Do you ever feel pressured to ‘cover the curriculum’? How often do you notice that some of your students have not understood much of what you just taught? What do you do when the pacing calendar indicates it is time to move on and your students are not ready to move on? Do you find that some of your students come to you without the requisite knowledge expected by the grade level standards? I have been teaching for more than 33 years and I have yet to meet a teacher who does not face the dilemma of how to manage both coverage and depth for the students in her class. This dilemma has existed since factory model schooling was implemented over 100 years ago and has been exacerbated by the pressure of high stakes standardized testing and the accountability movement begun under the Bush administration. What are teachers and principals to do?
Unfortunately, schools that serve populations of students who are behind when they enter the school—whether at the elementary, middle, or high school level—generally feel this pressure even more than suburban schools that serve students whose families (and schools) have more resources, and whose students have larger vocabularies and a wider range of experiences. I have worked in urban, suburban, and rural schools all over the USA. Surprisingly, the teaching in high income suburban schools is not necessarily any better, and in some cases, it is less sophisticated, than the teaching in some inner-city schools. Yet the results are very different. High income students usually do well and inner city students coming from high poverty neighborhoods often do not. When we use test scores to evaluate teacher effectiveness, the teacher in the urban setting is blamed for poor student test results and the teacher in a wealthy school gets credit equally undeserved. Clearly this is unfair and a misuse of standardized test results when there are far too many variables other than teaching that go into the test results.
Right now, we err on the side of coverage, and I’m advocating we tip the scale in the direction of depth. This means taking a risk. The risk of shifting our focus away from covering laundry lists of outcomes to engaging in learning processes that develop habits of mind that are more likely to serve our students throughout life. When we focus on the process of learning the content at hand rather than coverage of discrete facts and specific information, our students will learn to learn as well as learn the information they need to pass a test.
Take a Big Picture-Long Term Approach
How do we balance coverage and depth in a way that more students learn more deeply? We learn to think differently. Here’s what I mean. If we want students to retain what they learn and build upon it, then we need to invite them to think about and discuss the topic at hand analytically, from multiple perspectives, so that they will ultimately be able to articulate their understanding and/or perspective orally and in writing. This implies slowing down, covering less trivia, and focusing on the important underlying ideas.
I’m suggesting that we take a bigger picture, longer term view than unit or lesson planning. For example, if we were to really consider what the biggest, connecting concepts were in any content area, and where they repeatedly recur in the curriculum within a school year and across the grades, we would realize that we have less to cover and a few overarching ideas students have to own in order to fully grasp the subject. The old adage “slow down to speed up” would apply here.
If we took a big picture—big ideas approach, we’d need to review fewer topics from year to year and decrease the repetition that permeates the curriculum. We as educators would come to see that much of what we teach each year are examples of ideas at play. For example, adding, multiplying, and solving equations all employ the distributive property. Helping students own that property and analyze when it is used will give them solid grounding for many applications in mathematics through high school and beyond. By making explicit how the procedures you are using in the examples under study are the same as the procedures used in another context, and how those procedures are actually utilizing the same property (e.g. distribution) we would help students make more connections (generate more synapses in the brain) and help ourselves to slow down and dig deeper.
It is in large part the repetition we engage in that saps a great deal of time from year to year. I once read a study that showed that from 3rd grade on, 30-40% of what was covered in math textbooks was review from the previous year! This repetition is often advocated to combat lack of retention that is caused by failure to slow down and explicitly connect procedures to big ideas in the first place, and by failing to give students opportunities to articulate their understanding in writing so we could assess how much they fully grasped the ideas the first time. If we slowed down in the first place and really examined important ideas with our students, encouraged them to make connections among the ideas and their accompanying procedures every time they showed up in the curriculum, and asked students to articulate their reasoning and questions related to these ideas, they would own the concepts and we would have more time to teach new concepts and skills.
Using an example from elementary mathematics, let’s think about equivalence. Equivalence shows up in mathematics from basic counting through higher levels of mathematics, including algebra and geometry (Charles, 2005). Yet, we rarely focus on equivalence as an overarching theme in mathematics when developing number sense, operations, place value, or even geometry. Instead we offer a diet of worksheets requiring oodles of computations. The repetition of solving equations does not necessarily result in an understanding of equivalence. We see this in the ubiquitous tendency of students as late as middle school to think that the equal sign means, “What comes next?” rather than equivalence. For example, when faced with 7= 4 + __, many students choose 11 as their answer.
