How Do We Teach Math Well?
Over the past few weeks, I’ve been discussing the ongoing debate about how to teach math well. As the science of reading continues to gain awareness and is increasingly adopted by US schools, attention has now shifted to how to best teach math.
This conversation is a really important one for parents to be aware of because it has a direct impact on the way our children are being taught math at school.
In my first post on the topic, I discussed the substance of the disagreement (aka the Math Wars) and what the research says about this disagreement. (You can read that here.) I then shared an overview of the five interwoven and interdependent strands of math proficiency (here). Understanding the components of math proficiency is essential in the next step of this conversation: How do we actually teach math well?
It is at this point that I want to pause and be very clear on the current state of the research on how to teach math: Much of the current debate about how to teach math well stems from a belief that the research on the progression of math understanding over time and the cognitive science research on how the brain learns are in competition with each other. This is not my belief. My understanding of both the math concept progression research and the cognitive science research is that we can use what cognitive science tells us about how the brain learns in order to more effectively teach math concepts.
I continue to build my own understanding of both of these fields of research, and will continue to share what I learn here. In this post is my current working model for what must be included in effective math classes based on 15 years of working in math education, coaching and training hundreds of math teachers, and reading dozens of math education books and as much research about math and how we learn as I can find. I’ve been in a lot of great math classes and I’ve seen a lot of math classes that weren’t effective. It is the combination of all of these experiences that have led me to my list of non-negotiables in math class.
Why a list of non-negotiables and not a step-by-step explanation of what every math class should look like?
The best math teachers I’ve ever worked with move fluidly between a structured version of inquiry that I describe here and direct instruction (when the teacher shows the student how to do the problem and the student repeats this process).
As Adding It Up states, “Much debate centers on forms and approaches to teaching: “direct instruction” versus “inquiry,” “teacher centered” versus “student centered,” “traditional” versus “reform.” These labels make rhetorical distinctions that often miss the point regarding the quality of instruction. Our review of the research makes plain that the effectiveness of mathematics teaching and learning does not rest in simple labels. Rather, the quality of instruction is a function of teachers’ knowledge and use of mathematical content, teachers’ attention to and handling of students, and students’ engagement in and use of mathematical tasks. Moreover, effective teaching—teaching that fosters the development of mathematical proficiency over time—can take a variety of forms.” (pg. 314)
I’ve seen inquiry lead to confused and frustrated students, but I’ve also seen it lead to magical "aha" moments when the pieces click in students’ minds.
I’ve seen direct instruction lead to students who mindlessly repeat a procedure again and again without understanding why it works or when they should use it again, but I've also seen it lead to magical "aha" moments when the pieces click in students’ minds.
It is not that one way of teaching is right and the other is wrong. It is that there is a right time and a wrong time to use each of these approaches.
Because of this, I now have four non-negotiables in math class that help me determine the balance of when to be direct in my instruction and when to step back and let students make connections. I'll discuss two of those this week and the other two in next week's email.
(1) The complexity of math concepts must progress over time: Students need time to work with math concepts through concrete representations (like snap cubes or base ten blocks) and inefficient strategies first in order to make sense of the underlying structures and patterns in the math they’re learning about. From Adding It Up, “We show that students move from methods of solving numerical problems that are intuitive, concrete, and based on modeling the problem situation directly to methods that are more problem independent, mathematically sophisticated, and reliant on standard symbolic notation. Some form of this progression is seen in each operation for both single-digit and multi-digit numbers.” (pg. 181)
These concrete representations and inefficient procedures are just the first layer of understanding a math concept, and instruction should help students move past inefficient strategies towards more efficient strategies. For example, when first learning multi-digit multiplication, students use a visual called an area model to break apart the numbers by place value to better see what is happening in each place value as we multiply. But this procedure is not efficient. It’s designed to build understanding of the concept, not efficiency. And so, over the course of 5th grade in the US, students should stop using area models to multiply multi-digit numbers and start using the standard algorithm for multi-digit multiplication. But, because of their work with area models, their understanding of the standard algorithm will hold greater meaning.
