The changes in the math curriculum might seem small but they represent a very large philosophical shift.
The traditional approach to teaching math has been to focus on procedures. We taught students how to do the math but there wasn’t a big focus on when we would use the math or why the procedures work. Nor was there a focus on big Ideas or connections. We would teach the topic of the day by introducing it, showing the students how to solve and do some worked examples while they wrote notes. Following this, students would practice pages of questions. So, why are we moving away from this approach?
The research is clear and plentiful; most students learn best by seeing and doing and so using manipulatives and teaching for meaning helps students understand the math, engage in their learning and achieve success. We do not need to create human calculators, we need to create mathematical thinkers and problem solvers. We can teach them how to do procedures in a way that develops number sense, mathematical reasoning, problem solving and communication. We can also really engage them to become active participants in their learning- math is not a spectator sport!
How using manipulatives changed everything for me
Eight years ago I began researching about how children and adolescents learn math. Both brain-based research and educational research have been exploring this topic for decades. When I examined my traditional approach and compared it to what the research indicated, I decided to try this “new way” (it’s not really new at all but it was certainly new to me). I was hooked immediately, mostly because the level of engagement, confidence and understanding of my students skyrocketed. I have continually been amazed at the difference in the students’ attitudes, participation, mathematical reasoning and problem solving skills when they are taught with the focus on really understanding the math rather than just following procedures.
Many teachers might now be wondering ‘what does this new way look like in a classroom?”. First, the focus of the lesson has changed: my goal is for my students to develop fluency- both at a conceptual and procedural level, rather than solely at a procedural level. Here’s a brief example of how I now structure many of my lessons:
The sequence of my lessons
When I first introduce a topic, I don’t tell the students how to do any procedures at all. I start with a goal and it is usually a goal to develop conceptual understanding. Then I give them a task that I call APK or accessing prior knowledge (from the SMART learning framework). This is the time where I guide my students to the important connections we are going to make with previous learning. This is like glue for the new learning so it ‘sticks’ more firmly to the brain (really we are creating neural pathways). Using manipulatives, I pose different problems and ask my students to try to solve these problems. An example might be: show me what 4 x ⅓ means using the Cuisenaire Rods (I have taught them how to use the rods as a tool already). In order for a student to do this, they must understand what multiplication means (having their times tables memorized without any understanding will not help them here).
They need to know that multiplication means ‘groups of’ or repeated addition when one factor is a whole number. Through the exploration, students begin to develop a conceptual understanding. They are not necessarily writing anything down yet but I will write the symbolic (using numbers) on the board once they tell me their solutions. We do a number of these so that they are really understanding what this means and then they start to develop the procedure.
Our brains are natural pattern-finders so they will start to try to figure out what is happening with the numbers. I then ask “if we didn’t have these rods, how could we solve this numerically?”. This is towards the very end of the lesson (or sometimes the next lesson). They have not practiced yet, they have not been given notes, or a procedure but what they have been given is time to form meaning and understanding for themselves. According to Sousa (How the Brain Learn Mathematics), these are the first two factors needed for learning to be retained in the long-term memory. The focus is on understanding the concept not just the procedure. This alone shows a significant change from the traditional approach.
Students who are learning this way are learning through solving problems. I also use a lot of partner talk and reflections as this helps them to develop their mathematical reasoning and communication skills. It also unearths many misunderstandings that can then be corrected well before they start practicing. However, I would say that the most important aspects of my lessons are not the procedures at all but rather the processes.
Processes are the key
Mathematical Processes are the vehicle through which we teach concepts and procedures. The processes help students develop not only stronger conceptual understanding but also really valuable life skills. These processes are: communication, mathematical reasoning, making connections, problem solving, visualization, using technology, and mental math and estimation. If the focus is on the processes rather than the procedures, we will help our students develop strong number sense, flexibility in their thinking and solid problem solving skills. If a student has these, they can recreate or figure out a forgotten procedure (if you don’t use it you lose it). If a student has only ever memorized a procedure and they forget it after time, the chances of them being able to figure it out are pretty low because they never actually knew why they were doing the steps they were doing.
Is using manipulatives too time consuming?
It may seem time consuming to use the manipulatives and do the approach I outlined above, but there is far less practice needed once students actually understand what they are doing and why. There is also a lot less time spent on ‘reviewing’ (I often found I was not reviewing but re-teaching) previous topics. If we slow down and teach with meaning and understanding, students will have the foundational skills and knowledge to learn new and more complex math as they progress through the grades.
The manipulatives are not to be used as a once off, to tick off the box in a prescribed learning outcome, but rather they are used as an important tool to develop conceptual understanding and to help students solve problems. Even when we work solely in the symbolic, we need to be constantly questioning and reminding our students of the meaning and the deeper concepts so that the conceptual understanding deepens with procedural practice.
A parting comment on procedures…
Practicing procedures is easier for most and a lot less effort to learn, not to mention way quicker, but it does not translate to long-lasting learning or conceptual understanding for many students. It is also not what math is all about! Math isn’t just computations – it is a complex web of connections and relationships. Let’s give our students the opportunity to explore and perhaps even appreciate the beauty of mathematics!
Educating Now was created due to teacher requests to have Nikki as their daily math coach. The site has lesson by lesson video tutorials for teachers to help them prep for their next math class and incorporate manipulatives, differentiated tasks, games and specific language into their class. Teachers who use the site can improve student engagement and understanding, in addition to saving prep time, by watching a 10 minute video tutorial and downloading a detailed lesson plan.