When introducing a new math concept, begin by anchoring students' understanding in a concrete representation before progressing to a semi-concrete and then abstract representation of the concept.
concrete ---> semi-concrete --> abstract
For example, let's say you're teaching multiplication for the first time. Instead of beginning by showing students the times tables, you'll want to develop their understanding that multiplication is repeated addition. Start in the concrete stage:
CONCRETE REPRESENTATION OF 3 x 3
Have students manipulate counters, buttons or other objects, creating an array. An array is simply an arrangement of objects into rows and columns. This array has 3 columns and 3 rows, illustrating the multiplication fact 3 x 3. Students can count the total number of counters to reach the answer "9."
When students fully understand how to create arrays out of counters, move to the semi-concrete stage:
SEMI-CONCRETE REPRESENTATION OF 3 X 3
In this stage, students are moving away from using concrete objects they can manipulate, such as counters. Instead, they represent these objects on paper by drawing the array using circles, tally marks, stars, etc. Here you can begin introducing more mathematical language ("We are multiplying three times three...") and show how the array is represented by the equation 3 x 3 = 9.
Finally, after students demonstrate mastery drawing arrays, they're ready for the abstract stage:
ABSTRACT REPRESENTATION OF 3 X 3
When your students can draw arrays to show their understanding of multiplication, it's time to take away that support. In the abstract stage, students are dealing solely with numbers. Numbers are abstract because they're merely squiggly lines that don't mean anything until they're attached to an amount. It's in this stage that students begin to memorize their times tables and record 3 x 3 as an equation ("3 x 3 = 9) without the use of counters or other manipulatives.
The amount of time it takes your student or child to progress from one stage to the next varies. Some kid will take a few days, some may need a few weeks. The key is to give students the time to internalize the concept at each stage before moving into the next.
To summarize the progression when introducing a new math concept:
CONCRETE: using physical objects students can manipulate
SEMI-CONCRETE: making a drawing or other visual representation of the manipulatives without physically holding them
ABSTRACT: using solely numbers without the use of manipulatives or drawings to solve the problem
Consider how you might use this progression when teaching subtraction, adding fractions with like denominators, or comparing decimals to the tenths. How have you progressed from the concrete to the semi-concrete and then abstract with your own students?
I'm thrilled to be writing our very first blog post- welcome! We opened the doors to Luminous Learning nearly four weeks ago, after months of planning and researching and creating materials. With each passing day, we become more excited about the positive impact we can make in students' lives. I want to pass on some of that excitement to you by telling you a bit about why Luminous Learning exists and how you'll find our blog useful.
I worked with a teacher whose students would go to the corner deli during their lunch period. Midway through the school year, the teacher noticed that his students returned with suspiciously little change for the snacks they purchased. Accompanying his students to the deli, the teacher watched as his students went up to the cashier to pay for their snacks. One by one, the students received the wrong amount of change.
The students were affected by a myriad of learning disabilities-- ADHD, dyslexia, speech and language impairments, dyscalculia-- that hindered their mathematical reasoning skills and their ability to perform basic math calculations. Rather than struggling to decide how much change they should receive, the students trusted the cashier to hand back the correct amount. The cashier took advantage of this; five dollar bills became one dollar bills, coins disappeared. Fortunately, the teacher was able to intervene and prevent his students from getting shortchanged.
This is a common problem for students with learning disabilities but there won't always be a teacher or parent present to protect their students. We're here to help make sure your students don't end up in a similar situation.
Luminous Learning allows students to practice the math skills necessary to become confident math learners. Our worksheets are specially crafted with visuals and clear examples that support students' understanding of math concepts. Together, we can help your students and children navigate math in the real world on their own.
We're creating our blog to be a resource for teachers and parents of students with learning disabilities. We'll be summarizing the latest education news and highlighting the best instructional practices in teaching math, especially to struggling math learners. We want to give you practical tips and strategies that you can begin to implement in your classrooms or with your children today.
We hope you're as excited as we are to help your students and children become more confident mathematicians. Check back with us regularly to learn math strategies and tips! And please always feel free to leave comments about your own experiences in education.