Math Talk

Traditionally, math class was a quiet place. Kids alone in rows busy calculating with limited back and forth. A competition to see who could find the correct answer the fastest. The idea that discussion was a necessary tool for deepening and consolidating understanding was a foreign concept, given that much of mathematics was conveyed as symbols and numbers. There was a right and a wrong answer – so what was there really to talk about? Yet the kind of communication we want students to engage in is so much more than simply answering questions or reciting procedures. Of course these are a part of any math class, but they shouldn’t comprise most instructional time, as they often do.

Talking about math is not something that comes naturally to kids. 

There needs to be a shift from focussing on finding the answer to discussing the problem. When this happens there is a collective easing and the pressure is off of students who are reluctant to share their ideas for fear of getting it wrong. The potential embarrassment is not worth the risk. More than any other subject, math creates this anxiety among the less confident. As a result, executive functions such as working memory and regulating behaviour suffer and math proficiency is not fully developed (see research here). To alleviate this stress, teachers can redirect attention back to the problem. We’re in this together to find a solution.

With enough practice, these four simple questions will lead to profound math talk:

  1. Why?
  2. How do you know?
  3. Can you prove that?
  4. Can someone else disprove what’s been said?

Effective approaches to encourage math talk are Gallery Walks, Math Congress and Bansho (see here)In order to make students more comfortable sharing their mathematical thinking, the following strategies and sentence starters are a great way to scaffold dialogues. It is in these moments that some of the best consolidation of learning happens. Sometimes we don’t know what we truly know until we give it a voice.  


Characteristics of math communication to look for (see rubric here):

  • precise – relevant choice of method that has accurate calculations
  • clear – logical organization that is easy to follow and requires little inferencing
  • cohesive – reasoned argument held together through explanations, diagrams etc.
  • elaborate – justification of ideas and strategies with sufficient detail
  • appropriate – proper use of mathematical terminology, symbolic notations etc.

Things Come in Threes

Dan Meyer’s three acts are well known to math teachers. (For an explanation, see here. Excellent examples are here.) The tasks are engaging and get at the heart of mathematical modelling. According to Meyer, they follow a storyline structure:

  • Act One: Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible.
  • Act Two: The protagonist/student overcomes obstacles, looks for resources, and develops new tools.
  • Act Three: Resolve the conflict and set up a sequel/extension.

Structuring math in threes led to these observations on teaching it:

Three Challenges

  1. Conceptual: Math is inherently abstract. It is disembodied ideas that become increasingly difficult to grasp as you advance through the grades. (Think formulas, functions, proofs etc.) Connecting concepts to the concrete can be a struggle.
  2. Weird: Strange symbols and jargon (not to mention the amount of content teachers need to cover to unpack and make sense of all the oddness.)
  3. Anxiety: The association of math with phobias and stress. For many students, math is terrifying, boring or meaningless. It can result in shutting down and feeling overwhelmed. Which leads us to the next trio…

Three Audiences

  1. Traumatized: Students who have had a terrible experience with math that has turned them off the subject. Or maybe they hit a wall along the way with long division or linear algebra. The ‘I’m just not a math person’ group.
  2. Perplexed: Students who see math as pointless. They are a bit lost, but compensate for it by working hard, following directions and overcoming failures. This group is often the silent majority in any class.
  3. The Naturals: Students who have a feel for math. It makes sense to them. It gives them satisfaction and they have innate talent. They’re not always the ones with the highest grades, but they usually go into careers in related fields.

Three Routes

  1. Illuminations: The eureka moments that help students come to love math after searching in the dark. Whether it’s through explanation, visuals or practice, these are the times that progress is made. Breakthroughs that allow them to keep going.
  2. Connections: Tying any subject to what we already know is effective and math is no different. Sports, music, science, movies, business and nature are all great sources for math inspiration. The traumatized and perplexed benefit and math class is more interesting.
  3. Conversations: Talking about math is underrated. Because it is a concise subject, the more we can grapple with concepts through discussion, the better. Providing opportunities for students to express and argue about ideas is important.



Fermi and the Problem Solved

Word problems in math textbooks often give much of the information needed to solve them. There is no mystery. Students walk away from a math course with the only skill acquired being the ability to decode the textbook. They are just swapping numbers and plugging in different information. As a result, the so-called problems are no longer problems. They are routine and predictable. The problems are too scaffolded and the students realize that it’s an exercise in futility. An insult to their intelligence. While practice is indeed a fundamental part of math, when problems are variations of the same one, the motivation to complete them is lost. They don’t see the point of it all.

