Asking Questions That Elicit Higher Levels Of Bloom’s Taxonomy

ASKING QUESTIONS THAT ELICIT HIGHER LEVELS OF BLOOM’S TAXONOMY

Questioning essay – Mihai Catana

INTRODUCTION

“How can learners be encouraged to take initiative, to be active learners rather than passive receivers? Simply asking a lot of questions is not enough. What matters is the form and type of questions asked” (Mason, 2010). Therefore the relevance of questions asked by teacher in certain moments of the lesson is a key element of engaging students and promoting active and effective learning of Mathematics.

Despite the fact that I was aware about the Bloom’s taxonomy (please, see the Appendix 1) for categorizing level of abstraction of questions that commonly occur in general educational settings, I did not have a clear strategy on how to “ask maths questions mathematically” and how to “turn exercises into mathematical questions” until I have read the two Mason’s papers (2000 and 2010).

I can confirm that asking questions for checking knowledge, understanding and applying Maths (levels 1 to 3 from Bloom’s taxonomy) are occurring almost naturally in my lessons, being part of my everyday teaching routines. Reflecting now on my past teaching in the light of Mason’s ideas, I have to admit that I used in my lessons the higher level cognitive questions (levels 4 to 6 from Bloom’s taxonomy) only sporadically and inconsistently due to my genuine belief that this kind of questions are beyond the power of understanding of the majority of my students and not all of them would benefit. By attending the TAM course and becoming aware of the new researches and findings in the contemporary Maths education, I started to think on how to improve my teaching routines and how to progress from standard and safe Maths questions to “a range of typical questions posed by mathematicians” in order to expose my students ”to the breadth of typical mathematical questions” (Mason, 2010) and prepare them in this way to step up easier from the low levels of cognition accordingly with Bloom’s taxonomy to ”meta-cognition” (Garafalo Lester, 1985). Being aware of the fact that “questioning is so fundamental to a teacher’s practice that a change in questioning methods may require a fundamental shift in teacher beliefs about the goals and purpose of mathematics teaching” (Aizikovitsh-Udi et al, 2012) I started to investigate and experiment ways of including in my teaching questions promoting student higher-order thinking and to update my questioning practice.

In the first part of this essay I will describe this process of consciously moving on from my old questioning habits by consistently employing in my teaching a new repertoire of verbal and non-verbal questioning strategies that elicit the higher levels of Bloom’s taxonomy. In the second part of the essays are presented the challenges faced by myself and my students while implementing this process in a TAM observed Mechanics lessons. I have chosen to audio record and discuss in this essay this observed lessons as I consider the feedback and advices received from recognised expert teachers being invaluable for my own professional development.

In this essay I will be extended the idea of “questioning” from the conventional (ancient) maieutic method (used by Socrates to elicit truths from his students by asking questions) to a more general meaning including here posing verbal and nonverbal questions, problems or tasks.

Envisaged strategies for raising the cognitive level of my questioning and stimulate student’s higher level thinking

Inspired by Alexander about the need of a pedagogical repertoire for implementing effective teaching “the idea of teaching repertoire is paramount”(Alexander, 2005) I have decided to select carefully from a variety of available researches a list of methods, strategies and techniques for employing higher lever questioning which are appropriate to my teaching style and also, are suitable to my students. I have selected therefore this repertoire of approaches based on the following criteria: a) fitness for purpose in relation to my A level Maths and Further Maths learners b) are largely accepted or confirmed as stimulating and facilitating the high level cognitive processes and c) to be used comfortable by myself and presumptively by my students. For example, I excluded from my choices the Student Question Cards (SQC) technique which “could be incorporated into normal mathematics lessons to help students become more metacognitive about their learning” (Wong, 2012) because apparently this technique does not satisfy the criterion b) from above.

“The idea of repertoire can be extended infinitely, down to the finest nuance of discourse” (Alexander, 2005) but, in order to make my intended process of improvement manageable, I will stay focused only on the development of three areas of my teaching repertoires: the first one is aiming a change at verbal level with direct reverberation in student’s higher level thinking, the second area is intending to enrich the content of my questions by integrating fundamental Maths categories and processes and the last goal of improvement is concerning my ability of using the maieutic method for eliciting superior levels of thinking . More details of my intentions and a rationale of my choices are presented below.

Repertoire 1: I will incorporate in my questioning whenever is possible and appropiate, verbs that elicit higher levels of Bloom’s taxonomy. Verbs such as connect, elaborate, evaluate and justify prompt students to communicate their thinking and understanding, to deepen their understanding and to extend their learning (Ontario, 2012). Huinker and Freckman (2004) is also providing a list of verbs that elicit specific cognitive processes to engage thinking: observe, evaluate, decide, conclude, notice, summarize, identify, infer, remember, visualize, compare, relate, contrast, differ, predict, consider, interpret, distinguish, explain or describe.

