This is the homepage of Greg Ashman, a teacher, blogger and PhD candidate living and working in Australia. Everything that I write reflects my own personal opinion and does not necessarily represent the views of my employer or any other organisation.

I have written two books:

The Truth about Teaching: An evidence informed guide for new teachers

Ouroboros – an ebook

Watch my researchED talks here and here

I have written for The Australian about inquiry learning (paywalled):

Inquiry-learning fashion has us running in wheel

This is my take on the “Gonski 2.0” review of Australian education for Quillette:

The Tragedy of Australian Education

Here is a piece I wrote for The Age, a Melbourne newspaper:

Fads aside, the traditional VCE subjects remain the most valuable

Read a couple of articles I have written for The Spectator here:

A teacher tweets

School makes you smarter

Read my articles for the Conversation here:

Ignore the fads

Why students make silly mistakes

My most popular blog post is about Cognitive Load Theory:

Four ways cognitive load theory has changed my teaching

To commission an article, click here


Should cognitive load theory be taught to trainee teachers?

Perhaps prompted by the release of a new practice guide from the Centre for Education and Statistics in New South Wales, a number of teachers have appeared in my social media timeline commenting on how they first encountered cognitive load theory.

Most state they were never taught about cognitive load theory as a trainee. However, a few teachers have suggested that they were introduced to the theory at university and a number of enlightened tutors seem to be teaching it on their courses. This raises the question as to whether this is a good thing.

It is easy to understand why cognitive load theory has largely been neglected. As Kirschner, Sweller and Clark pointed out in their seminal 2006 paper, the implications of cognitive load theory for fashionable teaching methods are dire. Not only is it inefficient to learn via problem solving or by imitating the behaviours of expert practitioners, it is quite possible to learn nothing at all. Behavioural activity is not a proxy for learning. In other words, we cannot assume that learning will always accompany doing. Instead, we need to engage cognitive activity and at just the right level of complexity.

This is hard to take if you are a part of the long and broad educational progressivist tradition with its focus on ‘learning by doing’. This is also hard to take if you have been grounded in Freire’s critical pedagogy or one of the branches of postmodern ‘grievance studies‘ because cognitive load theory implies an important role for the instructional designer in making choices and structuring learning for students – a role that could be perceived as oppressive according to these outlooks.

It is therefore surprising that cognitive load theory features at all in teacher training and the fact that it does is a testament to the independent-mindedness of those involved.

You might expect me to be in favour of teaching trainees about cognitive load theory, and I am. However, I think we need to caution that it is not a finished theory. Some of its tenets, such as that we possess a severely limited working memory, are well established and accepted across the field of psychology and some of cognitive load theory’s effects, such as the worked example effect, have been replicated many times in different contexts.

However, cognitive load theory is not on the same footing as, say, the theory of evolution. We do not yet have a reliable way of directly measuring cognitive load – a situation similar to evolution prior to the discovery of DNA. There is also the issue around germane cognitive load that is wrapped up with a current controversy about whether ‘element interactivity’ is a thing. Sweller and others, including me in my research, use the construct of element interactivity to try to differentiate between occasions when cognitive load should be reduced, such as when first learning a mathematical procedure, and when it should be increased, such as when learning a list of names. However, the concept has been the subject of harsh criticism by educational psychology luminaries such as Jeffrey Karpicke.

We might also add that some of the explanatory ideas about the evolutionary underpinning of cognitive load theory are not as well established as practical findings such as the worked example effect. For instance, although I think it provides a helpful explanation of why academic learning proceeds differently to more natural forms of learning such as learning to speak your mother tongue, there are those who argue that Geary’s theory of biologically primary versus secondary knowledge is unfalsifiable. If true, that is a problem for a scientific theory, albeit the kind of problem that scientific theories tend to run into during the course of their development.

Fortunately, the most settled parts of cognitive load theory are the ones of the greatest practical significance to teachers.

It would be a shame if cognitive load theory were taught as if it were established fact. It is not. It is a provisional model with all the kinds of problems a developing scientific theory encounters. If we teach trainee teachers about cognitive load theory then we need to make this clear. We don’t want to repeat the mistakes of the past. Instead, we want to induct new teachers into a practical, sceptical professionalism that sits at the confluence of science and art.

Free cognitive load theory resources

There are a lots of free articles and other cognitive load theory resources available to teachers who do not have access to research papers, not least the video above in which John Sweller talks about the theory at researchED Melbourne.

