Leveraging variation of historical number systems to build understanding of the base-ten place-value system

Abstract

Prospective elementary school teachers (PTs) come to their mathematics courses fluent in using procedures for adding and subtracting multidigit whole numbers, but many are unaware of the essential features inherent in understanding the base-10 place-value system (i.e., grouping, place value, base). Understanding these features is crucial to understanding and teaching number and place value. The research aims of this paper are (1) to present a local instructional theory (LIT), designed to familiarize PTs with these features through comparison with historical number systems and (2) to present the effects of using the LIT in the PT classroom. A theory of learning (variation theory) is paired with a framework related to motivation (intellectual need) to illustrate the mutually supporting roles they may play in mathematical learning and task design. The LIT, a supporting task sequence, and the rationale for task design are shared. This theoretical contribution is then paired with evidence of PTs' changing growth in their conceptions of whole number before and after courses leveraging this task sequence.

Examining 4th Grade Gifted and Non-Gifted Students Understanding Levels of Place Value

“…They also stated that the place value makes the relationships among decimals, calculation, and measurement units are visible. Thanheiser and Melhuish (2019) stated that teachers are adept at using the rules of addition and subtraction, but they are unaware of the basic features that require understanding the base ten place value system. All these show that it would be beneficial to enrich the content of teacher training to understand the place value semantically.…”

Elementary pre-service teachers’ horizon knowledge for teaching addition and subtraction: An analysis of video presentations

“…In backward conceptual connections, teachers connect current learning topics with students' prior knowledge (Ma, 1999;Wu, 2011). For example, in the case of addition and subtraction, teachers can use base-10 concepts (Fuson, 2020;Thanheiser & Melhuish, 2019) and counting strategies (Fuson, 2020;Sun et al, 2018) to help students understand the conceptions of the whole number system. In comparison, forward conceptual connections focus on making connections between current learning topics and future learning topics (Ma, 1999;Wu, 2011).…”

“…Previous research suggested that making backward and forward conceptual connections to whole number addition and subtraction helps students develop a deep understanding when learning these concepts. For example, making connections to base-10 concepts while teaching whole number addition and subtraction helps students understand the value represented by each place and regrouping (Fuson, 2020;Thanheiser & Melhuish, 2019). In addition, despite that subtraction can be understood by two models: taking away and determining the difference (Usiskin, 2008), many students focus solely on taking away, being too onesided (Selter et al, 2012).…”

“…Specifically, although the importance of teachers' connection-making in mathematics knowledge (i.e., horizontal knowledge) has been increasingly recognized, most existing studies are mainly focused on examining the connection of whole number addition and subtraction to place value and regrouping (e.g., Matthews & Ding, 2011;Thanheiser, 2009Thanheiser, , 2012Thanheiser & Melhuish 2019). As a result, little is known about what other types of connections are used by PSTs and how well PSTs are using those connections.…”

Αλγοριθμικές Διαδικασίες Ως Κύρια Μαθηματική Γνώση

“…Η κατανόηση της δεκαδικής βάσης του αριθμητικού μας συστήματος είναι ένα από τα σημαντικότερα αλλά και από τα πλέον δύσκολα στη διδασκαλία και μάθησή τους μαθηματικά θέματα, που μελετώνται στο Δημοτικό Σχολείο (Thanheiser & Melhuish, 2019;Assiti, Zulkardi & Darmawijoyo, 2013;Price, 2001). Η θεμελιακή έννοια της θεσιακής αξίας των αριθμών επιβάλλεται, μάλιστα, να αποτελεί κεντρικό σημείο αναφοράς στα Προγράμματα Σπουδών των Μαθηματικών, από την Πρώτη κιόλας τάξη του Δημοτικού, επειδή δίνει την ευκαιρία στους μαθητές να κατανοήσουν και άλλες μαθηματικές ιδέες, αλλά και να κάνουν συνδέσεις μεταξύ διαφορετικών γνωστικών περιοχών του μαθήματος (NCTM, 2000;Price, 1999).…”