Posts Tagged algebra manipulatives

The Man Who Made Algebra Child’s Play

Published by — Dr. Henry Borenson wants kids not just to know algebra, but to like it.

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.

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 and 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 or contact her at or c/o GoLocalProv, 44 Weybosset Street, Providence, RI 02903.

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5th Graders Having a Blast with Algebra

Published by —  Making algebraic variables child’s play

I was forewarned, but still, it was a lovely surprise to spend a solid hour with a class of students constantly popping off with “I got it!”  And “Ooooohhhhh!”  And “I see!”

These yelps of satisfaction sound just like the ones you hear from boys playing video games.

Except that this is math class at Dawson Elementary School in Holden, Massachusetts.

No, worse:  it’s algebra, the academic subject that more than any other strikes fear in grown-ups’ hearts.  But these are 5th graders.  And some are struggling learners.

The lesson is plenty rigorous, as you’ll see.

But the kids get a useful assist from the “manipulatives,” concrete objects — pawns and blocks — that support a program called Hands-on Equations.  Kids use the objects to build and break down algebraic equations.

Six years ago, teacher Lori St. Germain jumped at her district’s offer to train her in Hands-on Equations.  She shrugs, “I was always interested in algebra.  So I took a one-day, easy training, and later showed it to the other 5th grade teachers who were able to pick up the rest on their own.  For a while we were sharing one set.”  Each now has her own.

St. Germain also acquired the digital version so she can demonstrate the day’s concept on a whiteboard.  Students talk her through moving the pieces to solve the problem.

They start with a warm-up:  2(x + 4) + x = x + 12.

On the board, and in a packet in front of each students, are red cubes showing the constants, or actual numbers.  An oversized blue pawn represents the oft-feared “x.”  Kids build their equations on a “scale,” really a laminated placemat with a picture.  The fulcrum is the equals sign.

Borrowing from game language, “legal moves” are the mathematical rules for how to make an equation balance. The equivalent of winning is solving for “x.”

St. Germain reviews a recent lesson about how to handle multipliers that are outside parentheses.  With the kids’ help, she builds two versions of x + 4, one on top of the other, on the left-hand side of the scale.  And adds the extra x.  On the other side, they put a pawn and a cube marked 12.  Kids whittle down the equation by subtracting equally from both sides, first a pawn and then the value 8, from each side.  What remains are 2 pawns on the left and a 4 on the right.

“So what’s x?”  The kids shout “2!”  The visual makes it painfully obvious.

“Let’s check our work.”  Again, the kids talk their teacher through plugging 2 into the places where the x had been.  It’s become a simple arithmetic problem.

After teaching the new concept, St. Germain cuts them loose to work the problems out on their own.  The worksheets each had four examples of the new problems, six from recent lessons, and ten review equations on the back.  Math smarties finish quickly and go back for sheets of word problems.

St. Germain explains, “Some kids can see the problem in their heads and don’t need the manipulatives.  Some just draw.  Eventually you want them all to solve without the manipulatives, making drawings or using letters.  But the one or two who still need the manipulatives in June will have them.”  The objects wean the kids from concrete to abstract, the way training wheels support learning to ride a bicycle.

Several kids struggle.  St. Germain hovers nearby, helping.  After she got a lollygagging, impeccably-dressed girl through the first problem successfully — with one of those big “Ohhh!”s — the little fashion-plate quit fooling around and got to it.  By the end of class, she was proudly booking through the practice problems.  She nudged her neighbors for clues here and there.  A little lazy, but she was getting it.

St. Germain’s colleague Marie Horton says “This program takes a complicated concept, like a variable, and turns it into something they can take apart and put back together.  Like a puzzle.  So this is incredible for 10, 11-year-olds because of how much it makes them think.  They get such a sense of pride instead of looking at a problem and rejecting it as ‘stupid.’  Kids go at their own pace, so the learning is differentiated.  But the best part is seeing the little light bulbs pop, that sense of happiness.”

That is the best part.

At one point, St. Germain firmly told a squirmy boy to look her in the face.  She had to ask him twice, but he finally swiveled around and complied.  Without impatience, she repeated a sentence she’d already said several times during the lesson.  This time, not only does he get it, but a chorus of eavesdroppers who’d been struggling with their own work bellowed their version of “Oooooohhhhh!!!”

That’s what algebra class looks like when we’ve got it right.

Next week we’ll meet the program’s inventor, Dr. Henry Borenson, to understand the thinking that went into this work, the only algebra manipulatives program to merit a patent.

Julia Steiny is a freelance columnist whose work also regularly appears at and 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 or contact her at or c/o GoLocalProv, 44 Weybosset Street, Providence, RI 02903.

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