Andrew, an academically advanced 4th grader, sat down at his desk at the start of math class. Opening his iPad, he began the required fluency practice in math facts with automated flash cards. He slumped in his seat, frustrated that he had to do this work again, even though he had mastered the same math facts in 3rd grade. He decided he wouldn’t complete it. Why should he? Andrew exited the app and put his head down on the desk.
Why Readiness-Based Differentiation Matters
Scenarios like this have become all too common in the aftermath of COVID-19. Gaps in students’ readiness levels for engaging with math content widened as a result of the pandemic because of disparate access to high-quality math instruction (Giorgio-Doherty et al., 2021). Teachers are tasked, now more than ever, with differentiating instruction in ways that respond to variations in students’ readiness levels. Teachers must “teach up” and design learning experiences that require students to work within their zones of proximal development—that is, at a level that is neither too difficult nor too easy (Vygotsky, 1978).
Because teachers typically don’t receive adequate preservice training or professional development in differentiation, they may struggle to differentiate effectively (van Geel et al., 2019) and end up “teaching to the middle.” In general education classrooms, students with varied readiness levels—those who need additional scaffolding to reach high-level goals, as well as those who have already mastered those goals—may not receive instruction within their zones of proximal development. Teaching to the middle is an equity issue because not all students have an opportunity to learn in meaningful ways. In addition, the pressures related to high-stakes testing may encourage some teachers to devote most of their attention to supporting their struggling learners, sometimes to the exclusion of other students’ needs.
The truth is, training teachers to implement readiness-based differentiation would likely yield better learning outcomes for all students because every student would have the chance to engage with higher-level material once they’re ready to do so. However, in the absence of training in differentiation, elementary math teachers may overlook the needs of students who are ready to engage with more complex material.
Some long-debunked misconceptions may result in fewer opportunities for advanced students to engage in meaningful learning. For example, teachers may believe that students who have already mastered grade-level content are “fine on their own” and don’t need teacher support (Berman et al., 2012) and that there’s little need for differentiated or accelerated instruction (Steenbergen-Hu & Moon, 2011). Another misconception is that advanced learners should serve as peer tutors once they’ve completed their own work. In fact, this type of peer tutoring may benefit neither the tutor nor the tutee, and rarely does the advanced learner experience academic growth (Berman et al., 2012).
Giving advanced students busywork may lead to underachievement or disengagement—the opposite of teaching up.
It’s not surprising, then, that many advanced learners are frustrated and bored in classrooms where readiness-based differentiation is underused (Hertberg-Davis & Callahan, 2008). The busywork these students are often given may lead to negative attitudes toward school, underachievement, or disengagement—the opposite of teaching up (Callahan, 2018).
Three Strategies for Teaching Up
Math teachers sometimes have difficulty determining what more rigorous or complex instruction for advanced learners looks like (Paige, Smith, & Sizemore, 2015). Therefore, in our work as university teacher-educators, we use Webb’s Depth of Knowledge (DOK) framework to show our pre- and in-service teachers how to make tasks more complex by considering the levels of cognitive demand required to complete those tasks. Webb’s DOK has four levels that are organized in ascending order, from lowest to highest complexity:
- DOK 1: Information recall
- DOK 2: Conceptual understandings and skill application
- DOK 3: Strategic thinking
- DOK 4: Extended thinking
At DOK 3, students should be challenged to think strategically about math concepts and engage in real-world problem solving. At DOK 4, students must extend their thinking to make connections among different math concepts or across different disciplines. However, learning at DOK 3 and 4 is not reserved solely for advanced students; all students should receive opportunities to engage with rigorous thinking. Advanced students may simply be ready to work at these levels sooner than their peers or may require less scaffolding.
Because not all students demonstrate advanced readiness levels for all math content at the same time, it’s important to look at multiple data sources—from classroom assessment scores to benchmark testing information to observations of students’ needs—and commit to flexible groupings to respond to students’ shifting needs (Tomlinson & Imbeau, 2010). Some students may quickly master identifying equivalent fractions using multiplication and be ready to move on to new material in a few days, whereas other students may require additional time and scaffolding to do so. With flexible groupings, advanced learners could work with fractions in a more abstract way. Other students could continue to use manipulatives to make the process of visualizing fractions more concrete, with their end goal eventually being to understand fractions more abstractly.
The three instructional strategies that follow will help you push your advanced learners to engage meaningfully with DOK 3 and 4 tasks.
Strategy 1: Differentiated Math Fact Fluency Games
Math fact fluency is essential for developing mathematic proficiency. However, academically advanced students like Andrew should not have to continually repeat fluency practice once they have mastered their addition, subtraction, multiplication, and division facts. Instead, teachers should shift practice opportunities from DOK 1 (information recall) to DOK 3 (strategic thinking). One approach is to select a math fact fluency game that requires students to strategically apply their knowledge of all mastered math facts to solve a problem. This strategy applies to any grade level.
