Saxon Math (use of manipulatives)

"The difference between young children and us is that we can switch freely from the manipulative mode of thinking to other modes we have learned to use. But very young children can't switch. They are tied to the manipulative mode."

-Ruth Beechik, An Easy Start to Arithmetic

While working to aid my children in understanding math concepts, my understanding and appreciation for manipulatives has grown exponentially. I originally thought manipulatives were simply objects which children (students) used to count or see three dimensional figures. I assumed they were to be used as a last resort, used only when they failed to grasp a particular math concept.

excerpts below from
Math Education: Manipulatives http://heatherpitcher.blogspot.com/

A mathematical manipulative is “an object which is designed so that the student can learn some mathematical concept by manipulating it” (Wikipedia, 2006). Students can use objects as counters, play with geometrical shapes to form other shapes, can build models to understand three dimensional figures, etc. In the manipulation of these materials, students are able to learn abstract concepts in concrete, hands on ways. These materials make even the most difficult mathematical concepts easier to understand for the student (Uttal, Scudder & DeLoache, 1997).

Manipulatives have been around for many years. One of the early versions of a manipulative, the abacus, can be dated back to 300 B.C. (About, 2005). This abacus, known as the Salamis Tablet, was used by Babylonians and was discovered in 1846 (About, 2005). Manipulatives have developed greatly from this early counting device.

A new push for the use of manipulatives occurred in the 19^th century when Pestalozzi lobbied for their use, eventually making manipulatives part of the mathematics curriculum in the 1930’s (Sowell, 1989). In the 1960’s, another resurgence of the use of manipulatives occurred, with a focus on the use of concrete objects and pictorial representations to help children better understand abstract mathematical ideas (Sowell, 1989). Now, manipulatives are available in almost every classroom around the world.

As previously stated, “manipulatives help children visualize abstract mathematical ideas (Heuser, 2000, p. 288). Students are able to use hands-on activities to create a knowledge base for mathematical thinking, allowing a greater understanding of the nature of mathematics, and some of its basic concepts, at an earlier age (McCarty, 1998). This is based on the findings of people such as Piaget, who helped prove that young children at the Primary and Elementary grades think at the concrete level. Therefore, the use of concrete objects in teaching abstract ideas would bring these abstract ideas down to the students’ concrete level, making problems tangible and tractable for these young learners (Uttal, Scudder & DeLoache, 1997).

The value of mathematical manipulatives can also be seen in the work of Driscoll, Sowell, and Suydam, who all discovered that students who use manipulatives outperform students who do not use them (Clements and McMillen, 1996). And this is not just true of students at the concrete level of thinking; students in all grade and ability levels, as well as students working in many topics, benefit from the use of manipulatives (Clements and McMillen, 1996). In the Driscoll, Sowell and Suydam study, retention and problem solving test scores were also improved if the students were exposed to manipulatives, and “attitudes toward mathematics [were] improved when students are instructed with concrete materials by teachers knowledgeable about their use” (Clements and McMillen, 1996, p. 270).

The Saxon Math program provides and develops its early curriculum around the use of manipulatives to teach math concepts. It is very careful to form a tangible connection between the manipulatives and the numbers, encouraging MEANINGFUL comprehension, rather than a rote learning. The program allows the child to engage with the manipulatives DURING the lesson. Yet AFTER the manipulatives are put away, the child must reflect on their learning and connect the manipulative with the mathematical concept being addressed. This is done through the worksheets (a morning worksheet and an afternoon worksheet) provided for each lesson.

Since beginning the Saxon Math program, Aiden has truly embraced and become more excited about math. Much of the manipulative use is just fun repetition at this point, as many of the concepts have been introduced to him over the course of the last two years. Yet, in contrast to years past, I DO NOT UNDERESTIMATE the value of manipulatives in early math.

In her book, An Easy Start in Arithmetic, Ruth Beechick discusses the different attitudes we need to be aware of when we teach math, the different ways children "see" math, and what we can do to promote effective learning in the early years. It was in this short BOOKLET that I came to the realization that there are different modes of thinking about arithmetic. Young children use the manipulative mode of thinking. Ruth Beechik says, "the difference between young children and us is that we can switch freely from the manipulative mode of thinking to other modes we have learned to use. But very young children can't switch. They are tied to the manipulative mode. They must become proficient in this mode as a preparation for other modes to follow. This thinking--this experience with objects--is the foundation upon which all later arithmetic understandings are built. THUS, we much teach young children in the manipulative mode." She believes that the failure to start them in this mode is the greatest single cause of children's arithmetic difficulties and anxieties.

So, math has become the hot subject in the McCarty household. And I have to give the credit to the curriculum switch and Saxon's fun use of manipulatives. (I realize, though, that Saxon isn't the only curriculum that applies the use of manipulatives.) For today, more linking towers and fun with geometric shapes.

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