Q: I can understand the desire to make learning algebra fun for the kids with Hands-On Equations®. Does this then carry over from that young age into the middle school or to ninth grade, when they are formally introduced to algebra? Do they know what the teachers are talking about?
In the ninth grade, teachers will use expressions like, "Whatever you do to one side, you must do to the other side." Well, let's look at an equation such as 4x + 2 = 2x + 10 from the kids' perspective. Remember, these symbols really have no meaning to the kids; they're abstract. So, let's try to follow the instructions, "Whatever you do to one side, you do to the other." First, you cross out the 2 on the left and the 2 on right—because you're doing the same thing to both sides, right? And then you cross out the x on each side … and then the plus signs … and what are you left with? 4 = 10. And the teacher is not too pleased with this result! Yet, the student has followed the teacher's instructions that stated, "Whatever you do to one side, you do to the other." Hence, we see how little help this statement can be to students who do not understand what the symbols in the equation mean. On the other hand, with HOE, the concepts are all visual and kinesthetic, and the child clearly understands what he's doing. In HOE, the child has pawns that are placed on a scale to represent the abstract equation. He then performs physical actions, such as removing a pawn from each side, to begin simplifying the equations.
Regarding carry over: If a young child in fourth grade has been taught with HOE, he will remember in ninth grade the essential concepts he has learned; the symbols aren't Greek or Chinese any more, the principles are no longer abstract rules to memorize—and, we find this to be true, whether the ninth-grade algebra teacher has any knowledge of HOE or not. Not only do the equations now have meaning but the statements the teacher makes, such as, "Whatever you do to one side, you must do to the other," also now have meaning for a child. The student now understands what the teacher means to convey with this statement.
Q: If HOE has been in use since the mid-1980s, you would know if it's also working in the higher grades by now, right?
Well, we know it's working because we begin every seminar—and we've done 1,000 public seminars so far—by having third- and fourth-grade students in to do a live demonstration in front of the participants. In 20 or 30 minutes, the children, who have had no exposure to our program, learn how to solve an algebraic equation such as 4x + 3 = 3x + 9, and to explain their solution process. The teachers can see that the kids can learn something that would normally take older students a week or several weeks to learn through traditional teaching methods. Of course, it would be helpful if these concepts were reinforced over the ensuing years prior to taking a formal course in algebra. But even if they are not, the concepts have been learned at a deep intuitive and kinesthetic level.
Q: Do you think it's true that if kids learn through play, if it's fun, they will retain it better?
Well, yes, it's fun. And they find the colorful physical pieces to be attractive to play with. But they also enjoy the challenge of solving for the unknown, the mystery number. So it becomes a game, rather than memorizing rules to get an answer. In algebra, so many kids say, "When am I ever going to need this?" But with HOE, because the kids enjoy and accept the challenge, they want to do it, whether they're ever going to use it or not. And, while it's good to get the right answer, that's not what motivates the kids. It's the challenge to solve the mystery, to play with the pieces, to use whatever strategy they would like to try, until the answer is obtained, and that's self-motivation.
Because algebra has traditionally been talked about as being difficult, the kids build up a great deal of anxiety about finally taking algebra in ninth grade. But we can turn this situation around.
Q: How is Seymour Papert's "Piagetian learning" incorporated into HOE? Why is it so essential to the learning system?
Seymour Papert was a student of Piaget. We touch upon "Piagetian learning" in our seminars for teachers, so they're aware of what makes HOE so effective. If you were to try to teach Chinese to a high school student, she would have a lot of hard work to do to learn the language: There are the unique sounds, the Chinese characters, the words, the grammar. But, if you were to take a five-year-old child to China and place her with a family and leave her there for several months—what would happen? She'd learn Chinese very quickly. She has been placed in what is called a "natural learning environment." Such an environment provides a very powerful arena for learning.
Likewise, when children are presented with HOE, they find themselves totally immersed in their learning. They are learning powerful algebraic concepts—such as the addition property of equality, the additive identity element, and other key algebraic properties—without even knowing it. The "natural algebraic learning environment" that we present to students makes it possible for them to succeed in their work, especially since they can use so many of their senses in their work. Learning algebraic concepts through the traditional abstract methods, on the other hand, is very difficult for most students. They try to memorize as best they can, but without understanding, they can only go so far.
Now you see why the student demonstrations are such a critical part of our seminars. We bring in four or five average-ability students, whom we've never seen before but who we know have not used HOE, and in half an hour or less, they're doing algebraic equations and enjoying it! And the professional educators present are able to witness first hand the student thinking that is used to solve the problems, and they recognize that thinking as valid and on target.
Q: What is it that you want the participants in your workshops to come away with?
