded clothing. The other team chose to replace one type of fabric with a more durable and cheaper variety, hoping to both reduce the price and make the shirt less fragile.

Telling kids they’ll need math one day is nothing new. But hearing that message from a professional in a field of interest can be more powerful than hearing it from a parent or a teacher. We like the sense of empowerment students will feel when they realize that they’re learning the kind of skills that will make them valuable additions to the workplace. We hope to see this promising interactive site expand!

Color-Coding Algebra

Algebra introduces students to ideas that were previously foreign to them. Suddenly, they are expected to work on both sides of an equal sign, calculate with negative numbers, and told they can’t add all the numbers in an expression together they way they’re accustomed to doing. To help kids navigate this confusing transition, try using color-coding to help them grasp concepts and notice details. 

Negatives vs. Positives

Prior to beginning algebra, most students have not had much practice with negative numbers. They are often not accustomed to taking a number’s sign into consideration. To help draw students’ attention to this important factor, ask them to highlight positive numbers in one color and negative numbers in another.


Examples:

-12 + 4 =

(4)(-6) =

Variables

Color-coding is a great strategy for helping algebra neophytes understand the idea of like terms. When adding or subtracting in expressions containing variables, ask students to highlight unattached numbers with one color and like terms with other colors before solving. Grouping blue terms with other blue terms will seem a lot more natural than grouping x’s with other x’s. Tip: Ask them to include the sign preceding each number in the highlight so that they will understand which numbers are positive and which are negative in tricky equations. When there is no sign, ask students to add in their own addition sign, then highlight it.


Example:

4n + 3x - 4 = 7x - 2n + 10      ->        +4n + 3x - 4 = +7x - 2n + 10