How can we make problems interesting and challenging?

Look to Enrico Fermi. The Italian physicist had a gift for making accurate estimates of seemingly unsolvable questions. Given little information, he was able to provide educated guesses that came very close to the actual answer. His most famous question, “how many piano tuners are in Chicago?” seems to make no sense, but through a series of questions, estimations and assumptions, he arrived at a reasonable answer. Legend says that Fermi calculated the power of an atomic explosion by looking at the distance his handkerchief travelled when he dropped it as the shockwave passed. He determined it within a factor of 2. For a discipline that is always looking for realistic applications, math class would do well to use Fermi problems. It doesn’t get more real-life than that!

(For more problems, see here)


Using Fermi questions in our math classes will remove the pseudo-contexts of much of the problems. They are actually used in real life, unlike many of the questions from a math textbook. Companies use Fermi problems during job interviews as they offer a window into a person’s ability to think on their feet and their creativity. Scientists, economists and engineers use them in their work to get a ballpark idea of the feasibility of their projects. This power of estimation is a key aspect of mathematical thinking. Without it, all the math in the world won’t mean a thing.

Fermi problems emphasize the mathematical processes and help students practise estimation and reasonableness when solving problems. They strengthen number sense, dimensional analysis, and are important in developing a quantitative understanding of the world around us. They allow students to ask the right questions and break down complex problems into smaller, solvable ones. The problems don’t have a definite solution, providing room for interpretation and multiple approaches to problem-solving. The questions are not grade specific and can be used in a range of classes. The open-ended quality of Fermi problems is one of their strengths. 

Encourage intuition in math class through Fermi questions. Give students something meaningful to solve. Tired, old word problems should be a thing of the past.

Depth Over Speed


Jo Boaler is a somewhat controversial figure in math education. The Stanford professor is in the ‘reform’ camp, arguing that new approaches to teaching math, that rely on a lot of group work, real-life examples and discovery should be emphasized over more traditional methods such as memorization, worked examples, repetition and the learning of key principles and facts. Back in 2012, she was accused by two academics (see here) of questionable research methods and inconsistent data in her Railside Report. For an excellent, in-depth post on the subject, see here. 

Despite the credibility storm that surrounded Boaler, she still has quite a large following. Her voice in math teaching is one of the loudest. Her opinions influence policy and make waves in education circles. When she suggested that memorizing times tables isn’t necessary for students to achieve success in math, it made headlines. It got people talking. It stirred the math pot. While many educators don’t agree with her philosophies, she continues to greatly influence the discussion on how to best teach math.

That said, like many things in education, it’s important to separate the politics and the egos from what works best. Education can sometimes suffer from too much self-righteousness. If Boaler can offer advice that will benefit math teachers, who really cares about the other noise. Leave the politics to the politicians. While you might not agree with everything Boaler says, she does offer valuable insights in her recent book Mathematical Mindsets. Sure, many are points that have been raised before, but they are worth repeating.

One of those ideas is depth over speed. The pressure to cover curriculum that many teachers feel leads to a rat race approach to math instruction. As a result, lessons are often a mile wide and an inch deep. Teachers get stressed out and students retain less as concepts are glossed over and enduring understanding is sacrificed. The train keeps moving down the track and if some get lost along the way, oh well.

This notion that mathematical skill is all about speed is just plain wrong. And yet, that’s the impression that most students have of math class. The best students are the fastest. Whoever can finish a problem the quickest must be the most capable. There is a beeline to the solution. We have been conditioned to look for easy answers, what Dan Meyer calls ‘impatience with irresolution’. To satisfy this dissonance, we rush through it to get it done. Maybe sitcoms are to blame, who knows. 

Boaler provides a telling example of her observation of a Chinese math class. With two of the top three PISA math scores, Shanghai and Hong Kong (along with Singapore) are the best in the world. It’s not even close. The assumption is that they use a lot of drill and kill instruction, where speed is valued, but the reality is much different. Students typically engage with no more than 3 questions per hour. Like mathematicians say, their work is done slowly and deeply. Justification and reasoning form the essence of math. And these take time.

We need to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant. 


What Is Mathematical Thinking?