Repertoire 2: In order to enhance the standard of my questions/tasks I will consistently incorporate in my lessons rich and relevant Maths categories and processes as suggested by Mason (2010):

· Mathematical themes: Invariance amidst change Doing and undoing Freedom and constraint

· Mathematical powers such as: stressing and ignoring specializing and generalizing imagining and expressing conjecturing and convincing ordering and classifying abstracting and instantiating

· Mathematical heuristics referring to different strategies that can be adopted to solve unfamiliar or non-routine Maths problems. An example of heuristic method in four steps is described by Polya (1945)

· Mathematical activities: characterizing, abstracting, generalising, particularising, classifying, negating or inversing Maths categories and statements.

From my point of view these principles used in lessons potentially can lead to mathematically enriched tasks, preparing in this way the ground for higher level cognitive questions.

Repertoire 3: I will update and improve my maieutic method or my “art of asking through provoking Maths questions” (Bellido et al, 2005). As expected my attention will be again concentrated on how and what to ask in order to stimulate and achieve students higher levels of thinking as presented below:

· Focusing students to listen and respond to others’ ideas in order to create an inclusive and “conjecturing atmosphere” (Mason, 2010). This type of questions is intending to set the class appropriately and ensure engagement and productivity.

· Supporting students to make connections (e.g., between a model and a mathematical idea or a specific notation) – this kind of questions will address to the level four questions from Bloom’s taxonomy.

· To guide students and encourage mathematical reflection and reasoning (e.g., make conjectures, state definitions, generalize, prove) – corresponding to level five from Bloom’s taxonomy.

· Extending students’ current thinking, and assessing how far they can be stretched – this type of questions are intended to elicit cognitive processes corresponding to level six from Bloom’s taxonomy.

In order to smoothly and effectively integrate in my repertoire these high level questions, I have designed a list of possible questions, presented in the Appendix 2, inspired from Ontario, (2012) and Bellito et al (2005).I will bear this list in my mind when addressing questions targeting specific cognitive levels.

Concerns and challenges

In essence my intended changes will raise the level of challenge of my questions making the tasks harder (or apparently harder) for my students. I am conscious that I need to anticipate the possible challenges and difficulties when employing the new type of questions/tasks in order to ensure that I will keep the students engaged. In fact, I am fully aware that my success on implementing the plan presented above depends very much on finding the right balance between the help/support offered to students and maintaining yet a high level of challenge. This dilemma of the appropriate degree of explicitness of the hints offered to students was flagged by Mason, (2010): “If everything is always explicit, students are supported in becoming dependent on the teacher to make things explicit but if too much is left implicit, many students may never encounter what is being made available to them”. Apart from this is also essential to organise learning in such manner to allow students to learn from the own (and others) mistakes and misconceptions and to use the mistakes as a starting point for a route to achievement and success. In order to overcome these potential difficulties of my enterprise I intend to use the following strategies:

· Encourage students to collaborate and explain their thinking in a “conjecturing atmosphere” as described above in the Repertoire 3 this approach should allow the transfer of ideas from a student to another and ensure the progress of learning

· Use “scaffolding and fading” when the students are stuck and especially for fading away I will use the method named by Mason, (2010) as Directed- Prompted – Spontaneous which consist in “moving from directing attention, through indirectly prompting attention, to the student spontaneously attending for themselves”. This strategy will provide me some flexibility and control on how much support is offered when this is necessary and then to fade away when the learners are able to carry on the task without the teacher’s support. Of course I am aware about the possibility of initiating a “funnelling sequence” while questioning and for avoiding this I will use the “focusing” strategy as suggested by Wood, (1998) which requires “the teacher to listen to students’ responses and guide them based on what the students are thinking rather than how the teacher would solve the problem”.

· Provide predominantly “process help” rather “product help” as “the process help leads to more mathematical level-rising in pre-university education”(PijlsDekker, 2011). On the light of this study I will try to support the learning by focussing on the process of interaction between students and explicitly stimulating them to present/discuss/criticise theirs work rather by giving explicit mathematical hints.

· Embed the use of ICT in my lessons allowing the students to represent/investigate/check the work by using appropriate technology.

Planning, preparing and delivering the tasks

At this point of my presentation I thing that is worthwhile to give the details on how I implemented the principles listed above (especially on the Repertoire 2) when I was planning the observed lesson.

