Currently making my way through the National Council of Teachers of Mathematics (NCTM) publication Principles to Actions: Ensuring Mathematical Success For All. It’s pretty much a how-to for teaching math. They discuss eight aspect of effective teaching and learning. For each, I’ll share thoughts, questions, and aha moments.
- Establish math goals to focus learning
- Implement tasks that promote reasoning and problem solving
- Use and connect mathematical representations
- Facilitate meaningful mathematical discourse
- Pose purposeful questions
- Build procedural fluency from conceptual understanding
- Support productive struggling in learning mathematics
- Elicit and use evidence of student thinking.
1. Establish math goals to focus learning
Both teachers and students should be able to answer the following:
- What mathematics is being learning?
- Why is it important?
- How does it relate to what has already been learned?
- Where are these mathematical ideas going?
I think these questions could be an option for what to write in their math journal blogs (starting those this year). Maybe the first few blogs I make it mandatory, but then as they get a feel for writing about math, they can sort of take it where they want.
When thinking through a lesson, don’t focus so much on what the students will be DOING, but rather what you want them to be LEARNING.
There was an emphasis on students being able to gauge and monitor their own learning progress. I pondered different ways to do that. I know the blogging does that in some regards, but I was planning on having them blog maybe once or twice per unit. That’s not really monitoring your learning as much. I’ve been reading a lot about formative assessments and the value of using them FOR learning instead of measuring what hasn’t been learned yet. More short quizzes, maybe one a week. Students are able to see what they’re getting and what they’re not and it helps me see where I need to move my instruction.
2. Implement tasks that promote reasoning and problem solving
Comparison of low-level demands vs. high-level demands:
Memorization– what is the rule for multiplying fractions?
Procedures w/out connections– 2/3 x 3/4 5/6 x 7/8
Procedures with connections: Find 1/6 of 1/2. Use pattern blocks. Draw your answer and explain your solution
Doing mathematics: Create a real-world situation for the following problem: 2/3 x 3/4. Solve the problem you have created without using the rule and explain your solution.
What I struggle with is finding great tasks for them. Even in that, giving them a high quality task doesn’t necessary mean they will engage in it at a high level if I don’t let them really struggle through it. The book gives an example of two teachers doing the same task but one teacher sort of just bailing her kids out as soon as they start to struggle while the other teacher asks more leading questions without giving them an exact way to solve it.
It’s the art of learning how to help them without taking over the process of thinking for them.
3. Use and connect mathematical representations
I struggle with knowing how to visually represent some abstract mathematical ideas. I want the students to be able to do it but they’ll have a tough time if they haven’t seen the concept visually represented before. I think they’re more able to show visual representations of a mathematical idea if you move from concept to procedure instead of the other way around.
There was an example of a 3rd grade teacher who was introducing multiplication and had his students model a problem that involved rows of chairs. Several students drew pictures, others made tables. Afterwards he wrote 7 x 20 on the board and asks the students to explain how that expression matched each of the models. I think this would be valuable in working with one or two-step equations. Having students model it before I explain what’s going on but then show them the equation and ask how it matches their models. It ties it all together to the concept you want them to understand. In the past I’ve had them model and solve things on their own but didn’t do that last step of helping them connect it to the more conventional representation of the concept.
4. Facilitate Meaningful Mathematical Discourse
I feel I had some excellent discourse last year in my classes but found that I need to be more mindful of the following:
- Anticipating student responses prior to the lesson
- Monitoring student’s work on and engagement with the tasks
- Selecting particular students to present their mathematical work
- Sequencing students responses in a specific order for discussion
- Connecting different student’s responses and connecting their responses to key mathematical ideas
I never put a ton of thought into the order in which students present. I did anticipate student responses in that I usually knew which misconceptions would probably arise, but I think it would be a good practice to write some of them down and think about how I could tactfully respond.
In the example given in the book, a teacher presented a problem with multiple solutions and allowed students to share at the board. The first student had the most basic way of finding the solution, the second was a little more complex, and the third being the one that took it to the level that hit on the concept the teacher was trying to teach (unit rate). After having them explain, he labeled the methods and asked students which they think would be most efficient and why. He also asked questions that led students to making connections between the methods and connections to the learning target. This is where I want to grow in; it’s the part that sort of ties it all together!
5. Pose Purposeful Questions
I feel this skill is woven into the other ones pretty consistently. It was more about learning to ask probing questions to get students to share their thinking. I made this spreadsheet full of questions to ask students in group settings as well as one-on-one.
6. Build Procedural Fluency from Conceptual Understanding
As I’ve learned more about having a inquiry-based math classroom, it almost seems like I was hearing ‘procedures are bad, kids don’t learn from them, they need to understand the concepts.’ It’s good to know that isn’t the case. My challenge is taking their conceptual understanding and bringing them into procedural fluency and having them gain an understanding of why the standard algorithms work.
A big part of that is having various visual models to help build that conceptual understanding and having students connect those visuals with the more conventional algorithms. Having students first struggle to create their own procedures will make sharing the more efficient and standard procedures a lot more meaningful.
7. Support Productive Struggle in Mathematics
I appreciated their distinction between productive and unproductive struggle. Unproductive struggle doesn’t lead to any progress in sense-making or conceptual understanding. I’ve found myself in that place before in class. You can feel it. You’ve given students something that they aren’t really ready to grapple with and it sort of leads nowhere. Their answers are all over the place and no one is even coming close. Finding that Goldilocks spot; not to easy, not too difficult, but just a little bit out of reach.
In the chapter, there’s a link to THIS video about a warm-up activity titled ‘My Favorite No.’ Give 1 or 2 problems to start out class with, have them answer them on index cards, collect them up, look through them super quick, sorting between yes and no (you can do this in front the class…saying out loud ‘yes…yes…no…no…yes…no…yes’), then pick your favorite ‘no’ and write it up on the board. You’re looking for the one that has a pretty common misconception that you’re excited to talk about. First ask what that student did right? Have students share. Then ask what misconceptions there may be. Ungraded formative assessment, quick, valuable, promotes growth mindset, shows value of mistakes, and just an awesome practice.
8. Elicit and Use Evidence of Student Thinking
One practice that I thought would be valuable is to have students solve a problem and then find someone in the class with an answer that is different from their own and compare/discuss their solutions. Probably works best with problems that have multiple methods or something with quite a few common misconceptions. Then it goes back skill 4 where you decide who to come up and share and what order. You’re eliciting student thinking both in the peer discussions and the group discussion.
The section of the book sort of ends with what I believe would make an excellent mathematics teaching philosophy in that they summarize the above eight aspects of effective teaching and learning. To solidify my own learning, I’ll summarize it in the form of my mathematical teaching philosophy:
I believe that learning happens in a math classroom when there is a clear vision of what we’re doing and why. I believe the responsibility of the teacher is to present tasks to students that would promote mathematical reasoning and problem solving where they have opportunities to think about numbers in a variety of ways. I believe a math classroom should be full of questions; questions from students to teacher, from teacher to students, from students to students; that a healthy and respectful discourse of mathematical ideas is consistently present. I believe that conceptual understanding will lead to familiarity and fluency with procedures and standard algorithms, not the other way around. I believe that students learn when they are having a productive struggle and are evaluating their own work as well as the work of their peers. I believe that the teacher should always be looking to learn what the students understand/comprehend in order to more effectively plan the next course of action. I believe all students can learn mathematics.