For example, take a game like Math 24, which requires students to determine ways to make 24 with four numbers shown on their game card. The card might show the numbers 8, 4, 3, and 7. Students can use any operation to formulate a problem that results in 24. Andrew, as an academically advanced 4th grader, might subtract 3 from 7, leaving him with a difference of 4. Then, he might multiply this 4 by the 4 provided on the number card, resulting in a product of 16. Finally, he might add 16 and 8 to find the sum of 24.
It is important to note, though, that Math 24 isn’t solely for advanced learners. Students of all readiness levels can benefit from these games that encourage numerical fluency and skill-building. However, an advantage of using these games is that their difficulty can be adjusted based on variations in readiness levels. For example, a teacher could require a group of advanced math learners to use multiplication and division in all their solutions, and ask another group not yet ready for those operations to use only addition and subtraction until they build up their other skills. The goal would be for each group to work in their zone of proximal development, using flexible grouping as needed.
Strategy 2: Multi-Solution Paths for the Problem of the Day
Many math classrooms start with a problem of the day that typically aligns with the current topic of study. Students like Andrew can often solve these problems easily. To meet the needs of advanced learners, ask them to find multiple ways of solving the problem by using multiple representations, such as concrete models, pictorial representations, and traditional algorithms.
Let’s look at an example:
Problem of the Day: Sally loved playing video games. On Saturday, she started playing Minecraft at 9:10 a.m., and she continued playing until 12:34 p.m. How long did Sally play video games on Saturday?
Initially, students might opt for an abstract approach using subtraction: 12:34 p.m. – 9:10 a.m. = [elapsed time]. Next, they might use an open number line to illustrate the problem: 9:10 a.m. + [elapsed time] = 12:34 p.m. Finally, they might get an actual clock and arrange the hands to 9:10 a.m., then move the hour hand ahead by three hours to 12:10 p.m. and the minute hand ahead by 24 minutes to 12:34 p.m. Students like Andrew might explain that the open number line represents a concrete solution, whereas the abstract approach of subtraction demonstrates the relationship between inverse operations. By asking students to solve the problem in multiple ways and compare the various solutions, you can challenge their ability to think strategically about elapsed time (DOK 3), not just engage them in the surface-level skill of solving the word problem using the correct algorithm (DOK 1 or 2).
Strategy 3: Adjusted Independent Practice
To reduce the repetitive and unnecessary practice of procedural problems (DOK 1), teachers might engage students in an ongoing project-based learning activity related to the topic of study (DOK 4). This can extend student thinking and require them to translate their mathematical knowledge into a real-world scenario and make connections across concepts or disciplines.
Project-based learning requires students to translate their mathematical knowledge into a real-world scenario and make connections across concepts or disciplines.
For example, Andrew’s 4th grade classroom is learning how to solve multi-digit addition and subtraction problems, including subtracting across zeros. Because Andrew can solve these types of problems easily, a teacher might give him an open-ended project-based learning activity instead, such as asking him to plan his ultimate vacation using 97,000 airline travel miles. Over several days, Andrew can research the number of miles between selected vacation destinations; determine the number of remaining airline miles; and calculate a budget for the cost of transportation, food, and the various activities planned. This project will require him to conduct numerous multi-digit addition and subtraction problems in a real-world application of mathematical skills.
Because project-based learning opportunities are less structured, they’re inherently more abstract. There isn’t one correct answer to the “ultimate vacation,” so the project provides more opportunities for students to engage in deep problem solving. In addition, teachers can structure the “ultimate vacation” project to make connections across a range of disciplines. For example, 4th graders in science who are learning about renewable and nonrenewable energy sources might calculate the amount of aviation fuel required to complete their “ultimate vacation” and describe the environmental effects of airline travel.
What Teachers Need to Teach Up
All students should have the opportunity to learn every day. For some, this means having instructional experiences with increased levels of rigor that may differ from what the traditional curriculum offers. Because creating these experiences requires intentional planning on the part of teachers, school leaders should ensure they have the time and resources needed for professional development that supports this work.
A good starting point would be for school leaders to plan a series of ongoing professional learning opportunities in which teachers are asked to:
- Unpack their beliefs about advanced learners to rethink any misconceptions they may have.
- Develop skills in collecting, organizing, and analyzing data, as well as expertise in instructional design, to enable them to differentiate for varied readiness levels.
- Add to their repertoire of instructional strategies that encourage deeper, more complex, or more accelerated engagement with math concepts.
To begin this process, school leaders and teachers can explore the Knowledge Center on the National Association for Gifted Children’s website, which offers strategies and best practices for teaching gifted learners. Math-specific resources guided by Webb’s Depth of Knowledge framework are also helpful (see here). With this knowledge in hand, teachers can begin their journey toward teaching up for students like Andrew so that advanced learners have equitable opportunities to engage in meaningful learning every time they step into the math classroom.
Reflect & Discuss
➛ How do you differentiate instruction to address the needs of your academically advanced learners?
➛ What math strategy for teaching up has been particularly successful with your elementary students?
➛ What professional development would best support you in meeting the needs of your advanced learners in math?
