Constricting the modern mind

I will be brief and rambling today.

As I was wasting a few minutes on “social media” I noticed post after post where a meme–often a very clever one–was offered as evidence for a political opinion. You know what I mean “X destroys opinion Y with one example” blah blah blah. I find it both irritating and a bit frightening.

I find it frightening because more and more it appears that memes are displacing newspapers and news broadcasts as the fundamental information for voters. I see memes blaming Prime Minister Justin Trudeau for Trans Canada Pipeline’s corporate decision not to further pursue the Energy East pipeline. I see local mayors tarred and feathered over I’m not sure what. I see that “the left” all agree on everything, and it’s all stupid. I see the “the right” all agree on everything, and that it’s all racist.

And I fear for the future of democracy.

One of the strongest responses could come from education. But I’m not sure that the will is there.

As I sadly looked at my Facebook page, a line from JS Mill’s On Liberty came to mind.

“He who knows only his own side of the case knows little of that. His reasons may be good, and no one may have been able to refute them. But if he is equally unable to refute the reasons on the opposite side, if he does not so much as know what they are, he has no ground for preferring either opinion… Nor is it enough that he should hear the opinions of adversaries from his own teachers, presented as they state them, and accompanied by what they offer as refutations. He must be able to hear them from persons who actually believe them…he must know them in their most plausible and persuasive form.”—John Stuart Mill, On Liberty (1859).

Mill knew better.

People who disagree with us are not all fools, nor are they all morally defective. (Some may be, of course.) For virtually any position worth fighting over, reasonable people can disagree on the details, and sometimes the fundamentals.

But do we take other people’s disagreements with us seriously enough?

What are the real issues behind athletes’ kneeling during the national anthem? They are not all fools; they are not all anarchists hell-bent on destroying a nation. You don’t have to agree with anyone to take him seriously. But you do have to have some intellectual and moral courage.

Can we teach students to take others seriously? I think we can. I don’t think we do it well enough. Classroom debates rarely get to the heart of the matter. Debates tend to quickly degenerate into glib contests of verbal cleverness.

Student writing (or oral, or visual representation, or film, or whatever) should always consider the strongest opposition to the point being argued. In fact, this should be one of the main points of assessment in the scoring rubric (that’s scoring guide, or rules, for non-teachers). Perhaps as much as 30-40% of the student’s grade should be contingent on whether she takes her opponents seriously and gives them fair and honest voice.

I’ll stop here. I’ve given this much thought over the years, but I’ve never tried to articulate it until now. Hopefully, you’ve noticed that I haven’t given fair voice to someone who disagrees with taking opposing views seriously. It’s a dilemma.


The Mighty “Do Now”

Today I have a simple idea for teachers: Have a routine where students begin working the second they get into your classroom.

Many (perhaps most) but not all teachers are aware of the “Do Now” but for reasons I cannot comprehend, not all make full use of it. I’m going to stick to the simplest form I know.

When students are coming to your class, write something for them to do on the board or project it on a screen.

Seriously. That’s it.

As a high school mathematics teacher, I put a couple of simple problems or exercises on the board. These will either directly review recent work, or will review material from previous grades that will be important in upcoming lessons. I expect students to spend 2-10 minutes on the problems (usually the lower end) and use the work as a starting point for today’s lesson.

When I taught English, I would give a short writing prompt. For example, I might ask students to write a paragraph describing something we recently read or observed in class. Or to respond to a statement. Or to do something silly and fun, such as to describe a kitchen appliance, or write a short monologue from the point of view of a dog. The big deal is that students will be thinking in terms of writing and expression before the lesson begins.

What’s the point?

There are two main ideas here. First, it provides a mental warm-up for the students. As soon as they begin the Do Now, they are engaged with their subject matter, their cognitive tools, and their communication. Second, it gives me (and the students) feedback about what they know and can do relative to the topic I have just given them.

The second idea is a surprising and wonderful side-effect of the Do Now. Students come to class earlier, and they settle into their work more comfortably than otherwise. In most middle and secondary schools, class break is a nuisance. Students slowly wander into their classrooms as late as they can get away with, socially catch up with their friends, and only open their books when instructed by the teacher to do so. If the teacher has to rush out of the room during the break (aging bladders and all) then the situation is even worse. With an established routine of Do Now, students actually come to class quicker. Settle into their seats more quickly. And they take care of their social engagements while opening their books and beginning the task. Seriously. They really do.

