Sunday, August 30, 2009

Framing Mathematics as Language

Mathematicians often refer to mathematics as a language. Math teachers often relate this concept to their students by saying, “Math is the language of science.” Unfortunately, it is often left at that. As a result, students often fail to see the connections between language and mathematics. They view language as only working with words and math as only working with numbers. I have had many students attempt to make excuses such as: “I was never good at Math, but I do well in English.” What teachers often overlook is making the appropriate connections between language and mathematics. I find that my students have rarely had the translational aspects, vocabulary, and syntax of mathematics framed in terms of language.

Translate Language to Math Through Intermediate Steps
The first time students have formalized translation of language into mathematics is when they study word problems. However, I have seen too many texts and teachers make too large a jump when going through this process. Let’s look at a simple example to illustrate the point: The relationship between distance rate and time in a word problem. Often, a formula is stated such as:
“Distance, d, is the product of the average rate, r, and time, t, or d = r* t.”
As experienced math teachers, we may view this as somewhat trivial and obvious, but for the inexperienced students, there is often a bit of anxiety about what the definition means. With my students, I insert an intermediate step in translation, a term I call “Mathlish” Basically, this involves stating the formula in terms of words before presenting the formula in terms of variables:
“Distance, d, is the product of the average rate, r, and time, t .
This can be expressed as DISTANCE = RATE * TIME
Or d = r* t ”

This process can be extended to other formulaic problems, and can be used by students to clearly identify the variables in more complicated formulas.

Frame Math Vocabulary in Terms of Familiar Concepts
When I first started teaching, I often had students express confusion about properties such as associative or commutative properties or the intersection or union of sets. What found was that if I could frame the words in terms of familiar concepts, this confusion became almost nonexistent. For example, when working with the associative property, I had students think of 3 people that formed a team. No matter which two people I saw first, they were still part of the same team since they were associated with each other: (Bob + Kris) + Tom = Bob + (Kris + Tom), Likewise in math, it doesn’t matter which two terms are combined through addition first, since the terms are associated with each other:
(A + B) + C = A + (B + C). For the commutative property, I had students envision a commuter going from home to work and then from work to home. The same trip is made, but the direction is different. In math this becomes A + B = B + A. Having the familiar framework, outside of mathematics, made the mathematics more understandable.

Order of Operations is the Syntax of Math
Another point that is not stressed enough is how the order of operations relates directly to syntax in mathematics. I will often relate this to my students by writing on the board, “important understanding word is order to.” A few minutes of puzzlement go by, and I write, “Word order is important to understanding.” Just as placing words in unexpected places creates nonsense in English, so too does improper use of the order of operations. By framing the mathematical concept with a more familiar concept, students gain insight into the relevance and are more likely to pay attention. I also stress that math is a non-linear language, and the order operations occur in is not necessarily left to right. I point out the difference between English and Spanish. In English, we may refer to two pencils as: “The blue pencil and the red pencil.” In Spanish, the modifier occurs after the noun, (lapis), and the statement becomes: “El lapis azul y el lapis rojo.” In mathematics, we may see the same need to adjust our perception of order when we deal with the distributive law: Pencil*(Red + Blue) = Red Pencil + Blue Pencil. At this point, I introduce the relevant mathematics: A*(B + C) = AB + AC

These are just a few of the ways language and mathematics can be related to one another. The table below can be used to show how the development of language proficiency and mathematical proficiency can possibly be related.



Recognition of spoken words

Recognition of quantities

Sight words

Math facts (Addition/Multiplication Tables)

Spelling/Phonetic Decoding

Elementary equations

(4 + __ = 10)

Simple sentences

Linear algebraic equations

Complex sentences

Higher order algebraic equations



Sentence Parsing

Order of Operations

Reading Comprehension

Analysis of Functions

Research Paper

Word Problems

(Particularly Related Rates)

Translation between Languages

Numeric Word Problems

Sunday, August 23, 2009

Let's Make Some Mistakes!