I was going to school at the Brooklyn Technical High School in New York City, and I enjoyed mathematics. I especially enjoyed the experience, when the instructor gave me the opportunity, to present in front of the classroom. So, later, when I was a student at the University of Southern California, trying to decide what career to pursue, I reflected upon the pleasure of teaching and the joy of mathematics and I decided that I would become a high school mathematics teacher.
Q: Hands-On Equations® (HOE) is described as "visual and kinesthetic"—What does this mean for students? Why is it important?
The way much of mathematics, and algebra in particular, is taught at the high school level is so theoretical and abstract, it involves memorization, and students are taught a series of steps to follow, and they're told if they follow these steps, they'll arrive at the right answer. But these set rules, and the symbols upon which they operate, have as much meaning to the kids as Greek or Chinese symbols would have. And therefore, to the children, the process of getting the "right" answer, as traditionally taught, is often devoid of meaning.
Instead of using symbols and set rules to memorize, in HOE, we give the students a concrete representation of the algebraic symbols and algebraic processes. The symbols are represented by game pieces. The algebraic processes are represented by physical actions upon these pieces. In other words, we have a counterpart for what's done on the blackboard, and it's done physically. As the equations are solved, the child can see what he's doing, can actually move the pieces, so he's now using his whole brain, and not just part of it to solve the problem mentally. It turns out that doing algebraic equations this way, even though many of these equations would not normally be presented until the ninth grade, is in fact easier than much of the curriculum we have in the elementary schools. So, a fourth grader can really be very successful in learning algebraic concepts presented this way, more so, for instance than in learning to divide fractions or do long division.
We believe that the seminar will be valuable to teachers, even if they never use HOE in their classrooms. But clearly, the seminar is intended for those who are considering using HOE in their classrooms, and the hope is that if they see how effective the program is, then they will want to use it with their own students. We have a number of objectives for the seminar: One is for the teachers to understand the concept of the teacher as a coach, or facilitator. Many teachers still think that the role of the teacher is to stand in front of the room and lecture for 50 minutes, that this is how knowledge is obtained by the student. And many teachers have difficulty with the child putting up the "wrong" answer, and they'll immediately "correct" it, rather than talk the child through the process of thinking about how he got the answer, and how he can adjust his thinking to achieve another more desirable answer. We'd like teachers to encourage kids, to let them interact with each other.
Another of the major purposes of the HOE seminar is to get teachers to motivate students—a better word is "inspire" them—to a higher level of self-confidence. You see, what happens is this: Because algebra has traditionally been talked about as being difficult, the kids build up a great deal of anxiety about finally taking algebra in ninth grade. But we can turn this situation around. At an early age, the child who uses HOE is successful in what looks complex or difficult, and their own perception of themselves as learners is enhanced.
We have many stories from teachers who tell us that kids don't want to go to lunch or out to recess because they're so engaged in the HOE game and finding the solution. They really enjoy the process, and they're motivated by the challenge. They consider it a fair task. Why is it a "fair task"? Because the way HOE was developed was through the children themselves. I took two years to work with children, a number of them with learning disabilities, some of whom were in fourth grade, who were getting D's, and they showed me how they would solve certain problems, how they would check their work—and those solutions became the program. So, because it evolved from their thinking, it's a program that is not imposed upon the children; it's not something they find artificial. Therefore, they find it reasonable, rather than arbitrary, and it's a fair task. Sometimes a teacher using traditional instruction methods will say, "Do it this way because when you take math five years from now, you'll need to do it this way." But the children don't find that reasonable; it's not intellectually honest, and they know it. But in HOE, they find methods that are reasonable, especially from teachers who attend the workshops and are encouraged to accept whatever solution the child comes up with that can be justified.
Q: Is HOE approved or recommended by the National Council of Teachers of Mathematics?
One of their publications provided a positive review, but NCTM does not really "recommend" programs. NCTM has guidelines, and the methods that we use in HOE are consistent with what they call the mathematical standards, such as encouraging student communication, creativity, and verbalization; encouraging multiple ways of solving a problem; enhancing self-esteem—that sort of thing.
Q: Accountability is a buzz word now … Do you have any hard facts, such as improved math test scores by users of HOE?