Popular myths to the contrary, students generally want to be successful. They want to feel clever. They want to achieve something in their school day. An established routine with established expectations helps this. Students know exactly what to do, when and how to do it, and how to get help. So they do it.

I never have to hector students to get into my class on time. I put the Do Now up (without fail) and I stand outside my door welcoming students. No shouting. No nagging. No cajoling. I’m glad to see you. And in a few minutes we discuss the Do Now and move on.

Can’t believe I didn’t do this for the first half of my teaching career.

Critical Thinking – Part 2

I closed the previous entry on critical thinking with the following picture.

critical thinkingNotice that critical thinking is underwritten by two sets of skills—cognitive and dispositional.

Cognitive Skills

This is the bit that probably comes to mind immediately when considering critical thinking. Critical thinking involves the mobilization of a number of concrete thinking skills. Let’s pick an example: you read in the paper that industrial emissions are implicated in the overall warming of the earth. Deciding whether you believe what you read is surely an exercise in critical thinking, isn’t it?

I don’t want to belabour anything, but let’s expand the skills list.

  • Interpretation
    • Categorization
    • Decoding significance
    • Clarifying meaning
  • Analysis
    • Examining Ideas
    • Identifying Arguments
    • Analyzing Arguments
  • Evaluation
    • Assessing Claims
    • Assessing Arguments
  • Inference
    • Querying Evidence
    • Conjecturing Alternatives
    • Drawing Conclusions
  • Explanation
    • Stating Results
    • Justifying Procedures
    • Presenting Arguments
  • Self-regulation
    • Self-examination
    • Self-correction

You can see that a number of these skills/sub-skills are generic. Assessing arguments, for example requires similar skills and understanding regardless of the content of the argument. Similarly Explanation and Self-regulation are (almost) independent of subject matter.

But content knowledge remains crucial to these activities. One cannot query evidence without understanding something of the standards of evidence. One cannot conjecture alternatives without some knowledge of which alternatives are possible.

In short, the skill dimension of critical thinking is both general and subject-specific.


The character virtues, or dispositions to think critically are often ignored. They seem to be the embarrassing ex who nobody wants to talk about.

You see, just having skills isn’t enough; you have to use them. And to use them, you need some motivation. And motivation comes from character. And this terrifies curriculum writers, disgusts assessment theorists, and worries teachers. But there it is, anyways.

  • Approaches to life and living in general:
    • Inquisitiveness with regard to a wide range of issues
    • Concern to become and remain generally well-informed
    • Alertness to opportunities to use critical thinking
    • Trust in the processes of reasoned inquiry
    • Self-confidence in one’s own ability to reason
    • Open-mindedness regarding divergent world views
    • Flexibility in considering alternatives and opinions
    • Understanding of the opinions of other people
    • Fair-mindedness in appraising reasoning
    • Honesty in facing one’s own biases, prejudices, stereotypes, egocentric or sociocentric tendencies
    • Prudence in suspending, making or altering judgments
    • Willingness to reconsider and revise views where honest reflection suggests that change is warranted
  • Approaches to specific issues, questions or problems:
    • Clarity in stating the question or concern
    • Orderliness in working with complexity
    • Diligence in seeking relevant information
    • Reasonableness in selecting and applying criteria
    • Care in focusing attention on the concern at hand
    • Persistence though difficulties are encountered
    • Precision to the degree permitted by the subject and the circumstance

Here’s the problem. Current educational thinking stipulates that all mandated student outcomes must be observable and measurable (either directly or indirectly). We can measure whether a student can deploy the Pythagorean Theorem to solve 2-dimensional problems. We can observe and measure a student’s ability to express the theme of a novel in prose (maybe not precisely, but we can do it). But how can we stipulate “approaches to life” and then measure them?

I’ll deal with this in a bit more detail in a future post. But let’s answer back with the question: why not?

Sure it’s uncomfortable to report that a student is not open minded about divergent world views. But if it’s a crucial component of critical thinking, then it’s fair game. Inquisitiveness? Alertness to opportunities? Honesty? Prudence in judgment? This is starting to look like student reports of a century ago!

And why not? Our grandparents and great-grandparents were not fools. They recognized that intellectual achievement is predicated on intellectual habits, though they wouldn’t have used those words. Yes, we’ve improved our educational thinking in many way, but we can’t change the way that responsible thinking works. And that’s a significant part of what makes thinking critical: it is responsible.

