Mathematics is one the subjects that generations of students have been struggling with and still do. Many students still find it difficult to comprehend concepts and the methods by which math teachers deliver these concepts in classrooms. So, why is this the case? Why do some math topics seem very comprehensible and some become like a nightmare for our students? Mathematics is challenging in all aspects whether it’s 3+5 or finding a discriminant, but it’s the ideology and methodology of teaching that makes all the difference, and that is the pathways educators take to facilitate math lessons. As a student teacher, I recently took a math methods class at the university as part of my degree requirements, which turned my whole knowledge of math upside down and challenged its practicality. The method we were introduced was called “CGI Framework”, which is the abbreviation of “Cognitively Guided Instruction” and is defined as a student-centered, student-driven approach to teaching math. So, what does this mean in classrooms and how does it apply to the modern math education? Let’s expand a bit more about CGI in couple paragraphs that took me a semester to explore only tiny part of the bigger idea.
The crucial schema of CGI is based on the students’ prior knowledge and builds on their natural capabilities in problem solving and number sense. The framework confronts the traditional way of teacher-centered teaching where the teacher lectures and students take notes, where the teacher provides the strategies by which students are expected to solve problems, and the students’ minds are framed in only the instructed ways. If you don’t see the problem yet, that’s ok, because I didn’t see a problem in this at first either. Have you ever solved a math problem never really knowing why it works, or what each component really meant in a real life situation? For example, we know that when you divide a fraction by another fraction, we do what we call KCF-keep, change, flip. But do you really know why you change the sign and flip the second fraction? Most probably not, and that’s because we’ve never been taught why division of fractions works that way. We were only taught to solve it that way, because that just works! The same understanding issue can also be observed in simple operation problems such as adding and subtraction of numbers, where kids are taught to carry or borrow the one with minimum or no explanation of the meaning of that one.
These types of problems create a false representation of “student-centered” classrooms because it’s not up to students to do the problem solving but rather solving the way that instructor teaches. With the CGI method which puts the emphasis solely on student intuitive problem solving skills that are naturally given to them, educators have witnessed crucial changes in their classroom culture. Instead of teaching kids solving strategies teachers rely on students’ thinking to solve math problems in a way that makes sense to them. Let’s consider this simple addition math problem:
There were 13 kids playing at the beach. Some of the kids went home. There were 6 children still playing at the beach. How many children went home?
If you’ve been taught in a traditional school, you’d most likely solve it by 13-6=7. This is exactly how the teachers teach students to solve similar math word problems. The issue that the CGI framework supporters see in the given solving strategy, is that the problem initially tells that some kids went home and only 7 kids stayed, while what teachers do to solve the problem is they “send” 6 kids home (13-6) to get the answer, where in reality, 6 is the number of students that stayed and not left. It might seem normal to an adult brain, but take a minute to be in a second grader’s shoes, who is probably going to use models to represent the problem and solve it from beginning to end direction, and not from end to beginning. Here’s a rough representation that I saw in a screen casting done by my professor and his colleagues where the student solved this problem.
Some went home, until there were 6 children left at the beach. So the kid starts crossing out children until he gets to 6. What’s phenomenal about this process is that the kid has never been taught a math strategy for these types of problems but still was able to solve the way that the problem was narrated and the way that made sense to his little creative brain.
Now consider the following problem:
“There were 13 kids playing at the beach. 6 of them went home. How many kids were still playing at the beach?”
The U.S. education system intervenes with student thinking by creating a sense that the first math problem is the same as the second math problem, where in algebraic language the first problem can be represented by 13-x=6, while the second problem is 13-6=x. Based on the child’s intuitive solving skills, the child would get an easier time to solve the second problem because all he/she needs to do is take out 6 and count the rest, while in the second problem he/she needs to take out as many until he/she gets to 7 and count and the seven again to make sure it’s the correct amount. Both of these math problems sound similar which is why many teachers consider them same level of difficulty and assign kids the first problem with the expectations that students can solve it like the first problem. The key lies within bring able to differentiate math problems based on the content, purpose, and difficulty and scaffold them as the student progresses.
Allowing kids to learn math with their own thinking, makes their work appreciated and meaningful in a way that now not only they have something to learn from the lesson, but educators as well have something to learn from our students, and that is how genius their minds can be if we just let them think over their thinking. You’ll witness not only 2-3 ways of problem solutions but so many more! This is where the beauty of teaching comes in, where as an educator you get to observe all the fascinating approaches students take solve a mathematical problem.
Learn more about CGI Framework at https://blog.heinemann.com/what-is-cgi
Big thanks to Dr. Jeffrey Shih, my math methods class professor whose teaching transformed my math viewpoints and made an enormous impact in writing this article.