Summary of Feedback from K-12 Participants

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Main Themes for Masterman Students from the Math Images Project

As we look at any of the feedback from the Masterman Math Images students, it is important to note how the students were selected for Math Images. This varied by the year involved. In the first year of the program, 2012, I presented about the Math Images project to all of the 9th-grade Geometry students, a total of 56 students. I made it clear that it was a voluntary extra credit project and that those who took it on would work on it primarily during class but would still be responsible for the regular curriculum. Of those 56, xx signed up to work on the project, one group of three, two groups of two, and nine individuals working alone. Two of the individuals did not complete their projects; these were the students who had struggled most with getting work done during the time before the Math Images project.

In 2013, the second year of the program, I invited a select group of students to lunch, where selected students who had completed the previous year’s work with Math Images spoke to the group about their experience of the project. I remember all or almost all of those invited ending up signing up for the project. Thus, the first year’s cohort were self-selected, the second year’s were invited, with the opportunity to decline the invitation. However, with the exception of a couple individuals from the first year who I would not have invited, the cohorts both consisted of strong, motivated students who completed work punctually and were active participants in the class, earning high grades.

During both years of the project, Diana Patton came in to the class one day per week to work with the Masterman students. During the 2012 project, she came in on her own during much of the project time, but I was also available since I had a student teacher that was teaching the rest of the class full-time. In 2013, I did not have a student teacher, so we chose to start the project later when the Swarthmore 2013 Math Images cohort was available. They came in twice per week on Tuesdays and Thursdays in late May and early June; each of the four Swarthmore students worked with the same Masterman students from week to week.

To summarize, the Masterman students who worked on Math Images during 2012 and 2013 were generally high-achieving, motivated students, ones who had already shown capacity to work independently and succeed despite the many challenges American adolescents face in completing high-level academic work. They had support from a Swarthmore upperclasswoman that was skilled in both the mathematics and the wiki language needed to put the project work up on the web. Either one of the participants from Swarthmore or I provided additional support. All themes that emerge from the interviews need to be understood in that context.

I. Positive Themes, not directly related to technology

Positive Theme, organized by number of persons who mentioned it

  1. of times mentioned out 7 projects

1. freedom of choice / play / not from grade 4+ 2. personal accomplishment 4+ 3. develop study skills—planning, using time well, keeping info in notebook 4 4. open-ended/divergent/discovery 3+ 5. developed higher-order and problem-solving skills 3 6. learning content that will help in future 2+ 7. group work / learning from others 2+ 8. grit / initiative 2+

There are three major categories the main themes fall into:

A. How the project was set-up

       	1.  freedom of choice / play / not from grade

4. open-ended/divergent/discovery 7. group work / learning from others

B. Positive results, cognitive

       	3.  develop study skills—planning, using time well, keeping info in notebook
       	5.  developed higher-order and problem-solving skills
       	6.  learning content that will help in future

C. Positive results, personal/psychological

       	2.  personal accomplishment
       	8.  grit / initiative

I was frankly thrilled in the interviews that there was so much positive feedback about the experience and that it spanned all three domains of Bloom’s taxonomy—cognitive, affective, and psychomotor. I consider the set-up of the project to be crucial to this. This was in many a model assignment—students had freedom to choose a topic that had meaning to them and then meet a challenge that required skills, persistence, and some help (scaffolding). In the process, they grew in all three domains.

I begin with the affective, the most fundamental of the three domains, as shown by the lowest of its five levels. Those levels, from lowest to highest order, are:

1. receiving—the student pays attention; without this, no learning can occur 2. responding—the student actively participates in the learning process 3. valuing—the student attaches value to the knowledge they acquire 4. organizing—the student can relate what they are learning to their own schema 5. characterizing—the student holds a particular value or belief that now exerts influence on his/her behavior so that it becomes a characteristic.