I’m going to leave off here. But if you’re with me this far, try to imagine how a teacher can assess and report on the items in the above lists without creating a disaster in class, school and home.

Returning to the classroom

After 20 years of teaching, I took a foray into bureaucracy. For five years, I conducted and coordinated research and evaluation in my school district, and for the past two years, I’ve been on secondment to the government, working in various areas of curriculum development.

And now it’s time to return.

This September I’ll be teaching high school mathematics (plus International Baccalaureate Theory of Knowledge). And I have to say that I’m stoked.

I remain convinced that high school mathematics is grounded in technique: arithmetic and symbolic manipulation are to high school mathematics as scales and arpeggios are to musical development. And I remain convinced that technique by itself is uninteresting, anti-motivational, and useless. Students need to deploy their technique to make their mathematical experiences come alive. This can be explored through practical application, through stereotyped problem solving and through exploration of abstract ideas that have nothing to do with the physical world.

The art of teaching is to find the right balance, to match students to the experiences that will matter the most to their developing technical skill, mathematical understanding and, yes, their aesthetic sense of learning mathematics.

Being out of the classroom has the wonderful benefit of giving you time to think, to reflect on what it is you do for a living. I have enjoyed this immensely. Being out of the classroom, I’ve had time to think about student assessment, about motivation, and about understanding.

But being out of the classroom has a tendency to lead to romanticized notions. When I’m thinking about all the wonderful things my imaginary students and I will do, well it’s pretty amazing. We’re brilliant together. But I will have to live with the uncomfortable friction that will arise when theory and reality collide in the classroom.

It’ll be fine. No. It’ll be great. I’m excited to be back.

Hannah’s Sweets—Anatomy of a Math Problem

The following problem has the internet all abuzz.

hannahs sweets

The problem comes from the Edexcel Maths GCSE paper—which, as I understand it is an examination for UK 16-year-olds with aspirations of studying at a university.

The buzz is fascinating for a number of reasons: some positive; some not so positive. But all are worth thinking about. First off, many students found the problem to be very difficult. I’m not sure why this is, but I’ll offer some speculations as we move through this post.

(I’ve had to convert my Word document into graphics because the equation objects wouldn’t paste into the WordPress format.)

Why I Like the Problem

There are a few very nice things about this problem. First, it asks students to demonstrate something, rather than to find a single solution. As a math teacher, I am always looking for demonstrations of understanding, rather than the consequence of a search that may or may not simply be a matter of “getting lucky”. On that note, I am pleased that this question requires a human marker. No multiple choice or numerical response item could replicate this. To demonstrate understanding, students must use natural language (i.e. English) and structured symbolic representation (i.e. numbers and letters with mathematical connectives) to convince another human mind that they understand. This is good.

Second, there are at least four straightforward methods of solution available to an average post-secondary-bound high school student. No one is bound to a single method. If I were to use this question in class, I would have choices for how to approach it, depending on the goals of the lesson.

Solution #1—Elementary Probability

This is probably the most straightforward way to approach the problem. We use two pieces of basic knowledge.


This is not hard, and it involves very easy manipulation. I’ll go over the next two solutions quickly, as they are very similar in form, but not in conception.

Solution #2—The Fundamental Counting Principle

This is a close cousin to #1. The idea is that if you have n ways to do one thing, and m ways to do another distinct thing, then you have n×m ways to do both. So if you have 3 pairs of pants and 4 shirts, you have 12 different outfits you can wear. We now see that Hannah has n(n-1) ways to pick two sweets out of n, (She ate the first sweet, so she only has n-1 left for her second choice.) By similar reasoning she has 6×5=30 ways to pick and eat two orange sweets. From here, the arithmetic and algebra are the same as #1.

Solution #3—Combinatorics


Solution #4—All Guts no Math


The Downside

One of the standard complaints about this problem (and others like it) is that it is so artificial. Sure it gives us a protagonist, a storyline and some talk of yummy sweets, but this is a million miles from real-world mathematics.

I hear this kind of objection all the time, and frankly I don’t get it. One of the other complaints about mathematics is its coldness and distance from human creativity. Well this is a silly story. It’s a flight of fancy; it’s a goofy narrative that paints a picture. Yes, very few people will do math when eating candy; we all know that. But no one ever complains that fiction is unlike real life, because it reminds us of real life and gives us something to think about. As do silly problems like this one.

