Julia Steiny: The Man Who Made Algebra Child's Play

Julia Steiny: The Man Who Made Algebra Child’s Play

by Julia Steiny Dr. Henry Borenson began his career as a math teacher at Stuyvesant High School in New York City. Like Boston Latin, Stuyvesant uses an exam to cream the best public-school students. For those smartie pantses, algebra was a breeze. Borenson’s biggest problem was the constant need to invent intriguing work to challenge [...]

by Julia Steiny

Dr. Henry Borenson began his career as a math teacher at Stuyvesant High School in New York City. Like Boston Latin, Stuyvesant uses an exam to cream the best public-school students. For those smartie pantses, algebra was a breeze. Borenson’s biggest problem was the constant need to invent intriguing work to challenge his kids.

Julia Steiny

Then he took a job as Math Supervisor in another state. As such, he descended from the lofty reaches of gifted-and-talented programs and became responsible for teaching, well, the rest of us. Like so many young students now as well as back in my day, I developed a profound algebra-aversion. It made me feel so hopelessly inept that I narrowed my college search to those that would not make me take math.

Borenson explains, “The way algebra was traditionally taught involved memorization without understanding.” Well, not understanding makes anyone feel stupid and totally turned off. No wonder many kids don’t like math.

Patricia Scales, the principal of the school I visited for last week’s column on this subject, explained, “We hurry kids along when we really need to slow down and teach process and understanding. Only by getting solid foundations of a skill can they get to the next level, which takes time. But if you have them do it by rote, they don’t understand and they’re not thinking.”

Especially with the testing insanity of the last decade or so, teachers want to help their students arrive at correct answers absolutely asap. So math instruction often regresses to teaching rules — algorithms, formulae, tricks, rote. Which is boring.

Of course, many math teachers don’t themselves have deep understanding that they can pass on with confidence. They too mainly learned the rules.

Borenson says, “The focus of my entire career has been on the teaching of math. Already 25 years ago, I was looking to make algebra more visual to support understanding. I wanted to demystify the meaning of equations by representing them physically.”

His first effort was a crude system of letters and pictures designed to help a disengaged 8th-grade class. “These visualizations allowed the weakest student in the class to solve advanced mathematics problem. To her it was instantly obvious. Clearly algebra needed to be more concrete so kids could get used to it and like it.”

That early work evolved into what became his life’s brainchild: Hands-on Equations. Designed for students grades 3 – 8, and struggling high-school students, the program has kids build equations, literally, with chess-like pawns representing the variables and numbered cubes. (A child demonstrates how to do it here.)

Borenson says, “Pawns and cubes are much friendlier than x and y. Kids can see that you can’t combine a constant (number) and x. Each lesson introduces only one more concept, and the sequence of lessons provides building blocks for young learners. Hands-on Equations is designed to give kids a head start before taking a regular algebra class.”

He adds, “When a kid is working on a video game, they don’t ask, when am I going to use this skill? The reason they always ask what algebra is good for is because it’s boring. They don’t understand what they’re doing, and they’re not successful. Video games require strategic thinking; Hands-on Equations does the same.”

Helping kids feel confident about their ability to think through a problem sets them up with good attitudes.

Hands-on Equations is not new, but it’s still too much under the radar. Over the years, tons of research has supported the program’s success with inner city kids, English language learners, special needs students, indeed, all kids. In video testimonials, math teachers and researchers both report the same experience I had at Patricia Scales’ school, watching light bulbs popping over the kids heads.

Hands-on Equations was voted the #2 most downloaded math program for the i-pad. Borenson argues that no other actually teaches algebra. “In most math apps, the child knows he’s right because the program says ‘Terrific!’ or ‘Good Job!’ or something. Scientific American gave one (program) a top rating that can’t teach algebra because there is no way for a child to check his answer. That’s enabling. The fancy graphics are not teaching a kid to solve the problem on his own.”

The program is gamelike, but without points, winning or racing. Kids learn math rules as “legal moves,” in the language of video games.

Borenson’s colleagues offer professional development for the use of the program. But honestly, he and the teachers I met believe that the manual supplied with the kits provides all a motivated teacher needs to know. The kits themselves are relatively inexpensive, and Borenson is negotiable when schools are seriously strapped. A book of word problems supplements each lesson, to keep the more advanced kids challenged.

It’s rare for me to laud a marketed product. But Hands-on Equations certainly would have cleared up my problems with algebra, perhaps opening up my college search.

Borenson says, “The point is to get kids used to algebra so they like it. It’s important that they develop positive attitudes towards math.”

Surely improved attitudes would work wonders on kids’ anemic math achievement.