Higher order levels imply lower; you can’t relate what you are learning to your own schema if you aren’t paying attention, actively participating, or attaching value to the knowledge you are acquiring. What was notable in the interviews was the number of times students used words that showed they valued what they were learning, level 3 of 5. And some of their self-reporting goes past that. When Anea said that “Math Images makes you realize that math is much more than what is in the textbooks,” this is an example of characterizing. Anea’s belief about mathematics changed through Math Images and it is now characteristic of her attitude about it.

Thus, with one notable exception, the open-ended nature of the project and the opportunity to discover mathematics on their own helped students in the affective domain. As shown by the table above, students also developed in the cognitive domain. I like the graphic below as a way to depict the cognitive domain categories in Bloom’s taxonomy.

Wyatt’s goal was to understand the advanced topic of polar coordinates and equations well enough to explain it to solve problems involving them and explain his work to others. Vicky wanted to apply what she had learned about intersections of concentric circles with lines to creating other designs and making sense of the mathematics involved. And Michael, Erin and her group, and Hana and her group wanted to create something new based on what they had learned about fractals and spirals. All groups (yes, even Victor!) grew in their knowledge and understanding of their topics and several noted how they grew in their problem-solving skills, all examples of higher-order cognitive work.

In the psychomotor domain, one used a new organizational scheme and a number of students grew in their technology skills, some of which they’ve put to use in courses since Math Images. This will be covered in more detail in Section III.

II. Challenges/Negative Themes

Challenge/Negative Theme

  1. of times mentioned out 7 projects

Challenge or Negative? 1. times of frustration 2+ Challenge, related to grit 2. limited background in math making many topics hard to understand; limited the choices (need for more scaffolding) 2+ Negative, needs more scaffolding 3. felt aimless at first; lack of help with ideas, picking the topic was hardest 3+ Partly Challenge, part of a divergent, open-ended project, and partly a negative, especially for kids like Victor who need more direction 4. Math Images site advanced, somewhat intimidating 2+ Negative; we need to pick out MI sites that are accessible. Implication/Strategy: Can be part of writing assignment for college students.

In my view, the “negatives” I initially classified the students’ remarks into can be seen as either challenges of the project that have educational purposes or as components of the project that merit attention with the goal of improving the instruction next time.

With open-ended, divergent projects such as Math Images, it should be expected that students have trouble at the beginning during the “research” phase. Masterman students are typically excellent at convergent thinking; they are excellent at taking multiple choice tests which are a large determinant of admission to the school. Many 9th graders at Masterman have had limited, often very limited opportunities to complete divergent assignments. Even large projects often come with carefully structured guidelines for how to complete the project and a rubric for how the project will be evaluated.

Open-ended, divergent assignments require very different skills than convergent ones. Handling times when there is no clear sense of direction and persevering when your work leads to a dead-end were experiences mentioned by students in the interviews. While these can feel very negative in the short-term, they are experiences we will all face at times in our lives. Projects like the Math Images assignment give students an opportunity to face this kind of situation with the guidance of teachers and older students who can help with problem solving in those situations.

The positive qualities that students have an opportunity to develop out of these experiences include initiative, tolerance for frustration, and problem solving. By providing too much structure and guidance at the beginning, teachers may deprive students of important learning opportunities.

On the other hand, too little scaffolding for the project task can lead to excessive frustration and a sense that a student is just “spinning their wheels,” making the project more a chore than an opportunity. This certainly seemed like the case for Victor. From the information in the interview, Victor had the most challenge with initiative and self-starting of any of the interviewees and so it makes sense that he would find this kind of project the least beneficial.

I have several thoughts about this. First, since the others were successful, it may be that Victor experiencing frustration and wanting more structure throughout is not a negative overall for him in the long run. However, unlike others such as Melina, Gabrielle, and Wyatt for whom the Math Images work has had very positive results in future courses, there is no evidence from Victor’s self-reporting that it has helped him at all with his future coursework. He admits to being a procrastinator and it makes sense that an extra credit project that had no negative consequences for a student who received an excellent grade in the course because of all of his other strong work, procrastination notwithstanding, would not have an effect on such a significant personal approach to completing work.

