Now in the world of performance tasks and student-centered math instruction, manipulatives are essential to getting the job done right.  We are no longer just teaching algorithms for computing answers to problems.  Students are expected to understand the underlying concepts that make the algorithms work.   Manipulatives help make this happen.

Manipulatives take on the form of paper folding, drawing models, using two color chips, attribute blocks, base-ten blocks, rulers, compasses, and even computer software.

There’s a great site on the internet called The National Library of Virtual Manipulatives. Here you will find a plethora of Free interactive manipulatives that you can be used to guide students into a conceptual understsanding of mathematical skills.

Virtual Manipulatives is a post from: Georgia Math Coach

By the time students reach upper elementary many have lost the idea that one item placed with another item became two items;  1 + 1 = 2.   This may be a slight exaggeration, but the idea is that students lost the means to comprehend what 1 + 1 really means.

They somehow forget that 3 x 4 represents three groups of four objects. The older elementary students lose the idea that two fractional amounts put together create a new fraction, whole number, or mixed number.

Starting with second grade, educators emphasized the use of algorithms in computation.  These algorithms for addition, subtraction, multiplication, and division were quick ways to guide students to an acceptable response.  But without realizing it, we destroyed number sense when teaching students to solve by algorithm.  Students lost the value of place in a number.  The lost the meaning of the symbols +, -, x, and / .  Furthermore, they become unable to explain with words what a simple computational expression represents.

Now, let’s not blame the educators.  We were only teaching the way we were taught in elementary school all the way through our college days.  We even earned a teaching degree and took the big certification test that simply required computation without understanding.

Today we understand that in order to maintain number sense, we must develop concepts before we teach algorithms.  Students must be able to verbally communicate what happens to numbers when objects are added, subtracted, multiplied, and divided.  They must be able to provide reasons that their computations work and are valid.  They must be able to use models (manipulatives) to demonstrate their understanding of concepts.  Today, math is much more than manipulatiing symbols.  To develop conceptual understandings that maintain number sense, we must engage students in learning with performance tasks that mimic real world applications.  Let students discover the algorithms with a little help from the teacher.

What have we done to Number Sense? Part 1 is a post from: Georgia Math Coach

When I entered education the big word was “developmentally appropriate”.  The word still sounds good today…if you maintain a strict level of definition the philosophy of “developmental appropriateness” works well. But somehow, once the idea got dumped into the real world of education, it became something that essentially meant refrain from pushing the kid too much lest we damage self-esteem and frustrate attitudes toward learning. Over the years, what actually happened was the creation of lazy minds with a false esteem.  The boredom of tiny incremental doses of instruction still frustrated attitudes toward learning and created a generation of people who to this day feel no loyalty or obligation to be productive contributors in a democratic society.  Instead, we’ve created a generation that believes they are exempt from correction and entitled to the benefits of the highest positions in society.

The new big word in education is “Rigor”.  Seems like somebody got smart about what the term “developmentally appropriate” had become and decided that we needed to do a little something about all the soft, mushy, lazy brains we created. “Rigor” really is a good idea, but as the years go by on this term I hope we don’t take it to the extreme.  Rigor refers to creating a challenging instructional environment that encourages kids to take their learning one step further. Students are encouraged to put some effort into there work, pursue deeper understandings and thoughtful investigations into the concepts they are learning.

There’s another big word floating around now too…”Differentiation”.  With differentiation, teachers don’t create simple lessons for students with less ability or hard lessons for those with greater ability.  Instead, students are expected to meet the same standards, but instruction and practice varies according to the way a student learns best.

Many teachers do not understand how rigor and differentiation are properly practiced in the classroom.  It’s certainly not because we are poor educators.  Instead, it’s because teacher preparation courses have not set the example for instruction.  Teachers (including those of us who are veterans) have never personally experienced rigorous differentiated instruction.   Furthermore, to get such instruction in a time of budget crisis requires the teacher to make sacrifices of personal time and personal finances to learn what this stuff is all about.  Also, once a teacher finds the needed staff development courses, they are often told what needs to be done…but not shown how.  In other words, the course work seems hypocritical…do as I say…not as I do.  I have taken a bunch of staff development courses…most of which have never been put into regular classroom practice.

Fortunately, I finally found a two-year elementary math endorsement program offered through a Math and Science Grant offered by the state of Georgia (pioneerresa.org) that really brought a change to the way I teach math.  The program was very rigorous, and it paid me…literally and figuratively. Three of the four courses in the program not only told me what to do, but they also showed me how to do it.  Get this…we didn’t discuss rigor or differentiation in the courses.  But somehow, the skills and concepts were so well presented that the rigor and differentiation were built in and any teacher looking for it could see it in action.  Yes, there were some moans and groans from the teachers taking the coursework (but don’t we all moan and groan when confronted with a challenge).   Yes, some of the teachers didn’t get it or even notice what was happening when the professors were teaching by example rather than just telling us what to do. Thanks to the program, I not only benefitted from learning more about instruction in the different math strands, but I also learned a great deal about rigor and differentiation through example (rather than direct instruction).  I believe the program worked very well for me.  In 2009,  11 of my 21 students exceeded requirements on the Georgia CRCT in Math (and Reading, too)!  Nine of them met requirements.  The one that did not meet requirements, didn’t receive instruction from me (and that’s another story).   I don’t want to think this is some fluke.  I want to see the same thing happen in the years to come.

“Rigor” and “Differentiation” are not frightening words to me anymore.  They are actually very redemptive!  I no longer have to feel guilty for being too easy OR too hard on the kids.

Rigor and Differentiation: Not so Frightening is a post from: Georgia Math Coach