(2) Students need to make connections among the mathematical concepts they've already learned: When possible, students should be given time to make leaps or connections in their understanding without a teacher directly modeling the connection. For example, if a student already knows how to calculate benchmark percentages of a number (such 10% or 25% of number) and they're fluent with fraction and decimal operations, then they should be given a chance to think about and make the connection to calculating 15% or 35% without a teacher immediately assuming they need to be shown how.
I often call this guided inquiry but structured inquiry would work well too. Students aren’t asked to come up with new math knowledge on their own, but they are allowed the time and space to make connections between mathematical concepts when they already have all the necessary prior knowledge to successfully make that connection. In the science of learning, this process is described as effortful thinking: “Teachers can prompt deeper, ‘effortful’ thinking with elaborative questions and tasks that cue thinking about relationships between ideas. These prompts often start with ‘how’ or ‘why’.” These tasks cue students to think about relationships between math ideas they already understand. In order to most effectively make these connections, students must have time to discuss and sharpen their ideas and understanding with their peers.
But students won't always make the connection on their own. Sometimes they start to put the relevant pieces together, but need help to see fully how it all connects. Other times when there isn't relevant prior knowledge to connect, the teacher should model the new math concept for students. It is logical that students cannot derive information or make connections when there is not yet knowledge to be connected.
Students should never be left struggling with no purpose, but they should be allowed to do the full level of mathematical thinking they are currently capable of doing independently. Understanding the difference between these two scenarios requires a deep understanding of the math content we are teaching.
(3) Regularly assessing what students know and addressing misconceptions is key: Data on student understanding should be collected daily so that teachers can quickly intervene when a student doesn’t understand and before instruction moves on to a new topic. As I mentioned in #1, math is hierarchical (built in layers) and so if instruction moves on to a new layer while a student is still not clear on a previous layer, the student’s struggle will grow. The most effective teachers I know check for student understanding regularly throughout the lesson. An essential component of assessing student understanding is to give a daily exit ticket (1 to 3 problems aligned to that day’s lesson) or look at one particular problem in the students’ practice problems to determine how well students understood the concept for the day. (This is a type of formative assessment.) Teachers then use this information to appropriately adjust upcoming lessons to respond to students' level of understanding.
(4) Practice matters and some types of practice are more effective than others: Students need a lot of time to practice math, but not just on the day a concept is first taught. A concept called “retrieval practice” is extremely effective in ensuring that students solidify their understanding of math concepts and store that understanding in long term memory. Retrieval practice states that students need practice that is both spaced and interleaved. Spaced practice means that the practice of a skill doesn’t just happen on the day it’s initially taught. Instead, the skill is first practiced on the day it’s initially taught and then practiced again, repeatedly, spaced out across the school year.
Interleaved practice is practice that mixes up multiple, related concepts. An example of interleaved practice is solving word problems that involve a combination of addition, subtraction, multiplication, and division scenarios rather than just addition word problems. This allows students to build stronger understanding because they aren’t able to just replicate a single procedure over and over, but rather have to spend time discerning differences among the problems and determining the appropriate strategy to solve. Finally, students should get regular feedback on their practice so that they are able to correct mistakes and adjust their thinking rather than repeatedly practicing the skill incorrectly.
My hope is that understanding these four non-negotiables in math instruction can empower both parents and teachers in the conversations they have when a child is struggling. For example, if a parent feels the strategy their child is using to solve a problem is really cumbersome or inefficient, they can ask the teacher, “What is this strategy designed to help my child understand?” or “When should I expect my child to start using the standard algorithm for this concept?” If a teacher notices that the curriculum the school uses doesn’t have opportunities for spaced, interleaved practice, they can discuss this with their principal or instructional coach and find the right way to make sure students are getting this necessary mixed review practice.
Finally, as a parent, this is why math conversations and games at home can have such a big impact–they often provide opportunities for exploration of math concepts and continued spaced, interleaved practice. I’m working on a resource for parents that includes my favorite math games and activities for each grade level (toddler through middle school) and will share that soon!