Typically, before introducing students to the traditional way of solving math problems, I use manipulatives (ones cubes, tens sticks, hundreds blocks, etc) or printed models (pictures) to help students think about and understand what the computation really means. Using hands-on activities and guiding students through different types of questions helps me assess the depth of their understanding. My goal is not to simply teach students to memorize steps and procedures, but to ensure that they understand why they are doing what they are doing in each step.
When we think about learning skills and concepts, we should imagine those skills and concepts on a continuum of learning with children at different levels of readiness. Given a specific concept, your child may be at different levels at different times. I found the following descriptors from Kathy Richardson (she's a math guru that develops materials for assessing students' true understanding and misconceptions):
Ready to Apply (A) – The student can already do a particular task and is ready to use this skill in other settings. (This student receives more challenging work).
Needs Practice (P) – The student can do a particular task with some level of effort but still needs more experiences to develop facility and consistency. (This student typically receives work at grade level that increases in difficulty as his/her readiness increases.)
Needs Instruction (I) – The student has some idea of what a task is about but needs support. (This student receives direct instruction that begins with the lowest level of their individual understanding and builds up to problems with increased difficulty to help meet grade-level expectations.)
Needs Prerequisite (N) – The student does not yet understand the concept and needs to work with
mathematical ideas that precede the concept being assessed.
When I discovered these descriptors, I really wanted to jump up and down--I was excited because these categories of learning really capture how I think about individual students' understanding of a math concept and how I choose materials and create groups for focused instruction.
It is also important to consider the size of the number(s) when placing students at these levels for a given concept--the size of the number with which they are independently successful needs to be taken into consideration. I often find that concepts that students may seem to have "mastered" are merely in the process of truly being understood when they are presented with larger numbers. As we know, accuracy also becomes a larger issue as the size of numbers increase. To instruct students having difficulty (say learning the procedures for how to multiply 23 x 67), I begin by taking a step back and instructing them on how we would solve a simpler problem, like 23 x 7. Once the student has consistently demonstrated that they are able to complete problems at this level, we move on to adding a digit in the tens place for the second factor.
Next up...
I'm going to share examples of problems that are below, on, and above grade level based on NC state standards for math. When you see your child's math work, you can use these levels to have a better understanding of what they have accomplished. If they are successfully completing below and on grade level problems, but having difficulty with above grade level, this means that they are where they are supposed to be and that they are being challenged to push beyond the average 4th grade expectations. If they are having difficulty with "on grade level" problems, rest-assured that they are being served in a small group that meets them where they are and works to help them build up to solid grade-level abilities.
By the way, I'm ready to reveal a secret...I'm a bit of a "math nerd" and I think that's cool!
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