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The following article is by Robert M. Berkman who has worked in mathematics and science education since 1984. He publishes educational materials under the name SamizdatMath, which can be found on Teachers Pay Teachers.  His also has a blog is entitled Better Living Through Mathematics. He currently lives in Brooklyn, New York.

I've always found Robert's forum posts and blog articles interesting because they contain a great deal of depth along with a bit of humor. He was kind enough to agree to write an article as a guest blogger for my blog. I think you will find the following article (it's part of a full day workshop called “Wiring the Brain for Mathematics Neuroscience and Numeracy") very thought provoking. 

Thinking About Skills, Context and Neuroscience

Many years ago I was hired to coordinate a mathematics program at a private school in Manhattan: I had been a classroom teacher for the previous 15 years, and my views definitely tilted towards the “progressive” end of the educational philosophical spectrum. I had no choice: I had seen the results of both “traditional” and “progressive” educational practices, and while neither was perfect, I definitely saw that progressive practices aligned better with what I wanted to see happen in the classroom. I remember the “sage on the stage” practices from my own school years, and while I responded to well to them (I was that kind of learner), many of my classmates were left in the dust. At the same time, I was familiar with the criticisms of progressive math practices, and was ready to modify practices in the classroom to address them.

Of course, although I had a thorough understanding of what progressive math looked like, this did not mean that everyone with whom I worked shared that same comprehension. I had two “bosses” at this school: the first was the head of my division, which spanned from kindergarten through 4th grade; she was impressed by both my philosophy and how I intended to translate it into practice.

My other “boss” was the chair of the math department, who was ten years my junior and lacked any kind of experience or understanding of what a K - 4 mathematics program looked like. She held some very ignorant views of progressive education, including the idea that using these methods, students would not be required to learn “basic facts.”

This came to a head during a meeting where “Jen” (I changed her name to protect those with similar names) described a situation where a 7th grade student she was tutoring for several months had forgotten the answer to 6 x 8. She recounted how she prompted the student to figure out the answer for himself and then watched in dismay as the student made 6 rows of 8 dots per row, and counted them one by one. Initially, I wanted to say the following: “Jen, you must be a pretty cruddy tutor if the parents are paying you all this money to help their son, and you wasted 10 minutes watching him draw and count out all these dots. Why didn't you just tell him that 6 x 8 is 48?”

However, Jen was my boss, so I activated my internal editor and I sadly shook my head and agreed that this was a sad state of affairs. But I also understood that this colleague was inadequately informed about many aspects of educational philosophy, especially the difference between “practice” and “outcomes.” Unfortunately, engaging her in discussions to tease out the difference inevitably led her to recount yet another story of a “progressive education failure.”

So let’s begin at the beginning: progressive education, in which I firmly believe, has nothing to do with the outcomes of that practice. There is nothing in the practice of progressive education that states that students don’t have to learn how to add, subtract, multiply or divide. This is because progressive education has nothing to do with outcomes: it has to do with methodology. As a progressive educator, my goals are fairly anodyne: I want my students to master mathematics with a balance of conceptual understanding factual knowledge (like computational facts) as well as the application of the latter to problem solving. This doesn’t sound particularly “radical” to me, and I would expect that it probably sounds reasonable to even the most traditional mathematics educator.

I’ll repeat this again: progressive and traditional mathematics educators seek exactly the same outcomes. Where we part ways is in the methodology: when I think of “progressive,” I don’t long for some long-lost 60’s era where students wrestled with “new math” or counted on their fingers in 5th grade. As a “progressive,” I’m sensible enough to understand that the “good old days” never really existed, and that students of today struggle with the same learning issues that they did in bygone years. I also don’t buy into the fact that students today are “different” from those of 10, 20 or even 100 years ago: students are students, and except for the fact that they’re more likely to come from economically impoverished households, I haven’t really seen much of a change in my three decades of teaching.

As a progressive educator, I do believe in one thing: scientific research. As a progressive, I’m interested in what advancements have been made in understanding how the brain comprehends the world and how it learns new things. My particular interest is in the neuroscience of numeracy, which has led to great insights into the learning of mathematics during the last two decades.

