If primary students cannot add or multiply properly, they will not be able to add fractions or find common denominators. If secondary students are unable to graph a set of coordinates on the Cartesian plane, it will be impossible to understand how to graph a linear function or any function for that matter.
When I was a student studying the unit circle in a trigonometry course, I used my understanding of fractions to correctly fill out the radian measures around the unit circle. Admittedly, this is also the time I finally arrived at a mastery level of understanding fractions. Before trigonometry I merely understood the rules that I needed to follow in order to correctly add, subtract, multiply, divide, and compare fractions. It wasn’t until I used fractions in a different context that I gained this higher level of understanding.
The acknowledgement of prerequisite concepts is important for students (not just teachers) so they can understand what needs to be understood in order to be successful in a new mathematics course. This leads us to the question, “How do we assess/know what a student understands?” I would argue that readiness/placement exams are important, but only if they are used properly. The validity of what one of these assessments (usually filled with multiple choice questions) can tell us has been overestimated. Because of this overestimation, critics claim that these assessments shouldn’t be allowed or even that they are racist because of the assessments’ results. So the outcry has become, “Abolish them!” However, the need for such an assessment is still there, and prerequisite knowledge is about much more than gatekeeping students from coursework based on the results of a single assessment.
The weight placed on single-shot assessments needs to be lessened because they may not accurately reflect a student’s readiness for the course work. Two important reasons for this include test-induced stress and the last time the student engaged with the topics (for example, Algebra 2 comes after a full course in Geometry, which means there are a handful of Algebra 1 concepts that are prerequisite to Algebra 2 concepts that were not worked on by your average new Algebra 2 student for almost a year and a half!). We must consider the other factors (e.g., grades in previous courses, teacher recommendations, etc.) that can help a student understand what is informationally needed for success in a course. But this is a topic for another post.
Students need to have this information because study time is finite. It must be rationed. Contrary to popular belief, the challenging mathematics course is not the only thing happening in the student’s life at that moment. Knowing this, it would be a crime to not inform students of what they need to know before they take a course. Educating students on where they stand in relation to the course content also helps students avoid feelings of failure that can inhibit their desire to go further in their studies in mathematics. Basically, instead of blocking students from taking a course, they should be informed of their current position on the trajectory of math content so that they can make the choice on whether they can afford the cost of time needed for them to be successful.
Knowing where students are in their understanding can help make an educator aware of the mini-lessons that might be required in the current subject’s lessons for a student to be “caught up.” Like I said earlier, some prerequisite concepts might have been taught and learned earlier in the student’s education, but they might need some reminders. Most of the time a quick reminder of the skills needed for that prerequisite concept is all that is needed for the student to be able to study the concept at hand in that current subject of study. In other words, when teaching an Algebra 2 student how to find roots in order to graph a polynomial function, a mini-lesson (reminder) on how to factor might need to happen for 2 minutes in the middle of the lesson. Mini-lessons make a good educator.
When thinking through these ideas, I realize there is more that can be expanded on, like “How many mini-lessons are enough?” or “How do we inform students of their position before taking a mathematics course?” Yet we must acknowledge the fact that prerequisites for a course are important. We should not ignore them. Math does, in many ways, build on itself. It is true that some concepts do not, but that does not diminish the importance of the concepts that do. Let’s not throw out the baby with the bath water.