The result of three research studies involving a total of 22 classrooms and 418 students is summarized above. The percentages shown are the grade group averages for the given test. We note that at each grade level 1) the students had a large gain from the pre-test to each of the post-tests, 2) the students did equally well using the pictorial notation (last column) as they did in using the game pieces (middle column), and 3) the program is age-blind: the 4th graders did as well as the 6th and 8th graders. Pre-test questions: a) 2x = 8, b) x + 3 = 8, c) 2x + 1 = 13, d) 3x = x + 12, e) 4x + 3 = 3x + 6, and f) 2(2x + 1) = 2x + 6. Click here for the full report.
The table above shows the gain the 4th graders made in only seven lessons. We note that 79% of these 4th graders, 60% of whom were inner city students, were able to successfully solve the equations 4x + 3 = 3x + 8 after seven lessons of instruction. It is worth noting that the retention test was administered three weeks after the instruction for Lesson #7, with no Hands-On Equations instruction in the interim. Furthermore, the retention test was taken without the use of the game pieces. Click here to view the full report.
Excerpts from Published Papers
"By using this approach, proceeding from the concrete to the abstract, U.S. students can exceed their age/grade counterparts in high achieving countries on this goal (of representing multistep word problems using a letter for the unknown)." (Page 25)
Borenson, Henry (2012). "Are the common core state standards for mathematics in grades three and four reasonable? Rethinking word problems using a letter for the unknown." Newsletter of the National Council of Supervisors of Mathematics (NCSM), Volume 42, Number 4, 24-25.
"Hands-On Equations is an algebraic learning environment in the sense that students are often able to develop some abstract concepts on their own." (Page 27)
Borenson, Henry (2011). "Demystifying the Learning of Algebra Using Clear Language, Visual Icons, and Gestures." Newsletter of the National Council of Supervisors of Mathematics (NCSM), Volume 41, No, (3): 24-27.
"Experience with nonsymbolic equivalence problems can actually lead to improvements on subsequent symbolic problems" (Page 97)
Sherman, Jody and Jeffrey Bisanz (2009). "Equivalence in Symbolic and Nonsymbolic Contexts: Benefits of Solving Problems with Manipulatives." Journal of Educational Psychology, Volume 101, No. 1, 88 - 100. American Psychological Association.
"Gesturing can thus play a causal role in learning, perhaps by giving learners an alternative, embodied way of representing new ideas. We may be able to improve children's learning just by encouraging them to move their hands" (Page 1047)
Cook, Susan Wagner, Zachary Mitchell and Susaan Goldin-Meadow (2008). "Gesturing Makes Learning Last." Cognition Volume 106, Issue 2, 1047- 1058.Elsevier B.V.
"...algebraic reasoning is associated with and embedded in many differrent representational systems....students need to be introduced to mathematical notations in ways that make sense to them" (Page 34)
Brizuela, Barbara and Analucia Schliemann (2004). "Ten Year-Old Students Solving Linear Equations." For the Learning of Mathematics, Volume 24, Issue 2, 33-40. Kingston, Ontario, Canada. FLM Publishing Association.
"Our results show that the balance model is especially suited to the study of how to solve equations. In fact, the isomorphism between the object itself and the mathematical notions implied allows students to form a mental image of the operations that they have to apply. They are able to reactivate this self-evident image at any moment" (Page 355)
Vlassis, Joelle (2002). "The Balance Model: Hindrance or Support for the Solving of Linear Equations with One Variable." Educational Studies in Mathematics 49, 341-359. Netherlands: Kluwer Academic Publishers.
“I want ALL my students to discover math for themselves, and Hands-On Equations does this. It guides them to a higher and more sophisticated level of mathematical thinking and understanding” (From "Marylin's Reflections," page 299)
Raymond, Anne M. and Leinenbach, Marylin (2000). "Collaborative Action Research on the Learning and Teaching of Algebra: A Story of One Mathematics Teacher's Development." Educational Studies in Mathematics 41, 283-287. Netherlands: Kluwer Academic Publishers.
"These children are not only able to solve correctly these ninth-grade algebraic linear equations, they are also able to verbalize what they are doing and the logic behind it. In other words, through these concrete experiences the children are developing a sound mathematical sense about algebraic liner equations and how they work" (Page 55)
Borenson, Henry (1987). "Algebra for Gifted Third Graders." Gifted Child Today, Volume 10, No. 3, 54-56. GCT, Inc.