One of the best books on the subject is Brian Butterworth’s book,What Counts: How Every Brain is Hardwired for Math.  Although this book is 15 years old, it presents the basics of how the brain works with numbers quite clearly and with a minimum of jargon. My favorite chapter is where Butterworth demolishes “neuromyths” like the idea that the ability to work with numbers is localized to the left side of the brain. How this pernicious piece of factual idiocy got indoctrinated into our educational culture is beyond me, but it still remains pervasive, perhaps because some people make a living perpetuating it as a “fact.”

In the course of his book, Butterworth describes a finding which would have helped Jen’s student who had forgotten the solution to 6 x 8. As it happens, multiplication facts are stored in a particular part of the brain that works with language, particularly words that are remembered as associations. This would include things like song lyrics, nursery rhymes and prayers. In essence, they are linguistic phrases that we repeat over and over again with little thought. The remedy was not to ask the student to “figure it out,” but to teach him the fact and help him create a “linguistic hook” that would help remind him of the answer when it appeared again (such as “6 x 8 is really great because the answer is 48....”) To me, this is what progressive education is about: I too want that student to have factual knowledge, but I want to use what science has shown me to help troubleshoot and correct the student’s deficit.

Of course, this brief treatise does not begin to cover the complete belief system of the progressive educator, but it should give you a better understanding of its depth and complexity. In fact, you may be using the technique I described in the previous paragraph, in which case, congratulations and welcome to our ranks!

-Robert Berkman



Yes or No? Stay or Go? Solving for "x".


My basic algebra classes have just begun solving equations containing one unknown. As I tell them, we are inquisitive detectives looking for the unknown.

My students' greatest difficulty is deciding what stays and what goes in an equation. In other words, which term should be cleared by using the inverse operation and which term should stay where it is?

Hands-On Equation
 Balance Beam
www.borenson.com
I always start this chapter using Hands-On Equations®. I have used them for years because it provides a visual for those concrete learners. I also refer to the written equation as a teeter-totter or a see-saw which must always stay balanced. In other words, the equal sign is the pivotal point and both sides of that = sign must be the same.  (Notice that Hands-On Equations® uses a balance beam.) We also discuss the importance of the"Whatsoever thou doest to one side of the equation, we must doest to the other". (Out of necessity, I admit that I was with Moses when he received the Ten Commandments, but it "fell upon me" to convey The First Commandment of Solving Equations to future mathematicians.)

One Unknown
After much practice with the Hands-On Equations®, we move to actual written equations such as: x + 9 = 12. Here's the rub; a few of my students know the answer and do not want to show any of their work. Maybe some of you have this type of student as well. Since, after 30+ years, I am still unable to grade what is in their minds, I insist that all steps are written down. I explain that it's like riding a tricycle to ride a bicycle to ride a unicycle.

First, I instruct the students to look at the equation and determine which terms are out of place. (Side note: Because my students are easily confused, at the present, we keep all of the unknowns on the left side and all of the numbers on the right side of the equal sign.) Let's go back to our sample of x + 9 = 12. Because the x is already on the left side of the equation, the students write a "Y" over it for the word, "Yes". The 9 is on the wrong side of the equal sign, so the students write a "N" over it for "No".  Finally, they write a "Y" over the 12 since it is the correct place. They now have exactly what they want, a Y and N on the right side and a Y on the left side. They now must clear anything that has a "N" over it.  The students recognize they if they use the inverse operation of addition, they can clear the 9. They therefore subtract 9 from each side of the equation resulting in an answer of 3.

Many algebra teachers will have the students write the step x + 0 = 9.  You may wish to include this step in the process, but since my college students readily see that +9 and -9 make zero, they put an X over the two opposites to show that they cancel each other out or when added together, they equal zero.

What if the equation is: 3 = y - 4? This always freaks my students out; yet, if they do the yes/no process, they will discover that they have two "no's" and one "yes", not a yes, no = yes.  This means they can rewrite the equation as y - 4 = 3 to get a yes, no = yes. The problem can now easily be solved like the one above.

Unknown on both sides
of the equation
The next step is what to do when an unknown appears on both sides of the equal sign.  Usually, my students are sure they are incapable of solving such a difficult problem, but let's use the yes/no method and see what it looks like. 

