Using a Conceptual Understanding and Procedural Fluency Heuristic to Target Math Interventions with Students in Early Elementary

2015 ◽  
Vol 30 (2) ◽  
pp. 52-60 ◽  
Author(s):  
Matthew K. Burns ◽  
Christopher Walick ◽  
Gregory R. Simonson ◽  
Lauren Dominguez ◽  
Laura Harelstad ◽  
...  
Keyword(s):  
Conceptual Understanding ◽  
Early Elementary ◽  
Math Interventions ◽  
Procedural Fluency ◽  
Fluency Heuristic

Multiplication by 10 base-5: Making Sense of Place Value Structure Through an Alternate Base

2015 ◽  
Vol 3 (2) ◽  
pp. 83-98
Author(s):  
Jodi Fasteen ◽  
Kathleen Melhuish ◽  
Eva Thanheiser
Keyword(s):  
Preservice Teachers ◽  
Conceptual Understanding ◽  
Sense Of Place ◽  
Number System ◽  
Place Value ◽  
Mathematical Activity ◽  
Making Sense ◽  
Procedural Fluency ◽  
Whole Number

Prior research has shown that preservice teachers (PSTs) are able to demonstrate procedural fluency with whole number rules and operations, but struggle to explain why these procedures work. Alternate bases provide a context for building conceptual understanding for overly routine rules. In this study, we analyze how PSTs are able to make sense of multiplication by 10five in base five. PSTs' mathematical activity shifted from a procedurally based concatenated digits approach to an explanation based on the structure of the place value number system.

scholarly journals Developing The Mathematics Conceptual Understanding and Procedural Fluency through Didactical Anticipatory Approach Equipped with Teaching AIDS

2018 ◽  
Vol 3 (2) ◽  
pp. 367
Author(s):  
Asmida Asmida ◽  
Sugiatno Sugiatno ◽  
Agung Hartoyo
Keyword(s):  
Conceptual Understanding ◽  
Junior High School ◽  
Mathematics Learning ◽  
Posttest Score ◽  
Mathematics Textbook ◽  
Teaching Aids ◽  
Procedural Fluency ◽  
Mathematical Problems ◽  
Epistemological Obstacles ◽  
And Mathematics

The students’ conceptual understanding and procedural fluency have not been yet integrated into the mathematics learning as the teachers’ common mathematics textbook has not explicitly explained the conceptual understanding and procedural fluency in solving the mathematical problems that the teachers have not yet connected it to the mathematics learning. The interview result shows that the students only memorize the procedures without understanding. If the procedure is continuously applied, it is predicted that the students may face the epistemological obstacles in solving the mathematical problems. This research aims at developing the students' mathematics conceptual understanding and procedural fluency through the Didactic Anticipatory Approach equipped with the teaching aids in learning the operations of integer multiplication at Junior High School in Grade VIII. This pedagogical action research involves 14 students. The research data are collected using tests, interviews, voice recorders and cameras. The result shows that learning mathematics through the Didactic Anticipatory Approach equipped with teaching aids may develop the students' conceptual understanding and mathematics procedural fluency marked by the reduced students’ epistemological obstacles. However, they are not yet been completely resolved. The students' conceptual understanding and mathematics procedural fluency also supported with the average posttest score higher than that of the pretest score.

Collaborative Planning as a Process

2017 ◽  
Vol 22 (7) ◽  
pp. 406-419
Author(s):  
Justin D. Boyle ◽  
Sarah B. Kaiser
Keyword(s):  
Student Learning ◽  
Learning Outcomes ◽  
Conceptual Understanding ◽  
Collaborative Planning ◽  
Learning Goals ◽  
Procedural Fluency ◽  
Cognitively Demanding Tasks

All students should be provided with opportunities to develop conceptual understanding prior to procedural fluency (NCTM 2014; CCSSI 2010). To develop students' conceptual understanding, teachers must learn such skills as how to select, plan, and enact cognitively demanding tasks (CDT) (Lambert and Stylianou 2013; Smith, Bill, and Hughes 2008) and to evaluate evidence of student learning (Hiebert et al. 2007). Therefore, teachers need opportunities to develop these skills to maximize their students' learning outcomes. Starting with a well-designed CDT is essential. In other words, before planning the Justin D. Boyle and Sarah B. Kaiser enactment of a task, teachers should analyze the task and make revisions to align it with student learning goals that promote conceptual understanding (Hiebert et al. 2007; Smith and Stein 2011).