‘Mathematics is the music of reason’ – James Joseph Sylvester

Mathematical thinking is a lot more than just being able to do arithmetic or solve algebra problems. It is a whole way of looking at things, stripping them down to their essentials, whether it’s numerical, structural or logical and then analyzing the underlying patterns. Math is about patterns. When we are teaching a mathematical method, we are showing something that happens all the time, something that happens in general. Getting students to see these underlying structures, whether it’s in a math problem, in society, or in nature, is one of the reasons that studying mathematics is so worthwhile. It transforms math from drudgery to artistry.

As Steven Strogatz says, ‘Math is not just what we heard about in high school, the known and straightforward part of the subject. There are all sorts of interesting theoretical and applied problems out there.’ Helping students to see the beauty and wonder in math will go a long way in getting them to think more mathematically. Let them know that what makes mathematicians is the quality of their creativity and the quality of their technique. Combining these two is what leads to great mathematical thinking.

The problem lies in the fact that students usually have no idea about their strengths and weaknesses in these two areas. Schools emphasize the procedural side of math. There is the misconception that it only involves mechanical, robotic thinking. What pervades is a ‘we didn’t cover that’ mentality when they encounter a novel problem. 

Once you have identified a task or situation to explore, mathematical thinking involves these steps that are often done together and simultaneously:

  • break task down into components
  • identify similar tasks that may help
  • identify appropriate knowledge and skills
  • identify assumptions
  • select appropriate strategy
  • consider alternative approaches
  • look for a pattern or connection
  • generate examples

This all leads to formulating a conjecture (solution, generalization or relationship) that can then be tested for counter examples or special cases. Similar to the design process, if it doesn’t work, it’s time to iterate and try again. Proving the solution, making connections, considering limitations, and extending it further to ‘what if’ questions are additional aspects of mathematical thinking. Spending the time to think of and ask really good questions is at the heart of math.

John Mason’s questions and prompts provide a nice framework for ensuring and promoting students’ mathematical thinking. The verbs across the top of the chart highlight what is involved when we ask students to think about math. The specific examples are practical and useful.



Open Questions

Differentiation in math is a relatively new idea. While it has become an integral part of any literacy program, it is scarcer in math. DI requires more planning and thinking through the logistics. Teachers may sometimes be reluctant to use it, because the majority of us experienced math as a subject in which everyone was on the same page. Deviating from the textbook was not encouraged. The class moved forward in lockstep. There was little room for varying the content, process or product.

Many of the questions we traditionally ask students call for a single number, figure, or mathematical object. These kinds of questions are closed because the expected answers are predetermined and specific. Conrad Wolfram estimates that they make up around 80% of math instruction! Students are sent the message that math is about simply finding the right answer. It is black or white. A static subject with no creativity that only relies on a series of endless computations. Motivation to learn math is lost. No wonder students tune out beyond a certain point.

Open questions, an approach popularized by Canadian math researcher Marian Small, offer an alternative. Creating a question that allows for multiple entry points gives all students the opportunity to find something meaningful and appropriate to contribute. They offer a variety of responses based on the students’ level of content knowledge and understanding. This gives students the confidence that is sometimes lacking in math class. Levels of anxiety are reduced as open questions close the various ZPD gaps among the students.

Math is seen as multi-faceted. The subject comes to life. 

The best open questions have a low floor and high ceiling. This effectively levels the playing field and gives more students a chance to engage in math. Open questions should focus on big ideas and curricular goals. They need just the right amount of ambiguity, ensuring that the question is broad enough to meet the needs of all students. Strategies for creating open questions:

  • turning around a question (think Jeopardy)
  • asking for similarities and differences
  • replacing a number with a blank
  • creating a sentence using certain concepts/definitions/numbers
  • using ‘soft’ vague words for flexibility

The first thing a mathematician has to do is pose an interesting question. These are virtually absent in most math classes today. Open questions are a way to change this.

The Dewey Deception

‘Educators, parents, and students must be induced to abandon the educational path that, rather blindly, they have been following as a result of John Dewey’s teaching.’

– Dwight Eisenhower

John Dewey’s title as the father of modern education is widely accepted by most people. He is regarded as one of the founders of progressive education and you’d be hard pressed to deny his rightful place among the intellectual giants of the 20th century. Many think he was a genius. Few educational theorists have equaled his widespread impact on societies throughout the world. Walk into any school and you will see Dewey’s influence at work. His thoughts and contributions have even extended to courts, laboratories, the labor movement, and politics. For someone that only taught two years of high school, his views on education tower over all others. No one is even close, then or now. Indeed, history has been kind to the revered educator.