Julia Steiny is a freelance columnist whose work also regularly appears at GoLocalProv.com and GoLocalWorcester.com. She is the founding director of the Youth Restoration Project, a restorative-practices initiative, currently building a demonstration project in Central Falls, Rhode Island. She consults for schools and government initiatives, including regular work for The Providence Plan for whom she analyzes data. For more detail, see juliasteiny.com or contact her at juliasteiny@gmail.com or c/o GoLocalProv, 44 Weybosset Street, Providence, RI 02903.

Comments

The Man Who Made Algebra Child’s Play « Standing With the Kids

February 14, 2013 at 9:22 am

[...] by EducationNews.org — Dr. Henry Borenson wants kids not just to know algebra, but to like [...]

Reply
- Cindi Crabtree
  
  February 25, 2013 at 7:05 am
  
  I really liked your comparison to video games–”when am I ever going to use this skill”. I agree that the reason students ask this is because they are bored because they don’t know why they are doing what they are doing. They are simply trying to memorize procedures and are not being successful. That is why I have never heard one of the my students say this when we are using Hands-On Equations. My Title I students were so thrilled that they finally understood why the traditional math was saying to do this or that. Also, like I have said before, the Title I students were so excited that they were sharing their understanding with other students in their “traditional” math classes and helping them. I believe one of the truest understandings of anything is when you are able to teach it to someone else. That is when it will be stored in long term memory.
  
  I liked the article because it was written by someone who was one of the students that we are missing by just using the traditional math of the early 1900′s. Sure, I agree that the “old rote traditional math” worked, but for how many? Only a select few were the ones who made it through the classes and went on to calculus. I remember when I attended college (the early 1970s) I was the only girl in my calculus I, II, and III classes because girls were being told that they were not mathematical thinkers. Because of Hands-On Equations and Everyday Math where we teach and give an understanding to ALL students we now have a larger number of girls in the “higher” math classes.
  
  As far as the comment ” what about all of the students who are not quite so active. What does the curriculum do to not let them fall through the cracks. “. That is where the traditional comes in. We need to be teaching both, so that ALL students have a chance at success. I never understand why some people are so threatened by different methods. I do know that if you are a traditional math person you have to think outside the box to understand the different methods, which stretches some mathematical people out of their comfort zone. In some of the Hands-On Equations workshops the high school and some middle school traditional math teachers are the ones who have the most difficulty-because they are having to think differently than their brain naturally thinks. Welcome to the way our struggling students feel when they are made to learn contrary to the way they think. You know how much I believe in Hands-On Equations because it does let students work equations in a way that makes sense to them. Just a couple of weeks ago I was at church on a Wednesday night. I noticed a young man and his mother working with a set of Hands-on Equations. I introduced myself and helped them through a couple of problems that were sent home by his teacher for practice. The next day the mother called to thank me. Her son was autistic which I realized when I was working with them but didn’t think it was important to ask about because he was able to learn and make sense of algebra with Hands-On Equations! Also, I told her about our apps for IPads and our website with many resources. She was really excited about the apps because she said that autistic children learned better with IPads. Again thanks for your creation of Hands-On Equations making it possible for EVERYONE to learn how to solve equations not just traditional mathematical thinkers.
  
  I did want to comment on elementary teachers. As you may recall, I was one for 15 years before moving to middle school math. I believe a lot of elementary teachers teach elementary because of their fear of math because we didn’t use Hands-On Equations and other methods to help them understand the foundation. Like I said earlier a true understanding is being able to teach to someone else. Several of these elementary teachers are being asked to “trust” the textbook because that is all they know because they were taught to just memorize procedures. On a whole different way of thinking– maybe we need to re-evaluate why in elementary school we have “specialists” (Physical Education, Music, Art) who teach these subjects so elementary teachers don’t have to. But we just pretend that everyone can teach math!!! Why don’t we have math specialists that go around to each elementary class for 30 minutes a day to give our students a solid foundation in math? What makes PE, art, and music so important? Do we really want ALL students to understand?
  
  Reply
SteveH

February 14, 2013 at 11:54 am

So how did all of those “smartie pantses” convert that mean old rote traditional math into the understanding needed to get to calculus and STEM careers? Why is it that the dominant pathway of Algebra, Geometry, Algebra II, Pre-Calc, and Calculus is mostly taught with traditional textbooks and a steady diet of homework sets? Do students who get a 5 on AP calculus do it with just memorization? This rote argument is just so old and wrong. There are other things happening here, but many educators never look past their rote biases.

