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 a 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.
A 'Hands-On' Approach
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 student, 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.
Like Riding a Bike
“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 10 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 EducationNews.org. She is the founding director of the Youth Restoration Project, a restorative-practices initiative, currently building demonstration projects in 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 email@example.com or c/o GoLocalProv, 44 Weybosset Street, Providence, RI 02903.