Now it is often easy to create a problem that can be stated with or without the narrative. Indeed, it is often the case that the narrative obscures the mathematics. That isn’t the case here. A scenario free statement of this question would be quite difficult to write without absolutely obscuring the clear and simple question being asked.

So why did students find it hard? My guess is because the question is surprising. Students are often lulled into calculating numbers, solving equations, concretely answering problems. In this case, they were given a concrete problem and were asked to show that the problem leads to an unexpected equation. The work isn’t difficult, but the interpretation and strategy-selection were surprising. At least that’s my supposition at this time.

Whew. This is more concrete mathematics than I aim for in this blog. Hope it brought some thought and (dare I say it?) some fun to your day.

If any students who wrote the paper happen to stumble onto this blog, I’d love to hear your reflections on the problem.

Finnish Education—the latest gossip

The West loves Finnish education. And it’s easy to see why. Finland runs schools in a way that we value and gets results that are just as good as the Asian countries who do things, well, differently. In a way, Finland delivers on the promise that the West keeps trying to fulfil, providing hope that our reforms will one day be fully implemented and our ideas will be vindicated. The following graph shows the fascinating tale of the tape from the second most recent (2009) Programme for International Student Assessment (PISA) tests, administered by OECD to almost half a million 15-year-olds worldwide. (Note that the graph only indicates countries. For historical and political reasons, some smaller jurisdictions such as Shanghai and Hong Kong participate independently. These are high-scoring jurisdictions.) You can well imagine the shock when on March 20, 2015, the Independent ran the headline Finland schools: Subjects scrapped and replaced with ‘topics’ as country reforms its education system. The article went on to claim that Finland–heretofore the darling of Western education–was eliminating the study of subjects–the heart of modern education. But because Finland has been so successful, the tone was largely incredulity rather than critique.

Subject-specific lessons – an hour of history in the morning, an hour of geography in the afternoon – are already being phased out for 16-year-olds in the city’s upper schools. They are being replaced by what the Finns call “phenomenon” teaching – or teaching by topic. For instance, a teenager studying a vocational course might take “cafeteria services” lessons, which would include elements of maths, languages (to help serve foreign customers), writing skills and communication skills.

For many Western reformers, this was fantastic news. Dreamers of educational utopia have long advocated for project-based learning, where students no longer learn fragments of disciplines in isolation, but rather become deeply immersed in topics of their interest, and learn the relevant mathematics, science, politics, etc. in the context of these projects. The Independent article, it looked as though the project-based learning advocates were vindicated. My initial reaction was simply disbelief. Either the Finnish authorities had lost their minds, or the reports were wrong. My objection is simple—a system predicated entirely on project-based learning would be gruesomely inefficient. Much of what we learn must be mastered, and mastery does not come in context. I would never advocate for a drill-only system, but neither would I ever claim that no drill, no repetition, no out-of-context exercises would be an appropriate way to organize K-12 education. I like projects, and I think that meaningful projects can and should be an important part of education; I simply don’t believe that meaningful projects could be accomplished without significant, structured subject-based learning in the background. Turns out I was right. The Finnish authorities are not crazy.

Pasi Sahlberg. PASI is an anagram for PISA. Coincidence?

Six days after the Independent story rattled around the world, the Washington Post published the contradictory No, Finland isn’t ditching traditional school subjects. Here’s what’s really happening. The Post notes that Pasi Sahlberg, the smooth-talking, good-looking ambassador for Finnish education, explained in a blog that Finnish students will continue to study subjects, but will have a mandatory project to complete in each year of schooling. He further notes that education in Finland is highly decentralized, with most of the important decisions about the details of each students’ school year being determined at the local level. The Finnish National Board of Education had corrected the record one day ahead of the Post.

Subject teaching in Finnish schools is not being abolished

The news that Finland is abolishing teaching separate subjects has recently hit the headlines world-wide. Subject teaching is not being abolished although the new core curriculum for basic education will bring about some changes in 2016.

The subjects common to all students in basic education are stipulated in the Basic Education Act, and the allocation of lesson hours among school subjects is prescribed in the Decree given by the Government. However, education providers have had a high degree of freedom in implementing nationally set objectives for more than twenty years. They may develop their own innovative methods, which can differ from those in other municipalities.

The new core curriculum for basic education that will be implemented in school in August 2016 contain some changes which might have given rise to the misunderstanding. In order to meet the challenges of the future, the focus is on transversal (generic) competences and work across school subjects. Collaborative classroom practices, where pupils may work with several teachers simultaneously during periods of phenomenon-based project studies are emphasised.

The pupils should participate each year in at least one such multidisciplinary learning module. These modules are designed and implemented locally. The core curriculum also states that the pupils should be involved in the planning.

So what’s the lesson from Finland? Local autonomy seems to be pretty darned important, but centralized curriculum and a broad outline of the conditions under which teacher and learning are to occur are indispensable as well. And, yes, disciplinary learning remains on the agenda.

High Stakes Testing—The Science

A while back I was challenged by reader Jack Shalom regarding student testing. In response to Timeliness and Grading, Jack wrote

IMO, there are exactly two reasons to give a test:

To sort students.

To help students learn more.

I believe reason #1 is the main reason tests (in particular, standardized tests) are given. We know this because for most standardized tests, teachers and students get no feedback at all about what items have been missed and why. Certainly by the time results of any kind are received, the student has moved on to a new teacher.

If the purpose of a test is to learn more, then it needs to be designed as such, and teachers need to treat them as such. Why, then, would there then need to be a score? When was the last time your tennis coach gave you a precise grade on your backhand? Would that have helped you play tennis better?

The same can be said for end of term grades.

At the time I promised Jack that I would respond to his challenge.

The opportunity presented itself afresh in an OP ED in the Edmonton Journal by my colleague Dr. Jacqueline Leighton.

Jacqueline wrote

The science behind high-stakes testing is based on giving all students the opportunity to show what they know under the same conditions by writing the same test — a test that will count significantly toward their final grade and has been developed to be reliable and valid in providing information about student learning and mastery. This is important because students working with different teachers, and completing different assignments and assessments during the year can end up with the same teacher-awarded grade at the end of the year — say, 85 per cent — but actually possess very different levels of preparedness, learning and mastery. Committees of content, technical and assessment specialists, composed of highly experienced educators and scientists, create high-stakes tests. These committees, using the latest educational, scientific and technical methods for test design and development, make sure that (a) the content material taught in classrooms is adequately covered in the high-stakes test, (b) new test items are reviewed and field-tested before they appear on the final operational test to make sure the wording is understandable, does not bias or offend students, and conforms to the technical standards of previous items, (c) test items are double and triple-checked using a variety of technical analyses to ensure that the results are consistent within a pattern or trend — for example, students who respond correctly to one item are also responding correctly to items measuring the same material; this is done to ensure that items are not underestimating or overestimating what students know, and (d) test results are constantly monitored so that the test continues to measure the appropriate content and skills in students who have learned the material well and achieved mastery. Advancements in the science of testing are continuously integrated into the design and development of high-stakes tests.

If we read this uncharitably, perhaps Jack is right. Leighton says that the test does, indeed, sort students, but, she adds, it does it well. I’ll grant this as trivially true. On the other hand, well-constructed exams provide a second level of assurance that the student has met some standard or other, that the student is competent. Further, it sets the standard of competence in a publicly understood ways. Yes, we are separating those who meet the standard from those who do not, but what alternative do we have?

I think that there is another important function of these examinations: they provide evidence of system-wide performance. That is, it is crucial that a publicly sponsored and funded system of education provide curriculum that is attainable by the students, that the resources appropriately support the curriculum, that teachers teach the curriculum to students and that students learn the curriculum. Any individual student can have an especially fortunate or unfortunate day. The test does not guarantee that the student has learned (or not learned) the material; but it shouldn’t be too gruesomely off. Because randomness and luck travel in all directions, a complete class of students should have a test average that is not too far off an accurate measurement of all their learning. This is important information for teachers, schools, and jurisdictions.

All the above is predicated on the sort of high-quality, curriculum-referenced test that Jacqueline Leighton is talking about. Test construction is a highly technical business, and it should not be taken lightly. No matter how high you make the stakes, a badly constructed test provides little to no valuable information about individuals or groups. Similarly, generic ability tests cannot measure the curricular achievement of students.

So if you’re going to have a high stakes examination, make sure it’s professionally constructed, validated and relevant to the curriculum taught and learned.

Leighton and Gierl, eds.

Oh, and if you want some fun reading in your spare time, here’s a book Jacquie co-edited and in which I co-authored Chapter 3.