Sunday, May 26, 2019

3 reasons why it's unwise to draw Bloom's Taxonomy as a pyramid

In my last post I looked the different ways Bloom's Taxonomy is presented in diagrams, and by far the most common is the famous Pyramid of Bloom's Taxomony. Here are 3 reasons why using the pyramid form can lead us to confused thinking about teaching and learning.

Problem #1.  A misunderstanding of the role of knowledge leads down the path of technological dependency.  

Pyramid thinking has led to this type of "21st Century Learning" discourse:

We remove the human element from "remembering", and indeed as we increasingly see in student work, there is an over reliance on Google and Wikipedia for analysis and evaluation.  I don't think we are far off from the next phase where Google Assistant, Siri and other tools do most of this thinking for us,


and finally it we just throw in the towel:

Yes, this may be an exaggeration, but people are now seriously considering that sophisticated software, built on a large knowledge depository (hmm, funny that) will soon be as creative, if not more so, than humans.  Anyone who thinks by focusing on creativity we will "future proof" students is in for a rude shock as the AI software goes to the next level.

Problem #2 : Thinking there is just one pyramid

Pyramid thinking can quickly become monolithic - the concept that there is one, all embracing pyramid which integrates all knowledge and all cognitive processes.

The ONE TRUE "integrated" pyramid of cognition

In this view, there is one large pool of knowledge to "remember" and learning how to "understand", "evaluate", "create" is a transferable skill that can be applied to all the knowledge in the "one pyramid". All the cognitive science papers I have read suggest this is just not true. Developing creative skills in music isn't going make me a creative essay writer or a creative mathematician. Both knowledge and cognitive skills are domain specific.

Problem #3 : The pyramid completely misrepresents the revised Bloom's Taxonomy

The 2001 revision of Bloom's Taxonomy by Anderson & Krathwohl involved so much more than changing the words from nouns to verbs. The 2001 work corrected a major flaw in Bloom's original taxonomy : the conflation of knowledge (bottom level of the pyramid) and cognitive processes (the rest of the pyramid).

Two explicitly delineated dimensions
in the 2001 revised Bloom's Taxonomy.
Nothing like a pyramid!

The pyramid representation only provides a one-dimensional view of the new taxonomy, completely omitting the knowledge dimension. Without the knowledge dimension explicitly in view, we are at risk of making category errors focusing too much on the development of skills at the expense of building knowledge.

Pyramids are just so 26th Century BC
it's time for a new diagram.

In terms of graphic design and correctness, this diagram is far better:

In my next and final post on this topic, I'll consider how Bloom's (revised) Taxonomy and indeed Bloom himself, is worthy of being considered a key player in the development of the Knowledge Based Curriculum.

Saturday, May 18, 2019

How do we visualise Bloom's Taxonomy?

I've been thinking a lot about Bloom's Taxonomy lately and just how influential it is to teachers' thinking. The more deeply I look into  Bloom's Taxonomy, the more surprising things I find.  However before we go there, here's a question for my teacher friends : If I say "Bloom's Taxonomy", what image do you form in your mind?

Now compare your mental image to the great Google mind....

Result of a Google Images search for "Bloom's Taxonomy"

Based on this search,  I'm predicting your mental image is a pyramid. With labels like "Facts", "Knowledge" or "Remember" at the bottom rung, and perhaps "Synthesise" or "Creativity" at the top.  And there's an implied, or maybe even explicit, upwardly pointing arrow.  And that's how I remember Bloom's Taxonomy. 

I couldn't help myself and did a frequency analysis of the first 60 images thrown up by Google. Just under 50% were pyramid versions, either the classic version:

Source: Learn NC, “Bloom’s Taxonomy,” used under a Creative Commons license.

or the updated 2001 Anderson & Krathwohl version:

Source: Learn NC, “Bloom’s Taxonomy,” used under a Creative Commons license.