Too often, students see mistakes as tantamount to failure; however, mistakes are an inevitable and necessary part of the learning process. Marvin Minsky, the Toshiba Professor of Media Arts and Sciences at MIT has noted, “We tend to think of knowledge in positive terms—and of experts as people who know exactly what to do. But one could argue that much of an expert’s competence stems from having learned to avoid the most common bugs.”1 Innovation also hinges on making mistakes and learning from the errors, be it realizing that a process will not work, or examining how an unanticipated result could be used to solve another problem.

In education, the reliance on high stakes testing has created a climate of absolutism. Students tend to believe that for any problem, there is one correct answer, and to reach this answer, there must be one unique process. While this may be useful in developing easily scored standardized tests, it does little to promote discovery, creativity, and learning. Once outside of academia, very few problems have a unique path to a correct result. A simple example I use is choosing a route to a destination. The majority of people will stick to the main roads. It’s simple and usually relatively efficient; however, there are times, particularly when traffic is heavy, when taking some of the side roads can save time and aggravation. To learn where these side roads are requires either a willingness to experiment, to take risks and possibly get lost, or a guide to show the way. Even if the wrong road is taken, one can usually backtrack and try a different road. Once these paths are learned, the driver has a wider array of choices and can choose a course that increases efficiency.

The same concept of multiple paths to a solution often holds true in education. One of the goals I set for my students is to allow them to see there are multiple ways to solve problems. After showing my class several ways to approach a problem, I typically ask, “Which is the correct way to do the problem?” This generally provokes some discussion with some students selecting the first method and others selecting different methods. My answer is always the same. “The correct method is the one you can both understand and do efficiently.”

I also try to have my students realize that mistakes are part of the learning process, and that there is no failure until one stops trying. Many of my students are athletes, and I often use analogies from sports to illustrate this point. I may ask students, “If you were one on one against an opponent and he burned you on a play, would you guard him the same way next time? If he burned you again, would you keep trying different things until you could defend him? And, when you do manage to stop him, what happened?” Typically, the students will affirm that they would try different methods until they learned what worked. At this point, the connection usually hasn’t sunk in, and they need one more prompt, “So, you made a bunch of mistakes, and you didn’t think you failed. You experimented with new methods, learned from your mistakes, and ultimately succeeded. Do the same thing in this class, and you will succeed here. Do the same thing in life and you can be successful there, too.”

Ultimately, the goal of education is having students realize that the road to success has many paths; they will make wrong turns occasionally, but learning which routes work and which do not is what will allow them to reach their destination.

1 Marvin Minsky Essays, OLPC Project,

Sunday, August 16, 2009

Math... It's Changing!

If you haven’t heard yet, math is changing. Don’t worry, 1 + 1 will still equal 2, (or 10 for those who think in binary code.) However, the way math will be taught, learned, and applied is evolving. Recent technological innovations, in particular Internet-based applications, have sparked this change, and teachers from the college level through elementary schools are grappling with the most effective ways to use these innovations.

With the advent of online courses, teachers found a need to be able to connect with students and for their students to connect with each other. Forums, blogs, and Learning Management Systems (LMS) such as Blackboard and moodle became the means to accomplish this. As these online courses matured, teachers began to incorporate the technological innovations into their regular classes to form a hybrid course, which is best described as a regular class with a significant online component. This worked quite well for discussion-based courses in the humanities; however, the need to express mathematical formulas such as:

required learning formatting languages such as LaTeX or having to post an image for each expression or equation. Simultaneously, a growth in social networking, such as Facebook, Twitter, and YouTube was taking place. Cloud computing and web-based applications gained a solid foothold on the Internet, email and text messaging became the communication media of choice, and students began to exploit these social technologies in large numbers.

Currently, more and more math teachers, particularly at the college level, are recognizing the synchronism that exists between Internet-based instruction and social media. This fusion of technology is being largely embraced as a means to promote collaborative work among students and as a means to engage students by using the technologies they are most familiar with.

This past June, Wolfram Research, the creators of Mathematica, rolled out a new website, On this site, users are able to access numeric data from a wide variety of sources, such as sociology, science, economics, and mathematics. Users can input an equation and the website will provide a solution, often with the steps of solution listed. Needless to say, this generated much discussion among math teachers at every level regarding its implications. Some worry that students will not learn basic mathematics if the computer can do the math for them. Other instructors are excited that students will be able to apply mathematics to actual data. The one fact that both camps agree on is that students will use the site.

