Teaching Place Value — Challenges and Solutions
Place value is a fundamental building block of primary mathematics education. The place value system relates to arithmetic capabilities. Researchers have found several areas that relate to developing place value understanding. Experts believe that the understanding of the place value system predicts math performance and math difficulties.
Understanding place value means getting a hold of position, base-10 rules, and multi-digits. Place value requires children to understand the principle of position. This is important as it applies to base-10 rules. In other words, when numbers change position, they have different magnitudes. In their example, researchers identify that in 65, 6 and 5 have different positions. These positions are “ordered values of different magnitudes”. Understanding position precedes the understanding of multi-digits in place value. Ansari and Cheung intended to understand when children acquire “positional knowledge of multi digits”. In other words, they attempted to find out how children related position and value. They found that learning number names helped to understand the positional property of place value.
Sari and Olkun explored how a sound understanding of the place value system was essential to solving arithmetic and performing well in mathematics. Fourth grade students were observed for the purpose and results indicated that regardless of whether the student was a male or female, place value was a definite milestone in the arithmetic and mathematics learning process. Place value serves as a foundation to develop an understanding of fractions, operations on large numbers, decimals, measurement, and many more advanced mathematical concepts. Gaps in learning place value might make it difficult for learners to understand complex algorithms.
A Review of Existing Approaches to Teaching Place Value
Metonymic Approaches (Ordinality) with Traditional Metaphorical Approaches (Cardinality)
Place value is an integral aspect of developing a good number sense. Coles and Sinclair highlight the difficulty faced by children when getting on to the hundredth’s place. They narrate a common incident where the child refers to a hundred as “tenty”, as it is followed by “ninety” (a child may deduce this language pattern as ten comes after nine, so tenty comes after ninety). They indicate the need educators feel about helping children get the right “sense” of the hundred. Only when the child knows what a hundred is, is it possible to understand the tens and ones that follow it to make bigger numbers. Researchers observe that counting and patterns in language is one aspect that children may rely on to advance in the different digits in place value, such as in the case of ninety and tenty indicated above. Children may sometimes map nine and ten of the units place to ninety and tenty of the tens place during the initial stages of teaching tens digits.
From a neuroscience perspective, numbers have a visual code, verbal code, and magnitude code (digit representation, spoken words, and analogical quantity). Educators have used ordinality and cardinality, or sometimes a combination to build this sense. Place value is often associated with understanding the whole quantity as the sum of digits (additive property), positions of digits in the number (positional property), the value obtained from multiplying the face value of the digits with the value of the position (multiplicative property), increase in value of positions in the power of ten when viewing from right to left (base ten). Researchers have indicated that the general focus of place value education is metaphoric, focusing on cardinality, rather than metonymic, focusing on ordinality.
Another popular approach is the structuring of the numbers on a Gattengo chart, which supports visuo-spatial numbering patterns. Learning with the Gattengo chart involves calling out number names while tapping or pointing at the numbers. Researchers believe that the Gattengo chart and other such tools (including handheld electronic touchpads) engage learners with the metonymic representation of numbers and help them form connections between “symbols, sounds, names, touch, and gestures”. Focusing on metaphoric and metonymic approaches may create the right balance between cardinality and ordinality approaches for learning during the early years to minimize problems during the later stages.
Modeling and Structure Approach
Brendefur and colleagues recommend a “modeling perspective” to understand the concept of place value. They utilized the bar model to show quantity and units. The model helped students differentiate units from tens. They were also able to show an understanding of teen numbers. Their research proved the effectiveness of visual models to understand place value.
Modeling helps develop mathematical thinking and mathematical reasoning. The bar model consisted of thinking of one cube as a “unit of 1” and then drawing the required number of cubes, one on top of the other to get to the desired number. To show tens, ten cubes lined up in a row were covered by tape and presented as “1 unit of size 10”. Educators conducted an interactive activity to understand different perspectives. Representations of teen numbers consisting of tens and ones models were also shown to students. They also demonstrated jumps of tens and ones on the number line to model larger numbers. Therefore, modeling and structure are effective teaching approaches. Students were able to understand differences in “relative size of quantities” and differences between number and amount represented. They were able to compare “number length” and digit positions. Students learned the skill to create “proportional representations of numbers”. Finally, they were able to use language to describe place value in the right context.
Use of Language to Strengthen Place Value Understanding
Mathematical language and vocabulary can be an important tool in learning place value. Disney and Eisenreich explain how to use discourse to “make sense” of mathematics. They advocate student-centered classrooms to promote autonomy in young children. Effective communication in the math classroom is essential to engage students in “explanation, justification, questioning, and sense-making”. Discussions and activities in the classroom and interaction with peers is the path to creating new knowledge. Discourse in the form of oral and written communication supported by visual representations promotes asking questions, making clarifications, developing literacy, comparing problem-solving strategies, and mastering speaking and listening skills.
Hands-on models serve as great tools for grouping and making sense of tens and ones. These models act as starting points to promote discourses and understand the relation between tens and ones. Discourses not just encourage student engagement but adding a context to explain place value also promote mathematical reasoning. Researchers indicate that the context of a candy shop serves as a reason for students to group by ten. Place value understanding through the use of different objects representing hundreds, tens, and ones, accompanied by inquiring students about approaches used to make sense of problems promotes a deep understanding of place value. The language approach (using a candy shop context) proved to be effective in deep learning complex aspects of place value.
Place Value Challenges for Children with Math Difficulties
Place value is particularly challenging for children with mathematics difficulties. Lambert and Moeller investigated strategies used to process place value. They studied third-grade students to understand how they computed two-digit addition. They found that children with math difficulties made errors, especially in carryover. Scientists indicated that this pointed at an impairment in understanding place value. They found a relation between computing place value and its strategy and working memory.
MacDonald and team used a constructivist approach, engaging students through the use of mathematical materials to explore childrens’ understanding of multi-digit numbers. The purpose of their study was to identify the precise methods that work well to support low achieving learners with a sound understanding of multi-digits. Researchers assessed the understanding of the study group by working with counting questions, forming groups of tens, decomposing, and regrouping. They also tested number relationships and comparison. Students found it difficult to infer digits according to their place value positions (for example, researchers indicated that children treated 3 as 3 in 34 rather than 30). Such an example refers to a difficulty with transitioning from single-digit to multi-digit numbers by “interiorizing the concept of ten”). They enforced the use of concrete materials to help learners “coordinate two levels of units”. The use of concrete material is also essential to develop an understanding of three levels of unit coordination. Interiorizing two or three levels of units was essential for students to find the number between two specific multi-digit numbers. Being unable to find numbers in between two multi-digit numbers, therefore, referred to the learner’s difficulty with interiorizing different levels of units. Furthermore, being able to perform operations on numbers required students to develop sophisticated and flexible part-whole relationships.
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