- Edith Graf
- Senior Research Scientist
- DEVELOPMENT AND EMPIRICAL RECOVERY FOR A LEARNING PROGRESSION-BASED ASSESSMENT OF THE FUNCTION CONCEPT
- ETS Educational Testing Service
- Frank Davis
- Evaluation & Research Consulting
- DEVELOPMENT AND EMPIRICAL RECOVERY FOR A LEARNING PROGRESSION-BASED ASSESSMENT OF THE FUNCTION CONCEPT
- Cheryl Eames
- Associate Professor
- DEVELOPMENT AND EMPIRICAL RECOVERY FOR A LEARNING PROGRESSION-BASED ASSESSMENT OF THE FUNCTION CONCEPT
- Southern Illinois University Edwardsville
- Chad Milner
- National Director of Tech & Media
- DEVELOPMENT AND EMPIRICAL RECOVERY FOR A LEARNING PROGRESSION-BASED ASSESSMENT OF THE FUNCTION CONCEPT
- The Algebra Project
- Maisha Moses
- Executive Director
- DEVELOPMENT AND EMPIRICAL RECOVERY FOR A LEARNING PROGRESSION-BASED ASSESSMENT OF THE FUNCTION CONCEPT
Public Discussion
Continue the discussion of this presentation on the Multiplex. Go to Multiplex
Edith Graf
Senior Research Scientist
Welcome to our video presentation! We are interested in approaches to instruction and assessment of the function concept.
1) How are approaches you have taken to instruct/assess the function concept similar or different from the approach showcased in the video?
2) What are your thoughts about learning progression-based assessment in mathematics in general and for the concept of function in particular?
Rebecca Vieyra
Project Manager
Dear Edith and the project team,
This video showcases such a great, active way to teach functions! As a prior physics teacher who is now working on a computational modeling in physics project, I am also familiar with students' struggles with functions, so I see the critical need for this kind of work.
As I watched the video, I was curious to know if this project is more focused on the experience of teachers (as curriculum/pedagogy learners and implementers) or students (as learners of functions). Regardless, I'd like to know some more specifics about the kinds of struggles / naive conceptions that teachers and students display with regard to functions, and how the five-step approach explicitly addresses them.
I also saw that the five-step process is briefly listed on the board in the video (I'm otherwise not very familiar with it), but I am curious to know if it is the underlying framework of a pre-existing set of curricular resources that teachers were asked to learn and adopt, or if there is a developmental component in which teachers were asked to come up with novel experiences using the approach. In either case, what challenges and/or opportunities did teachers find with the approach?
Sincerely,
Rebecca
Edith Graf
Senior Research Scientist
Hi Rebecca, thank you for your questions! This response reflects collective input from the project team.
The project addresses both the experiences of teachers (as curriculum/pedagogy learners and implementers) and students (as learners of functions). One of the major efforts in learning progression work is to understand what one might call a misconception as part of a pathway of learning that builds a more comprehensive understanding of a concept—in this case the concept of function.
In this research, we piloted a learning-progression based assessment for the concept of function. The learning progression (and the assessment it is based on) address some of the conceptions that students have about functions early on—for example, the notion that relationships that do not have one-to-one correspondence cannot be functions.
With respect to the experiences of teachers, the Road Coloring Unit takes a different approach in that it is experiential—students learn about functions by walking between the buildings in the cities. The cities are constructed in accordance with certain rules, and one of the rules is that the red roads and the blue roads each represent a function. Teachers have the opportunity to reflect on how aspects of the function concept addressed by traditional instruction are realized in this new context. For example, when students walk between the buildings, they are encouraged to walk them simultaneously, to emphasize the idea that the function (following the red roads or the blue roads) is applied to all inputs at once.