Notice in the sample on the left that we have a yes, no = no, yes. We start by clearing the "N" on the left hand side of the equation by using the inverse of -9. We then go to the right side and clear the y by using the inverse operation of addition. (Yes, I am aware both can be cleared at the same time, but again simple and methodical is what is best for my mathphobics.) We then divide each side by 4 resulting in the answer of 3. When the problem is completed, my students are amazed and proud that they could solve such a long equation. (You might notice in the illustration, a dotted line is drawn vertically where the equal sign is. This helps my visual students to separate the two sides of the equation.)

If any of you try this approach with your students or have a different method, I would love to hear from you. Just leave a comment and a short statement of how this process worked for you or what process you use that is even better. That way, we can learn from each other.

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Hands-On Equations® is algebra for the visual and kinesthetic learner. This system, developed by Dr. Henry Borenson, enables students (even those in 4th or 5th grade) to easily learn essential algebraic concepts and skills. Dr. Borenson received a U.S. patent for his teaching invention.

Spelling is Important!


Noah Webster, an American lexicographer (one who compiles a dictionary) was the first person to write a dictionary of American English. It may have taken him more than 25 years to do so, but this book permanently altered the spelling of American English by offering a standardized way to spell and pronounce words. He learned 26 languages, including Anglo-Saxon and Sanskrit, in order to research the origins of our country's tongue. You may not know this, but Webster used the Bible as the foundation for his definitions.

Before his dictionary, Americans in different parts of the country spelled, pronounced and used words differently. To create uniformity, Noah used American spellings like "color" instead of the English "colour" and "music" instead " of "musick". He also added American words that didn't appear in English dictionaries like "skunk" and "squash". When he finished in 1828, Noah's dictionary contained 70,000 words.

During Webster's lifetime, American schools were anything but productive. Sometimes 70 children of all ages were crammed into one-room schoolhouses with no desks, poor books, and untrained teachers. The textbooks came from England. Noah thought Americans should learn from American books so he wrote a spelling book for children. Known for generations simply as The Blue-back Speller, millions of American children learned how to uniformly spell and pronounce words. Webster also established a system of rules to govern grammar, and reading. Clearly, he understood the power of words, their definitions, and the need for precise word usage in communication. Without a common oral and written language, he felt the country would remain divided.

Fast forward to today with the use texting. Writing skills have turned into sentence fragments while spelling consists of numbers, symbols or abbreviations. These habits carry over when students are at school; consequently many really don’t know how to spell or write well. No longer can students punctuate correctly since text messages often contain run on sentences with no punctuation, In addition, with the constant use of lowercase letters, students fail to use capital letters where they should. How do I know? I teach at a community college where about 60% of our students are in remedial English which involves sentence structure, basic grammar and spelling. When assigning a written assignment, I must include how many words a good sentence contains and how many sentences are in an acceptable paragraph. Even these requirements do not guarantee a complete sentence.

It seems we have moved away from standard spelling to inventive spelling (an abbreviated, expedient form); yet customary spelling has not gone out of style. It is required at school, in business, at work and in just everyday life.  In addition, the correct spelling of words affects academic success. Students are frequently assessed on their skills in written language because it is considered a strong indication of their intelligence.

Spelling is an indication of a number of things when a person applies for a job.  When correct spelling is used, words are readable and communication is clear. This convinces a prospective employer that the job applicant has been well educated. It also tells them that they take care of detail and take pride in what they
present.  Let’s face it, university applications and job resumes littered with spelling errors don’t make it very far becuz badd spilleng is hrd two undrstnd wen yuu reed it.

Furthermore, good spelling streamlines communication. By following the identical rules for spelling words, we can all understand the text we read. Likewise, good spelling avoids confusion. In a way spelling is similar to football. It is up to the person passing the ball to make sure the receiver actually catches it. The same goes for spelling. If you write with intent and proper spelling, the receiver of that text will understand it.

As teachers and parents, we should care about the fundamental part good spelling plays in our language and everyday lives. We owe it to our students to give them the necessary skills and essential spelling tools for learning and communication so they can be successful.

Spell Down

Spelling Shake Up
If you liked this article and would like to purchase some useful spelling resources, check out these two games. Their purpose is to help and encourage students to practice spelling words in a non-threatening way.