Around 33% show the taxonomy in a grid with clear hierarchy:

Many of these diagrams come with arrows and labels to reinforce the visual message of hierarchy:

Less than 10% of the images present the taxonomy as components without a particular order - or more interestingly, as an integrated view:

Much to my surprise, the more deeply I read into the history of Bloom's Taxonomy, the critique of the taxonomy, it's 2001 reformulation and the more nuanced commentary by supporters and critics of the taxonomy, the more I realise my mental image of Bloom's Taxonomy is just plain wrong - and I think Bloom would say that too.

Scrolling further down the Google Images search result, much further down, image #76 reveals something very different which hints at the important (and much neglected) aspect of the revised 2001 taxonomy:

SOURCE: Anderson, L. W., & Krathwohl, D. R. (2001). A Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom's Taxonomy of Educational Objectives. New York, N.Y.: Pearson.

No more pyramid, no more hierarchy (mostly).  Knowledge has been pulled out of the list and turned into a separate dimension. The other parts of the taxonomy have moved into a cognitive process dimension.

I find this summary from Julie Stern very helpful in understanding the change:
Few educators, including those who criticize the taxonomy, have considered the other major change to Bloom’s Taxonomy: the knowledge dimension. Anderson and Krathwohl (2001) have taken “knowledge” out of the cognitive domain and added it as a separate dimension, recognizing four distinct types: factual, conceptual, procedural and metacognitive. ... that instead of six ways to think about one type of knowledge, there are now six ways to think about four distinct types of knowledge. 

Here's a very nice attempt to show both dimensions from the 2001 model in one diagram without too much hierarchy

The most serious problem with the pyramid view of the first iteration of Bloom's Taxonomy is that knowledge is right at the bottom and seen as something we just build upon, a "low order" thing (the lowest in fact).  Versions of the 2001 pyramid omit the knowledge dimension, focusing solely on the cognitive processes, leaving most teachers with the impression nothing changed except the words, with knowledge now just called "remembering (facts)".

In the next blog post, I'll be looking at some of the issues resulting from using an oversimplified view of Bloom's Taxonomy. For now it's enough to point out many of us, myself included, have been guilty of lazy thinking. Maybe it's not our fault - it's the brain's wonderful design to simplify ideas so we can cope with them efficiently.

To finish off, here are two surprising 21st Century Learning versions of the taxonomy which really had me scratching my head.  Observe that well motivitated and exciting as they are, both of the them have completely lost the knowledge dimension.

Ron Carranza's "Bloom's Digital Taxonomy".
I'm very pleased to see that by writing a blog post I'm at the top of the grid...
In the 21st Century we're not even "remembering", we're "bookmarking".

The "Flipped Learning Bloom's Diamond"

Sunday, May 12, 2019

Three ideas for defusing the pedagogy wars

There's been a battle for teachers' (and parents') hearts and minds over the last few decades that, now appearing on twitter, can on some days seem particularly virulent. No, I'm not talking about the reading wars, but a broader dispute between what is often called "traditional" versus "progressive" approaches to teaching and learning. Or to use a catch phrase often used, pejoratively I'm afraid, as a choice between the teacher being "the sage on the stage" versus "the guide on the side".  

In the Australian context, it feels this debate is pretty much a one-sided affair, with almost unquestioning acceptance that we need more "future focused, student centred" learning, but for those following the broader international scene, it's remarkable to witness the resurgence and reformation of traditional approaches into "knowledge based, explicit teaching" informed by findings from cognitive psychology. For the mathematics teachers among us, it's been fascinating to see some high profile teachers in Britain completely change their view on how to teach mathematics - switching from very innovative "student centred discovery learning" approaches to embracing their inner "teacher as the subject expert".

Craig Barton, a highly respected UK mathematics teacher, writes in 2018 about
his almost 180 degree change of view on how to teach mathematics. 