With that said, it’s time to question and reassess the thoughts of Dewey. What is the real nature of our inherited education system? If we want to improve a system that many feel is broken, we need to take a closer look at the original architect. Seldom has an educator drawn on such diverse viewpoints as Dewey did. While most assume he had the best of intentions, what if they’re wrong? History has a way of telling only one side of the story. Painting the famous in only a glowing light. Perhaps it’s worth digging deeper into the untold story of this famed educator.

Many education systems have been built from a model designed by Dewey, one which rejected the classics, any emphasis on rhetoric and logic, or rote memorization. Instead, the pragmatist Dewey valued experience over facts, logic or debate. In his 1899 book School and Society, Dewey writes: ‘The mere absorbing of facts and truths is so exclusively individual an affair that it tends very naturally to pass into selfishness. There is no obvious social motive for the acquirement of merely learning, there is no clear social gain in success thereat.’ What was taught was no longer relevant thanks to Dewey. The how and why were the main considerations. The watering down of curriculum had begun.

In those two sentences Dewey relegates all art, science, mathematics, philosophy and history to the trash heap of irrelevancy. For those that believe there has been a dumbing down of education thanks to progressives, there’s your smoking gun. Dewey was not primarily concerned with teaching knowledge. He was concerned with instilling new attitudes and indoctrinating educators with the experimental psychology of Wilhelm Wundt. John Dewey was a social engineer. He wasn’t interested in education as you and I understand the term. In fact, Dewey intended to subvert and diminish traditional education.

faculty_img19_lrgIn 1896, Dewey created his famous Laboratory School allowing him to devise a curriculum that broke from traditional schooling. The key element that held the entire existing system together was high literacy. It gives the individual the means to seek knowledge independently, to question the status quo and to exercise one’s own judgement. Literacy allows us to think for ourselves. If you’re wondering why the look say method of reading was pushed in schools, leading to a generation of illiterate students, look to Dewey. According to Dewey, ‘It is one of the great mistakes of education to make reading and writing constitute the bulk of the school work the first two years. The true way is to teach them incidentally as the outgrowth of the social activities at this time.’ Dewey knew that the reading program he was suggesting was not as effective as traditional methods. Unfortunately, his social agenda overshadowed effective instruction.

In Dewey’s seminal essay The Primary Education Fetich is a sentence that is telling: ‘Change must come gradually. To force it unduly would compromise its final success by favoring a violent reaction.’  He knew that what he was doing would have a negative impact on education. Dewey and the progressives have wreaked havoc on the learning of millions. It’s time we wake up and face the truth.

Dr. Seuss and Dyslexia

Theodor Seuss Geisel, better known as Dr. Seuss, wrote some of the most famous children’s books of all time. The Cat in the Hat and Green Eggs and Ham have more in common than just being popular. Both were written almost entirely with sight words. Most people are unaware that Dr. Seuss books were created to supplement the majority of whole word reading programs in schools. In 1957, Seuss was commissioned to write a book using only 223 sight words supplied by the publisher. The publishers believed that if kids could memorize the words in the book, they would be better prepared for reading instruction at school. Dr. Seuss books have been categorized with the ‘look say’ movement, a method of teaching beginners to read by memorizing and recognizing whole words, rather than by associating letters with sounds. It was invented in the 1830s by Thomas Gallaudet, the famous teacher of deaf students. For some strange reason he thought it could be adapted for all readers.

Because the books are so simple you would think they were easy for Dr. Seuss to write. The reality was much different:

Ted_Geisel_NYWTSThey think I did it in twenty minutes. That damned Cat in the Hat took nine months until I was satisfied. I did it for a textbook house and they sent me a word list. That was due to the Dewey revolt in the Twenties in which they threw out phonic reading and went to word recognition, as if you’re reading Chinese pictographs instead of blending sounds of different letters. I think killing phonics was one of the greatest causes of illiteracy in the country. Anyway, they had it all worked out that a healthy child at the age of four can learn so many words in a week and that’s all. So there were two hundred and twenty-three words to use in this book. I read the list three times and I almost went out of my head. I said, I’ll read it once more and if I can find two words that rhyme that’ll be the title of my book. (That’s genius at work.) I found “cat” and “hat” and I said, “The title will be The Cat in the Hat.”

