Abdur:

You may also want to look at this video that gives current insights into students' thinking processes: https://www.youtube.com/watch?v=N8DEeR1sraA

Best regards,

Debra

One thing you might consider in the interview process is the degree to which we are conscious of and able to articulate our own thought processes.

I have undertaken work in a similar field where the respondents explained their actions in a team problem solving task in highly systematic and logical  processes they called  "Computational thinking". CT. They "remembered" a linear process resembling what they had learned of CT. Accounts were given of systematic analysis, construction of algorithms, evaluation etc.

Observation records did not provide evidence of this systematic process of 'mechanical problem solving'. Instead it showed an intuitive, anarchic and successful process which was far from machine logic.

It would be a good investment to ascertain the degree to which what learners say about their approach actually relates to what they do.

छात्रों की सोच और विचार प्रक्रिया को पढने का कारगर उपाय अध्यापकों का छात्रों के संपर्क रहना ही है। कोई किसी व्यक्ति को अचानक पढ नहीं सकता है, उसका संपर्क, अनुभव, आकलन और विश्लेषण क्षमता जबरदस्त है तो छात्रों के सोच और विचार तक पहुंचा जा सकता है। इस कार्य में सफलता पाना अध्यापकों के लिए तीसरी आंख का खुलना होता है। यह एक प्रकार का कौशल (स्कील) है उसको पाना अध्यापकों की प्राथमिकता होनी चाहिए।

Maybe you can read The Theory of Planned Behavior, Icek Ajzen to indentificate behavior of students. There is three determinant antecedent is attitudes toward behavior, subjective norms, and perceived behavioral control to show up behavior.

You may explore the faculty of Cognitive Neuroscience to get an insight how our brain processes the information, gives consciousness and triggers for the best alternative option in decision making process. there is book "Principles of Cognitive Neuroscience by Purves et. al., 2008"

Hello Abdur,

Your say that you want to study students’ thinking processes while solving mathematical problems. Although it seems not to be case, your question is a timely question. Suffice it to say that even though the concept of “thinking or mental processes” is a pervasive concept in the majority, if not, psychological domains, we, psychologists, often refer to this concept without realizing that psychological processes, such as thinking or mental processes are internal, and hence, cannot be observed directly, but only inferred with basis on some external manifestations. As Wittgenstein once cogently remarked inner processes -- be they mental or neuronal – are in need of outer or external criteria. When you say that you are appealing to a “think aloud” procedure (not process) to get access to such processes you are following – I think-- a correction direction or track, a track also followed in Piaget’s clinical method with its appeal, for example, to judgments plus explanations, not only to judgments (e.g., “Why do you know that in a set of 10 flowers, 8 being roses, and 2 daffodils, there are more flowers than daffodils?”).  A typical 5-year-old child “correctly” answers that there are more flowers than daffodils. However, when asked to justify his/her “correct” answer s/he usually says that there are more flowers than daffodils because there are 8 roses and only 2 daffodils. Thus, an apparently correct answer is truly incorrect when we know of the mental processes at issue.

In short, we can have access to individual’s thinking or mental processes, for example, when we asked them to justify the answers they present to solve mathematical and logical problems. When this is the case, an apparent “correct” answer may turn out to be clearly incorrect. To ask children to justify their answers is a kind of think aloud procedure. In a nutshell, individuals can give answers to logical or mathematical problems that appear to be correct at a first glance, but, in the end, they are really incorrect when we know of the mental, thinking processes underlying those answers.

 Let me substantiate this idea with further examples within the field of conditional reasoning. There is empirical evidence that shows that, while solving, for example, Modus Tollens problems (e.g., If Mary is at school (p), then John is also at school (q). Not q is the case (John is not at school). Hence, not p has also to be the case (Mary is not at school either), 8-9-year-olds or even younger perform “better” than adolescents or even adults.  However, when one tries to grasp the underlying processes underlying such “correct” solution by asking children why this is so,  it is often the case that children’s apparently correct solution is based on what is called “matching bias” (If p, then q;  p is the case. Hence, q is also the case; Modus Ponens. If p, then q; not q is the case. Hence, not p has to be necessarily the case; Modus Tollens), and not based on an idea of logical necessity. Actually, if John were not at school and Mary was there, then it would be false to say that if Mary is at school, John is also at school.

As adolescents and adults do not appeal to the “matching bias” process, they tend to perform worse than children, while solving Modus Tollens logical problems. It is because of this “matching bias” that children also say, for example, that if Mary is not at school, John is not at school either (Denial of Antecedent), which is an incorrect answer.

In short, your think aloud procedure is, among others, a good way of grasping the underlying mental processes in individuals’ solutions of mathematical and logical problems. Repeated questioning, justifications, and counter-suggestions a la Piaget are also good ways and procedures to get access to individuals’ mental processes while they solve, for example, mathematical or logical problems.  

I hope I has go your question, and that this helps.