I thought it would be a good idea to try and make a list of these resources and I intend to update this over time. I haven’t included blog posts because there are too many.

What is it?

If you want a general overview of cognitive load theory there are a number of places to go. I wrote a piece for The Conversation that gives a very basic summary. John Sweller has written an article that discusses the development of the theory and it is a good place to go to see a brief summary of all of the cognitive load theory effect. The Centre for Education Statistics and Evaluation, part of the government of New South Wales, has taken up the cause of cognitive load theory with determination. They have produced an overview which they have turned into an audio recording and they have recently released a report on putting cognitive load theory into action. All three resources are available here.

The Times Educational Supplement asked John Sweller to write an article last year. It is paywalled but you can login using your Facebook account.

Human cognitive architecture

If you want to delve a little deeper then cognitive load theory has some interesting things to say about human cognitive architecture and how it evolved. When I first read about cognitive load theory, I saw the limitation of working memory as a design flaw. I now understand it as a design feature.

This understanding comes from a deeper dive into how the mind functions – human cognitive architecture. Through the thesis writing process for my PhD, I recently discovered that the Sweller and Sweller paper on the evolutionary basis of the theory is not paywalled. This provides a good summary of biologically primary and secondary knowledge as well as other explanatory features of the model.

Implications for teaching

I first encountered cognitive load theory due to the paper Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. This paper, by Kirschner, Sweller and Clark, argues that many fashionable approaches to teaching are at odds with cognitive load theory. It also popularised Paul Kirschner’s definition of learning as a change in long-term memory.

This paper was not well-liked by supporters of the teaching methods that it criticised. Three rebuttal papers were written (here, here and here) and the authors then responded to these rebuttals. Eventually, the controversy led to a conference and then a book.

The same three authors also wrote an accessible piece about the idea that teaching general problem solving skills is no substitute for teaching maths. This has wider implications for other subjects and for other posited generic skills.

Is the discussion about school behaviour as polarised as it first seems?

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In March 2017, the British government released a report, authored by Tom Bennett, called Creating a Culture: How school leaders can optimise behaviour. In one section of this report, Bennett discussed the different reasons why it may sometimes be better for a student to be withdrawn from a standard classroom and he wrote about what he called ‘internal inclusion units’:

“Once available strategies have been exhausted, it can be necessary and in the student’s best interests to be somewhere their needs and behaviour can be better provided for.”

At the time, Dr Linda Graham, an Australian researcher, took a dim view of Bennett’s report and, in particular, the concept of inclusion units. And yet, in a recent piece for The Conversation, Graham writes the following:

“Sustained bullying (cyber or otherwise) is another example where suspension may be appropriate. But in-school suspension, where students are removed from their regular classes and required to complete their work in a supervised setting, is a better option than out-of-school suspension.”

This comes immediately after a paragraph outlining what Graham believes are justifiable reasons for exclusion from school:

“There are times when suspension is appropriate, such as when a student brings drugs or a weapon to school, or engages in physical violence resulting in injury. Hitting a teacher is never OK. But even here, it’s important to make sure a frightened five-year-old accidentally connecting with a teacher mid-meltdown is not construed as a deliberate act of violence.”

I can’t help wondering whether the opposing sides of this debate are as far apart as they first seem.

We can perhaps understand the situation better by looking at the main thrust of Graham’s article.

Graham argues that exclusions have increased in Queensland after a change of rules that allowed principals to exercise more discretion. She notes that these increases are not necessarily large, but the implication of her argument is that the numbers have moved from a more appropriate to a less appropriate level. I am not sure about this. It is possible that some incidents that according to Graham’s own criteria should have led to an exclusion in previous years, did not do so due to the tighter regulation. That would account for any increase. We would need to know what the natural level of exclusions should be and this would be extremely difficult to determine.

Graham then goes on to link exclusion with increases in antisocial behaviour and contact with the criminal justice system. There is clearly a link, but it is hard to determine that this is because exclusion causes these outcomes. Some of the evidence to support this case comes from matched studies, but consider this: If two children are matched in every way, but one of them does a terrible thing at school, which one do you think is more likely to go on to have contact with the criminal justice system? It is highly likely that there is an underlying factor that causes both the behaviour that leads to exclusion and contact with the criminal justice system.