The five-step curricular process is an underlying framework for several Algebra Project curriculum units that describes a way that a mathematical idea (content) can be accessed through learning (a pedagogy). It starts by grounding the idea in everyday experience. This is followed by developing a picture of the event, which in turn is followed by an everyday language description of the event in the picture. The language is then formalized, and finally captured as a mathematical representation (e.g., a graph or equation). For example, in the road coloring unit students use arrow diagrams, ordered pairs, and matrices as representations.
KRISTEN BIEDA
Maisha Moses
Executive Director
Hi Rebecca, I just wanted to add that the 5-step process provides an opportunity to democratize knowledge and to foster engagement in the classroom, as each person's experience is a valid point of view on which to ground discussion, to make meaning, and build understanding. The challenge lies in being able to facilitate the conversations and structure the activities so that shared meaning is developed and so that students, individually and collectively, can relate their experientially based "grounding metaphors" to the formal mathematics. This work takes time and can feel like a departure from "the math," particularly when instruction is focused on test preparation. Our experience and research shows that it is well worth the time, both from the perspective of student engagement and participation, and from the perspective of student learning and understanding.
Gina Greenidge
Rebecca Vieyra
Project Manager
Dear Edith and Maisha,
Thanks to both of your for your responses!
Understanding that teachers are using the Road Coloring Unit (which I think is a great example -- I'm starting to think about programming connections!), where does this fit into the "traditional" math curriculum? Additionally, given that it is a unit, I'm curious to know if the approach or the concepts tend to bleed over into other topics across a course, and what that looks like.
Lastly, I've seen reference to "a progression-based assessment." Can you elaborate on what you mean by this, and what the status is of that assessment?
Thank you!
Edith Graf
Senior Research Scientist
Hi Rebecca, good questions!
The Road Coloring unit is focused on finite-to-finite functions, or mappings from finite sets to finite sets. “Traditional” curricula usually give an introduction with finite-to-finite functions but tend to focus on real-valued functions. However, the uniqueness property of a function (that every input is paired with exactly one output) is a foundational idea that is common to functions addressed by both curricula. Translating among representations (though different representations may be optimal for different types of functions) is another important skill that both curricula emphasize, as is conceptualizing domain and range. We’ve referred to the finite-to-finite and traditional as different “strands” in understanding the big idea of function, but they have shared foundational concepts.
In the provisional learning progression, finite-to-finite and traditional are realized as different strands. Each strand theorizes levels of student thinking with respect to the concept of function. We then developed tasks aligned to the learning progression, and rubrics to score each response using a level of the learning progression. First, students worked on the tasks in paper and pencil format as part of cognitive interviews, and the tasks were revised accordingly. Later, the tasks were computer delivered as part of a pilot, and were further revised in light of statistical results.
Rebecca Vieyra
Project Manager
I'd love to see this assessment (or a sample) at some point! Thanks so much.
Edith Graf
Edith Graf
Senior Research Scientist
Hi Rebecca, we have some presentation slides that include sample tasks as well as some more background on the project, shall I email you those? Also see Greg Budzban's post below if you are interested in the Road Coloring curriculum module.
Willmary Rose
Thank you for sharing this video! My school also works with the Algebra Project. This is such a helpful approach for our students as we have different types of learners. Students who struggle with computational math are now able to learn it through experiences in this unit. This is a fun and interactive way for our students to not only learn about functions, but they are able to apply them to the real world.
Edith Graf
Jonathan Margolin
Principal Researcher
Hi Willmary, it is great to hear directly from a teacher about experiences with these curriculum materials. The project team is developing an assessment of student learning progressions for the concept of functions. What has been your experience with evaluating student learning progressions?
Edith Graf
Senior Research Scientist
Hi Willmary, thank you for your feedback about use of the Road Coloring unit! Jonathan, we are still in the process of evaluating the learning progression itself but we have learned a lot about the challenges in designing and revising tasks and rubrics that are based on a learning progression. We have taken an iterative approach to the design and revision of these tasks. Following the development of the learning progression, the team developed tasks which students responded to in cognitive interviews and focus groups. From these efforts we learned a lot about the use of language and vocabulary in the tasks, as well as whether the tasks were relatable. We made revisions, and the tasks were then programmed for computer delivery, after which we conducted a pilot. The pilot statistics provided more insight into which tasks were assessing thinking with respect to the learning progression as intended and which tasks required revision.
