Walk into an early elementary school classroom and you are likely to see lots of manipulatives. From Popsicle sticks to Cuisenaire rods we have a strong intuitive sense that these objects should help children learn mathematics.

Although manipulatives can increase students’ attention, this attention may not benefit their learning. In fact, the very aspect of manipulatives that capture students’ attention—bright colors, visual appeal, realistic features—may be their downfall. Manipulatives that are more visually interesting or realistic can increase off-task behavior, such as building or sorting (1). This is especially true if students interact with that object in other contexts, such as during play time or outside of the classroom.

Students who learn with manipulatives can become too reliant on the object and context, and as a result, have difficulty transferring their knowledge to new contexts, different testing formats, or to abstract representations (e.g., algebraic expressions) of the problem

“Often derided as “drill and kill” or making children “parrot” what is being taught, the UK report and other research suggests that memorization and rote learning are important classroom strategies, which all teachers should be familiar with.”

Turn to chapter 39 of Steven Smith’sThe Great Mental Calculators and you will read about Arthur Benjamin. Benjamin teaches mathematics at Harvey Mudd College. He is also extremely skilled at mental arithmetic. Here is a video of him in performance:

And here is a video of Benjamin talking about the development of his skills:

“Humans and nonhuman animals share an approximate number system (ANS) that permits estimation and rough calculation of quantities without symbols. Recent studies show a correlation between the acuity of the ANS and performance in symbolic math throughout development and into adulthood, which suggests that the ANS may serve as a cognitive foundation for the uniquely human capacity for symbolic math. Such a proposition leads to the untested prediction that training aimed at improving ANS performance will transfer to improvement in symbolic-math ability. In the two experiments reported here, we showed that ANS training on approximate addition and subtraction of arrays of dots selectively improved symbolic addition and subtraction. This finding strongly supports the hypothesis that complex math skills are fundamentally linked to rudimentary preverbal quantitative abilities and provides the first direct evidence that the ANS and symbolic math may be causally related. It also raises the possibility that interventions aimed at the ANS could benefit children and adults who struggle with math.”