But meanwhile the slanging matches continue - it can be quite ugly some days to read the ad hominem attacks and to see the emotive grenades being tossed over the trenches. What is a teacher to do? Especially if they have formed a view that isn't the currently dominant view? How can we move forward?

#1: Respect and recognition for our colleagues ("niceness")
Right from the outset, I think we need to set a much much better example to the outside world as to how educated people can have a proper and respectful debate. It really disturbs me to see teachers write messages on twitter that exhibit behaviour we would not accept in our students.  It's essential we recognise that even if we 100% disagree with a fellow educator, even if we think they are naive/partisan/ignorant/bigoted, that we recognise they are motivated by the very best intentions: the well being and care of young people.  I may not concede that motivation for some others in the education debate (especially people with products to sell), but it's axiomatic to me that I respect and recognise that anyone who signs up to be a teacher and stays with it really has the best motivation. Let's remember that and start every discussion with the right tone. My mum called it "being nice".

#2: Remember context, context, context
Just as the real estate agent reminds us it's all about "location, location, location", as teachers it's essential we keep "context, context, context" at front of mind. Each time we  are about to say that something "for sure is correct", or "definitely doesn't work", we need to remind ourselves just how much context matters. What works for a Year 8 mathematics teacher with a specific class for a specific year for a specific group of students may well not work for a Year 4 primary teacher or a Year 12 music teacher in a different year, at a different school.  It's so easy to get caught up in your own certainty, your lived experience as reality, and forget context. Am I a "relativist"? No I'm not, it is my view now that there are some universal things we can confidently say about teaching and learning. But on a broader scale, education truly is a "wicked problem" - there aren't many simple answers that work everywhere.  We need a lot more caveats in our heated debates to explain our context and to recognise different contexts.

#3: Can we change the words? Mode A and Mode B
Tom Sherrington, in his wonderful book "The Learning Rainforest" takes an interesting approach to defuse the angst: just stop using the words "traditional" and "progressive". Instead use the generic terms "Mode A" and "Mode B". 

A language reset : "Mode A" and "Mode B".
Tom Sherrington's book is my #1 reading pick for 2019.

This is a surprisingly powerful technique - it reduces the emotion and short circuits the automatic, non-thinking responses. It allows Tom to make statements like "for my subject, for my group of students, I think 80% Mode A and 20% Mode B is a good mix". And then we can calmly look at the ideas, strengths and challenges inherent in each different Mode A and Mode B teaching technique, without getting bogged down in polemics.

So does that mean there shouldn't be a debate?
Certainly not! There's lots to debate about when it comes to teaching and learning and it's important, especially if we believe, as I do, that well educated young people are an essential component to addressing challenges such as climate change, socioeconomic inequality and role of technology in our future.  But we need to deescalate. It's not an arms race - we need diversity in our education systems and every school will have a blend of Mode A and Mode B. Different students will respond differently to different teachers, to different teaching and learning strategies. Diversity is the strength of our system.

Sunday, March 3, 2019

The ultimate formative feedback? Tales from the whiteboard classroom.

The idea is extremely (and deceptively) simple: Cover your classroom with whiteboards, then think about how to get students to spend most of their time standing at the whiteboards, rather than sitting at a desk listening to you talk from the front.

A "fish eye" view of my whiteboard classroom - Dec 2017. 
There are 7 whiteboards (not all are shown here) and the room comfortably fits 16 students, and 24 at a pinch. In mid 2018 I removed half the desks to make more space.

This was the challenge posed to me by Tricia Forester (University of Wollongong) at the 2017 MANSW Conference, referencing the "Building Thinking Classrooms" work of Peter Liljedahl.  Well actually, Tricia was more clever than that - she posed interesting maths problems to groups of teachers, providing a "vertical non-permanent surface" space to try it out.  It didn't take long to see the power. When I got back to school, my principal gave me permission to try it out for real, transforming a smallish classroom no-one really liked into a fully fledged whiteboard room with 7 large boards.