These innovations have great implications for how math will be taught in the future. Students will become more technologically savvy, more electronically connected, and have greater resources available to assist them. As educators, we must not only be aware of these trends, but we must learn how to leverage them to provide better instruction. We must examine how and why we present material. Are we trying to provide basic understanding of mathematical facts and procedures, or are we trying to develop an understanding of the underlying mathematical principles? Is the homework we assign designed as drill and practice, or are we trying to get our students to think about how math is applied to make decisions in the real world? Do we as instructors provide the information, or do we allow students to discover the how various concepts relate to one another? All of the above methods are valid in the correct context. However, we must think critically about how we teach mathematics if we want our students to develop critical thinking skills about mathematics.

Tuesday, August 11, 2009

Welcome To Bearded Math

Welcome to ‘Bearded Math’! This blog hopes to explore mathematics, teaching, and maybe give a little slice of life around Bridgton Academy. In the weeks to come, I will share some of my teaching techniques, and interesting sites and programs involving math, technology, and education. Hopefully I’ll spark some dialog with you, my readers. If you were wondering where the title came from, then you obviously haven’t met me yet. I’ve been teaching at Bridgton Academy since January of 1986, and while my beard used to vanish during the summer, it has been a permanent feature for the past fifteen years. Aside from the occasional old photograph, my wife and kids have never seen me without my facial fur.

Bearded people tend to evoke reactions in others. Some may recall the pleasant times around Christmas, and the joy of Santa Claus. Other people may picture a sage ready to impart truth or wisdom, such as Solomon or Lincoln. Others may envision a wizard possessing arcane knowledge, inaccessible to others, such as Merlin or Gandalf. For some, the sight of a beard produces fear or mistrust as they imagine Satan or Bin Laden. A bearded person chooses to make a bold statement to the world. In today’s society beards challenge convention; they force people to look at something that is different from the norm, and they disrupt the status quo.

So it is with mathematics. People tend to either enjoy math and see the possibilities in the numbers and equations, or they look for any way possible to avoid it. This polarization is even more apparent when people face higher mathematics such as Probability and Statistics, Calculus, or Differential Equations. As a math teacher, I see this polarization everyday. Those students who enjoy mathematics will diligently do their work, grasp new concepts relatively easily, and continue to make steady progress. The remainder will shuffle in, already convinced that the concepts will be beyond their understanding, and hoping they will survive the class. My task is to provide bearded math to confront the preconceptions of those who seek to avoid math.

The first day of class can often set the stage for the remainder of the year. Many students nervously shuffle into my class, dreading another foray into mathematics. Some have had any enthusiasm for the subject “drilled and killed.” Others have consistently struggled and can’t understand why they can’t comprehend the material. More often, I see the students who spent high school filling space for the required time and hoped to learn through osmosis rather than effort on their part. For all of these students, before I write anything else on the board, I inform the class that there is one particular rule they must follow for the entire year if they are going to succeed in my class. With a bit of flourish and over dramatization I write in large, capital letters, two words:

Hopefully, this is not what they were expecting. I find that if my students are more relaxed about being in class, they are more receptive to learning. Furthermore, if I can catch them a bit off guard, they are more eager to discover what comes next. With that, the first few whiskers of bearded math begin to sprout.

Peter Horne has taught both mathematics and computer science, and he has coached golf, skiing, and cross-country teams at Bridgton. Currently Peter teaches Calculus and Pre-Calculus, and he coaches the golf team. Peter earned his Bachelor of Aerospace Engineering from Georgia Institute of Technology, and he earned an appointment as Lecturer of Mathematics from the University of Southern Maine. Peter lives midway between Bridgton Academy and the golf course, which he rarely visits, because of his busy life off-campus. Peter and his wife, Laurie, are the parents of seven children ranging in age from four to twenty-one. In his free moments, Peter enjoys reading, cooking, golf, and spending time with his family.