We have some information posted at our Web site www.borenson.com and I invite anyone to access the Validation page there and look for him or herself. We do a participant survey at the beginning of each seminar. We ask the participants for a show of hands of how many of them are confident that they would be able to successfully teach the equations 2x + x + x + 2 = 2x + 10 and 2(x + 4) + x = x + 16 to the large majority of all their students, in all of their classes, using the traditional methods by which they were taught algebra in high school. Occasionally, one or two hands will go up. The rest of the participants, however, simply know from their own experience that there are far too many students they are not able to teach these concepts to using the traditional teaching methods, even in the eighth or ninth grade. We then proceed with the morning portion of the HOE seminar. At the conclusion of the morning session, we again ask the same question to the participants, but this time, we would like to know what their response would be if they had available a class set of HOE to use with their students and the methods presented at the seminar. Now, the numbers are completely reversed. Close to 100 percent of the participants feel they would be successful in presenting the above concepts using HOE, even to their third or fourth graders, or "lower ability" middle school students. The teachers take cognizance of the dramatic change in self-perception of their ability to teach these concepts that has taken place within the span of a few short hours. And since we use the coaching model in our seminars, the participants have actually experienced HOE themselves, and they see how it simplifies the concepts for them. They leave the seminar with actual skills they can use.
This change in teacher attitude is dramatic and consistent across seminars conducted throughout the United States. Any participant attending a seminar is able to witness this transformation in teacher self-perception of their ability to teach algebraic concepts using HOE vs. the traditional methods.
It is important and essential for teachers to realize that it is up to us, the educators, to develop the techniques and the means by which the children can learn.
Q: What is next for you? What do you want to accomplish that you haven't done yet?
I'm spending more of my time learning and teaching Talmud. I am in Israel a significant portion of the year. More and more of my time is taken up with these intellectual and spiritual pursuits. I'm still very involved with my business, but I'm trying to move more in the direction of this other type of learning and teaching.
Q: Who are your heroes, your mentors?
Well, my mentor really has been someone named Dr. Edward P. Gottlieb, who is now 97 years old. I have known him for more than 35 years. He was a nationally recognized educator, quite innovative—for example, he is the one, to the best of my knowledge, who is responsible for a number of innovations in schools, including movable tables and chairs in classrooms, libraries located in schools, breakfast programs for students. Over the years, I've always consulted him about issues I've had. I met him originally in the anti-war movement in the 1960s. At that time, he was President of The War Resisters International, a pacifist group, which interested me. The educator Paul Goodman had suggested I contact him, and a friendship developed between Eddie and myself. He has had a number of interests—he has been a poet, who has written about 3,000 poems; and he was very active in the anti-war movement, the civil rights movement and of course in education.
Whenever I needed encouragement, or whenever I was not sure about the course I was pursuing, Eddie was there to give me a lift.
Q: How would you want your children to remember you? What is your legacy to future generations?
We don't have our own children. However, in Judaism, we say that those students to whom we teach Torah are like our children. So, from that sense, I have many children. In terms of my work in mathematics education, I would like to think I would be remembered for making the lives of millions of children more pleasant in the classroom through HOE, because a lot of terrifying experiences were avoided. I would say that's a significant contribution.
Q: One more question: Have I not asked you a question you wanted to answer?
Whenever we do seminars, each instructor is asked to come up with an inspirational close to end the seminar. And the one that I give is the one about my teacher at the Brooklyn Technical High School in New York City, Isidore Glaubiger. I remember him for many reasons. One was, whenever he gave a quiz, and a student received less than a 100% grade, he would let the student retake it after school in order to try to obtain a higher grade! So, I often ended up taking the same quiz or test several times, seeking a higher mark. I accomplished a higher mark, even though it took a lot of work, but I did it on my own time and of course, I learned more doing so. Many years later, when I was working toward my doctorate in educational administration at Teachers College in Columbia University, I called Mr. Glaubiger to invite him to my graduation. He couldn't attend, but he agreed to talk to me about his secret of success as a teacher over the phone. I got my pen and paper to take notes, and this is what he told me: "If you really want the students to learn, you will do whatever is necessary. You will find new methods, or you will change your approach in the middle of the lesson, so that they will learn." And that was always in the back of my mind, so when the children had trouble learning algebra, I remembered: If they're not learning, I need to change my approach, and that is what lead me to search for a way to simplify these concepts. When I began, I did not know this search would last about 1000 hours. But the outcome of this work is the teaching system knows as HANDS-ON EQUATIONS®.
The reason I tell this story is this: It is important and essential for teachers to realize that it is up to us, the educators, to develop the techniques and the means by which the children can learn. It is not satisfactory for teachers to simply say, "Well, they don't have the ability, they can't cut it," or "I can't do anything about that, the curriculum is set." The teacher's task is to see what they can do to help the student be successful.
When I was working on my master's and teaching at Stuyvesant High School in New York City, I had a supervisor named Henrietta Midonick. One day I showed her a test I had prepared for my students, and she looked at it and questioned me: "What's this question doing on this test? This is a very difficult, challenging question—why did you include it on this test? Are you trying to trick your students? Are you trying to show them what they can't do?" It stayed with me. Unfortunately, very often, teachers want to stump the students, like a game, rather than to ensure their growth or to help them discover what they can do. That should be our goal.