It's now been nearly one and half years that I've been working in a whiteboard room for my senior classes (unfortunately I can't fit 30 junior class students in the room) and there's no looking back! There are many things to share about the experience and why it supports such powerful learning, but for now I'll confine myself to a few big ideas.

The power of the whiteboard room is the way it provides continuous formative feedback - to both the teacher and the student. In fact, for the teacher, it's overwhelming - you will be bombarded by feedback. You instantly see what students can and cannot do, you see the approaches they take and you hear many simultaneous conversations discussing the problem. For the whole lesson. It's almost too much feedback. In the whiteboard room, we have a "live stream" answer to the "How do I know what my students learned?" question. I don't have to give them a diagnostic test every day - I can see it every minute. This continuous formative feedback allows me to dynamically alter the lesson according to needs of the students. It's a bit of a high-wire act some days, and often full of unexpected surprises, but it makes for exciting and rewarding lessons that students really enjoy and demonstrably produces learning. I'll write some more soon about the strategies I've developed to dynamically respond to so much continuous feedback.

Students working in "interleaved" mode on a problem.  The students worked out on their own that harder integration questions are easier done in "parallel" by two students - but they still end up correcting each others work! Different colour markers let me quickly see how (or if)  students are working together.

The student engagement factors at play in the whiteboard room, especially when combined with the Visibly Random Groups technique, are truly astounding. I have seen students who previously had difficulty making friends,  difficulty communicating, who were disengaged, lacked confidence, transformed by the whiteboard room experience. The combination of the social interaction, the risk free "non-permanent" writing surfaces and physical movement in the whiteboard room seems to work magic on students. One day a learning support teacher came into the room and asked a notoriously lazy student why he liked the room so much, he replied "Because I can't sleep in the back of the classroom any more!".  Everyone, without exception, is engaged in the whiteboard room - there's no option to not be involved *.

We also use the whiteboard room for our before school "Year 8 Algebra Workshop" - a weekly session for students who need a little more support with algebra. The whiteboards allow them to work together while developing fluency. The social interaction and support created by the whiteboard room is an essential component of this program which seeks to build confidence and a growth mindset attitude.

Is this "student-led" or "teacher-led" teaching? I think it's a blended approach, and a good example of why we need to get beyond using labels.  However I do think I need to be clear: while students are doing most of the work, the lessons for my senior classes are very strongly guided by me. The student feedback steers the lesson according to what they need or where they are ready to go, but I'm still the "guide", it's just I don't "teach from the front". Several times during the lesson I will regroup the class in front of a student whiteboard I selected because it has a "teachable moment" and then "microteach" to summarise or clarify, and then teach a small amount of new content as required so students can continue with the next set of problems.  Other times, a carefully selected sequence of problems means I don't need to do that - the learning follows naturally, and I can "spot teach" to the groups who need more scaffolding. My lesson designs have changed significantly. Because the students are doing so much work, the whiteboard room forces me to seriously consider the "What will my students be doing?" question. I spend much more time on thinking about examples and problems than on what I will say. 

It's scary to change how you teach, and even after all this time, I sometimes worry I'm doing something too different from the other classes at my school. This year, the whiteboard room is less available to me (because I'm encouraging other teachers to use it!), and due to timetabling, while I still have the whiteboard room for my Year 12 class, I can only use the room with my Year 11 class for four out of every thirteen periods - so it's been interesting to see what happens when you mix things up.  Sometimes old habits resurface, and I'm tempted not to use the whiteboard room when it's available - because it is easier to stand at the front, or you just haven't had the time to plan a whiteboard room lesson sequence. But much to my (pleasant) surprise, my Year 11 students complain and insist we go to the whiteboard room, so we just do it. And you know, when we finish the period, it's so clear they were right - it's a better lesson, even if I hadn't planned it as a whiteboard room lesson.