Moreover, if we take a look at Graham’s list of acts that she believes should lead to exclusion, to which I would add sexual assault and verbal threats or racist abuse, perhaps this point is moot. Do we really want to keep an abuser in a classroom because the life chances of the abuser might be affected if he or she is excluded? What about the teacher and the other students and their rights to a safe working environment? At some point, we must all concede there is a trade-off.

If we can establish the points that we agree upon, perhaps there is room for a fruitful discussion around the details. If we agree, for instance, that a school has a role in teaching pro-social behaviours, we can discuss the best ways of doing this. At one point, Graham states, “For students who have language disorders or attention difficulties, teachers can adopt proactive strategies that benefit all students.” She then lists a series of strategies that seem entirely sensible and overlap a great deal with the strategies I promote in The Truth About Teaching in the chapter on classroom management.

Some of the social media debate becomes particularly emotive around so-called ‘no excuses’ schools (a term I dislike). However, one of the key features of these schools in the establishment of routines that act to preempt and prevent behaviour issues from arising and Graham recommends both ‘clear and consistent routines’ as well as ‘well-designed seating plans’. This is the kind of antecedent control supported by extensive research. There is common ground here if we can avoid the temptation to divide everyone up into bewhiskered, romantic rebels versus sadistic authoritarians.

It is worth seeking such common ground because the details are worth discussing. For instance, Graham makes the point above that, “it’s important to make sure a frightened five-year-old accidentally connecting with a teacher mid-meltdown is not construed as a deliberate act of violence.” I would agree. I would add that a student swearing in the course of talking to a teacher is not the same as a student swearing at the teacher. Let’s thrash out some of what we mean and how, as a community, we believe we should deal with a range of challenging behaviours. It is this practical level that most engages teachers and schools.

One issue we might even label as a kind of meta-antecedent is that of reading instruction. As you might expect, there is a clear correlation between those students who are excluded and those with poor reading ability. It seems understandable that a child who cannot read – something that schools require students to engage in every day – will start to dislike school and misbehave. Can we not at least all agree that we should use the most effective, scientifically based approaches to teaching reading and that we should intervene heavily with those students who struggle? That’s a cause for which we can all carry a banner.

And perhaps the remaining differences between positions are not as stark as Twitter hyperbole generated in school shaming incidents suggests. Perhaps it is the case that we all agree there are circumstances where exclusion is appropriate and we all agree there are circumstances where internal suspension is appropriate (although we dislike certain terminology). Perhaps the real difference is one of emphasis: a different perception of appropriate rates of exclusion and a different perception of the impact of exclusions on the excluded and the extent this should be weighed in our decisions.

A discussion around these points would likely be more productive and less heated than the current debate.

Yes, the vocabulary gap is unfair. What are you going to do about it?

Writing in The Age, Brendan James Murray, author and teacher, laments the vocabulary gap between advantaged and disadvantaged Year 12 students sitting this week’s Victorian Certificate of Education Literature exam. He notes that students are not allowed to take a dictionary into the exam, but mentions that they nonetheless possess an invisible dictionary:

“…studies have found that children from high socio-economic backgrounds hear many thousands more words than their less advantaged counterparts.”

Murray is concerned that this will further disadvantage students like the ones he teaches:

“Will the lack of a dictionary further disadvantage Victoria’s most vulnerable students? A selection of vocabulary from the unseen portion of last year’s examination gives a clue. Anybody who has taught disadvantaged children will know the challenges they face when confronted with words like ambiguity, archetypal, and transience.”

Although never made explicit, we can infer Murray thinks that at least part of the solution is to provide students with dictionaries.

Dictionaries are no substitute for vocabulary

We know from research on students learning words that Murray’s proposal may not work. It can be challenging to understand dictionary definitions that often require an understanding of related vocabulary. In addition, dictionary definitions cannot offer the rich range of contexts that learning a word by repeated exposure can offer.

E D Hirsch describes a study by George Miller and Patricia Gildea in which students were asked to use a dictionary to help them write sentences. Some of the examples Hirsch cites include:

“Mrs Morrow stimulated the soup.” [i.e. stirred it]

“Our family erodes a lot.” [i.e. they eat out]

“I was meticulous about falling off the cliff.”

A Year 12 Literature student may do a better job with a dictionary, perhaps, but it seems unlikely that handing over a dictionary in the exam is going to go anywhere near close to compensating for words missing from a student’s vocabulary.

