Reflection: Learning the "HOW" and "WHAT" learners are thinking

Technology is a powerful tool that enables us to "see" and "document" what's going on in our mind, hence creating a much enriched learning experience, for self and others!

Well, the above seemed too broad and generic, especially the second half of the statement. Isn't it a given? Some of us might think. Well, what assumptions do we make if we say it's a given?  That is, the first half of the statement is TRUE for the learners!

Episodes of the learning activities flashed across my head when attending (and mentally preparing to attend) the recent webinar. Indeed, value this opportunity as I hardly had a chance to sit down and think about T&L more thoroughly and making good connections across my practices during this assessment period.

Sometimes, we tend to claim (and therefore generate excuses) like due to the nature of subject discipline, there is little room for technology to come into the picture more often that we could. Well, often it's a "yes" response if we look at what's expected of the students in the syllabus document. E.g. They are expected to carry out proofs in Mathematics. The application of knowledge from various units and the writing to present their train of thought - all to be written on the paper. On other hand, how do we know if the students are not regurgitate what they had memorised?

• Drawing from my personal experience, sad to say, that's how I cleared my Physics at "O" Levels and went through the same when  I did Calculus in my university days! Glad that I joined the teaching profession and over the years,  I gain better insights to understand how learners learn and how teachers could do differently to enable learning. I believe, if I had learnt or shown how to learn the abstract topics differently (or if  my teachers had approach teaching differently), my learning experience and perspectives would have been rather different today. The teacher/lecturer had taught; but I had not learnt well.
• From another perspective, this experience has helped me to better understand the struggle that learners in similar situation as I did have to go through. It helped me to recognise and appreciate how important it is for teachers to possess the necessary and relevant knowledge and skills to bring about positive experiences for the learner.

So, how did we know how well students have learnt? By doing the questions on papers correctly, is that enough? How do we assess the thinking behind the set of solution? Many very experienced educators would share that the design of the assessment items must be clear, on which aspect of learning are we trying to assess? That is the clarity of the intent and the design of assessment items. The next step is, how often or regular, and at what juncture should we carry out the 'checks'? How tedious is the process? How big a group can we reach out and cater to (using tradition methods, without technology) in a timely manner? The entire purpose of formative assessment is about timely intervention.

• To conclude this paragraph: It's about the type of assessment item (to draw out what we want to know about our learner) and the timeliness of our response to need the need.

What are some ways that enable us to find how the learners think, or ways that enable us to probe their thoughts? This is one challenge that I notice students who just joined the Sec 1 classes would face. Not because they are not thinking - They had already proven themselves after braving through the Primary 6 Mathematics problem sums! They are great problem solvers who applied heuristics that would surprise many of us! The next question is, how well can they articulate how they think? or are they simply applying some "sure work" method that they learnt?

Let's refer to the Singapore Mathematics Framework (Syllabus document, p14) - which aspect(s) do we try to develop/ inculcate in our students? We are good in some areas outlined in the 'pentagon' (in particular, the skills, concepts and processes). Metacognition is an area that we should work harder on; and some aspects of "processes" could be further strengthened by riding on the affordances of technologies!

Making thinking visible is one approach that enable us to "see" how students think, how they process data/ information and definitely enable us to diagnose any misconceptions. It does not stop here! Because we could "see" the misconceptions, it enable us to surface the difficult points (Three Point Framework. (Yang & Ricks, 2012)) which we, as educators, might not be aware of! There are many generic thinking routines that we can pick and choose. One way of implementation is to make the routine "really" visible - flag it out each time it is used. Another way is to have it embedded more seamlessly in the way the problem/ question is being put across - which I think it's more 'natural'. 

To probe how they think, often we get them to elaborate the thinking process, which includes articulating:

• What are the key words they identify - which frames the direction or way they will approach the question/ problem?
• What are the info/ data they would use or need to solve the question/ problem?
• What strategies would they consider? How to do they decide which to use if there is more than one way? (What's the criteria)
• How do they articulate the steps clearly - where the concepts are explained?
• How would they check that the answer is reasonably correct? 

Well, does the above sound like Polya's 4 steps to problem solving?