Best regards.

Semi-structured interviews.

The Robert Wood Johnson Foundation has a Qualitative Research Guidelines Project. I recommend checking it out. Here is the link to some basic info on semi-structured interviews:

http://www.qualres.org/HomeSemi-3629.html

students are changing

tools: modes of representation and translation between them (Duval)

Operators and control structures (OUvrier-Buffet).

Metodology: case stuy

In studying students' thinking processes in solving mathematical problems, it is important to consider the theoretical framework that informs the research design and the approach to the analysis of the data. The following book illustrates various research designs and approaches to the analysis of data:

Approaches to Qualitative Research in Mathematics Education: Examples of Methodology and Methods

http://www.springer.com/gb/book/9789401791809

As a suggestion, after you specify what you will be focused while students are solving the problems such as metacognitive thinking or self-regulating behaviors or both of them, you may look out such frameworks that will help you better analyze your data (e.g. Artzt & Armour-Thomas, 1992). As a methodological concern, it is important that your students are adapted to the think-aloud process because sometimes it may interfere with the thinking process and may affect the quality of data It could be another strategy to follow retrospective video analyses in which you collect students'work sheet and video record your students' problem solving behaviors and you investigate students' problem solving behaviors together while watching the record.

You need to ask yourself fs the two students are representative of the larger student population you are interested in studying and reaching,  Are they of different genders, cultures, mathematical abilities, educational backgrounds, ages, developmental levels, socio-emotional skills, and other individual differences variables shown theoretically or empirically to contribute to successful thinking processes involved in mathematical problem solving.  Qualitative studies are tricky and only add to our knowledge base if the data collected not only include think-aloud processes but also a control group that does not think aloud processes (per the comment above).  Your research design and methodology needs to be grounded in a theoretical framework (per the suggestion offered by Keith above) that leads you to your own burning questions and hypotheses about the students you want to study and why.

Please be aware that students are actually good at carrying two conflicting models in their heads, and switching back and fort between them when one starts to fail.  That is normal.

Your question makes me think of discussions about realistical math endorsed among others by Stephen Wolfram, and of work by Polya, three tomes in the library that I still have to read, and of the teach back protocol in psychology, where you check understanding by asking to explain it, and check explaining by first establishing what the other thinks. .

Define Thinking.  

I do so in my dissertation titled: A Grounded Theory Investigation of Thinking and Reasoning with Multiple Representational Systems for Epistemological Change in Introductory Physics

Moreover, I provide definitions of, and ways to measure: Thinking, Reasoning, and Understanding in terms of the way that concepts are actually constructed.  It is called the TRU Learning Theory.

Given that it took an entire dissertation to explain, I would gladly provide guidance or even collaboration on a study using the theory and its models.  Even if you choose not to use the theory, my literature review of thinking and reasoning will provide some useful articles for how to capture student thinking.

http://gradworks.umi.com/10/02/10027568.html

Nielson's point is important in that there will be model/model interaction taking place and the conversation will be about the interaction of these two models.  Another issue is that the student will in many cases shift the meaning of the words or symbols they use and it is this shift in meaning that makes this type of study difficult.  

To make your study easier I would suggest that you separate thinking from reasoning in the following way.  Thinking is a process of concept construction and reasonings are those processes that operate within or between models.  In general, students will be using an interpretive frameworks to construct models.

Another issue to be aware of is that each student will have his or her own epistemological framework that structures their thinking and reasoning.

Also I would strongly suggest you add Dick Lesh and Alan Schoenfeld to your reference list.

Thank you very much everybody for your excellent responses. I get many ideas from your answers. However, there is one more aspect that I need to know for sure. This is related to saturation of the data. 

Having asked the participants to think aloud during solving a problem (one problem only), followed by in depth interview about his/her arguments, I can see the thinking processes of this participant . My point is: "Is it ok to draw a conclusion about thinking processes through describing participants reaction toward one problem only?".

I am in doubt of my conclusion quality.  I think that what I just have is merely a kind of "hypothesis". I dont know for sure whether the participants will use the same or similar thinking processes when they are given similar problems. 

What do you think? Am I correct? 

Herbert Simon had a Nobel price for studying thinking aloud protocols with Goal Oriented Method Selection. Probably did some more too. That book is over 900 pages. One of many sources for needed reading in this subtle and difficult area.

Abdur,

Drawing from my experience with Lillian McDermott's Tutorials, I think that it is appropriate to draw tentative hypothesis from interviews from a sample of students.

The McDermott Physics Education Group at the University of Washington uses such interviews to develop essay sorts of questions for their college level students as a later tier in their study. 

Sometimes those results send them back to the interview level.  Surprises are common, and occur at every level.

I know one Grad student who wanted to write Tutorials for the Modern Physics topics.   But his first effort at Special Relativity (Einstein- adding velocities near the speed of light)  gave total surprise. 

He spent his entire PhD digging deeper into student difficulties. 