Factoring for Roots

2018 ◽  
Vol 111 (7) ◽  
pp. 528-534
Author(s):  
Adam Clinch
Keyword(s):  
Conceptual Understanding ◽  
New Technique ◽  
Procedural Fluency ◽  
A New Technique

Justifications, methods, and results compare two classes of students who used a new technique that ties together procedural fluency and conceptual understanding in a manner unlike other current strategies.

Spotlight on the Principles: Creating an Environment for Learning with Understanding: The Learning Principle

2005 ◽  
Vol 11 (1) ◽  
pp. 35-39
Author(s):  
Emily Fagan
Keyword(s):  
Prior Knowledge ◽  
Conceptual Understanding ◽  
School Mathematics ◽  
Factual Knowledge ◽  
Procedural Fluency ◽  
New Knowledge ◽  
Principles And Standards ◽  
Knowledge Learning

The learning principle in NCTM'S Principles and Standards for School Mathematics (2000) states: “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.” Learning with understanding is defined as “being able to apply procedures, concepts, and processes” (NCTM 2000, p. 20). This view of learning represents a departure from a view that emphasizes a student's factual knowledge and ability to apply procedures. Although facts and procedures are important, they will not, in and of themselves, result in learning with understanding. Instead, factual understanding, procedural fluency, and conceptual understanding must coexist so that students reach learning with understanding. The extent to which a student can apply his or her learning to a new problem or situation is often an indicator of this understanding.

Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician's Perspective

2003 ◽  
Vol 9 (6) ◽  
pp. 322-327
Author(s):  
Hyman Bass
Keyword(s):  
Conceptual Understanding ◽  
Research Council ◽  
National Research Council ◽  
Strategic Competence ◽  
Computational Fluency ◽  
Procedural Fluency ◽  
National Research Council Report ◽  
Mathematical Proficiency ◽  
Mathematical Algorithms ◽  
Council Report

In recent years, few aspects of mathematics education have been as much discussed and debated as the notions of computational fluency and algorithms. A National Research Council report, Adding It Up: Helping Children Learn Mathematics (Kilpatrick, Swafford, and Findell 2001), offers an image of what it means to have skill with mathematics, or mathematical proficiency. This concept is helpful for moving beyond these debates. Mathematical proficiency includes five components: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (Kilpatrick, Swafford, and Findell 2001, p. 116). That these components are not separate but fundamentally intertwined is important to note. This article illustrates some of the ways in which the goal of computational fluency and an appreciation of mathematical algorithms are related to this larger concept of mathematical proficiency.

scholarly journals Measuring Conceptual Understanding, Procedural Fluency and Integrating Procedural and Conceptual Knowledge in Mathematical Problem Solving

2020 ◽  
Vol 8 (05) ◽  
pp. 1334-1350
Author(s):  
Thi Minh Phuong Ho
Keyword(s):  
Problem Solving ◽  
Conceptual Understanding ◽  
Structural Equation ◽  
Conceptual Knowledge ◽  
Mathematical Problem ◽  
Linear Functions ◽  
Equation Modeling ◽  
Mathematical Problem Solving ◽  
Procedural Fluency ◽  
Mathematical Proficiency