One of the learning objectives of this recorded lesson was: “The students will be able to sketch, interpret and use displacement-time graphs, velocity-time graphs and acceleration-time graphs for constant acceleration motion”. In order to achieve this objective I decided to prepare one activity which will help students to clarify firstly the Mathematical side of the problem and then, based on these clarifications to move on to the Mechanical part. Instead to start discussing about how to represent the graphs of v=u+at and s=ut+12

at² as functions of time, I decided to start the lesson by proposing to my students to analyse the graphs of y=ax+b and y=12 ax²+bx taking into account all possible signs combinations of the two parameters a and b and at a later time to ask them to connect the findings for the two functions with the displacement-time and velocity-time graphs.

I considered here two possible options: the easiest one was to give to the learners a sorting table as presented in the first part Appendix 3 and ask learners to place the graph cards in the correct position or the harder option, to ask learners to sketch the graphs of y=ax+b and y=12 ax²+bx on the provided grids as presented in second part of the Appendix 3. Of course, I opted for the second option as this is involving more higher level Maths thinking especially abstracting, generalising and analysis. In order to overcome possible difficulties I decided to ask learners (working in pairs) to sketch first only the graphs of the linear functions, then the graph of quadratic and then to ask them to sketch the both graphs on the same grid. In this way I intended to “scaffold” the task giving to learners implicitly a “process help” by gradually increasing the difficulty and complexity of the task.

The graph of the linear function was sketched correctly by the all pairs. At this stage I was using probing questions:

Me: Why this line looks in this way? (I pointed a line from the box corresponding to a<0, b>0).

Sabrina: Aaa …the gradient is negative … so the line goes down…

Me (asking another student): This is right but, this line is going down as well (I pointed the box corresponding to a<0, b<0). What is the difference then?

Rui : Aaa… this one has the y-intercept negative and another one has the y-intercept positive.

Me: Excellent so far… Please, move on and do the same thing with the quadratic graphs.

Apart from a pair who has done the task partially, the quadratic graph was also sketched correctly by all other pairs. Here was remarkably to see the pairs negotiating different heuristic approaches for dealing with the task pairs who have replaced the values of a and b by different particular numbers and from here, by generalising they have deduced the general shape of the graph in the given circumstances for a and b. Also, I was pleased to see a pair reasoning abstractly using only the signs of a and b for completing the task without using particular values. This was indeed a sample of high level thinking supposing the analysis and synthesis of the data involved within the task.

I was very careful on giving hints to the pair who has encountered difficulties on doing this task:

Me: I cannot see any graph sketched so far …

Lang: If we don’t know a and b we cannot sketch the graph…

Me: But you have done this for the linear function with no values given for a and b. Please, discuss it between you and agree a strategy for the quadratic. You need to remember first what is needed for sketching the graph of a quadratic

At this stage I refrained from giving to the pair any hints (content help) and I focused the two students to the process they need to undertake.

After few minutes:

Me: Any progress here?

Zheming: We decided to take a=1 and b=1 and draw the graph as here (pointing a graph from the box a>0 and b>0).

Me: Is this solving the general problem?

Lang: No, but the graph will look similarly …

Me: You are right, the graph will look similarly, but you need to explain why this is happening. Please, discuss with your pair and find a way to show it.

At this point is very interesting what the observer has noticed: ”While the students were working on their activities, you stood back and intervened very little. In general, I think this is a good thing – teachers tend to be far too quick to step in – but perhaps you took this a little too far in this lesson? There was one pair in particular who would have benefited from a little guidance in the initial activity to set them on the right track”. (Feedback received from Susan Whitehouse). Indeed I took it deliberately a little bit too far at that stage as I considered that the students will benefit more from their struggle. Later on in the lesson I have addressed to this issue by using technology as the observer has noticed: “You used dynamic imagery very effectively with a graph-drawing facility from the internet, changing the values of a and b in the equations, to show how this affected the linear and quadratic graphs”. (Feedback received from Susan Whitehouse).

In the last part of this activity I asked the students to represent the graphs of the linear and quadratic functions on the same grid first and then to spot connections between the two graphs. This task has required high level thinking according with Bloom’s taxonomy as the students were required to use the old knowledge about linear and quadratic functions and to create new ones, generalise from given facts and then to connect knowledge from algebra and calculus and apply them in Mechanics. Also, the “mathematical themes” were present in the task: different values for a and b but the x-intercept of the line was the turning point for quadratic in all four situations (Invariance amidst change), the linear function was derivative of quadratic (Doing and undoing).

Only one pair out of four has produced the correct graphs but no one realised the expected connection between the two graphs and this was quite frustrating:

Me: What connection do you notice between the graph of the linear and quadratic function?

Rui: When the linear graph has a positive gradient it is also true that y→∞ as x →∞ for the quadratic curve

Me: Rui this is a valid point and a good observation… What else do you notice about the two graphs? (addressing to the class in the same time)

???? (No answers)

Me: This is about important points from the graphs of the two functions.