Michael Haney
Administrator, Educator
The video very clearly explains and justifies the 5-step approach to teaching functions. The teachers appear to be committed and the students are certainly engaged in learning a difficult concept through a concrete example. Probably all students would benefit from this approach and would better understand and have a more workable use of algebra had they had this foundation. Having taught Algebra in high school, I struggled to help the various students who separately needed this more deliberate approach.
As a former STEM HS principal, I firmly believe that mathematics is the most difficult discipline to change, largely because deep change must begin by changing how teacher learn and understand mathematics, as your video states. So eventually, the ideas embedded in this project must reach teacher preservice education. Is it your belief that data from the Algebra Project can provide enough evidence that purveyors of preservice education could be convinced? Is there interest in such changes within the math associations?
Melanie West
It is great to see that the Algebra Project and the Young People's Project are still going strong and evolving. I love them & learned so much from working with the group years ago.
Mathematics and the connections necessary to make us whole human beings in the process of learning make these programs so incredibly impactful. Are you doing any work online with youth given this pandemic situation?
Chad Milner
Chad Milner
National Director of Tech & Media
Hi Melanie - How are you? To answer your question yes we have some emerging work in the virtual learning space as a result of the current pandemic restrictions.
In the short term we are adapting our annual National Flagway Tournament to the virtual space and expanding it to a series of curriculum related games that students can play independently and collaboratively through mobile apps and online communications platforms such as zoom and google meet. You can find a more detailed description of the upcoming Virtual Festival at typp.org/flagway_festival.
Another project we are working on -- which is an offshoot of our NSF-supported work with ETS on establishing a learning progression for the concept of function -- is adapting the cognitive interview protocol for our Flagway Curriculum and then adapting those to the virtual space by conducting them via Zoom. We have completed the design of the Flagway cognitive interviews and are looking to pilot them in Broward County, FL Cambridge, MA and Flint, MI in the upcoming months.
Thanks for checking our project out and for the question!
Maisha Moses
Executive Director
Melanie, thanks for your comments! It's great to hear from you. We are thinking deeply and learning about how to transfer some of our work to online forums while still maintaining our core values of connection and relationship building, collaboration, and experientially grounded and meaningful learning. Most, if not all of our work for the rest of the school year and summer will take place online.
Frank Davis
Evaluation & Research Consulting
Dear Michael,
While this work is primarily about validating a learning progression about the concept of function, it has taken on the challenge for the LP to represent the way a diversity of students in public schools learn about functions. The video illustrates one of the goals of the partners -- that this work will benefit teachers in their work in classrooms with often under represented students who do not achieve in mathematics, and will include both students and teachers who are often do not participants in such research.
The partners in the work are part of a larger effort also supported by NSF (its INCLUDES program) to build alliances of partners nationally and locally that are needed to tackle complex problems of broader participation in STEM, particularly those groups currently underserved by our public education system.
In this effort, partners in a developing national alliance called “We the People for Mathematics Literacy for All”, are enlisting university teacher education programs to educate potential new teachers who can teach all students effectively and who will use assessment tools to improve their impact in classrooms.
In the video you are observing, a local alliance is developing in the community of a major school district. The local alliance includes a higher education institution that services a large number of students from the school district, and in the video, teachers are providing their thoughts about a curriculum they are using to teach concepts of function. That institution has an interest in focusing their pre-service education programs so as to produce teachers who can be effective in the school district, and students who will not in need remedial mathematics programs. I believe that this has occurred because the leaders of such institutions have been able to see first-hand the impact on students and teachers of a program that works, and the possibilities of partnering with school districts with shared goals.