However, given that other students noted that that existing Math Images pages were often daunting to make sense of, it seems appropriate to provide greater scaffolding for that part of the project. The pages were written by talented college students who typically have four more years of experience with mathematics, both with the content and processes such as writing proofs. The Math Images instructors did attempt to alleviate this challenge by having later Swarthmore students in the program write at least one page appropriate for high school students, so there should have been at least some pages that are appropriate for 9th graders, particularly the advanced students selected to work on the project.

So one of the important results from the interviews is the need for greater scaffolding in introducing the project. Components of this should include: 1. finding and listing the pages that are good for the 9th graders to start with 2. improving the instructions for writing the wiki pages 3. checking in with students frequently at the beginning of the project to help them problem-solve with topic selection

III.  Themes Related to Technology


Tech Positives Themes, organized by number of persons who mentioned it

  1. of times mentioned out 7 projects

1. discovery aspect of gsp 3 2. pride; positive identification with finished product 2-3 3. able to find web resources to make own gifs 3 4. Diana's help 1 5. Math Images site as resource (Polar Rose) 3 6. skills learned applied later, like making gifs 2+


Tech Challenges Themes, organized by number of persons who mentioned it

  1. of times mentioned out 7 projects

1. challenges of working with wiki 3 2. challenges of transferring gsp to wiki 2 3. problems with gsp; Swat students didn't know it either 1 4. MI website non-intuitive 1

I chose to analyze the positives and challenges together because there were relationships between the two. In fact, the scaffolding we provided grew out of either our expectations or experiences of the challenges the students faced. We knew in advance that the Masterman students did not know the wiki language they would need to use to input their Math Images work on the website and so that was something that Diana provide instruction about early on. I was the go-to person for Geometer’s Sketchpad and so tried to make time available during class to provide that help.

The two main elements from the positives that I chose to include under the category of “technology” involved the kind of assignment—one of discovery—and the scaffolding provided for the students. It was notable to me how many students expressed how much they enjoyed the freedom to discover on their own using Geometer’s Sketchpad (gsp). It’s important to note that by this time of the year, the students had learned many techniques in gsp, so that plenty of scaffolding had been provided. Hana specifically mentioned the Car Jack project, one in which there was a balance of scaffolding as well as open-ended challenge.

Diana was clearly a huge positive for the students; not only did she have skills in both mathematics and technology but she was also an excellent teacher who worked very well with adolescents. For those who wish to replicate the Math Images program at either the collegiate or secondary level, finding staff such as Diana is critical to a successful program.

� IV. Themes Related to Developing or Retaining Student Interest in STEM subjects (Science-Technology-Engineering-Mathematics)

STEM Connections Themes project matched well her way of solving problems project pushed her to think about how things are constructed; take apart and put together open-ended didn't have one focused question but instead allowed students to explore and examine patterns like to discover; we definitely discovered always had basic interest; like things that explain; STEM is an interest and I want to get into a stem field. Recently it’s been information technology. I’m strong in math so I want to do something connected with it.

The first note is that I did not get time to ask several of the interviewees about their interest in STEM subjects, making the already sample size that much smaller.

The verbs in this category included “matched,” “allowed,” and “pushed.” I’ve chosen the order to represent how the Math Images experiences facilitated the work of the students, from being a good “match” for someone already with that learning style to one in which the students was “pushed” to try something she may not have done otherwise.

� V. What Students Remembered About the Project, without other prompts

What Do you Remember About the Project? fun, (mentioned 3 times) tough to get started, no real clear goal; hard to come up with idea social component Diana's help personal stuff about Swat students—pimple, same train station interesting; helped me think about how math works started with something I had done for fun Really prepared me well for future course in pre-calc Helped me learn how to model a problem

This was the first question asked during the interview. Given its open-ended nature and lack of other prompts, it gets at some of the core memories of the students. Overall, they remember it as a fun experience that taught them valuable skills and content and resulted in a successful final product.