2/2 David Computational Literacy and Scientific Modeling

Throughout Nersessian’s article a few of the NGSS seem prominent. First, developing and using models was most evident, as this was practiced throughout the case study. The scientists observed during the study obtainined, evaluated and communicated information with one another, a practice also listed in the NGSS. Throughout the development and use of models and the obtaining, evaluating and communicating information the models in use were continued to be revised amongst the scientists in the study. I’m curious where NGSS would categorize revision in their practices? Certainly the specific revision of models would fall under the development and use of models, but how about revision as a practice of asking questions, revision of planning investigations and revision of engaging in argument from evidence?

diSessa says of computational literacy, “Programming turns analysis into experience and allows a connection between analytic forms and their experiential implications that algebra and even calculus can’t touch,” on page 40. I interpreted Nersessian’s ideas about scientific modeling as such: to create a model so that we are able to accurately predict a phenomenon. This is best summarized when she says, “The primary investigative practice in many areas is constructing physical models that adequately exemplify the phenomena of interest so as to be able to conduct controlled experiments with the models and transfer outcomes to in vivo phenomena,” on page 21. While these statements between the two authors are not explicitly similar, I think that the ideas share an inquiry thought. Comparing to our in-class experiences with ViMAP, computational literacy includes an analytical experience that incudes making a model to predict a phenomenon. Programming is a critical but not inclusive part of computational literacy. As discussed in class, programming involves inquiry thought and problem solving, which is also a critical part of effective and efficient scientific modeling.

2/2 Steve: diSessa Round 2

1)      The Narcessian paper addresses ‘asking questions and defining problems’ when it discusses how the bursts were initially considered noise but turned out to be meaningful.  This was a case of an improperly defined problem which modeling helped resolve.  The scientists thought that bursts were the problem, but they were actually part of the solution.  The practice most stressed by the Narcessian paper is ‘developing and using models’.  This is the whole focus of the paper.  The scientists used several different models and combined them in the end after making numerous iterations that improved each time.  The models not only communicated certain aspects of the network to others, but they helped the scientists understand what was actually going on in the network.  The paper addressed ‘analyzing and interpreting data’ by showing how the models continuously evolved based on new data coming from the neural network.  As information flowed from the network, it was used to improve the models.  ‘Computational thinking’ was stressed when Narcessian discussed how the computer model was made by one student who preferred to think in a more mathematical way about the network.  This model ended up being extremely useful in benefiting from the study.  ‘Constructing explanations’ was a skill used by the scientists throughout.  They were continuously bombarded with new information both from the models and the network itself, and they had to keep trying to find explanations that were consistent with all of the available data.  ‘Communicating information’ was discussed in the context of sharing models.  When the scientists came together to combine the benefits of each individual model, they had to carefully communicate to each other all of the aspects of their model.  Working as a team collaborating together was an important aspect of the study’s success. 

2)      diSessa thinks that computational literacy is useful because it could allow students to see motion and other concepts in a new way that would make it much easier for them to understand kinematics. He shows how computer programming is well suited to describe motion, particularly with the help of software like Boxer, and the reader can well imagine there are other concepts than motion that would be similarly aided by computer software.  DiSessa speculates the same thing that Narcessian anecdotally chronicles: computers allow us to model scientific phenomena in ways that pen and pencil cannot.  Traditional static models or even video animations cannot encapsulate the variation present in complex systems like kinematics or neural networks.  To properly understand complex systems, we need to engage with models that we can not only see move, but that we can change and modify and poke and watch the results.  In diSessa’s piece, he references the intractability of computer models when he mentions children changing the slider for the variable a which affects the direction of the motion vector v.  In Narcessian’s piece, the same thing is demonstrated in the way that the computer model continues to be changed based on new pieces of information gathered from the real neural network.  Both of these practices involve complex systems, modeling, and using the power of computers to understand how the system responds to change.  The students in Narcessian’s paper would not have been able to understand the neural network had they not possessed the computational literacy to be able to make a working model of the network.  Because they possessed that literacy, they were able to make a breakthrough discovery, and that is the kind of thing that diSessa claims would happen much more often in a world where computer science was taught to all children.

1/12 Steve - learning progression and ngss practices

1/12

Summaries:

            In their paper Developing a Learning Progression for Scientific Modeling, Schwarz and colleagues propose guidelines for effectively using modeling to aid scientific education.  After defining modeling and discussing the different ways that modeling benefits students, the authors outline two major ways that student modeling can be evaluated.  The first criteria is how the model is used as a “tool for explaining and predicting”.  The second criteria is how students understand model as changeable in the face of new data or feedback. 

            The chapter on Scientific and Engineering Practices defines 8 practices used by both scientists and engineers in their work and suggests that science education should include chances for students to engage in all 8 of those practices.  Engineering and Science are continuously compared and contrasted, engineering being repeatedly described as solving human problems, and science as figuring out how things work.  The authors list a variety of modeling skills that high school graduates should possess. 

Relevant Themes:

·         As an engineer I have always wanted to bring more “real world” practices into my teaching and modeling seems like an effective way to do that.

·         I like the idea of students creating and critiquing models because that can help them become more critical of models or graphs they see in the news and other non-school situations.

·         I like the messiness of models – how they change and how they are never perfect.  The quest for perfection leads to so much good inquiry and reasoning.

·         I appreciate the inclusion of engineering in the chapter 3 piece.  Pure science is taught in schools and engineering is rarely taught, but there are more engineers than scientists in the world.

Relationship among the readings:

            The chapter 3 piece was more of an overview of all of the scientific practices and what students should be able to do and why it is important that they be able to do those things. The Schwarz article was more of a practical guide to using modeling in the classroom.  It included examples of teachers using models and how students progressed with the models, and what challenges the teachers faced when implementing the modeling unit.  I would be interested to see a Schwarz style article about modeling engineering challenges.  I would have liked to see more concrete examples of student work in the chapter 3 piece to illustrate how students could engage in all the practices.  I wonder if authentic audiences for models are really necessary or if the combination of peer judgment and grade incentives can serve to provide the same level of motivation for students.

1/12 Dan: NGSS and Modeling Practices

Summaries:

The Schwartz et al article discusses scientific modeling, specifically practices that are meaningful and productive for science education and the pros and cons of using modeling practices in the classroom.  The researches address the learning progression on two fronts - the creation of models as tools to explain and predict scientific phenomena and the understanding that models are tools that can be changed and developed as knowledge advances.

The NGSS Framework article discusses the differences between science and engineering, and the importance of teaching both practices in schools.  Specifically, the authors outline 8 practices for K-12 science classrooms for both science and engineering.  The article emphasizes the importance of teaching students to think critically and scientifically about questions and problems in the world.

Relevant Themes:

Science is a dynamic process that is constantly being evaluated and re-evaluated, especially in the presence of new information and knowledge.  It is important that students understand the creating models is a process that is never complete.

The scientific process, specifically creating models, is a critical component of solving real world problems and answering questions and one that is not well-emphasized in typical K-12 science classrooms.

Science and engineering, while often considered to be one in the same, use similar processes to address very different goals.

Relationships:

I appreciated both articles emphasis on the scientific process, specifically modeling, as a continual process.  I know from my own science education, even in high school and college, students were trained to accept information from teachers and textbooks as indisputable.  It creates an inability to reconcile new information that may not completely fit within an existing model.  I also enjoyed the comparing and contrasting of science and engineering.  Too often science classes are taught in the abstract, or “in a vacuum”, and fail to teach students how to solve real world problems.  Teaching skills and practices specific to engineering, especially using modeling would help students, would help students learn to seek out new information and ideas to solve problems and receive feedback and critique as a part of the process.  The Schwarz article left me wondering if it is reasonable to expect the average student  to progress to “Level 4”, where they would “construct models spontaneously” and seek changes to their models to enhance their explanations. The NGSS Framework article did not address any specific challenges or obstacles in implementing this type of framework, so I would be curious to explore the practical restrictions.

1/12 Jenna - Practices in Practice

The NGSS Framework chapter defines the need in K – 12 classrooms for students to be engaged in authentic scientific and engineering inquiries by employing the practices that professionals would use in their own investigations. The Framework identifies eight general practices that students should develop competence with, and distinguishes how these practices would look when employed by a scientist and an engineer. For each practice, the authors of the chapter list “goals” (what a student should be able to do by grade 12) and briefly describe a trajectory of “progression” (in a student’s fluency with or sophisticated use of the practice).

In contrast to the general foundations that the NGSS Framework is building for K – 12 education, Schwarz et al. are specifically developing a “learning progression” for modeling as a scientific practice. In devising their learning progression, they also incorporated opportunities for metamodeling, sense-making, and communication. Schwarz et al. then provide an analysis of a study of elementary and middle school students that had used scientific modeling practices in their classrooms to help illustrate elements of their learning progression.

I appreciated that both of these texts are taking a situative approach to science and engineering education (and I take this to be both a relevant theme and a relationship between readings). I think this is an especially important philosophy to hold because our knowledge of the world is limited by the questions we ask and the technologies we have available to use in inquiry. Scientific “facts” can break down in light of new evidence and interpretations, but the fundamental practices by which professionals create and communicate scientific knowledge will always be relevant. Both readings are clarifying how educators might successfully engage students in participating with these practices and, consequently, how the next generation will come to be prepared for facing future challenges.

One thing I noticed with both articles is that there is a prominent focus on visual/spatial representations and observations in constructing models. I wonder how students with visual disabilities might fare with this particular practice. Any learning progression, if it is to be implemented in a classroom, needs to be inclusive of all kinds of student needs.

Response: Caitlin Farney 

I agree with you on how it is important that students learn about how science is communicated within a community, and how scientists and engineers ask and answer problems. Furthermore, these practices, including models, can be used in other disciplines as well. Mathematics help create models, but models can also be used in math classes. Explaining and communicating is important from science, to political science, to journalism, to business, and more. I had not thought about how most models are visual. Mental models may help here, but what about when something needs to be demonstrated?

Response: Laura Cummings 

It’s interesting to me that you see the two articles in contrast over the concept of progression, as I think that both clearly acknowledge the way in which scientific practice grows and evolves in the young mind.  Because the next gen and modeling focus is new in science education, I think that we will often encounter students at varied stages of familiarity with modes of independent scientific thinking and progress in their mastery of these practices.  Thus I think it is really valuable to consider how students may be interacting with these activities at various stages in their academic career and appropriately support them in reaching mastery.

1/12 Laura Mindful Participation

 This week’s readings introduced the contemporary strategies for incorporating authentic scientific practice into the classroom.  The NGSS chapter from A Framework for K-12 Science Education highlighted a cyclic framework, which combines inquiry, modeling, revision, and argument with the more traditionally, though not necessarily effectively, highlighted practice of experimentation.  The chapter goes on to define goals and progression for each practice in both science and engineering classrooms, a distinction which is at times tenuous.

       In their 2009 article “Developing a Learning Progression for Scientific Modeling: Making Scientific Modeling Accessible and Meaningful for Learners,” Schwartz et al focus specifically on the practice of modeling in the classroom.  Through classroom studies, Schwartz et al define progressive levels of engagement with modeling along two dimensions- models as generative and models as dynamic- in order to understand students’ level of facility with both the practice of modeling and the concept knowledge at hand.

The themes and questions that were especially striking to me were:

- What is the value of separating the study of science and engineering?  Are engineering courses on the horizon for schools? Or would it make more sense to incorporate engineering into existing courses? (i.e. chemical engineering into chemistry, etc.)

- Are the levels presented in Schwartz et al necessarily step-wise?

- In our quest for authentic practice, are we asking too much of young students?  Schwartz et al reference young minds’ tendency toward literal over abstract, what are students able to reasonably achieve?

- Understanding that knowledge is not fixed is a significant phase shift in our experience of learning and existence.  How can we encourage our students to consider information sources more critically and scaffold their experience such that this realization is safe and not crisis inducing?

       Both readings highlighted a movement toward more meaningful participation in the study of science and the pursuit of a minds-on curriculum, elements which were attempted over the past few decades with an emphasis on laboratory science, but not often effectively presented.  By focusing on more student driven, mindful practices surrounding experimentation, I think that it is much more likely that students will have a more valuable understanding of both content and what it means to do and know science.  Along that vein, I think it is also valuable to help students see that scientific knowledge is not a fixed entity, however I think it will be important for us as teachers to be especially deliberate when discussing this concept so that our students can successfully make that mental shift in steps without crisis.  

1/12-Elizabeth-the framework...your new best friend

In the Schwartz et al article, the researchers discuss the components of effective and successful modeling within science education, including those of argumentation, explanation, and revision, alongside the understanding of the purpose of modeling and why it is important.  Schwartz et al also pointed out that modeling is a dynamic and interactive practice, where students continually change their models as their own understanding develops.

The Conceptual Framework for New K-12 Science Education Standards is a new framework focused on improving science education, striving to create an educated and active generation of individuals.  This framework emphasizes the importance of science and engineering practices, how successful practices are achieved, and how these skills can be used to produce enlightened students who critically think about the science around them.

Major Relevant Themes:

·      Science is NOT a static subject but an activity-focused endeavor:  I find it is always important to remind myself that science education is not about learning “isolated facts” as the Framework called it, but instead practice or “activity” driven, where students engage in argumentation, explanation, and revision of their ideas about scientific topics.

·      Making science meaningful is critical for student success: I have learned over the course of my time here that making information relevant to students and letting them know why it is important that they understand science it critical to their motivation and grasp of the concepts.

Both articles discussed the practices of modeling, argumentation, explanation, and revision are essential for successful science education.  In the framework, this is seen in the 8 practices listed for the K-12 curriculum, where the authors have outlined a potential student’s progression through their grades and where they should be in terms of understanding and practice.  In the Schwartz et al article, the researchers discuss similar practices of modeling yet provide specific examples of this practice-in-play within a 5^th and 6^th grade classroom.  Schwartz et al also emphasized this idea of progression and he ever-changing nature of student modeling.  While these articles did have similarities, the framework delved into the differences between engineering and science, an idea I never really thought about.  I had always considered engineering a science and therefore made little distinction between the two.  However, I did always view engineering research as a more “hands-on” science, or, as the framework puts it, one that has “immediate practical application” (47).  Thus, how would an engineering classroom differ from a science classroom?  Furthermore, I liked how the reading talked about the two varying purposes of argumentation.  While science argumentation focuses on coming up with a simple, single coherent theory for a wide range of phenomena, engineering argumentation focuses on coming up with the best productive designs tailored to certain specifications and choosing among them based on other reasons.  The framework mentions using “models” in both science and engineering classrooms, but what are some specific models for each?  How much “practical application” would students be able to achieve in an engineering classroom and what would these practices look like in the younger grades? 

1/12 Joey: NGSS Foundational Framework

Week 1 Memo- Joey Tedeschi

The NGSS Framework reading focused on the importance of students engaging in scientific inquiry by using knowledge and skill together.  The chapter explains why learning science and engineering practices is important for students and which practices are essential through K-12.  Asking questions (for science) and defining problems (for engineering), developing and using models, planning and carrying out investigations, analyzing and interpreting data, using mathematics and computational thinking, constructing explanations (for science) and designing solutions (for engineering), engaging in argument from evidence, and finally obtaining, evaluating and communicating information are the practices mentioned.  The chapter then discusses the similarities and differences between science and engineering practices. The importance of each practice is explained, and then tied back to the importance of students exploring how science is done in the real world and how these communities communicate.  The hope is students will not only become more knowledgeable in content and skills, but will also be more critical consumers of scientific knowledge.

The second reading by Schwarz et al. was a research article that explored scientific modeling and how one might develop a learning progression to make modeling accessible and meaningful for learners.  The article mentions two dimensions, the first being metaknowledge and the second being elements of practice.  Basically, the authors want students to not only construct, use, evaluate, and revise scientific models, but also understand the nature and purpose of models.  The article then presented and explained the research they conducted for elementary and middle school students.  They found that students were able to use models to make predictions, evaluate and compare important aspects of models for a consensus model, and they were able to revise models to accommodate for new information.  The article then discussed some of the challenges that arise when modeling in the classroom.

Both readings commented on the importance of students constructing, using, evaluating, and revising scientific models.  The readings also mention how it is important for students to not just memorize or be taught facts, but rather explore scientific phenomena and engage in practices actual scientists do.  I completely agree with these ideas and think that students will get a lot more out of classes that take a modeling approach.  We often get information from a media source or the Internet and many people blindly accept what they see as truth.  By having students engage in modeling and learn how scientists actually practice, they will be much more critical of the source of the information they hear.  Students will also be better at asking questions, communicating, and understanding science.  One of the challenges mentioned by Schwarz was motivating the need to revise models.  By telling students when to revise their models we take the decision about whether the model is “good” or not out of their hands.  How do we know when a model is done being revised? (Aka no further revisions are necessary)

1/12 - Kim - Engaging & Evaluating Scientific Processes

The first half of Chapter 3 in the NGSS Framework discusses what scientific practice means and how that can be compared to engineering.  General science mostly seeks to develop explanations for various phenomena, whereas engineering seeks to find a practical application for some need or want.  Some themes I noticed in this text were the emphasis on keeping students engaged in the scientific process along with the importance of letting students make mistakes so that they can better understand real scientific inquiry that involves a constant process of evaluations, critiques, and revisions.  I really appreciate the point of knowing why the wrong answer is wrong can be more beneficial for students to develop a deeper understanding of a topic than just knowing why the right answer is right.  I think this is so important for students because to explain why something is wrong makes them not only know the correct reasoning but also engages them in argumentation and forces them to back up their thoughts and reasonings with evidences.

The Schwarz et al article discusses the importance of modeling in relation to practicing science and scientific literacy.  The article stresses the need for scientific modeling to occur at all age levels so that learners can be engaged in gathering the knowledge needed to explain and represent phenomena.  A theme I noticed in this article also corresponds to a theme within the NGSS Framework chapter we read; involving learners in modeling practices can help them build topic expertise, epistemic understanding, as well as expertise in the practice of building and evaluating scientific knowledge.  I think this article has really important considerations for elementary and middle school science teachers.  Introducing modeling at an early age is key for students to develop a scientific mind where they use evidences to back up claims and are able to reflect on why or why not a model works well for any given phenomena.  This process needs to be taught and used often so that as students delve into more complex topics, their basic understandings and schemas do not need to be drastically altered like many students currently have to do.  Simplifying or altering science for young students is too tempting for teachers but in the long run, it just hurts students’ abilities to deeply understand various phenomena.  I think both of these articles are great references as to why the term “minds-on learning” is much more important to consider in lesson planning than just “hands-on learning.”  Lessons need to be meaningful and productive for students, and modeling is a great way for students to fulfill that.

1/12 Caitlin Practice Modeling

The readings for this week were a selection from chapter 3 of the NGSS Framework, and an article of a study conducted by Schwarz. NGSS Framework encourages scientific and engineering practices to be emphasized in primary and secondary curricula. Modeling should be included as often as possible in instruction plans, and students should be able to use the models to solve problems. Schwarz studied the progression of understanding the creation and use of models in elementary and middle school students.

The Framework recognizes the importance of students learning how science is practiced, but also encourages that engineering practices, such as using scientific knowledge to solve problems, should also be included in the classroom. Modeling is how they suggest teaching students these practices. I agree with this as scientists and engineers use modeling to analyze data and communicate their findings with their respective communities.

Schwarz focuses on the practice of modeling as a scientific tool, but the idea of teaching students how modeling is used in scientific practice is still a major theme in the article. Schwarz talks about how students should learn how to use modeling, not just for explaining phenomenon, but also to predict other systems and events as well. NGSS also includes this idea in the standards. Models can be used to predict other events in the scientific community, and help solve problems in the engineering field.

Last semester we looked at how science is a process of building upon, or revising, prior knowledge. The NGSS and Schwarz readings seem to support this process. Schwarz noticed how students, with the right guidance, progress to understand more about modeling as a tool. NGSS also discusses this, but building of knowledge process seems to be the basis of the curriculum standards in general.

The common theme between the readings seems to be how students should learn how to use models to help them understand concepts and continue building that understanding. Both readings emphasize that the models should not be stagnant after they have been made, but should be revised and improved upon. Even other models can be created, if needed, to help in the progression of understanding. Neither article specifically talks about how modeling should be used in a high school setting. It would be interesting to how Shwarz’s study would have turned out if it was conducted with high school students.

1/12 David NGSS and Modeling Practice

Next Generation Science Standards Chapter 3 and Schwarz et al

Chapter 3 of the Next Generation Science Standards begins with encouraging the practice of science. Practicing science in the class including inquiry and modeling is important to a students knowledge. Eight different but related practices are described and then explained how they are fluid in practicing science. 

Schwarz et al focused more specifically on the modeling practice of Science Education. The importance of modeling is to incorporate the knowledge of theory and reasoning behind models so that learners understand why models are used.

Describe themes for both

Between both readings, use of effective modeling was strongly urged in Science and Engineering education. Effective modeling includes discussing how the model relates to what it is representing. Learners should be thinking about how the model is similar and different from the theory of the representation. Both readings discuss the importance of learners discussing models including explaining and arguing about the representations. Explanation and argumentation are critical to effective model use for Science learning; elaboration of ideas creates a deeper understanding of the theory.

Both readings used similar pictures to describe the role of models in Science Education. First, the Next Generation Science Standards described three columns labeled investigating, evaluating and developing explanations and solutions. Investigating in the real world involves asking questions, making observations, experimenting, measuring, collecting data and testing solutions. Developing explanations and solutions involves theories and models with imagination, reason, calculating, predicting, formulating hypotheses and proposing solutions. Connecting these two areas is evaluation where both are argued, critiqued and analyzed. Schwarz et al uses a similar picture to describe sense making and communicating understanding. On one side of the picture a table called elements of the practice lists constructing models, using models to explain and predict, evaluating models and revising models. The other side labeled meta-modeling knowledge lists models change to capture improved understanding built on new findings and models are generative tools for predicting and explaining. 

It is so easy to accept information from a media source and blindly accept what we were told. Engaging in practices creates good habits for students such as asking questions, communicating and understanding science. I think that models are never done being revised because learners are engaging in the models differently, unique to their perspective. Models can always be revised to help a learner gain a better understanding of the theory or knowledge. Teachers should be cautious telling students when to revise their models. Rather, the teacher should guide a class discussion about why a model should or should not be revised. Students should construct their knowledge as a whole, bringing value to their ideas.

1/26-Elizabeth-technologically challenged

In her article, Modeling Practices in Conceptual Innovation, Nersessian focuses on the understanding of scientific concepts as dynamic, continually changing through “model-based reasoning,” the formation of new concepts from various domains, and the role of these scientific ideas in the active process of investigation (2).  Nersessian discusses a novel and innovative study of hers in the field of ethnographic research, where the researchers sought to “recreate the brain,” therefore creating the dish-model system, which involved a neuron culture and the simple networking characteristics of the brain, to further understand the interactions between neurons when learning is taking place.  This study was a huge step in neurological research and understanding because it took what was unavailable for scientists to study, enabled them to make a simplistic model, and used the data collected to further understanding, make necessary changes, and add further complexity to the model.

In her book, Changing Minds, diSessa discusses the various meanings of the word, literacy and how they have changed over time.  She makes an important distinction between computation literacy, “an intelligence achieved cooperatively with external materials” and computer literacy, which alludes to the simple ability to work with a computer (6).  She begins by describing her three pillars of literacy: material (physical representations), cognitive (the mental component that is necessary alongside the material), and social, using the development of calculus to show the influence of this third component to change this mathematical literacy from merely a "pleasurable success for a few” to an “infrastructural assumption,” something that is necessary to drive the educational process (13).  Finally, diSessa uses the example of Galileo and his concept of uniform motion to show the extent to which literacy, specifically mathematical literacies, are infrastructural, breaking it down into the material, cognitive, and social components.  

Both authors incorporated various components of the 8 practices outlined by NGSS as critical for student success in the science and engineering classroom.  In the Neressian article, all 8 practices, from asking questions to constructing explanations, were used.  This is so mainly because it was a novel study that was conducted and there was a lot of room for investigation, explanation, and revision.  First, they started out with a question regarding the communication between neurons when learning takes place.  Then they developed the dish-model system and carried out a four year investigation.  Throughout those four years data was continually collected and additional computational models were created.  This was evident with the “bursting” component of their study, which prevented the detection of learning, which was critical for data to be collected.  Thus, the scientists had to come up with a way to prevent bursting (designing solutions and constructing explanations) and included one researcher creating a computational model of the system to promote progress and understanding of the neurons and their interaction.  Throughout the whole process, positive and productive argumentation took place from the obstacles they hit to the data that was collected.  While this study was new in its field, the information collected from this study has made a huge innovative step towards further understanding neurons and their interaction when learning. 

In the diSessa article, there was less emphasis on the experimental components of the 8 practices, such as developing and using models, planning and carrying out investigations, and analyzing and interpreting data, and more on using mathematical and computational thinking and communicating information.  Her emphasis on mathematical literacy is evident in the beginning of her book where she discusses the three pillars of literacy and their importance.  She also makes note to distinguish between computer literacy and computational literacy, deeming the latter to be more complex, involving the collaboration between mental ability and outside materials.  To me, her view that this literacy is incredibly important comes from her use of the word “infrastructural,” the idea that something is critical for the success of, in this case, the educational process (5).  Thus, mathematical and computational literacy is critical, or will be infrastructural, in the educational process.  While she does discuss the models of Galileo and Newton and their importance, diSessa does so in a way that focuses or emphasizes the importance of communication and clarity.  This is primarily evident in the Newton-Leibnitz debacle where, while Newton came up with the same concepts, his notation was clearer and therefore readily accepted.  It is what distinguishes something from being merely fun and pleasurable to necessary and infrastructural.