By reading Dr. Seuss books children entered grade one already having mastered a sight vocabulary of several hundred words. The hope was that reading would be a breeze. However some parents started asking: how is it that my child is showing signs of dyslexia before even having had any formal reading instruction? Because they memorized Dr. Seuss books! The children developed a block against seeing words phonetically, with some developing dyslexia. They became sight readers with a holistic reflex rather than phonetic readers with a phonetic reflex. The problem is that this approach ignores the letter-sound association of reading. This sort of practice produces the symptoms of dyslexia: reading words backwards, reversing letters when writing, gross misspellings, word guessing, word skipping, leaving out words, and putting in words that aren’t there. The reason why dyslexia is so hard to cure is because the child has acquired a holistic reflex, automatically looking at words in their whole configurations.

Once the words get more complex the sight reader has no strategy to sound out the words. By third or fourth grade, where the reading demands are much greater, the sight reader’s overburdened memory cannot handle decoding. This explains much of the ‘fourth grade slump’. There is a breakdown in learning. The reading disability becomes evident. No wonder many students struggle to comprehend what they are reading when they have trouble even decoding the words. The phonetic way is a method used for thousands of years with an unparalleled track record of success. Why did educators try to reinvent the wheel with the sight method?

Dr. Seuss knew that ‘killing phonics’ was a cause of dyslexia. But somehow that insight, made by one of the most famous writers of children’s books, has escaped educators.

Learning That Sticks

Why don’t most of us remember what we learned in school? It seems a lot of what we were taught failed to stick. In Make it Stick: The Science of Successful Learning, there are useful suggestions for improving our ability to retain things. Learning is a three step process – initial encoding, consolidation and retrieval – and it’s deeper and more durable when serious effort is made. Spacing out practice and interweaving it with other learning is more effective than mass practice methods like cramming the night before an exam. Retrieval practice (known as the ‘testing effect’) involving the recalling of facts and concepts from memory through self-testing is more productive than just rereading notes or a textbook. The reason is we are often poor judges of when we are learning well and when we’re not. This is due to illusions of mastery. While all of these strategies are the individual learner’s responsibility, there are ways teachers can make learning stick too.

Most of the following principles are common sense, however there is something known as ‘the curse of knowledge’ that can prevent teachers from making their lessons stick. Once we know something, we find it hard to imagine what it is like not to know it. Our knowledge has ‘cursed’ us. It can sometimes be difficult for us to share knowledge with others, because we cannot easily recreate the state of mind of our students. Every year we bring new knowledge to our class. Given that we’ve taught it before, teachers need to ensure that we are transforming our ideas into a language that is understandable to our students. Always being aware of the differences between expert and novice is important. It can have a significant impact on making learning stick.


The ability to decide what is most crucial is at the heart of simplicity. It is an essential skill for all good teaching. When planning a lesson or course outline, some concepts are more critical than others, so we ask ourselves: what should be left in and what should be taken out? To get the ideas we are teaching to stick with students, we need to identify our core messages. We need to cut extraneous details. We need to trim the fat.

Teaching with simplicity does not mean dumbing down the curriculum, but instead choosing what content is worthwhile. A good way to communicate lots of information succinctly and make it stick is through anchoring it to what students already know. Teachers use this simple principle of anchoring all the time. Another effective way to make the learning memorable is analogies, the bread and butter of teacher explanation. 


Sparking curiosity is one of the holy grails of teaching. According to George Loewenstein’s gap theory, this inquisitiveness arises when we feel there is a hole in our knowledge. One of the keys to the theory is that we need to open gaps before we can close them. Teachers must make students realize they need to fill these gaps. If we want to know something but don’t, it’s like an itch that needs to be scratched. The trick then is to sometimes deliberately withhold information for awhile in order to pique curiosity. This struggle to find the answer can aid in the development of long-term memory.

Some unexpected approaches to use are: create mystery out of ordinary topics, construct your lesson around an interesting story, and ask students to make predictions and estimations at the beginning of lessons. This last technique, also known as ‘concept testing’, forces students to commit to an answer and gets them more invested and curious about the outcome.


Many aspects of a curriculum (especially in math) can seem abstract and mysterious. If we can make concepts more concrete and link them to the tangible through real-world examples, the learning will be more enduring. The Velcro Theory of Memory says that the more sensory ‘hooks’ we can put into an idea, the better it will stick. Using the five senses to provide concreteness helps to etch ideas into our brains. It’s the difference between reading about something in a textbook and experiencing it firsthand through experiments, manipulatives, field trips and other sensory experiences. Think of how much easier it is to remember a song than your credit card number, even though the song has much more data. The same rules apply to learning at school.