Similarly, that Alliance nationally has involved organizations such as NCTM and its affiliated organization the Benjamin Banneker Organization in this question of improving mathematics education for all students.
Chad Milner
Michael Haney
Administrator, Educator
Thank you for the thorough reply, it is very encouraging. Preservice is the most efficient way to scale, I hope you make inroads into teacher education institutions broadly. Algebra is not only a gatekeeper for many courses in STEM that follow, but a basic understanding of Algebra helps one understand so much more about the world in which we live...a deficiency too many people have. This is a wonderful project.
Kate Belin
Like Willmary, I am also a teacher at Fannie Lou Hamer Freedom High School in the Bronx. I agree with what she is saying about students at our school being able to learn the concept of function through experiences. As highlighted in the video, many functions that we commonly teach in schools are abstract representations. Even if what is being represented by the function is a tangible human experience, I personally often forget that the function itself is not concrete. What is so wonderful about the Road Coloring is that the function itself is physical and calls for the students themselves to move in it.
Gina Greenidge
Edith Graf
Edith Graf
Senior Research Scientist
Kate, thank you for the post and insightful observations! I definitely agree that the opportunity to experience these representations concretely is a defining characteristic of the unit.
Kate Belin
Danielle Bassie
Another Fannie Lou Hamer Freedom High School teacher here! I've had the absolute joy of teaching Road Coloring to a group of students through 3-4 iterations now. We follow an Algebra Project curriculum throughout, and his is a student favorite unit always. Students find themselves creating their own functions, debating their solutions, and championing different representations without the mathphobia I have sometimes encountered in the past. Students come back to me years later and still remember the experiences of creating cities, walking them, solving (or not solving) the problem - and then seeing it all come together at the end. I love this unit so much and am so happy to see it being successful across the states.
Gregory Budzban
Kate Belin
Chad Milner
Michael Haney
Edith Graf
Edith Graf
Senior Research Scientist
Thank you for the post Danielle! I think your observations that this unit lends itself to mathematical debate and may provide memorable mathematical experiences are especially interesting and raise questions about what features of curriculum units may support these goals.
Chad Milner
Gina Greenidge
Excited to see the continued expansion of the Algebra Project! I was engaged with the Project in South Florida for 4 years and can attest to how the Road Coloring unit functions as a method to have students physically interpret math. I am happy that this work is still being done throughout the US.
Edith Graf
Shellie Banfield
Gina, How can we add this math strategy to our project? This could work well with Rocketry, Robotics, the Guitar curricula. Learning about functions could be a great learning objective for each of our courses.
Jennifer Ward
I loved seeing students use their bodies to walk and study math! There aren't many opportunities for students to make physical connections with a subject to make a muscle memory, if you will. I don't have the data to back up my hunch, but I feel that actually moving and exploring math will make the concepts stick better for the students.
Also, beautiful video.
Edith Graf
Edith Graf
Senior Research Scientist
Thank you Jennifer!
Maisha Moses
Executive Director
Thank you Jennifer.
I agree with your comment about the efficacy of physical connections and muscle memory for learning. Do you know about the emerging body of research on embodied cognition and math learning? It's quite interesting!
Joan Wynne
This is a remarkable quick look at the serious work being done by all of you. Kudos to all.
Edith Graf
Edith Graf
Senior Research Scientist
Thank you Joan!
Maisha Moses
Executive Director
Thank you Joannie!
Leanne Ketterlin Geller
It is great to hear more about your research. I especially appreciate seeing the classroom activities and hearing more about how the intervention and teacher professional development are grounded in the learning progression.
Edith Graf
Edith Graf
Senior Research Scientist
Thank you Leanne! As you saw, the classroom activities are experiential. One of the challenges in developing the learning progression-based tasks was thinking about how to assess experiential learning in a computer-delivered context. As a result, many of the tasks are interactive--students use tools to construct arrow diagrams and directed graphs, for example.