??????(Still no answers)

At this stage I realized that I have initiated a “funnelling sequence” and I decided to stop it by telling to the class exactly what I am looking for as suggested by Mason (2010).

Again is interesting what the observer has noted about this sequence of the lesson: ”After this (asking about the connection between graphs) you did start to funnel the students towards the answer you wanted, but I think you realised what you were doing, because I was very pleased to see that you stopped yourself and asked the students directly what they noticed about the x -coordinate of the turning point. I think this was the right thing to do”. (Feedback received from Susan Whitehouse).

Also, regarding the observation offered by Rui in the above dialog Susan has noted on her feedback: “This was an interesting observation, and I felt that it deserved a little discussion, even though it wasn’t the observation you were looking for. When asking questions of your students, genuinely listen to what they have to say, even when it is not the answer you were expecting or hoping for”

Now looking back I completely agree with Susan about being genuinely interested on students answers as “The secret of effective questioning is to be genuinely interested not only in what learners are thinking, but in how they are thinking” (Mason,2010). This was one point of improvement on which I reflected very seriously as the time spent on exploring the students thinking and ideas can be beneficial for learning and create that inclusive and “conjectural atmosphere” as detailed in the Repertoire 3.

1) Mason, J. (2000) Asking mathematical questions mathematically, International Journal of Mathematical Education in Science Technology, 31 (1), pp. 97-111.

2) Mason, J. 2002, Minding Your Qs and Rs: effective questioning and responding in the mathematics classroom, in L. Haggerty (Ed.) Aspects of Teaching Secondary Mathematics: perspectives on practice, RoutledgeFalmer, London, p248-258. (extended version accessed 7/3/13)

3) Mason, J. (2010), Effective questioning and responding in the Mathematics classroom, revised version of Mason(2002), http://xtec.cat/centres/a8005072/articles/effective_questioning.pdf

4) Aizikovitsh-Udi, E., Clarke, D. And Star, J. (2012). Good questions or good teaching: An essential issue for effective teaching. Proceedings of CERME8 6-10 February 2013 Manavgat-Side, Antalya - Turkey. Accessed http://cerme8.metu.edu.tr/wgpapers/WG17/WG17_Aizikovitsh_Udi.pdf.

5) GARAFALO, J, LESTER, F., 1985, Metacognition, cognitive monitoring, and mathematical performance, Journal for Research in Mathematics Education, 1985, Vol 16, No.3, 163-176

http://olms1.cte.jhu.edu/olms/data/resource/2083/math_compmon_research.pdf

6) Alexander, R.,(2005) CULTURE, DIALOGUE AND LEARNING:NOTES ON AN EMERGING PEDAGOGY Education, Culture and Cognition: intervening for growth International Association for Cognitive Education and Psychology (IACEP) 10th International Conference, University of Durham, UK, 10-14 July 2005

7) WONG, K(2012), USE OF STUDENT MATHEMATICS QUESTIONING TO PROMOTE ACTIVE LEARNING AND METACOGNITION, 12th International Congress on Mathematical Education, 8 July – 15 July, 2012, COEX, Seoul, Korea

8) Huinker, D., Freckmann, J. L. (2004). Focusing conversations to promote teacher thinking. Teaching Children Mathematics, Ontario Ministry of Education. (2006). Toronto, ON: Queen’s Printer for Ontario

9) Ontario Ministry of Education (2012). Asking Effective Questions, Capacity Building Series, Special edition # 21, http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/cbs_askingeffectivequestions.pdf

1)10) Polya, G.(1945), How to Solve It (Cambridge, MA: Princeton University Press), https://notendur.hi.is/hei2/teaching/Polya_HowToSolveIt.pdf

11) Bellido, C, Walker, U., Wayland, K. (2005), The Art of Asking Thought Provoking Questions in the Mathematics Classroom, published online http://hub.mspnet.org/media/data/TheArtOfGeneratingGoodQuestions.pdf?media_000000006087.pdf

12) Wood, T. (1998), Alternative Patterns of Communication in Mathematics Classes: Funneling or Focusing?” In Language and Communication in the Mathematics Classroom, edited by Heinz Steinbring, Maria G. Bartolini Bussi, and Anna Sierpinska, pp. 167–78. Reston, Va.: NCTM, 1998.

13) Pijls, M. Dekker, R.,(2011), Students discussing their mathematical ideas: the role of the teacher, This article is published with open access at Springerlink.com

14) Megan Loef Franke, Eham Kazemi, 2001, Learning to Teach Mathematics: Focus on Student Thinking, THEORY INTO PRACTICE, Volume 40, Number 2, Spring 2001 Copyright (D 2001 College of Education, The Ohio State University 0040-5841/2001), http://faculty.washington.edu/ekazemi/theory%20into%20practice.pdf