Some starting resources and ideas:
Laura Wheeler's @wheeler_laura blog post Building thinking classrooms is an excellent starting point.  Here's a handy link to all her posts with the "thinking classrooms" tag.

Can't set up whiteboards in your room? Try out Magic Whiteboard to transform any wall into a vertical non-permanent surface.

Peter Liljedahl's @pgliljedahl papers "Building Thinking Classrooms (pdf)" and Visibly Random Groups (pdf) are highly readable and motivating!  Liljedhal's "Thinking Classrooms" framework is much more than just vertical non-permanent surfaces (VNPS) and visibly random groups (VRG). For me this is start of a journey.

* If a student really wants "to be alone", I'll give them a break and let them work solo for a while, hoping they will change their mind later, or next period. They always do.  Also at the end of a really hot day, we might decide to sit on chairs for a change - sadly the whiteboard room doesn't have air conditioning.

Some lessons work best when you do (almost) nothing

Returning to my blog after many years absence, I found this Feb 2017 post in my "drafts" folder, waiting for posting.  I think it's still worth posting now in March 2019 (!).

It's a new school year, for me the start of the seventh year teaching. And this week I think I taught my best lesson ever. The surprising thing is that I barely taught anything. In the words of one of my mentors, a teacher with 35 years experience, "you know it's a great lesson when you do almost nothing. You sit back*, close your eyes, and you hear the student conversation taking place - they are talking about mathematics - and you'll hear how they think - and then you know what you need to teach, specific to each student."

For this lesson, I handed out "A/B quizzes" to the students which looked liked this:

They are designed to test skills in way that encourages students to help each other. In the A/B quiz lesson design, different students get different papers (see the letter in the bottom right hand corner of the paper). They can help each other if they like, but since the questions are slightly different, they have to actually teach each other rather than just share answers. Half way through the period, the students form into groups with those who did the same paper and compare their work, teaching each other anything they need to. Then they repeat the process for a second round, using a different version of the "A/B" quiz.

It was amazing to watch this group of students work together. I saw them struggle through the harder integrals with negative fractional powers, debating with each other what the correct answers were. Every single student was involved, no-one was left out.  I mistakenly had some indefinite integrals on the quiz, which I hadn't taught yet, but the students who had worked ahead taught the others how to do them. And they loved the lesson, "Integration is so much fun, Sir!" I think they packed a week of learning into forty minutes. I only taught for the last ten minutes, reinforcing some of their findings and explaining a few finer points of the setting out and reasoning required to provide the highest quality solutions.  It felt like one of the best lessons I had ever taught - but strangely I had barely done any teaching.

OK - I'm exaggerating when I said I did nothing. The lesson happened as a result of many months working with this group of students, building trust, confidence and openness.  It does take a fair bit of training to get students working effectively in groups for this process - to ensure they actually help each other and look carefully at each other's work, rather than just "looking for an answer". I did design a very specific learning sequence for the topic, selecting items in the diagnostic quiz designed to elicit discussion and to expose any difficulty students had executing the required skills. And it does take many years of experience teaching a topic such as Integration to anticipate what difficulties students will have, how to diagnose them and how to provide the necessary support.

And that's the joy of the Art and Science of Teaching - sometimes the very best lessons have very little 'visible' teaching. But if you look closely, you'll see a lot of visible learning - and it will take many years of experience to feel like you got the best out of a particular lesson plan.

Would this work for every lesson? Absolutely not! It's my view, after trying many different approaches, that when teaching mathematics, four out of every five lessons should follow more traditional, explicit instruction supported with ongoing and regular formative assessment. And for the fourth or fifth lesson - try lots of different things - the more student-centred the better. "A/B" quizzes are a lesson design that works really well - the gift that keeps on giving, year after year.

* Update for 2019: I've found an even better way to run a lesson like this! Coming soon...

Sunday, August 2, 2015

"Why are we learning algebra?"

It had been several weeks since my Year 7 class had the discussion of why we were learning algebra, so I was taken off guard when the perennial question came up again: "Mr Zuber, why are we learning algebra (again)?" 