Cliff Freeman
In 2017, while in grad school and working at YPP, I had the opportunity to work with Algebra Project colleagues on a NSF EAGER project located at Kennesaw State University with PI Alan Shaw. Although I’ve been part of YPP since 10th grade, it was my first time learning Road Coloring. I wish I had it in Highschool. Anyway, we developed a mobile and web app that can be used in AP classrooms that are learning road coloring. The implementation was fun and obviously more fun for the students who got to use the app to help them with their classwork. If you want to see and use the app on your free time or in your class, follow this link!
http://ksuweb.kennesaw.edu/~ashaw8/RoadColoring...
Edith Graf
Senior Research Scientist
Thank you for posting this link, Cliff. I like the exploratory nature of this tool.
Shameeka Browne
Hello,
I recently read Mr. Moses' book about the Algebra Project. I am a fan of the 5 step approach, and I try to replicate it with every unit that I teach. I am very interested in attending your professional development series. In regard to teaching functions, I liken the idea of a function to a vending machine, whereas each input has one output (push F2, the item in F2 falls out the machine. However, it is possible for the same items to be located in both F2 and F3). I wear many hats in the field of education, and I would like to attend your trainings.
Kate Belin
Hello Shameeka. I am a high school teacher and also love the approach of using the idea of a vending machine for functions. We've developed some materials around this and would be interested in getting feedback and sharing ideas for other approaches.
Gregory Budzban
Hello Rebecca,
Thanks for your comments and questions! As the primary author of the Road Coloring unit, I'm going to try to address some of your questions. The Road Coloring unit is the Algebra Project's curriculum introduction to the concept of function emerging from a famous problem in Math that was still unsolved when the unit was originally written. It's solution actually was announced in various venues, including USA Today.
https://en.wikipedia.org/wiki/Road_coloring_theorem
There is still significant amount of research being done in this area in Mathematics (see Cerny's conjecture) and exposing students to the idea that Math is still being "discovered" and that they can participate in this process is an important revelation.
Where does this fit into the "traditional" math curriculum?
Most Algebra 1 texts have a treatment of "arrow diagrams" in a section on finite-to-finite functions, but then quickly move on to a "traditional" treatment of functions via equations and formulas. The Road Coloring unit digs deep and introduces all of the traditional representations of a function in a coherent setting, showing students how to translate between them. In addition, it introduces matrices as the "mother of all representations" (pun intended), and introduces the connection between matrix multiplication and composition of functions. After introducing standard function f(x) notation, it circles back to investigate the axioms (associative, commutative, identity elements, inverses) from a matrix perspective.
...if the approach or the concepts tend to bleed over into other topics across a course?
As indicated above, the unit does a great deal with matrix algebra and shows students that not every operation is commutative and that not every element has an inverse. In addition, like all Algebra Project curriculum units, student discourse is highly valued, and students are constantly being asked to articulate their ideas in group and classroom settings, and to present these ideas to their class. The complexity of the problem (even small examples can create significant issues) helps to "normalize" mathematical struggle, but in a fun and engaging setting.
Hope this helps!
If anyone want to see the full curriculum module, send me an email at gbudzba@siue.edu.
Best,
Greg Budzban
KRISTEN BIEDA
I am impressed by the 5-step process you describe above, Edith, and by an approach to building understanding of function that supports students' generalizing the qualities of arbitrariness of functions that is necessary for flexible functional thinking (as opposed to just focusing on examples of functions represented in algebraic symbols). I'm wondering about the schools you've been working with. Have they been primarily charter schools, or also public schools? I'm wondering about how teachers and administrators in public school settings have reacted to using this kind of unit to dive deeper into functions rather than focus on "covering" a curriculum.
Frank Davis
Evaluation & Research Consulting
Hi Kisten,
The 5-step process have been a process that has been used to link a pedagogy to the design of curriculum materials. It very much depends on students using an experiential learning process to build understanding of functions as the video depicts. The schools and school districts in this research and previous NSF work building the curriculum materials are members of a developing alliance called “We the People for Math Literacy for all”. That alliance that has a goal of bringing learners who are typically having difficulties with mathematics to high school graduation ready to continue postsecondary education or jobs without the need to be remediated In mathematics. Please visit its website at https://mathliteracyforall.org/. This work is happening primarily in public schools. As you noted this work does involve a deeper dive into functions as well as how to effective teach students. This is a challenge for many teachers requiring what some call productive struggle. It is also a challenge for administrators who have to take a longer term view of how to bring their students to mathematics literacy.
Frank
Frank Davis
Evaluation & Research Consulting
Hi Shameeka,
Thank you for your post about our project. The trainings around Algebra Project curriculum materials are usually arranged by schools and schools districts that are part of a developing Alliance called “We the People for Math Literacy for all.” Please visit its website at https://mathliteracyforall.org/. The organizations in the alliance have a goal of bringing learners who are typically difficulties with mathematics to high school graduation ready to continue postsecondary education or jobs without the need to be remediated In mathematics. Joining these trainings would require permission of the schools or school districts. If interest in upcoming training in school districts please send your contact information to Ben@algebra.org
Frank
William Zahner
This is a great video and summary of the work you are doing. Thank you for sharing! I am wondering about your learning progression for functions and how you merged that with the Algebra Project approach. Did you find that you needed to modify either the prior research on LPs in function, or the AP's 5 step approach to be mutually compatible?
Edith Graf
Senior Research Scientist
Great question. The learning progression (LP) has three strands: one of these strands assumes a traditional approach to the instruction of functions (i.e., a focus on real-valued functions) and the other two strands are linked to Algebra Project curriculum modules. So in other words, the curriculum modules are sufficiently different from each other and from traditional curriculum that the LP needed three strands. However these strands are also linked by foundational ideas that span all three, for example the uniqueness property of a function, translating among representations, and attention to domain and range. The 5-step process is more general than the LP and could be applied to any concept in mathematics, but is aligned with what we found in the research literature on functions (a progression from concrete to symbolic and abstract).
Eric Hamilton
This has been an exciting and inspiring line of work fr several decades. Please give my regards to Bob Moses and Frank Davis. Any chance that you have been able to apply the 5 step approach in PD with teacher outside of the US? Much of our work has involved teachers in sub-Sahara, and it is more technology related but heavily focused on mathematics.
Frank Davis
Evaluation & Research Consulting
Dear Eric,
Thank you for your post. I did see your video and can understand your passion for the work.
Historically the Algebra Project has focused its math literacy work in public school communities within the U.S. However, there are a couple examples of international exchanges.
One, in the late 1990s and early 2000s, Maria Keita Diarra, Director, together with a delegation of teachers from the Institute for Popular Education in Kati, Mali, West Africa, visited the Algebra Project classrooms and professional development workshops in Cambridge, MA as well as in Jackson, MS. In January 1999, an Algebra Project delegation visited L'Institute Pour L'Education Populaire in Kati.
A second, in the spring of 2015, a delegation from the Algebra Project and Young People's Project was invited to present and provide workshops for students and teachers at the University of Ireland - Maynooth in coordination with the National Council for Curriculum and Assessment (NCCA) in Dublin, Ireland. The NCCA then sent a delegation of teachers to participate in the summer 2015 student/teacher induction academy in Miami, FL.
The AP and its school partners are doing work via the internet due to the pandemic. After viewing your video, I couldn’t help but imagine similar types of interactions between young people in the AP across the country.
