Number Sense

CAREER: Reciprocal Noticing: Latino/a Students and Teachers Constructing Common Resources in Mathematics

The goal of this project is to extend the theoretical and methodological construct of noticing to develop the concept of reciprocal noticing, a process by which teacher and student noticing are shared. The researcher argues that through reciprocal noticing the classroom can become the space for more equitable mathematics learning, particularly for language learners.

Lead Organization(s): 
Award Number: 
1253822
Funding Period: 
Wed, 05/15/2013 to Mon, 04/30/2018
Full Description: 

The goal of this project is to extend the theoretical and methodological construct of noticing to develop the concept of reciprocal noticing, a process by which teacher and student noticing are shared. The researcher argues that through reciprocal noticing the classroom can become the space for more equitable mathematics learning, particularly for language learners. Thus, the focus of the project is on developing the concept of reciprocal noticing as a way to support better interactions between teachers and Latino/a students in elementary mathematics classrooms.

The project uses a transformative teaching experiment methodology and is guided by the initial conjectures that to make mathematics classrooms intellectually attractive places, Latino/a students and teachers need to learn to develop common resources for teaching and learning mathematics, and that reciprocal noticing as a process supports teachers and students in developing these common resources for teaching and learning mathematics. The project design centers around two research questions:How do teachers and Latino/a students tune to each other's mathematical ideas and explicitly indicate to one another how their ideas are important for discourse that promotes mathematical reasoning in classrooms characterized by reciprocal noticing? What patterns emerge across four classrooms when teachers and Latino/a students engage in reciprocal noticing?

The concept of reciprocal noticing can significantly enhance emerging research in mathematics education about the importance of teacher noticing. Further, this revised concept of noticing can transform mathematics classroom to better support English Language Learners.

The PI will incorporate project findings and videos into methods courses for preservice elementary teachers.

CAREER: Fraction Activities and Assessments for Conceptual Teaching (FAACT) for Students with Learning Disabilities

This project is studying and supporting the development of conceptual understanding of fractions by students with learning disabilities (LD).  Rather than focusing on whether students can or cannot develop conceptual understanding, the project is focused on uncovering the complex understanding students DO have.

Award Number: 
1708327
Funding Period: 
Tue, 07/01/2014 to Tue, 06/30/2020
Project Evaluator: 
Dr. Mary Little
Full Description: 

Dr. Hunt, a former middle school and elementary school mathematics in inclusive settings in a state-demonstration STEM school, works with students deemed to be at risk for mathematics difficulties or labeled as having disabilities. Hunt contends that research and pedagogical practice for children with disabilities should begin from a respect for children's ways of knowing and learning. Rather than focusing on whether students can or cannot develop conceptual understanding, research should attempt to uncover the complex understanding students DO have. She argues that teaching based in learning theory that positions children's learning as adaptation advances reasoning, sense-making, and co-construction of meaning.

The goal of Hunt's project- "CAREER: Fraction Activities and Assessments for Conceptual Teaching (FAACT)"-  is to study and support the development of conceptual understanding of fractions by students with learning disabilities (LD).  Dr. Hunt is re-conceptualizing intensive intervention as children's knowing and learning in "Small Environments". This approach suggests a redirect of research and instructional practice in mathematics for an underserved population of students. The project has the potential to offer a transformative approach to mathematics instruction for students with LD, bringing together expertise on learning disabilities and mathematics education to address an area in which there is very little research. 

The main outcomes of the project include (a) a theory of knowing, learning, and teaching connected to students with LDs in the small environment of supplemental and intensive intervention, (b) selected research-based trajectories specific to the conceptual understandings of fractions evidenced by students with LD presented in case study format, and (c) a set of practices and tools for teaching in the small environment (e.g., explicated knowing and learning framework; a set of learning situations to be used for teaching and/or formative assessment in fraction concepts, and suggestions for instructional decision making to aid teachers in designing student-centered instruction both in small groups and individualized formats).

This project was previously funded under award #1253254 and 1446250.


Project Videos

2019 STEM for All Video Showcase

Title: Fractional Reasoning: Students with Learning Disabilities

Presenter(s): Jessica Hunt, Andy Khounmeuang, Kristi Martin, Blain Patterson, & Juanita Silva


Assessing Secondary Teachers' Algebraic Habits of Mind (Collaborative Research: Stevens)

This collaborative project is developing instruments to assess secondary teachers' Mathematical Habits of Mind (MHoM). These habits bring parsimony, focus, and coherence to teachers' mathematical thinking and, in turn, to their work with students. This work fits into a larger research agenda with the ultimate goal of understanding the connections between secondary teachers' mathematical knowledge for teaching and secondary students' mathematical understanding and achievement.

Partner Organization(s): 
Award Number: 
1222496
Funding Period: 
Wed, 08/15/2012 to Sun, 07/31/2016
Full Description: 

Boston University, Education Development Center, Inc., and St. Olaf College are collaborating on Assessing Secondary Teachers' Algebraic Habits of Mind (ASTAHM) to develop instruments to assess secondary teachers' Mathematical Habits of Mind (MHoM). These habits bring parsimony, focus, and coherence to teachers' mathematical thinking and, in turn, to their work with students. MHoM is a critical component of mathematical knowledge for teaching at the secondary level. Recognizing the need for a scientific approach to investigate the ways in which MHoM is an indicator of teacher effectiveness, the partnership is researching the following questions:

1. How do teachers who engage MHoM when doing mathematics for themselves also bring MHoM to their teaching practice?

2. How are teachers' engagement with MHoM and their use of these habits in teaching related to student understanding and achievement?

To investigate these questions, ASTAHM is developing two instruments: a paper and pencil (P&P) assessment and an observation protocol that measure teachers' knowledge and classroom use, respectively, of MHoM.

The work is being conducted in two phases: (1) an instrument-refinement and learning phase, and (2) an instrument-testing and research phase. Objectives of Phase 1 are to gather data to refine the project's existing instruments and to learn about the bridge factors that impact the relationship between teachers' knowledge and classroom use of MHoM. Specific research activities include: administering the pilot P&P assessment to 40 teachers, videotaping Algebra instructions of 8 teachers, performing initial testing and refinement of the instruments, and using the data to analyze the bridge factors. Phase 2 is a large-scale study involving field-testing the P&P assessment with 200 teachers, videotaping 20 teachers and studying them using the observation protocol, collecting achievement data from 3000 students, and checking P&P content validity with 200 mathematicians. With these validated instruments in hand, the project will then conduct an investigation into the above research questions. Lesley University's Program Evaluation and Research Group (PERG) is the external evaluator. PERG is assessing ASTAHM's overall success in developing valid and reliable instruments to investigate the extent to which a relationship exists between teachers' MHoM and their classroom practice, as well as student achievement. Evaluators are also investigating whether users' coding guides for both instruments enable field-testers to effectively use and adequately score them.

This work fits into a larger research agenda with the ultimate goal of understanding the connections between secondary teachers' mathematical knowledge for teaching and secondary students' mathematical understanding and achievement. The MHoM construct is closely aligned with the Common Core State Standards-Mathematics (CCSS-M); especially its Standards for Mathematical Practice. For example, both place importance on seeking and using mathematical structure. Thus the instruments this project produces can act as pre- and post-measures of the effectiveness of professional development programs in preparing teachers to implement the CCSS-M. Mathematics teacher knowledge at the secondary level is an understudied field. Through analyses of the practices and habits of mind that teachers bring to their work, ASTAHM is developing instruments that can be used to shed light on effective secondary teaching.


Project Videos

2019 STEM for All Video Showcase

Title: Studying Teachers' Mathematical Habits of Mind

Presenter(s): Sarah Sword, Eden Badertscher, Al Cuoco, Miriam Gates, Ryota Matsuura, & Glenn Stevens

2017 STEM for All Video Showcase
Title: Assessing Secondary Teachers' Algebraic Habits of Mind

Presenter(s): Sarah Sword, Courtney Arthur, Al Cuoco, Miriam Gates, Ryota Matsuura, & Glenn Stevens

2016 STEM for All Video Showcase

Title: Assessing Secondary Teachers' Algebraic Habits of Mind

Presenter(s): Ryota Matsuura, Al Cuoco, Glenn Stevens, & Sarah Sword


Learning Mathematics of the City in the City

This project is developing teaching modules that engage high school students in learning and using mathematics. Using geo-spatial technologies, students explore their city with the purpose of collecting data they bring back to the formal classroom and use as part of their mathematics lessons. This place-based orientation helps students connect their everyday and school mathematical thinking. Researchers are investigating the impact of place-based learning on students' attitudes, beliefs, and self-concepts about mathematics in urban schools.

Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1222430
Funding Period: 
Sat, 09/01/2012 to Mon, 08/31/2015
Full Description: 

Learning Mathematics of the City in The City is an exploratory project that is developing teaching modules that engage high school students in learning mathematics and using the mathematics they learn. Using geo-spatial technologies, students explore their city with the purpose of collecting data they bring back to the formal classroom and use as part of their mathematics lessons. This place-based orientation is helping students connect their everyday and school mathematical thinking.

Researchers are investigating the impact of place-based learning on students' attitudes, beliefs, and self-concepts about mathematics in urban schools. Specifically, researchers want to understand how place-based learning helps students apply mathematics to address questions about their local environment. Researchers are also learning about the opportunities for teaching mathematics using carefully planned lessons enhanced by geo-spatial technologies. Data are being collected through student interviews, classroom observations, student questionnaires, and student work.

As the authors explain, "The use of familiar or engaging contexts is widely accepted as productive in the teaching and learning of mathematics." By working in urban neighborhoods with large populations of low-income families, this exploratory project is illustrating what can be done to engage students in mathematics and mathematical thinking. The products from the project include student materials, software adaptations, lesson plans, and findings from their research. These products enable further experimentation with place-based mathematics learning and lead the way for connecting mathematical activities in school and outside of school.

Developing Teaching Expertise in K-5 Mathematics

This project designs materials and an accompanying support system to enable the development of expertise in the teaching of mathematics at the elementary level. The project has four main components: online professional development modules; practice-based assessments; resources for facilitators; and web-based technologies to deliver module content to diverse settings. Three modules are being developed and focus on fractions, reasoning and explanation, and geometry. Each module is organized into ten 1.5 hour sessions.

Project Email: 
Award Number: 
1118745
Funding Period: 
Thu, 09/01/2011 to Fri, 08/31/2018
Project Evaluator: 
American Institutes for Research
Full Description: 

Developers and researchers at the University of Michigan and the University of Denver are engaged in a project to design materials and an accompanying support system to enable the development of expertise in the teaching of mathematics at the elementary level. The project has four main components: a set of online professional development modules; practice-based assessments; a set of resources for facilitators; and web-based technologies to deliver module content to diverse settings. Three modules are planned: one focused on fractions and one focused on reasoning and explanation designed by Deborah Ball, Hyman Bass and the University of Michigan development team; and one on geometry developed by Douglas Clements and Julie Sarama at the University of Denver. Each module is organized into ten 1.5 hour sessions. 

Each module goes through a two-year design and development process that includes initial design, piloting, revision, and dissemination. Modules are piloted in a variety of settings, including university based courses for practicing teachers and district based in-service activities. These contexts include face-to-face professional development, real-time distance learning, and combinations of the two. Data are collected on participant engagement with the modules, on teacher classroom practice, and on mathematical knowledge for teaching.

The modules and associated materials will be widely available and will be free to schools. The materials can be imported into any learning management system, such as Blackboard, Moodle, and others.

Teachers Empowered to Advance Change in Mathematics (TEACH MATH): Preparing Pre K-8 Teachers to Connect Children's Mathematical Thinking and Community-Based Funds of Knowledge

This project will modify the teacher preparation program for preK-8 teachers. The program is designed to help pre-service teachers learn mathematics well, learn to access students' cultural funds of knowledge, and learn to encourage students' mathematical thinking. The developers are designing (a) modules that can be used in teacher preparation courses, (b) a mentoring program for new teachers, and (c) on-line networks to facilitate collaboration among participating teachers and institutions.

Lead Organization(s): 
Award Number: 
1228034
Funding Period: 
Thu, 09/01/2011 to Thu, 08/31/2017
Project Evaluator: 
Research Institute for Studies in Education
Full Description: 

This research and development project will modify the teacher preparation program for preK-8 teachers at six universities located in different regions of the U.S. The new program is designed to help pre-service teachers learn mathematics well, learn to access students' cultural funds of knowledge in ways that will help them teach mathematics, and learn to encourage students' mathematical thinking. By integrating these important bodies of knowledge, pre-service teachers should be better prepared to teach mathematics to the variety of students in their classes. The developers are designing (a) modules that can be used in teacher preparation courses, (b) a mentoring program for new teachers, and (c) on-line networks to facilitate collaboration among participating teachers and institutions.

The project includes a study of how pre-service teachers learn to apply the knowledge they have gained in the program. The research team has planned a longitudinal collection of data that will track the pre-service teachers into their careers. Their goal is to document teachers' understandings of children's mathematical thinking and children's cultural funds of knowledge and to understand the relationship between teachers' understandings and the learning and disposition of preK-8 students. The study will be implemented at all six universities with staggered start dates allowing for analysis and revisions between cohorts.

These research and development efforts have the potential to impact preK-8 teacher preparation through (1) the development of modules that integrate several relevant proficiencies in mathematics teaching, and (2) the research that studies the impact of such a program on the mathematical learning and disposition of preK-8 students.

Evaluating the Developing Mathematical Ideas Professional Development Program: Researching its Impact on Teaching and Student Learning

This is a 3.5-year efficacy study of the Developing Mathematical Ideas (DMI) elementary math teacher professional development (PD) program. DMI is a well-known, commercially available PD program with substantial prior evidence showing its impact on elementary teachers' mathematical and pedagogical knowledge. However, no studies have yet linked DMI directly with changes in teachers' classroom practice, or with improved student outcomes in math. This study aims to remedy this gap.

Project Email: 
Lead Organization(s): 
Partner Organization(s): 
Award Number: 
1019769
Funding Period: 
Wed, 09/01/2010 to Fri, 08/31/2012
Project Evaluator: 
Bill Nave
Full Description: 

This is a 3.5-year efficacy study of the Developing Mathematical Ideas (DMI) elementary math teacher professional development (PD) program. DMI was developed by staff from Education Development Center (EDC), SummerMath for Teachers, and TERC, the STEM research and development institution responsible for this research. DMI is a well-known, commercially available PD program with substantial prior evidence showing its impact on elementary teachers' mathematical and pedagogical knowledge. However, no studies have yet linked DMI directly with changes in teachers' classroom practice, or with improved student outcomes in math. This study aims to remedy this gap.

The research questions for the study are:

1) Does participation in the Developing Mathematical Ideas (DMI) professional development program lead to increases in reform-oriented teaching?

2) Does participation in DMI lead to increases in students' mathematics learning and achievement, especially in their ability to explain their thinking and justify their answers?

3) What is the process by which a reform-oriented professional development program can influence teaching practice and, thus, student learning? Through what mechanisms does DMI have impact, and with what kinds of support do we see the desired changes on our outcome measures when the larger professional development context is examined?

The dependent variables for this study include a) teachers' pedagogical and mathematics knowledge for teaching; b) the nature of their classroom practice; and c) student learning/ achievement in mathematics.

The study uses experimental and quasi-experimental methods, working with about 195 elementary grades teachers and their students in Boston, Springfield, Leominster, Fitchburg, and other Massachusetts public schools. Volunteer teachers are randomly assigned either to PD with DMI in the first year of the efficacy study, or to a control group that will wait until the second year of the study to receive DMI PD. Both groups of teachers will be followed through two academic years. Analyses use OLS regression, hierarchical modeling, and structural equation modeling, as appropriate, to compare the two groups and to track changes over time. In this way, the project explores several aspects of a conceptual framework hypothesizing relationships among PD, teacher mathematical and pedagogical knowledge, classroom teaching practice, and student outcomes. There are multiple measures of each construct, including video-analysis of teacher practice, and a new video-based measure of teacher knowledge.

The study tests the impact of DMI in a range of districts (large urban, small urban, suburban) serving an ethnically and economically diverse mix of students. It provides much needed, rigorous evidence testing the efficacy of this reform-oriented professional development program. It also directly explores the commonplace theory that teachers' understanding of content and student thinking and their encouragement of rich mathematical discourse for student sense-making lead to improvement on measures of mathematics achievement. Findings from the study are disseminated to both research and practitioner communities. The project provides professional development in mathematics to about 195 teachers to improve their ability to teach important concepts. If the evidence for efficacy is positive, then even larger-scale use of this PD program is likely.