Psychology has shown that stories are ‘psychologically privileged’ in the human mind. We seem to be primed to remember narratives more and have a natural affinity for stories. Personal anecdotes, stories of past students or even ones taken from today’s headlines can all be used to help make learning stickier. The stories don’t have to be overly dramatic, captivating or entertaining, as long as they allow students to experience the mental simulation of placing themselves into the story. The mental simulation that stories provide is the next best thing to actually doing something. Stories are like flight simulators for the brain. A class can be transformed through their power, as stories are the currency of our thoughts.


Making our teaching stick is easier than it seems. It just takes a bit of time and focus. Keeping these four principles in mind when lesson planning is a good step towards ensuring that what we teach endures in the minds of students.


The End Of Bloom’s Taxonomy

Bloom’s taxonomy has acquired a mythological status in education, being one of those reference frameworks that teachers adhere to with some sort of blind allegiance. It has been around for so long that educators take it for granted. While some question and criticize its validity, it is still being widely used by teachers as a tool in the analysis of learning objectives. The levels appear on tests, and factor in lesson planning and curriculum design, with the taxonomy being ubiquitous in schools. A Google search offers over a million results. Despite its widespread use, I’ve never understood its appeal.

Originally designed by Bloom and a team of psychology graduates in the 1950s (updated 2001 by Anderson and Krathwohl) as a method for the development of college test questions for WWII veterans, the six levels of cognitive domain have dominated education for the past half century. However, the hierarchical pyramid was never meant to be used as an evaluative tool and does not claim to measure ‘effective teaching’. With that said, Bloom’s still serves as the backbone of many teaching philosophies, in particular those that lean more towards skills rather than content. Unfortunately, it’s time has come and gone. The taxonomy no longer serves a useful function.

As Brenda Sugrue states here, Bloom’s taxonomy is not supported by any research on learning. Developed before advances in cognitive science (applicable to education), the taxonomy is little more than a best guess by some knowledgeable people of the time. It has led to several misunderstandings among educators as outlined in this post by Grant Wiggins. While it may help some teachers with questioning and checking for learning, for students it’s just a triangle. It lacks clarity. Students find it difficult to chart their progress. To them, more or less, it is meaningless.

What’s the alternative? I think there is real merit in the SOLO taxonomy. For some teachers, especially in the UK, this taxonomy is well known. Although it has received criticism, most notably from David Didau in this post, I believe it has a lot of positive aspects. Like many things in education, the more practical, the better. SOLO provides that pragmatic approach that Bloom’s always lacked.

SOLO (Structure of Observed Learning Outcomes) offers a structured framework to help students progress in their thinking and learning. As its creators Biggs and Collis (1982) state, ‘it provides a simple and robust way of describing how learning outcomes grow in complexity from surface to deep learning.’ While it is similar to Bloom’s, it has one major advantage: a user-friendly, common language of learning that allows students to explicitly understand the learning process. SOLO has none of the confusing overlaps of Bloom’s (i.e. ‘identify’ appears in knowledge, comprehension and analysis).



  • Prestructural – Students don’t have any real knowledge or understanding of the topic. You will typically see blank stares. They will usually answer, ‘I don’t understand it.’
  • Unistructural – Students have limited knowledge of the topic. They may only know one isolated fact or aspect of the topic. They have some basic understanding.
  • Multistructual – Students responses focus on several relevant aspects but they are seen as independent and little connection is made. They are unable to link ideas. Assessment at this stage is mostly quantitative.
  • Relational – Students start to integrate concepts into a coherent whole with the development of higher order thinking. They link together and explain several ideas on a related topic. The pieces of the puzzle fit together.
  • Extended Abstract –  Students reach the most complex level. Ideas are linked together, extended to the bigger picture and then looked at in a new and different way. This is the holy grail of learning.


The benefits of SOLO can be found in its ability to make progress more visible. Ideally, it empowers students to take more control of their own learning and gain greater independence. The forward momentum found in the taxonomy can be a powerful motivator. As well, the idea of progression between the levels is more clearly defined than in Bloom’s. It is something that students can plainly see and experience. SOLO allows students to reflect meaningfully on what their next steps in learning are. Hopefully, it also gives them a sense of purpose in their learning.

In the end, it’s what every teacher wants for their students.