Traci Higgins
Great project! One of the key comments I noted when watching the video is that many PD programs show teachers a different way to teach content, but that an important goal of this project is to work with teachers to build a different understanding of the content not just a different way to teach it. There was also an administrators that said your PD had been transformative for teachers in ways that previous PD had not. This made me think of my experience doing research on the PD program Developing Mathematical Ideas. I will always remember at the end of one DMI inspired project hearing the teachers talk about what that experience meant to them. They spoke of it as a transformative experience both personally and professionally. Thinking back to this, your comment really resonates--providing experiences that help teachers understand the material in a different way is extremely empowering for them. And I think the modeling approach does so many good things to support reasoning and problem solving. Once teachers have had this experience they can begin to create similar experiences for kids. Nice work!
Jeffrey Ram
By providing hands-on experience in the field collecting and analyzing organisms, teachers in our Professional Development program (imaged on our video, and including a section where one of the teachers comments about how it transformed her feelings about teaching science). Right now, we are grappling with: how do we provide that same experience this summer when our school system has said: no in-person PD, because of the COVID-19 epidemic. How has the epidemic affected your project? Your visitors from Mali came a long way! Wouldn't it be great if they could get similar experiences on-line?
Edith Graf
Senior Research Scientist
Thank you, Jeffrey-- and I enjoyed your video. In answer to your question, after completing a pilot, we had started (but not completed) field-testing tasks for the learning progression-based assessment, so yes, the project was affected. Virtual PD has potential I think, but its implementation, especially for hands-on curriculum units, needs to be thought through and these are discussions we need to have among members of our project team.
Frank Davis
Evaluation & Research Consulting
We just wanted to give a bit more context for Cliff Freeman’s post about the App used to simulate the road coloring problem.
The App was developed as part of an NSF EArly-concept Grant for Exploratory Research (EAGER) grant (award #1651092) entitled, “Incorporating Computer Programming Into Middle School Curricula to Enhance Learning for Low Performing, Underserved Students”. The PI of the grant was Alan Shaw at Kennesaw State University and the grant period was 8/29/2016 – 9/30/2018.
The Road Coloring App was developed through this EAGER study, in collaboration with the Computer Science Dept. at Kennesaw State University, the Algebra Project, the Young People’s Project, and a number of Algebra Project teachers across the country. The App is intended as an extension of the Algebra Project’s Road Coloring curriculum module, and, to enhance the experiential learning process.
For a web-based version of the App, please see: http://ksuweb.kennesaw.edu/~ashaw8/RoadColoring/index.html
You can also download the App for android phones at: http://ksuweb.kennesaw.edu/~ashaw8/RoadColoringDownload/
If you would like to read more about how the Road Coloring module has been used in additional research, please see, High School Students' Understanding of the Function Concept, Ed Dubinsky and Robin T Wilson, Journal of Mathematical Behavior, Vol. 32, Issue 1, March 2013, pp. 83-101. For abstract, please see: https://doi.org/10.1016/j.jmathb.2012.12.001
Iris Wagstaff
Great video! I like that you include both students and teachers as leaners. I think this 5-step framework is definitely applicable to other areas of STEM beyond math concepts.
Edith Graf
Senior Research Scientist
Thank you Iris for the comment, and yes, I would agree.
Danielle Espino
I want to concur with Iris' comment-- those were also my immediate thoughts after watching the video :) I'm especially interested in collaboration, and took note of one of the comments in the video that this work also helps students to build understanding together. Kudos to Edith, Frank, Cheryl, Chad, Maisha and the entire project team!
Edith Graf
Senior Research Scientist
Thank you Danielle for visiting and for the comment!
Frank Davis
Evaluation & Research Consulting
Hi Traci,
I do remember you from TERC.
Here is a response to your post from Bill Crombie who is the Director of Professional Development at the Algebra Project.
Right on! I think it is important to note that this type of
transformative learning for and by teachers is one of the main aspects
of teacher participation in the We the People - Math Literacy for All
Alliance that this work is connected to. The Alliance recognizes that
broadening participation in STEM requires the best efforts of those
who are closest to these issues: namely teachers and ultimately
their students.
Traci Higgins