Finally he discovered that the problem was that students really didn't get Gallilean relativity.  (Throwing baseballs from a moving truck.)  His entire PhD was spend digging deeper, and only produced one 4 page tutorial.

I must say that having students trust you with their real thinking is a profound gift, which I am careful to honor and dignify.

Your question is: 

Having asked the participants to think aloud during solving a problem (one problem only), followed by in depth interview about his/her arguments, I can see the thinking processes of this participant . My point is: "Is it ok to draw a conclusion about thinking processes through describing participants reaction toward one problem only?".

The short answer is no, since we are not mind readers. But you can study their expressed explicit arguments. I suggest that instead of 'thinking', you focus on reasoning as defined in Lithner's framework (line of thought) and then you analyze student's expressed arguments for choices that they have made when solving the task:

 http://link.springer.com/article/10.1007/s10649-007-9104-2

I like Lovisa's answer.  While we really want to know about their thinking, that is not so directly observable.

Agree, also, it is not so easy to define thinking, or understanding. People speak about gut feeling.. makes you wonder if we even want to know it!

Abdur,

https://www.nngroup.com/articles/thinking-aloud-the-1-usability-tool/

reports of 19+ years of experience with that technique. 

Commognitive aproach from anna sfard

Interesting term - commognitive. What does it mean?

Many thanks,

Debra

PLEASE CHECK THE RESOURCES

http://link.springer.com/chapter/10.1007%2F978-0-387-35352-4_10#page-5

http://www.utc.edu/walker-center-teaching-learning/teaching-resources/ct-ps.php

@Nirmala -- excellent resources!  My question is, as a researcher, can someone succinctly say what re the best ways to study critical thinking in students in developmentally appropriate ways?

I agree with the explanation provided by Salvatore Nasisi.

Regards

SM Najim

Salvatore's comments made me think of the technique used at the University of Washington to design instruction in physics.  It might be adapted to meet a wide variety of study purposes, perhaps even yours.

After they have selected a topic which gives students troubles, they interview a bunch of them, hypothesize their difficulties, and test the hypothesis with interviews and essay questions.  (These are college students.)  Once they have some confidence that they have a valid hypothesis, they develop strategies for addressing the difficulties.

They assess how students respond to the strategies again with essays and interviews.  As their confidence grows, they wean themselves from interviews and depend upon essays. 

As a result, when their materials survive this process and get published, they are very good.   I was impressed enough that I took a sabbatical to learn more about them.  They had room that summer and allowed me to attend their summer in service for practicing K-12 teachers as a guest.

See https://depts.washington.edu/uwpeg/  

These links may help you

http://www.criticalthinking.org/pages/how-to-study-and-learn-part-one/513

http://www.cdl.org/articles/how-to-increase-high-order-thinking/

http://www.ronritchhart.com/Papers_files/UNCOVERING%20STUDENTS%20THINKING_Pub.pdf

https://dl.sps.northwestern.edu/blog/2015/09/using-technology-to-develop-students-critical-thinking-skills/

http://education.byu.edu/sites/default/files/TELL/documents/TELL%20440%20TP%20Social%20Studies.pdf

http://success.oregonstate.edu/sites/success.oregonstate.edu/files/LearningCorner/Tools/what_students_can_do_to_improve_information_processing.pdf

http://www.hawaii.edu/intlrel/pols382/Reflective%20Thinking%20-%20UH/reflection.html

https://www.australiancurriculum.edu.au/f-10-curriculum/general-capabilities/critical-and-creative-thinking/

You can rely on educational scaffolding strategy or the Marzano model.

Dear Joris Verrips , good link, Thanks.

After teaching "Physics by Inquiry" for quite a while, I am going to suggest an interesting answer to this question: Ask the students. Why can't they be more than the subjects of this experiment? Why can they not be co-Principle Investigators? They have a great interest in the outcome.

Yes, I have learned that we cannot always take what they say at face value -more than a decade of teaching this course confirms that. However it is surprising how often we can take it at face value! This is particularly true if we consistently model our own thinking out loud, including the fits and starts, and set that expectation.

If we quickly pass judgement on their answers, they soon learn to mostly shut up.

If we guide their thinking with leading questions, and allow them to reach their own conclusions, I think that we can then give great value to what they say they are thinking.

You can read this article to have a grasp of literature review and to see an example of studying problem solving phases of the students. A Case of Teacher and Student Mathematical Problem-Solving Behaviors from the Perspective of a Cognitive-Metacognitive Framework . Here is the link Article A Case of Teacher and Student Mathematical Problem-Solving B...

There are several approaches to teaching thinking, including: Teaching thinking to think, and this is done through relying on programs designed to teach thinking, or education about thinking, which is that activities are incorporated into the curriculum content aimed at developing thinking among students.

Here is this paper on how to teach thinking according to the second approach:

Article Students’ thinking process in solving mathematical problems ...