The biggest problem in K-6 math is that many curricula ignore mastery “at any one point in time”, as Everyday Math likes to say. They tell teachers to keep going and to “trust the spiral”. I call it repeated partial learning. I know that from first hand experience with my son. I had to ensure mastery of the material at home. When he was in fifth grade, his teacher had to NOT trust the spiral because many bright kids still didn’t know the times table. So much for trusting the spiral, critical thinking, and understanding.

One of the key points made by Hands-On-Equations is that students do well on post-tests. They seem to acutally care about mastery. Good for them, but that is the issue going on here, not rote versus critical thinking and understanding. Does their post test results offer any more believable feedback on understanding than traditional tests? Besides, why is there such a need to focus on algebra in the early grades when there are so many other skills one has to learn and apply. We hear nothing about that.

Also, why don’t you look at Singapore Math’s bar models.? Many claim they work wonders. However, like Hands-On-Equations, math moves on to more complex forms, and understanding must come more from definitions, identities, and proofs. Understanding comes from work on individual homework sets. There are different levels of understanding and one can NEVER pass math classes with just rote memorization unless the teachers are incompetent.

Please move on from the rote pedagogical idea that this is a battle between understanding and rote learning. This is about mastery and higher expectations. This is about not letting kids slip through the cracks. Go ahead and use Hands-On-Equations, but you better learn how to evaluate curricula and teaching methods based on something more that a rote understanding of rote.

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TMW

February 17, 2013 at 5:43 am

Having used this program, 4-7 and for struggling HS kids, I saw elementary kids strategically solving problems that TRADITIONALLY had stumped Alg I students. And they told me why!! it was correct! They understood the Distributive Property, they understood that balancing the scale (equation) must be maintained at all times- and they were having a good time figuring out word problems – and asked for more!!
Their conceptual understanding began with the concrete, thus it was solid and it was theirs! We don’t get that with rote rules, and confusion comes as more rules are given… which rule do I use now???
Go see a class using the program- there is excitement and mathematics being discussed, at their appropriate level, AND it’s fun for them! It does carry over to the HS course, because it’s understood!

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Dr. Henry Borenson

February 17, 2013 at 12:50 pm

I just received this note, “I work with struggling math students involved in Closing the Math Achievement Gap research. I have never been able to have my students master two step Algebra problems before. They are so successful with hands on equations and most of my students prefer working on IPads. They do not see it as work, but as play. They love it, I am getting achievement. This program is a win-win.”

Yes, this is about high expectations and “not letting kids slip through the cracks.” Experiencing success enhances the expectations a student has for himself. So does having a sense of understanding.

Ms. Steiny made a simple point: Had she had Hands-On Equations as a young student, and therefore understood how to work with algebraic equations, she might not have developed her profound “algebra aversion” and she might not have needed to narrow her college search to those that did not require additional math courses.

There are other students, too, who have the perception that school mathematics is something that involves memorization of procedures. But they find such memorization difficult and disheartening. To them, learning mathematics is not much different from trying to memorize Chinese symbols — a task that is next to impossible. (Of course, there are others who would be delighted by the challenge.)

An examination of the types of errors that students make in trying to solve equations in traditional math classes shows very clearly that they are trying to memorize procedures without understanding the meaning of the procedures. Hence, they will add 3x and 5 and get 8x. Or, they will add a negative 3 and a negative 5 and get a positive 8 because they are trying to remember what rules to use when (they mixed up “the rule” for addition of integers with multiplication of integers.)

With proper instruction, neither of the misconceptions noted above could occur. The fact that they do occur, and that so many students have math phobia, I suggest, can be placed at the doors of poor instruction.

But even with poor instruction, those students who want to make sense of the mathematics will use their own reasoning to do so. They will have a conceptual understanding of what negative means and will never add two negatives to get a positive.

I remember as a young math supervisor visiting a high school math teacher of second year algebra who expected her students to memorize all the trigonometric identities. The idea that with a simple diagram the students could derive them all as needed –which I had learned from Mr Glaubiger as a young student at Brooklyn Tech– was foreign to her. Yet it would have empowered her students and she would not have had the high failure rate that she had– and which is typical of most high school algebra classes in the United States.

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SteveH

February 19, 2013 at 5:14 am

There are two separate issues: whether to introduce algebra early and whether Hands-On-Equations works better than some vague other things that could have problems with the pedagogy or the competence of instruction. The problem arises when people try to claim that they know what those other things are.