The main aim of this paper is to meassure students’ mathematical proficiency on conceptual understanding and procedural fluency, and their ability of integrating procedural and conceptual knowledge in problem solving. Based on the PCK taxonomy (Ho 2018), we design a questionnaire consisting of 12 questions with 22 tasks whose content is focus on linear functions and equations. The collected data is analysed by the statistical software IBM SPSS Statistics 22. Moreover, we use the structural equation modeling (SEM) to study the correlation between these two components of mathematical proficiency and the ability of integrating procedural and conceptual knowledge in problem solving, implemented in IBM SPSS AMOS 24. The findings show that students’ mathematical proficiency on procedural fluency on linear functions and equations is higher than that of conceptual understanding, and their ability of integrating procedural and conceptual knowledge is very low. Moreover, these categories have a bi-directional relationship, in which the affection of mathematical proficiency on conceptual understanding to the ability of integrating procedural and conceptual knowledge in problem solving is stronger than on procedural fluency.

scholarly journals Using crises, feedback and fading for online task design

2014 ◽  
Vol 8 (4) ◽  
pp. 127-138
Author(s):  
Christian Bokhove
Keyword(s):  
The Netherlands ◽  
Conceptual Understanding ◽  
Design Principles ◽  
Task Design ◽  
Online Environment ◽  
Recent Discussion ◽  
Procedural Fluency ◽  
Quantitative Results

A recent discussion involves the elaboration on possible design principles for sequences of tasks. This paper builds on three principles, as described by Bokhove and Drijvers (2012a). A model with ingredients of crises, feedback and fading of sequences with near-similar tasks can be used to address both procedural fluency and conceptual understanding in an online environment. Apart from theoretical underpinnings, this is demonstrated by analyzing a case example from a study conducted in nine schools in the Netherlands. Together with quantitative results of the underlying study, it is showed that the model described could be a fruitful addition to the task design repertoire.Uso de crisis, realimentación y desvanecimiento para el diseño de tareas en líneaUna discusión reciente implica la elaboración de posibles principios para el diseño de secuencias de tareas. Este documento se basa en tres principios, descritos en Bokhove y Drijvers (2012a). Un modelo que comprende las componentes de crisis, realimentación y desvanecimiento de secuencias con tareas muy similares puede ser utilizado para abordar tanto la fluidez procedimental como la comprensión conceptual en un entorno en línea. Además de estar fundamentado teóricamente, esto se demuestra mediante el análisis de un ejemplo de caso de un estudio realizado en nueve centros educativos de los Países Bajos. Junto con los resultados cuantitativos del estudio subyacente, se muestra que el modelo descrito podría ser una incorporación útil en el repertorio del diseño de tareas.Handle: http://hdl.handle.net/10481/31597Nº de citas en WOS (2017): 1 (Citas de 2º orden, 0) 

scholarly journals Students’ Algebraic Proficiency from the Perspective of Symbol Sense

2020 ◽  
Vol 5 (1) ◽  
pp. 86-94 ◽  
Author(s):  
Al Jupri ◽  
Ririn Sispiyati
Keyword(s):  
Mathematics Education ◽  
Pilot Study ◽  
Conceptual Understanding ◽  
Education Students ◽  
Learning And Teaching ◽  
Symbolic Representations ◽  
Procedural Fluency ◽  
Symbol Sense ◽  
Solving Equations ◽  
Set Up

Algebraic proficiency, including procedural fluency and conceptual understanding, is widely discussed worldwide. Algebraic proficiency refers largely to proficiency in symbolic representations which can be investigated through a framework of symbol sense. This research, therefore, aims to analyze students’ algebraic proficiency in terms of symbol sense. We set up a pilot study, involving 22 Indonesian mathematics education students (18-19 year old), in the form of two weeks teaching that combine a conventional approach and the use of a camera calculator in the learning and teaching of quadratic and related equations. The results showed that more than half number of students lacks of symbol sense in the sense that they tend to use procedural strategies rather than symbol sense strategies in solving equations. From the perspective of symbol sense, we concluded that the students acquired more on procedural fluency than on conceptual understanding.

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