I have a whole range of answers I like to offer to this favourite question but this time something unexpected came out of my mind.  "Have you seen those amazing new pictures of Pluto that came in this week from the Horizon spacecraft?"  I was pleased to see many students in the class start to get excited - they certainly were inspired by those photos.  

Global mosaic of Pluto in true color (NASA) July 2015

"Well", I said, "that was algebra. Algebra brought us those pictures. Very complicated algebra, and physics and engineering worked out by smart people helped get that spacecraft just at the right place, at the right time above Pluto, millions of miles away from Earth, to get that photo and send it back to us. That's why we're doing algebra."

I think that was the best answer I gave in class all week - and the students seemed to like it. Thank you NASA!

Here are four reasons for learning algebra that I like to offer students when I start the introductory algebra topic.  

Firstly we have some utilitarian reasons:

Algebra is a tool to help solve problems.
We use it to find values of something we don't know.

Algebra allows us to record information about relationships between numbers in a formula.
We can then put values into those formulas to find related numbers. This could be the area of a triangle, or the dosage of medicine to give a child based on their weight.

At a deeper level, algebra has an important place in our discovery of the world:

Algebra allows us to describe how the world works.
Students like this image. The picture in the centre is a matter-antimatter collision and the formula is Heisenberg's Uncertainty Principle.

and in supporting our exploration and representation of mathematical ideas.

Algebra allows us to represent and explore mathematical ideas and mathematical objects.
At least some students in your class will have seen the Mandelbrot Set and know how complicated it is - they will be very surprised how 'simple' the algebra looks.

Putting all these ideas together, I like to summarise with the one big idea: algebra is a language.

So - for those people who say "but I will never use the quadratic formula in my future work", I would respond: "Wouldn't you like to learn this amazing language? It will open up so many career possibilities to you (a utilitarian argument) and it's also a fascinating and rich language that will let you access a whole new level of knowledge and ideas (a sheer pleasure argument)"

What's even more amazing about this language is that it's an international language. I can speak algebra with a Russian or a Chinese mathematician. Somewhere out there in space, a class of Year 7 students with green skin and three eyes is also learning algebra.  Can you think of another subject you're learning at school which is also being taught in Alpha Centuri?  (OK - science... but let's pretend that's the same as maths :-)

Postscript: Should I have mentioned that algebra helps us develop reasoning skills? Possibly.... but I'm not sure most students buy the "it's good for your thinking" argument. So I take the "what algebra will offer you" line, and make sure I give emphasis to its role in abstract thinking as well as in 'practical' applications.

Thursday, July 9, 2015

Wormholes and Tesseracts in the Classroom - Part 2

In Part A I looked at some inspiring ideas about teaching in the movie Interstellar. Here are some ideas for using some Interstellar content in the classroom.

Exploring Dimensions: whether you're doing lessons on 2D and 3D solids, or just having a discussion why we say "x-squared" and "x-cubed" but "x-to-the-fourth", it's time to bring out The Tesseract.  In the Interstellar version, it's an object that has spatial and time dimensions: as the main character Cooper moves through space, he's moving into different time "rooms".

"Time is represented here as a physical dimension"
Warning: SPOILER for the film!

My Year 7 students liked this, and were very fast on their internet devices to find more traditional mathematical representations of the tesseract (the hypercube) which made for a good discussion.

Permutations and Combinations: I haven't worked it out yet, but there's definitely a perms and combs activity to do with the CASE and TARS robots! Might link in well with a Quadrilaterals exploration too.

Watch CASE at work on Miller's Planet: 

Some good resources:

An exciting way to introduce circular motion:

A nice adjunct to the more classic and sedate circular motion sequences in my other all time favourite movie, "2001: A Space Odyssey".

And for something different, you may like to point your music teacher friends at this mini documentary about the making of the soundtrack: