Analysis: Falk's research provide a framework to understand how to achieve a deeper level of understanding and awareness and learning
Step 1: Choice of second article (2%, due Friday 2/12)
The article you choose must be a scholarly article related to the Falk article in some substantial way, but
need not be an article “on the same subject” except that it must have a direct connection to the subject of
infinity. We will discuss in class, and in particular with our librarian partner Morgan Stoddard, what
indicates an article is scholarly, but a basic requirement is that there be a bibliography and in-text
citations to sources.
Your second article will need to be approved by Dr. Abrams; you must submit a .pdf copy of the article
(NOT a link) on Blackboard no later than 9am on Friday 2/12. If the article is not acceptable, you will
have 36 hours to submit a replacement. It’s a good idea to submit your choice of second article a day or
two early and send Dr. Abrams an email alerting him to the fact that you have done so.
Step 2: Essay Overview (20%, due Sunday 2/21)
The Pearlman-Carillo framework has five basic components -- Analyze, Question, Evaluate,
Complicate, Conclusde -- each of which refers to certain general questions that you need to be able to
answer when you are truly engaging with sources and your own ideas. The overview of your essay will
be a document containing five clear, precise, concise, coherent and cohesive mini-essays, each of which
addresses questions corresponding to one of the five components of the Perlman-Carillo framework.
These questions should be applied to the Falk article, the second article you chose, and the point you
intend to argue, but your mini-essays must be one coherent and cohesive whole, so you cannot simply
address each of these separately. Furthermore, the requirement for coherence and cohesion means you
cannot simply list questions and answers. Suggested approximate length for each mini-essay: 250 words
A page giving the required structure for this sketch appears on page 3 in this document. Copy that
template into a separate document. Note that this overview should definitely NOT be an outline of the
essay in the traditional sense, and you should have NO expectation that the ordering of sections and
paragraphs in your essay will line up exactly with the five categories.
Step 3. Draft essay with framework-questions (25%, due Sunday 2/28)
The essay should argue for one, and only one, assertion. The draft essay must satisfy all the “Specific
Requirements” listed below. Suggested approximate length for the draft essay: 1,300 words. At the
beginning of each paragraph put in brackets [ ] and ALL CAPS the question/questions from the
Pearlman-Carillo framework that that paragraph addresses.
Step 4: Meeting with Dr. Abrams (8%, Monday 3/8 – Wednesday 3/10)
I will meet with each of you individually to discuss your draft, helping you crystallize and/or further
develop your ideas, and generally providing individualized guidance. You must be prepared to discuss
your work, which requires having done a “legit” job on the draft.
Step 5: Annotated version of draft essay (20%, due Friday 3/12)
An exact, i.e. completely unchanged, copy of the text (but not layout) of your draft essay, but with
NOTES IN ALL CAPS interspersed in the text itself indicating what will need to be revised and why.
Do not include the revisions. There must be notes on all sections, on almost all paragraphs, and on many
sentences. These notes must not refer to grammar, punctuation, spelling or other mechanics of writing.
Step 6. Final version of essay( 25%, due Monday 3/22)
A percentage of the grade will depend on the extent to which you responded to the feedback you
received on your overview from Step 2 and your draft essay from Step 3. Be sure to fulfill all the
“Specific Requiements” specified below. Suggested approximate length for the final version of the
essay: 1,750 words.
Specific Requirements
The draft essay and final essay must satisfy/include the following:
12 point font; double-spacing; 1 inch margins all around; PDF format
a separate title page with the title of your essay, your name, and the date
an introduction and a conclusion
a bibliography at the end of the essay listing, in proper MLA bibliographic format, the two sources you
used. YOU MAY NOT USE MORE THAN TWO SOURCES.
proper in-text citations, as appropriate, in MLA format
descriptive section headings (note: you may NOT use the terms “Analyze”, “Question”, “Evaluate”,
“Complicate” as section headings, since they are not adequately descriptive)
an argument for one, and only one, assertion.
utilization, at least implicitly, of the Pearlman-Carillo framework.
Please take some time reviewing the two original readings. And please pay close attention to the instructions. Thank you
Course
Instructor
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Draft Essay
Key point:Falk's research provide a framework to understand how to achieve a deeper level of understanding and awareness and learning
Background and Analysis
Falk evaluates discourse on learning infinity and what differentiates those who grasp the concept of infinity from those who do not. The infinity of integers implies that there is a never-ending possibility of numbers, but young learners are firsts exposed to finite numbers. Beliefs and surroundings influence behavior and behavior in turn affects perceptions including conceptualization of infinity in mathematics and the sciences. Some of these learners may still face challenges trying to understand infinite sets. Asking learners about their understanding and representation of concepts can provide insights about their interpretations, even as there are divergent perceptions about certain concepts. Inconsistencies in understanding concepts make it difficult to evaluate the learners’ understanding when they attach multiple meanings to concepts.
The interview approach is useful to understand students’ understating of concepts, but the interpreter may misunderstand the responses. As such care should be exercised when evaluating responses so that there is a clear understanding of the learners' intention. Different learning and intellectual practice do not have the same effect on learners’ understanding of concepts. Having interviewed the respondents so they can give detailed explanations and reasons can provide insights about their learning (Falk 8). To Sfard (211), interpreting Falk's keywords is necessary to understanding Falk's research and her understanding of the infinity concept. In other words, language usage affects interpretation, and Falk mentions "concepts," "infinite," and "infinity." Explicit use of words and offering alternatives are evaluable to researchers to avoid misunderstanding (Sfard 210). Acquisition of concepts is partly dependent on age, but without a clear understanding of how this occurs interviewing students is necessary to uncover how understanding can vary depending on the context and age.
Assessing ideas on concepts and student understanding
Sfard (210) points out that to understand research studies, it is necessary to evaluate the use of keywords, getting meaning from context, and her own understanding and interpretation of the text. Falk does not explicitly describe the meaning of concepts or include keywords. Still, there is an explanation of how the research was carried out to evaluate students’ understanding of the concept of infinity (Sfard 212). There is a need to distinguish concepts where readers can then evaluate the different meanings and concepts in various contexts. Falk (6) also addressed how children of different ages understand the infinity of numbers. This approach helps understand how the meaning of concepts changes over time and where the teaching and learning processes influence a deeper understanding.
Evaluating students’ conceptions can indicate how teaching and learning influences their understanding. A greater understanding of concepts, but few studies have focused on the potential barriers to understanding certain concept. Motivating and meaningful tasks are useful to determine a child’s best learning ability and their understanding of certain concepts (Falk 5)
Thinking and learning are intertwined and students' assumptions can affect how they learn, and interviews are one way to uncover the learners’ experience in their own words. In any case, interviewing allows the researchers to pose new question when there is a need to get clarity. There is increased knowledge for both the interviewer and the interviewees about how students conceptualize and how this relates to the learningprocess.
Falk's research highlights that people understand the finite set that represents actual infinity, but some face difficulties conceptualizing the potential infinite, which never ends. The difference between potential and actual infinity is that the latter is completed and definitive, while the former is non-terminating. Knowledge about infinity and other mathematical concepts partly depends on how learners were exposed to the concepts and mathematical thinking. People can understand finite sets or sequences, but the concept of infinity lacks concrete representations. Falk's research helps understand how people conceptualize infinity, and Sfard highlights that Falk's language use of conceptions and infinity is best recognized when considering the language use and the context.
Falk acknowledges that children go through different learning stages, which may explain their understanding of the concept of infinity. Children may be exposed to the idea of the "infinite," but their understanding is partly linked to their environment. There is a need to evaluate what children have learned in infinity discourse and how this influences their understanding. Falk focuses on how people understand the infinite concepts from early childhood. However, she does not address how theoretical assumptions, perceptions, and intuition influence how they understand the concept of infinity contexts. Sfard rightfully notes that interpretation is based on how people understand things. Falk's experiments validate this as the respondents were asked to explain their understanding of infinite partly based on their game strategies that focused on the concept of potential infinity.
Evaluating the researchers’ ideas and concepts
Sometimes there are difficult concepts for students to understand where they can evaluate and compare new concepts with what they already know. Investigations about an understanding of concepts that consider the learners’ experiences are helpful to identify gaps in teaching and learning. Falk (7) conducted experiments to investigate children’s understanding of potential infinity, actual infinity, and the infinite gap. Since the findings were based on the learner’s responses and performances, Falk’s approach helps uncover the students’ reasoning and learning at different ages. Falk’s demonstrated that children are more comfortable performing tasks but are not necessarily well placed to explain how and why they choose certain steps in the tasks. The research findings are ...