Effort and time spent in developing the ability of critical thinking is fruitful when students learn to think and

think to learn. A critically thinking students usually develop:

1. deeper engagement and understanding.

2. greater independence and self-regulation.

3. stronger competence with mathematical processes.

Present math education standards focuses on the mathematical processes such as problem solving, reasoning, representing and communicating. If a student acquires these qualities, then they are said to have Critical thinking ability.

Effort and time spent in developing the ability of critical thinking is fruitful when students learn to think and

think to learn. A critically thinking students usually develop:

1. deeper engagement and understanding.

2. greater independence and self-regulation.

3. stronger competence with mathematical processes.

Present math education standards focuses on the mathematical processes such as problem solving, reasoning, representing and communicating. If a student acquires these qualities, then they are said to have Critical thinking ability.

A student first  need to understand the question, the terms appearing in question (the prerequisites) then the critical thinking on that question or problem is possible but if the it is a free environment where even his wrong answer will not put him in a cage in front of others.

I think that Problem based learning definitely will increase critical thinking as in this methodology a problem drives learning.

Kien Lim and colleagues of University of Texas at El Paso have developed some surveys on mathematical disposition, which may be similar to what you are looking for.

This is a good question.

In addition to the incisive, good suggestions already made, there are some other things to consider in assessing the critical thinking (and acumen) in doing mathematics.

One good way to do this is to challenge students to find direct proofs (instead of indirect proof or proof-by-contradiction) of theorems on different levels.   

Consider, for example, a direct proof that there are an infinite number of primes (Euclid).

Examples of direct proofs can be found in

Solutions of Exercises on Catalan and Related Numbers:

http://www-math.mit.edu/~rstan/ec/catsol.pdf

See, for example, a direct proof of enumeration of 321-avoiding permutations, first published in D.G. Rogers, Discrete Math., 1978 (see part (ee) in this paper).

Another neat theorem to prove directly is

Theorem   Transcendental numbers are uncountable.

For an indirect proof, see page 80 in

S.C. Huneke, An inductive proof of hex uniqueness, The American Mathematical Monthly 121, no. 1, 78-80:

http://www.lse.ac.uk/maths/PDFs/An-Inductive-Proof-of-Hex-Uniqueness-.pdf

Mathematics is considered to be one of the fundamental subjects to sharpen and augment critical thinking skills of students - critical in the sense of being critic, analytic, imaginative to investigate the truth or falsity of things, carefully looking at conditions why a statement may be false and why it is true only for the stated hypothesis? What happens to the converse of a mathematical statement ? Will it be true or false and what condition(s) may amend to be affirmative?

Mathematical statements are truths of abstractions of mathematics written in most cases as a one sided or two sided implications and proving them valid via sequences of logical arguments with in an existing mathematical structure and search to truth  requires the above mentioned abilities of critics, reasoning  and imaginations. 

Please read the paper at http://irep.iium.edu.my/37220/1/madihah2.pdf that was written in 2008 on "a brief look at teaching and assessing mathematical thinking at primary level". That was a preliminary study and I would very much like to expand on this and would be pleased to collaborate with you. Thank you

 I share your concerns about the attitude of your students to the teaching of mathematics. After such great strides forward in science with the help of modern technology, we must take qualitative changes in the content and method of teaching mathematics. In this sense in 2007 to 4. 04. 2007 audience my views of the Internet address https://annatomova.wordpress.com/. In 2010 - the younger brother of Stephen Wolfram - Conrad Wolfram made a great statement entitled: Conrad Wolfram:

Teaching kids real math with computers of address :http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers?language=en. To be able to evaluate critical thinking of students in mathematics, we must first create it by changing both the content and the manner of presentation of mathematical knowledge. Can not today, in the 21st century, to pass basic math of 17-19 century in the same way as before the advent of modern technology. Although it is believed that Euclid through its "Origins" has made the basis of the  geometry for the next  2000 years and in this sense is erected a monument like the Egyptian pyramids.I'd love to be included in an assessing critical thinking skills and disposition in a mathematics context for mathematics students and to joint  to develop these tools.

I expect your reply with interest. Sincerely Yours, Anna Tomova.

Anna and others,

I am happy that you have answered my question. I want to tell additional fact which lead me to ask the question.

About two months ago, I asked my students (undergraduate and postgraduate students) to respond to the following command "given that triangle ABC is a right triangle. Angle ABC is a right angle, and the measure of angle BCA is 60 degree. A line BD is drawn perpendicularly to AC such that the length of BD is one unit, and AB is three units. Find the perimeter of triangle ABD!"

I was shocked by the fact that they just follow the procedure how to find the length of each sides of triangle ABD using trigonometry. They come up with a result. They did not check the appropriateness of the problem. They did not realize that they should not directly apply the procedure. The given conditions are incorrect, so there is no need to follow up this problem.

