Recently, I began a MOOC (massive open online course) called How to Learn Math – For Teachers and Parents (2). More than 40,000 people took the last class – mainly teachers, parents and school administrators. 95% of people completing the end of course survey said that they would change their teaching or ways of helping as a result of the course. The course offers important new research ideas on learning, the brain, and math that can transform students’ experiences with math and is based on the work of and narrated by Jo Boaler. It’s divided into 8 sessions and each one ends with a prompt to write a one paragraph summary reflection on the ideas. My plan is to post those reflections here.
The seventh session is titled, Appreciating Algebra. There are 3 ideas that spoke to me here.
Students struggles in algebra stem from learning structural algebra vs procedural algebra. Structural algebra is thinking of a variable x, as any number, like in the following problem:
How many total tires are there on 10 tricycles?
Structural algebra is more about studying and describing patterns. It’s also about the most important part of algebra – learning to generalize or describe the general form of a pattern. Procedural algebra is thinking of a variable x, as one single number. So x in the equation, 2x+5=9 would be thinking of x as one single number. Procedural algebra is about solving for x. The difference between procedural and structural algebra seems to follow a common theme throughout this course. And it’s that we need to concentrate more on the creative aspects in mathematics not only because it promotes more engaging learning and conceptual understanding but also because it promotes a better mindset in learning mathematics. Research tells us that students find it very difficult moving from procedural to structural algebra. So if we spend a lot of time asking students to solve for x or think of one single number for x in their early school years and then move to describing patterns and generalizing them, that’s a big conceptual leap. It’s interesting though, the research says that studying structural algebra first and then transitioning to procedural algebra does not introduce the same struggles for students. This is not a significant conceptual leap.
Students struggle with the meaning of the equal sign. In the early years, if teachers focus their attention on addition and subtraction facts, students often come to believe that the equal sign is an instruction to do something. An equal sign signifies a symmetrical relationship between each side. So if students don’t recognize this relationship they run-on their answers attaching the next operation onto the last answer and the calculations end up on one line across the page – equal signs abound and equality, nowhere to be found. Knowing that the equal sign means equivalent is an important idea in helping students make sense of algebra.
Learning algebra should be about enabling learners to use algebra to make sense of the world. This isn’t a new idea in mathematics or education in general. When learning requires us to solve real problems that are meaningful to us, there’s more engagement with the learning. In teaching algebra we need to identify those situations that the students are excited to investigate and need to explain, predict, and model. Students should be encouraged to use varying representations of their patterns with visuals, graphs, tables, and descriptions. Collaborative problem solving is important in making sense of rich modeling situations as well as using new technologies to engage in and explain them. In an effort to help students make sense of a problem, we often use letters or variables with the same starting letter as the object being described. Consider the value, in cents, of a number of nickels and dimes as in V = 10d + 5n. (d represents the number of dimes, n represents the number of nickels) Research tells us that using the first letter of objects leads to a classic misconception in algebra. It leads students to think that we’re adding 10 dimes and 5 nickels. I can see how that could happen because in most Ontario grade 9 textbooks, operations with algebraic expressions comes before linear relations. So the procedural algebra comes before the structural algebra. It’s important to remember to use other letters in this case so as not to confuse the value of the coins with the total number of coins.