Math teachers tend to teach mathematics as discrete topics (e.g. addition, subtraction, solving equations) and focus lessons on correct answers. They tend to focus on the procedures that allow us to compute or find answers to equations or geometric situations, and sometimes tell students the important idea or property underlying the procedure. If instead, we were to focus on the centrality of ideas like equivalence and highlight equivalence in all its many guises, then we might find that students are making more sense of operations, equations, or congruence. (This is not a new idea, it is actually advocated in the CCSS and was part of the NCTM standards offered back in 1989.)
If we knew that equivalence is going to show up repeatedly throughout the year(s), then we can worry less about getting through every lesson in a unit of study or handing out every worksheet, and think more about the lessons that are essential for our students. This implies knowing our curriculum materials well—not just the materials used in our grade level, but the grade level prior and the one after the grade we teach.
Do you know to what degree your curriculum materials focus on procedures or big ideas or both? So often the curriculum materials we use are not designed to focus on big ideas, and offer hundreds of worksheets for practice instead. Teachers are sometimes prompted in the “teacher notes” section of the textbook to state an important idea during the lesson (as part of the script the teacher is supposed to follow). However, following a script is not an effective way to help students learn. Students need to discover and articulate the idea by being nudged to notice when it shows up in various contexts. These contexts (e.g. contextualized problems) need to be offered to students by teachers in thoughtful sequences that help students uncover important ideas. The ideas need to recur several times and the teacher needs to highlight these ides when the students start to notice and articulate them in their own ways. Telling students up front or writing ideas as an objective on the board is generally ineffective in getting students to own the idea and counterproductive if you are aiming for depth.
In too many math classes the focus is on the correct answer. The emphasis is on efficient algorithms to compute or formulaic ways to solve problems. This can be true even when teachers are employing “math talks” or other techniques that are meant to develop student reasoning and discourse. If the emphasis is still on the correct answer, the pedagogy may have changed, but the focus of the teaching has not.
Other than reminding students what we did yesterday, we rarely ask students to connect the major idea(s) running through the series of activities or problems that we may be working on over the course of a week or a unit or a year. In too many classes, we are still feeding students a diet of worksheets to practice skills and not emphasizing enough the need for students to analyze and make sense of why the procedures work. In other words, a superficial approach to content results in a lack of depth in understanding. Which leads to wasted time reviewing concepts over and over again. This is not just true in mathematics. It is true in all subject areas.
We can navigate the tension between coverage and depth by starting with the big picture and planning out our year around the major ideas, rather than thinking only about the individual lesson or unit. This allows us to select and adapt the most important lessons from our curriculum materials and to chunk standards in ways that revolve around the ideas.
We can further ameliorate the tension by building our capacity to assess in detail the specific issue(s) that each student is grappling with and cultivating a repertoire of strategies to meet each student where he/she is. This means cultivating a more conversational style of teaching and a wider variety of ways to gather information (e.g. moves like, “Tell me more about that,” during discussion, stop and jot moments, exit tickets focused on student self-assessment of what they’ve learned and what they still have questions about) about what our students understand and are wrestling with. Cultivating student discourse, asking students about their thinking, and listening to students as they express ideas gives us a window into how well they understand what we think we taught. We can then use that information to gage whether to dive deeper or move on.
While we would like a pat answer to the coverage verses depth tension, the truth is that it will persist so long as we have factory-like schedules in our schools. The challenge for all of us is to work within the structures we have in more sophisticated ways until we become brave enough or wise enough to create new structures that tilt the balance towards deeper learning.
Charles, Randall, Big Ideas and Understandings as the Foundation of Elementary and Middle School Mathematics. Journal of Mathematics Education Leadership, volume 7, number 3, 2005
Forschungszentrum Juelich. (2013, October 10). New theory of synapse formation in the brain. Science Daily. Retrieved December 9, 2016 from www.sciencedaily.com/releases/2013/10/131010205325.htm
Author Lucy West.