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?

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

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. 

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?

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? 

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.

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. 

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. 

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.

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.

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. 

This is a remarkable quick look at the serious work being done by all of you.  Kudos to all.

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. 

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...

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.   

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

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.

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

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

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? 

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.

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!  

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

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. 

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.