Building Place Value Proficiency: Key Insights from Educational Studies

Research highlights the critical role of visual representations in developing students’ conceptual understanding of place value.

Introduction:

Place value is a fundamental mathematical concept crucial for understanding our number system and performing arithmetic operations. However, research consistently shows that many students struggle to fully grasp and apply place value concepts (Fuson, 1990; Ross, 1989; Hiebert & Wearne, 1992). This overview examines key findings and recommendations from studies on teaching and learning place value, focusing on effective instructional approaches and common challenges students face.

The Nature of Place Value

Place value is the foundation of our base-ten numeration system, where the position of a digit in a number determines its value. This concept allows us to represent large numbers efficiently and perform calculations using a relatively small set of symbols. At its core, place value involves understanding that each place represents a power of ten and that digits in different positions represent different quantities (Fuson, 1990; Hiebert & Wearne, 1992). This understanding is crucial for developing number sense and computational fluency.

Common Misconceptions

When teaching place value, it is important to focus on three key ideas: understanding the base-ten system, recognizing the multiplicative nature of place value, and flexibly composing and decomposing numbers. However, research shows that students often struggle with certain aspects of place value. Common difficulties include viewing multi-digit numbers as concatenated single digits rather than as composed of different place values, misunderstanding the role of zero as a placeholder, and struggling with the concept of regrouping in addition and subtraction (Kamii, 1986; Ross, 1989). To address these challenges, using concrete materials, visual models, and activities that build conceptual understanding is helpful rather than just teaching procedural methods. Students develop a more robust and flexible understanding of place value by focusing on these areas.

Representations

Research strongly supports using multiple representations when teaching place value, including enactive (base-ten blocks), iconic (number line), and symbolic (place value charts and expanded notation). This approach encourages students to move flexibly between different representations, create their own, and explicitly discuss connections among them. Students progress through various levels of understanding when exploring these representations (Bruner, 1966; Van de Walle, 2007). Enactive models, particularly base-ten blocks, play a crucial role in developing place value understanding by concretely representing the relative sizes of different place values, facilitating regrouping, and supporting problem-solving across various place-value situations (Fuson, 1990; Hiebert & Wearne, 1992).

Iconic or visual models, such as number lines and bar or area models, can effectively address common misconceptions and challenges in learning place value, including difficulties with regrouping, understanding the role of zero, and extending place value understanding to decimals (Ross, 1989; Kamii, 1986). These visual representations adapt to various problem types, supporting students in developing a more robust and flexible understanding of place value. This approach ultimately equips them for success in advanced mathematics and real-world applications, helping students connect place value concepts to other mathematical ideas, such as operations and measurement (Fuson, 1990).

Studies suggest that place value charts that center the unit of one rather than the decimal point may provide a clearer understanding of the base-ten system. This approach emphasizes the multiplicative relationships inherent in place value, allowing students to conceptualize better how numbers are constructed and manipulated through powers of ten. By iterating or partitioning numbers based on the unit of one, students can develop a more robust understanding of how digits represent complete units, enhancing their overall numerical literacy. This design of place value charts, combined with carefully formulated questions that probe deeper comprehension, can significantly improve students' grasp of place value concepts.

Grounding in Real-World Contexts

Research strongly supports introducing and exploring place value concepts through authentic, real-life situations (Gravemeijer, 1994; Van de Walle, 2007). This approach helps students recognize the relevance and applicability of place value, develop intuitive understanding before formal procedures, and make connections between place value and other mathematical ideas. Examples of effective contextual learning activities include using packaging, measuring length or weight, and exploring large numbers in contexts like populations or distances in space.

Developing Benchmark Understanding

Studies emphasize the importance of establishing strong benchmark numbers as reference points, including 10, 100, 1,000, and their multiples (Hiebert & Wearne, 1992; Van de Walle, 2007). Activities that promote flexible thinking with these benchmarks, such as mental math exercises and estimation tasks, can significantly enhance place value fluency.

Language

Research has identified several conceptual challenges students face when learning place value, including understanding the multiplicative nature of the base-ten system, grasping the idea of regrouping, and flexibly moving between different representations (Fuson, 1990; Ross, 1989). While many students can follow rote procedures for place value calculations, they often struggle with explaining the reasoning behind their calculations and applying Place Value | Page 3 their understanding to new situations (Kamii, 1986; Hiebert & Wearne, 1992). A common misconception Kamii (1986) identified is the tendency to treat multi-digit numbers as single digits placed side by side, suggesting that traditional language focusing solely on digit positions may be inadequate for comprehensive place value understanding.

To address these challenges, researchers propose a shift in language that emphasizes the multiplicative relationships between place values (Fuson, 1990). This approach, combined with multiple representations and contexts such as base-ten blocks, number lines, place value charts, and real-world situations, can help students develop a more flexible and nuanced understanding of place value. By moving beyond the limited positional conception, students can better grasp the multiplicative nature of place value and apply this knowledge to a broader range of problem-solving situations, especially when dealing with larger numbers and decimals.

Research also supports the effectiveness of asking students about complete units, such as how many complete tens are in a number, rather than merely identifying the value of a digit in a specific place. This questioning strategy encourages deeper thinking about the composition of numbers and helps students recognize the significance of grouping in tens and ones. For instance, when students are asked how many complete tens are in 123, they see the number comprising 12 complete tens and 3 ones rather than just focusing on the digit in the tens place (Brendefur et al., 2024). This shift in questioning not only reinforces their understanding of place value but also aids in developing their computational skills as they learn to manipulate numbers flexibly and understand their structure more comprehensively.

Conclusion

Place value instruction is a cornerstone of mathematical understanding, requiring a multifaceted approach to ensure student success. Educators can help students develop a robust and flexible understanding of place value concepts by integrating real-world contexts, employing diverse representations, and utilizing precise language thinking (Fuson, 1990; Hiebert & Wearne, 1992; Van de Walle, 2007).

The research emphasizes moving beyond rote procedures to foster deep conceptual understanding. This involves centering place value charts around the unit of one, asking questions about complete units, and addressing common misconceptions head-on. By doing so, students can better grasp the multiplicative nature of the base-ten system and apply their knowledge to a wide range of mathematical situations.

As mathematics education continues to evolve, effective place-value instruction remains crucial for developing numerically literate individuals capable of tackling complex mathematical challenges. By implementing researchbased strategies and continuously refining our teaching approaches, we can equip students with the foundational skills necessary for success in advanced mathematics and real-world problem-solving.

References

Brendefur, J. et. Al. (2024). Developing Mathematical Thinking Institute. DMTI inc.

Bruner, J. S. (1966). Toward a theory of instruction. Harvard University Press.

Fuson, K. C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7(4), 343-403.

Gravemeijer, K. (1994). Developing realistic mathematics education. CD-β Press.

Hiebert, J., & Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for Research in Mathematics Education, 23(2), 98-122.

Kamii, C. (1986). Place value: An explanation of its difficulty and educational implications for the primary grades. Journal of Research in Childhood Education, 1(2), 75-86.

Ross, S. H. (1989). Parts, wholes, and place value: A developmental view. The Arithmetic Teacher, 36(6), 47-51.

Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching developmentally. Pearson Education.

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Why is place value often challenging to teach and learn? Research highlights the critical role of visual representations in developing students’ conceptual understanding of place value. Base-ten blocks and place value charts, in particular, have emerged as powerful tools that serve multiple functions, from illustrating the base-ten system to facilitating regrouping and calculation.

Moreover, the research suggests a shift in language use, moving beyond traditional digit position descriptions to emphasizing the multiplicative relationships between place values. This linguistic approach fosters a more nuanced comprehension of multi-digit numbers. By implementing these evidence-based strategies, educators can cultivate flexible thinking about place value, thereby equipping students for success in advanced mathematics and real-world applications.

Join us in exploring these powerful learning strategies and their impact on mathematical thinking!