“To them, learning mathematics is not much different from trying to memorize Chinese symbols — a task that is next to impossible. ”

This analysis is so common for educators who want to believe in some sort of easy path to math where engagement and motivation are all you need. I was taught many of the ideas expressed in Hands-On-Equations in my traditional math classes when I was young. I’ve used them with my son, but what about everything else that has to be learned? What happens when the equations become rational, exponential or logarithmic? What about all of the other basic skills where you haven’t found a nice model for understanding and fun hands-on learning? There are many different levels of understanding and you can always trade understanding for speed of coverage.

“Hence, they will add 3x and 5 and get 8x.”

This indicates extraordinarily bad teaching, no matter what pedagogy is being used. Look at a proper algebra textbook at the explanations for this concept. Look at the homework examples that move from simple problems to more complex ones. Look at how the textbook moves students towards more flexibility. Something else might be going on with Hands-On-Equations, but this example is not what it’s all about.

“I suggest, can be placed at the doors of poor instruction.”

I will agree with that, and Hands-On-Equations would fail without proper implementation. Singapore Math would fail without proper implementation. The key is to have schools take responsibility (using whatever curriculum they have) for ensuring that all kids master skills at each grade level. Unfortunately, there are curricula like Everyday Math that want to “trust the spiral”. You can also have traditional classes where teachers just go through the motions and kids desperately memorize things in the hope of getting something correct. It’s not that the textbook is telling kids to memorize math.

“…teacher of second year algebra who expected her students to memorize all the trigonometric identities.”

That’s not how my son’s “traditional” textbook taught the material. He can derive any of the rational expressions from a few concepts. Educators love to place the blame on some sort of “traditional” pedagogy of teaching because that appeals to their own sensibilitites. They then typically use that as an excuse to introduce their own pet ideas of hands-on group learning in class. It may work, but success is probably coming from some other source, like having teachers who are properly prepared and care about the success of each student.

I tutor lots of kids, and the major problem I see is that schools just go through the motions no matter what curriculum they use. It’s especially bad in K-6 because schools pump kids along and assume that they will “learn when they are ready”. Everyday Math is all about reintroducing material year after year with that assumption in mind. Then, when the kids get to more traditional math in 7th and 8th grades, teachers move right along and tell kids that they have to take responsibility for their own learning. They think they need to toughen them up for high school. The textbooks do a fine job of introducing the material and explaining concepts and understandings, but it’s hard for many students to keep up the pace after their poor K-6 training. Many never recover. Problems that occur in algebra have their roots in K-6. If Hands-On-Equations were just used starting in 7th or 8th grades, it would not be so successful. Something else is going on.

There is an education meme that thinks that all you need are hands-on engagement and motivation. This puts the onus for learning on the students. While those things are nice, schools still need to take responsibility for all of the students who don’t “get it”. It may be heart warming to see so much active hands-on learning in class (for some kids), but what about all of the students who are not quite so active. What does the curriculum do to not let them fall through the cracks. That’s the most important part of any curriculum.

Another issue is that some curricula seem to think that understanding drives mastery of skills; that skills are just a matter of speed. This is incorrect. There is a huge amount of linkage between all levels of understanding and mastery of skills. One cannot get to the next level of understanding without mastery of previous skills. One cannot simply know how to derive any trig function from a few basic rules. One has to master their application to understand what trigonometry is all about. Unfortunately, too many curricula start with concepts and hands-on learning, but drop the ball when it comes to mastery. The concepts are superficial and kids can’t think their way to a solution except with guess and check.

Reply
Dr. Henry Borenson

February 20, 2013 at 12:14 am

Our experience is that all students benefit from Hands-On Equations– especially those with learning disabilities or the so-calle weaker students. A number of students will quickly abstract the concepts and be able to solve linear equations such as 4x+3=3x+9 mentally.

They can then apply the concepts to solving verbal problems. Here is a video of a bright young girl solving what would be considered an advanced consecutive integer problem in the sense that this type of problem is not normally presented until the 8th or 9th grade. All students can benefit from solving this type of problem in this manner before trying to do it purely symbolically.

http://www.youtube.com/watch?v=zXRYrwD0W8w

In reference to “traditional instruction” the majority of teachers try to follow the textbook, provide homework exercises, etc., exactly as you advocate. Yet it is not unusual to have 50% of an Algebra 1 class fail algebra. What went wrong? Here you have to go to the kids to ask them. They do not understand what they are doing. They are simply trying to memorize procedures and are not being successful.

Of course there are those who are successful, who read the textbook carefully and make up for what the teacher may not have said, or what they might not have heard in class.

One cannot deny two simple statements: 1) students benefit from having an understanding of what they are doing; 2) visual illustrations and hands-on approaches have the capacity, when used properly, to clarify concepts and lead to greater retention.

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James Sgroi

February 20, 2013 at 10:00 am

Dear Steve,

I think you missed a key message of the article on Hands-On Equations. Memorization without understanding is a leading cause of anxiety and student failure. Todays students are being expected to handle algebraic concepts in elementary now as early as 3rd grade. The Common Core Curriculum that nearly all 50 states have adopted has an algebraic strand in the elementary grades. Students are expected to evaluate expressions at grade 4 and to solve linear equations in grade 5. We know that to wait to 8th or 9th grade to teach algebra is doomed with a high failure rate. For many students the abstractness of algebra and written steps are key factors that produce high failure rates on state performance tests.

Hands-On Equations is a program aimed at elementary and middle school students. Students who start Hands-On Equations in grade 4 and complete the program get a four year jump on mastery of linear algebra then the average 8th grader who starts the traditional algebra program for the first time.

When teaching younger students, Hands-On Equations demystifies the learning of algebra using concrete objects: colored pawns to represent variables and colored cubes to represent positive and negative integers. One of the important techniques used in this program is active engagement of students. They are constantly building models of abstract linear equations and using algebraic techniques called “legal moves” to solve the equation. This is not memorization out of context but understanding. Students are able to show and explain their thinking. This is a much better approach to help students learn algebra. You asked do the post-tests offer more believable feedback on understanding than traditional tests? When you ask a 4th grader and an 8th grader to solve the same equations and you find that statistically they perform the same (88% mastery) you come away with the belief that given the appropriate program with age appropriate techniques, students as early as grade 4 can do algebra. The goal of this program is not to supplant the way algebra is taught in grades 8th or 9th, but to prepare them to handle algebra using traditional written notation.

Steve you make several points advocating for rote traditional math for understanding needed to get to calculus and an AP Calculus grade of 5. I doubt very much that students in courses leading to Calculus and beyond just memorized their way to success. Successful practice based on understanding is key for students to cement to memory information for long term retrieval. Hands-On Equations uses practice at every lesson and students do out of class practice using classwork sheets of exercises.

There are many effective programs such as the Singapore Math’s bar models to teach fractional concepts. Hands-On Equations is an effective program to teach algebra to a wide range of students. For today’s teachers it is being familiar what works best in teaching mathematics and adapt these effective programs to the needs of children.

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SteveH

February 20, 2013 at 12:17 pm

“Yet it is not unusual to have 50% of an Algebra 1 class fail algebra. What went wrong? Here you have to go to the kids to ask them. They do not understand what they are doing. They are simply trying to memorize procedures and are not being successful.”

Problems in algebra don’t start in algebra. The problems start much earlier, and most of those kids had curricula like TERC and Everyday Math which just pumped them along and put the onus of mastery on the kids.

“1) students benefit from having an understanding of what they are doing;”

No traditional curriculum teaches just memorization. Look at the textbooks. Look at Singapore Math bar diagrams.

” 2) visual illustrations and hands-on approaches have the capacity, when used properly, to clarify concepts and lead to greater retention.”

I don’t disagree with that, but does the whole curriculum have to be driven by hands-on group work in class? That seems to be the goal for many educators who complain about traditional textbooks. Pedogogy comes first and they desperately try to find any way that it can work. We have a Harkness Table high school in our area that produces good results, but the real reason for their success is high expectations. Curricula like TERC and Everyday Math set low expectations and put the onus on the students. As I said, there are other things going on here than the old understanding versus traditional gambit.

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SteveH

February 20, 2013 at 12:31 pm

“Memorization without understanding is a leading cause of anxiety and student failure.”

And, as I said, that is a sign of bad teaching. It is not a fundamental part of a traditional approach to math. So what do we have now? We have curricula like TERC and Everyday Math that go on about understanding but never get the job done. Kids get to algebra and struggle and educators jump on the chance to blame traditional algebra, not what went on before. So try to use Hands-On-Equations only starting in 7th grade using kids coming up via TERC and EM.

“Todays students are being expected to handle algebraic concepts in elementary now as early as 3rd grade.”

CCSS standards are pathetically low and vague. Any algebra below 7th grade is “play” algebra.

“Steve you make several points advocating for rote traditional math for understanding needed to get to calculus and an AP Calculus grade of 5. ”

Reread my comments carefully. Traditional math is not rote by any definition. It’s only rote in the minds of educators who are bound and determined to find some way – any way – to make hands-on learning in class work, with the teacher as the guide on the side.

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Dr. Carolyn Talton, Professor Emeritus

March 6, 2013 at 6:16 pm

I enjoyed the article. Students do enjoy working with Hands-On Equations and develop success quickly. Good work Dr. Borenson.

Reply