Math Snacks: Addressing Gaps in Conceptual Mathematics Understanding with Innovative Media

This project is developing and evaluating effectiveness of 15 - 20 short computer mediated animations and games that are designed to: (1) increase students' conceptual understanding in especially problematic topics of middle grades mathematics; and (2) increase students' mathematics process skills with a focus on capabilities to think and talk mathematically.

Lead Organization(s): 
Award Number: 
0918794
Funding Period: 
Tue, 09/01/2009 to Fri, 08/31/2012
Project Evaluator: 
Sheila Cassidy WEXFORD INC.
Full Description: 

View a project spotlight on Math Snacks.

This project Math Snacks: Addressing Gaps in Conceptual Mathematics Understanding with Innovative Media, led by mathematics and education faculty at New Mexico State University, is developing and evaluating effectiveness of 15 - 20 short computer mediated animations and games that are designed to: (1) increase students' conceptual understanding in especially problematic topics of middle grades mathematics; and (2) increase students' mathematics process skills with a focus on problem-solviing and communicating mathematically. The basic research question for this project is whether the planned collection of computer-mediated animations and games can provide an effective strategy for helping students learn core middle grades mathematics concepts in conceptual areas that research suggests are difficult for these students.  A second question relates to types of delivery that are effective for mathematics learning using these tools including in classrooms during extended learning time at home or in informal educational settings. The project is developing and testing the effectiveness of a set of such learning tools and companion print materials, including student and teacher guides, and short video clips documenting best practices by  teachers using the developed materials with students. A pilot study in year 3 and a substantial randomized control trial in year 4 will test the effects of using the Math Snacks web-based and mobile technologies on student learning and retention of identified core middle school mathematics concepts, as measured by performance on disaggregated strands of the New Mexico state standardized mathematics assessments. Thus the project will produce animations and games using the web and new mobile technologies, and useful empirical evidence about the efficacy of their use. One of the key features of the Math Snacks project is development of the mediated games and simulations in a form that can be used by students outside of normal classroom settings on media and game players that are ubiquitous and popular among today's young people. Thus the project holds the promise of exploiting learning in informal settings to enhance traditional school experiences.

Introducing Dynamic Number as a Transformative Technology for Number and Early Algebra

This project operationalizes research in number, operation, and early algebra. It builds on the paradigm of Dynamic Geometry (the interactive and continuous manipulation of geometric shapes and constructions) with a new technological paradigm, Dynamic Number, centered on the direct manipulation of numerical representations and constructions. Using The Geometer’s Sketchpad as a starting point, KCP Technologies is developing new software tools to deepen students’ conceptions of number and early algebra in grades 2–8.

Project Email: 
Lead Organization(s): 
Award Number: 
0918733
Funding Period: 
Tue, 09/01/2009 to Fri, 08/31/2012
Project Evaluator: 
John Olive

Diagnostic E-learning Trajectories Approach (DELTA) Applied to Rational Number Reasoning for Grades 3-8

This project aims to develop a software diagnostic tool for integrating diagnostic interviews, group administered assessments, and student data in real-time so that teachers can enter and view student status information. This project would concentrate on rational number learning in grades 3-8. The design is based on a model of learning trajectories developed from existing research studies.

Project Email: 
Award Number: 
0733272
Funding Period: 
Sat, 09/01/2007 to Tue, 08/31/2010
Project Evaluator: 
William Penuel (SRI)
Full Description: 

This project aims to develop a software diagnostic tool for integrating diagnostic interviews, group administered assessments, and student data in real-time so that teachers can enter and view student status information. This project would concentrate on rational number learning in grades 3-8. The design is based on a model of learning trajectories developed from existing research studies.

The diagnostic system to be developed for teachers would be used in assessing their students' knowledge and would identify difficulties in understanding five key clusters of concepts and skills in rational number reasoning. It would also investigate the diagnostic system's effects on student and teacher learning in relation to state standards, assessments, and curricular programs. The five areas include understanding: (1) multiplicative and division space; (2) fractions, ratio, proportion and rates; (3) rectangular area and volume; (4) decimals and percents; and (5) similarity and scaling.

The diagnostic measures will include diagnostic interviews collecting data using a handheld computer, two types of group-administered assessments of student progress, one set along learning trajectories for each of the five sub-constructs and one composite measurement per grade. The diagnostic system will produce computer-based progress maps, summarizing individual student and class performance and linking to state assessments.

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