If I see this situation from critical thinking dispositions perspective, for me, these students do not showing critical thinking disposition. So, I tried to explore and find informations on how to assess critical thinking skills and dispositions for mathematics students and using mathematical contents (not general language). I was unable to find it, and I need help on how to get it.

OK... thanks for your contributions

                                      Dear Abdur Rahman Asari,

Thanks for the reply and for your example. I have the following question: sometimes asked you such provocative questions? Perhaps students blindly believe the conditions of mathematical tasks,Nevertheless, I always recommend a check of the answers: it's fascinating mathematical (reverse) task. I expect your reply with interest. Sincerely Yours, Anna Tomova.

Dear Abdur,

I discussed critical thinking in an advanced mathematical context in my recent article:

https://www.researchgate.net/publication/265058163_Techniques_for_capturing_critical_thinking_in_the_creation_and_composition_of_advanced_mathematical_knowledge

Firstly, as I remember it, my main point is that mathematics has to be right (e.g. Frege's treatise that was undermined by Russell's Paradox). Secondly, I asserted that critical thinking can be seen in mathematical creativity and problem solving in knowing what to do next, which aligns with Schoenfeld's control level of metacognition.

Happy to work with you further.

Article Techniques for capturing critical thinking in the creation a...

What is the sum of the interior angles of any n-sided polygon ? Justify your answer. Does this require critical thinking/question existing theories or simple reproduction or acceptance of existing knowledge. Perhaps for those who know the "answer" the question is routine with a ready answer, not to be questioned. For a novice mathematician (e.g 11 year old), being presented with the task  "What is the sum of the interior angles of a triangle?" for the first time,  this challenge, if welcomed, triggers some thinking. How do you assess critical thinking?

This is qualitative assessment. One needs to consider both the internal and external representations of the thinking process. Its not about the product of the thinking process only. The metacognitive processes and a whole host of other mental activities are involved. Can one put a quantitative value to all that?? Do we need to measure or to induce and promote critical thinking.

A recommendation:

Step 1: Define critical thinking end-state [e.g., analysis, evaluation and explanation of facts and theories that explain (answers "why"), discusses effects (answers "so what") and forecasts (answers what's next)].

Step 2: Determine whether the question the "critical thinker" is trying to answer can be operationalized in mathematical terms

Step 3: If yes, determine what logic techniques are being applied when answering the question (deduction, induction, abduction, modal logic, direct proofs, indirect proofs, etc)

Step 4: Determine whether the logic techniques can be tracked in mathematical terms (e.g., nominal or ordinal variables?)

Step 5: Determine whether there is a quantifiable correlation between what questions could be answered and logic patterns.

Hope this helps: best of luck with your inquiry.

 If we look at the concept of critical thinking as : “interpreting, analyzing or evaluating information, arguments or experiences with a set of reflective attitudes, skills, and abilities to guide our thoughts, beliefs and actions” then the most vital part of mathematical contents which can help in developing and increasing the critical thinking ability for students is mathematical logic where the student usually learns proposition calculus , main rules of inferences and deductions. If those subjects were tought properly, the student will be able to think critically.

You can try using worded problems in Math for students to solve that require them to think critically in finding the solutions. Please see link below.

Best regards,

Debra

https://www.google.tt/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwiFidXQ2NvVAhWLMyYKHdbVDusQFggsMAA&url=http%3A%2F%2Fbibliotecadigital.usb.edu.co%3A8080%2Fbitstream%2F10819%2F405%2F1%2F64580%255B1%255D.pdf&usg=AFQjCNEc5CErkYzJxlHJ9bp9qM5xfWuoMA

Dear colleague if you still research on critical thinking disposition and skills, there are two international valid and reliable instruments geared to assess both of them. They are California Critical Thinking Disposition Inventory and California Critical Thinking Skills Test. Both instruments can be used to assess disposition and skills at any field. Just search on them on google or google scholar, you will find. Also, you may check this website for more information: www.insightassessment.com.

Here are some rubrics for assessment proposed by Facione and Facione for critical thinking in general, not specifically in mathematics: http://oie.pittstate.edu/dotAsset/3acbca5d-3f0c-4004-8fd6-e8d0b8a3e102.pdf

I don't think that mathematical logic is the whole story about developing critical thinking skills in mathematics, but I certainly agree with Ahmed Ali Khammash to the extent that he highlights it as an important aspect. Learning how to develop strategies for doing formal deductive proofs and representing those strategies in simple algorithms, as well as learning how to express statements of everyday language in an ambiguity-free formal language, does enhance the ability to think more clearly, precisely, and critically in other areas of mathematics. So I would not be surprised that a test that measured such logical skills might also be a good indicator of critical thinking skills in mathematics more generally .

Dear Abdur, I would suggest taking a look at the following website:

http://www.macchinematematiche.org