How do we do this in a less elaborate but deliberate manner during lesson, and at the same time, reach out to as many learners in the class as possible?

I guess, that's where technology makes its entrance and presence felt to this learning ecosystem.
Its key is accessibility. It's not about physical accessibility (which no doubt, students are well provided with technological devices in Singapore schools). But, it provides a channel for teachers to access learners' thinking far much faster and easier. With the appropriate choice of tools used in a well thought through manner, we are able to "see" how students think!

One of my favourite examples is getting students to explain via viva voce  how they manage a problem (notice that I use 'manage' instead of 'solve'?): They

• are required to articulate the background (concepts)
• describe how they dissect the question (to understand what's given and what's required)
• list down the strategies can they apply and 
• articulate what considerations they need to take into account to make a choice of the method; 
• and eventually, how they would check the reasonableness of their proposed solution or answer!

Of course, we don't expect students being able to do all the above in the first attempt. It requires them rehearse the same process over and over again and improve with the feedback given along the way. That's where we help them to see their potential!

Students do not know and do not carry out the above steps naturally. It's not something that they are born with. Nevertheless, quite often, as teachers, we make assumptions that all learners are quick to pick that up, forgetting they need time to hone their stills and feedback is necessary for them to make positive progress. Scaffolding is a must to bring students through all these stages; and the guidance given in their first attempt would be far much more compared to subsequent attempts. Rubrics help. But we must explain the rubrics and give illustrations to that they become aware of the expectations and polish their crafts along the way. In summary,

Over the years, error analysis is introduced into the mathematics classrooms. It serves dual purposes. Indeed, it could be used as a strategy for differentiated learning, though I did not notice it when I first introduced it to the students.

The original intent, still remains as one of the primary intent, is for students to identify and explain what's wrong with the working (which is usually a result of misconception). Think more deeply, what is the underlying assumption behind this?

•  In order to identify the error, the assumption is students already know the concept quite well such that they could see or identify the step that went wrong. There is another group who could identify the error; however, are unable to explain the misconception - simply because through their own experiences - they might have made enough similar mistakes or similar mistakes had been pointed to them before which would help them to see the mistakes very quickly. It is possible that latter could make the same mistake because they have yet understood the concept behind the operation. To differentiate these 2 groups of students (on who really understand what's going on) would require them to explain what was the thinking behind the error. This is the first step to develop their metacognition. The ability to articulate demonstrate how well they have known the concepts (as concepts in isolation, as well as links to others). 

• On the other hand, error analysis-type of exercises would quickly filter those who are still trying to grabble with the basic or still functioning at the operational level. Likelihood is, the working speaks the type of misconception that these students already have. So, they would not be able to surface any mistake in the working at all! Hence, their conclusion is "nothing is wrong", but I guess that's what would worry them! 

• We also have route procedural learners who probably re-attempt the entire question using the 'correct' method. To surface the error in the given working, they "spot" the difference and try to describe the mistake. They are unlikely able to explain the thinking behind the mistake (i.e. the misconception), but able to tell operationally what has gone wrong.

Indeed, error analysis is a very powerful strategy to differentiate learner - as they interact and exchange their observations during discussions. On another note, for this to play out well in a classroom setting, it is necessary to maintain a safe learning environment and to cultivate a open mindset among the learners.

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How does technology come in to support the above?

It can be as simple as getting students to provide a brief explanation to support/ justify their answer. No sophisticated technology is required to facilitate the above processes. With the use of blog (comments) and Google Form/ Spreadsheet (embedded in a blog).

• E.g. 1: (S1) Algebra - Which is larger?
• E.g. 2:  6 AM Quiz - Which is larger? Question, Response 
• E.g. 3: Substitution: Can you tell what is wrong? 
• E.g. 4: Factorisation: Which method to use? Question, Response
• E.g. 5: What is the deciding Factor? HCF or LCM. Question, Response 
• E.g. 6: Viva Voce (posted and organised in Youtube).
•  In the recent years, viva voce clips are submitted via the Google Classroom
• When students posted their work in Youtube, we could provide feedback via the comments feature. Similarly, we can also post our feedback to students using the comments feature in Google Classroom.

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