A search on the internet for “CUBES math strategy” yields about 36 million results. Seeing a bright, cheery, and colorful CUBES poster on display in mathematics classrooms is a typical occurrence. Elementary and middle school mathematics teachers who are familiar with the Common Core State Standards for Mathematics (CCSSM) or their state’s version of CCSSM might know that the phrase “solve word problems” shows up 20 times in the grades kindergarten through grade seven standards, signaling its importance (National Governors Association, 2010).
In elementary mathematics education, problem-solving strategies like CUBES (Circle, Underline, Box, Evaluate, Solve) and keyword approaches have been staples in the toolkit of teachers aiming to guide young learners through puzzling mathematical tasks also known as story problems. According to the National Assessment of Educational Progress (NAEP) report, “the average fourth-grade mathematics score decreased by 5 points and was lower than all previous assessment years going back to 2005. The average eighth-grade mathematics score decreased by 8 points compared to 2019 and was lower than all previous assessment years going back to 2003. In 2022, fourth- and eighth-grade mathematics scores declined for most states/jurisdictions as well as for most participating urban districts compared to 2019” (NAEP, 2022).
This alarming trend again underscores the need for comprehensive reforms in mathematics education across the United States, emphasizing teaching methodologies grounded in research, increased support for educators, and targeted instruction to address learning gaps and promote mathematical proficiency among all students.
Structured and procedural methods, like CUBES, that are often offered as support to students as they solve word problems need to be eradicated. The ability to think critically and independently is essential for all students. In this blog post, we’ll explore why reliance solely on CUBES and keyword strategies may hinder the development of cognitive skills and why it is imperative for teachers to prioritize teaching students how to think instead.
The Pitfalls of Structured Strategies
Structured problem-solving strategies may offer a sense of security and direction for both students and teachers. However, such scaffolds come with inherent limitations that can impede genuine understanding and hinder the cultivation of critical thinking abilities. Recipe-like directives also prevent students from engaging in tasks that are of high cognitive demand and collaborative, for example.
1. Rote Memorization Over Conceptual Understanding
Structured strategies often encourage students to follow a predetermined set of steps without truly grasping the underlying concepts. This can lead to a surface-level understanding based on rote memorization rather than conceptual understanding and opportunities for making connections, seeing patterns, and thinking mathematically, in general.
2. Lack of Flexibility and Adaptability
While CUBES and keyword strategies may be effective for certain types of problems, they can be restrictive when faced with more complex or unconventional scenarios. Students may struggle to apply these rigid frameworks outside of the specific contexts they were taught, limiting their ability to adapt their problem-solving skills to novel situations. Powell, Namkung, and Lin (2022) analyzed 747 high-stakes released items across grades 3, 4, 5, 6, 7, and 8. Keywords were identified and then matched to the implied or commonly associated operation. The correct solution to routine problems was found less than 50% of the time when following the method implied by the keyword. The rate of success was less than 10% of the time with multistep routine word problems. Elementary and middle school students are asked to solve word problems categorized as directive, routine, or nonroutine. (Powell & Fuchs, 2018).
3. Stifling Creativity and Exploration
By providing a formulaic approach to problem-solving, structured strategies leave little room for creativity, exploration, and divergent thinking. Students may become accustomed to following prescribed procedures rather than engaging in the exploration and experimentation essential for developing innovative problem-solving skills. Keyword strategies such as CUBES remove opportunities for students to think while at the same time fostering the act of pulling numbers from the task and simply operating. The well-known example problem of the shepherd’s age illustrates this. When presented with the problem “There are 120 sheep and 5 dogs in a flock. How old is the shepherd?”, a common response of “125” is given. The correct response is “There is not enough information”. For more on this task, see “Informing the Need for Critical Thinking in Mathematics” by Vardeh (2020).
The Case for Teaching Students to Think
In contrast to the confines of structured strategies, nurturing students’ ability to think critically empowers them to tackle problems with depth, insight, and flexibility. Here is why prioritizing student thinking is paramount.
1. Building a Strong Foundation for Lifelong Learning
By focusing on cultivating critical thinking skills, educators lay the groundwork for students to become lifelong learners who can adapt to the challenges of an ever-changing world. Rather than simply providing answers, teachers equip students with the tools to analyze, evaluate, and solve problems independently.
2. Fostering Resilience and Confidence
When students are encouraged to think critically and grapple with challenging problems, they develop resilience and confidence in their abilities. They learn to persevere in the face of difficulties, knowing that they have the skills and resources to overcome obstacles.
3. Encouraging Deep Conceptual Understanding
Critical thinking goes hand in hand with deep conceptual understanding. By engaging students in meaningful explorations of mathematical concepts, educators foster a profound understanding that transcends memorization and enables students to make connections across various domains of knowledge.
Strategies for Cultivating Critical Thinking
How can teachers shift their focus from structured strategies to nurturing critical thinking in their students? Here are a few practical approaches:
1. Inquiry-Based Learning
Embrace inquiry-based learning approaches that encourage curiosity, exploration, and discovery. Provide students with open-ended questions and real-world problems that invite them to investigate, analyze, and propose solutions.
2. Incorporate Reading Comprehension Strategies
Visualizing, retelling, making connections, and asking questions are established reading comprehension strategies that teachers of all grade levels can use to support students in making sense of the context of any type of word problem. Many students learn these strategies in language arts classes and can easily apply them to mathematics problem-solving as well (Gallagher, Ellis, & Weiland, 2021).
3. Promote Metacognition
Encourage metacognitive reflection by asking students to articulate their problem-solving processes and strategies. Prompt them to consider why certain approaches were successful or unsuccessful and how they might approach similar problems differently in the future. Encourage dialogue among peers and provide opportunities for students to write or draw to represent their thinking.
Conclusion
In moving toward developing critical thinking in mathematics education, teachers should move beyond the constraints of step-by-step problem-solving strategies like CUBES and keyword approaches. By prioritizing the development of critical thinking skills, educators empower students to become resilient, confident, and independent learners who are equipped to tackle the challenges of the 21st century and beyond. If you are seeking valuable insights and strategies to enhance your teaching practices in elementary and middle school mathematics, I highly recommend “Elementary and Middle School Mathematics: Teaching Developmentally” (Van de Walle, Karp, & Bay-Williams, 2022).
References
Gallagher, M. A., Ellis, L., & Weiland, T. (2021). Making Word Problems Meaningful. Mathematics Teacher: Learning and Teaching PK-12, 114(8), 580-590. National Governors Association. (2010). Common core state standards. Washington, DC.
Powell, S. R., & Fuchs, L. S. (2018). Effective word-problem instruction: Using schemas to facilitate mathematical reasoning. Teaching Exceptional Children, 51(1), 31–42.
Powell, S. R., Namkung, J. M., & Lin, X. (2022). An investigation of using keywords to solve word problems. The Elementary School Journal, 122(3), 452-4.
U.S. Department of Education. Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2022 Mathematics Assessment.
Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2022). Elementary and middle school mathematics: Teaching developmentally. Pearson. One Lake Street, Upper Saddle River, New Jersey.
Vardeh, M. (2020). Informing the Need for Critical Thinking in Mathematics. Unpublished manuscript. Department of Mathematics, California State University, Turlock, CA.