You cannot always simplify the language of the assessment

One potential approach to leveling the playing field is to take careful control of the language used in the assessment. This has happened, to some extent, in one of the subjects I teach: mathematical methods. Until 2014, there would be a question every year about the exploits of ‘Tasmania Jones’, an explorer modelled on Indiana Jones, and his attempts to retrieve an emerald for a South American tribe or to build a ‘thrilling train ride’ in the Alps.

When I attended a feedback meeting for teachers in 2015, I was told that Jones would no longer feature in future exams because he potentially disadvantaged students for whom English is not their first language. Not only did Jones’s questions require lengthy blocks of text to set them up, he is an archetype familiar to members of a particular culture (even though, to be fair, most students found him baffling). It was a smart move to show him the (trap)door.

And yet it is hard to see the justification for dialing down the verbiage in questions on a Literature exam. And you have to wonder whether the vocabulary disadvantage actually becomes apparent far earlier i.e. at the point students first pick up one of the books they are going to study for the course.

No easy fix

Unfortunately, there is no easy fix for a lack of vocabulary in Year 12, especially in Literature where vocabulary is intrinsic to the subject.

Instead, the only way that this gap can be addressed is by incrementally building vocabulary throughout the entire course of schooling. Some children will still enter school with an advantage over others, but we can do plenty to build the vocabulary of the less advantaged.

Some of this work will involve the intentional teaching of words. However, a lot will come from regularly meeting words in the context of knowledge-rich domains such as literature, history, science, geography and the arts.

The problem that is faced by anyone making such an argument is that a knowledge-rich curriculum consisting of standard subjects is not a fashionable proposition in Australia, particularly at primary school. Instead, as epitomised by the Gonski 2.0 review of the curriculum, advocates tend to request an expanded role for the supposedly ‘general’ capabilities of critical thinking, problem solving and so on. These capabilities are not really general and are, in fact, highly subject specific.

In turn, literacy instruction shies away from knowledge-rich domains. There is a widespread belief that you can teach students how to comprehend a text by teaching them reading comprehension strategies. Yes, such strategies do have a limited positive effect, but repeated rehearsal does not magnify this effect because students still hit a limit based upon their vocabulary and world knowledge. Time would better be spent building this vocabulary and knowledge.

Vocabulary growth is a matter of social justice that should interest anyone who claims to be concerned about social justice. It is a matter of equality of opportunity. I do not claim that schools can completely level the playing field. They cannot. But schools can do more than they are at present and a purposeful, sequenced and knowledge-rich curriculum would be a good start.

Demand one.

More evidence that ‘metacognition’ is progressivism’s newest excuse.

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I have written about metacognition on a number of occasions. Specifically, I have questioned the Education Endowment Foundation’s toolkit strand of ‘metacognition and self-regulation’, suggesting it represents a bag of very different things, the evidence for which is mixed, at best.

I have also noted the way that a new fuzzy maths initiative in South Australia trumpeted its use of ‘metacognition’. Fuzzy maths is essentially educational progressivism applied to the teaching of maths and the evidence suggests that this is a harmful thing to do.

Progressivism tends to focus more on the process of learning rather than the content to be learnt. In fact, it often conflates process with learning. When applied to the maths classroom, progressivism prioritises ‘rich’ tasks and activities, usually without making exactly clear what the objectives are.

Clear objectives are a problem because progressivism sees students as unique individuals who respond to tasks in their own way and whose choices, to a greater or lesser extent, are meant to drive the learning process. Advocates can avoid this difficulty if they frame their objectives much more vaguely, and so we have claims that certain tasks will develop ‘critical thinking’, ‘creativity’ or, in maths and science, ‘deep understanding’. Who is to say that these tasks don’t develop these abstract and poorly defined qualities? Such is the value of an unfalsifiable position.

Ten years ago, the same activities may also have been held up as promoting the skill of ‘learning to learn’. Today, a similar end seems to be served by appeals to ‘metacognition’.

My perception is only enhanced by an article I have just discovered from earlier this year in the Times Educational Supplement. We read about a maths lesson focused on improving metacognition:

“The pupils are working in groups as part of a summary of their unit of work in mathematics. The groups, of four to five pupils, are working on some challenging problems which have been set by group facilitators… aged about 10.

There are two strands to this learning: the correct completion of the mathematical problems using the strategies and language previously taught; and the development of metacognitive thinking.

Towards the close of the lesson, the pupils begin to evaluate their learning… At the same time, the facilitators meet to discuss the performance of their respective groups…

You might be wondering what the teacher was doing in this example. She was moving around the groups and modelling, through dialogue, how to ask questions which promote thinking and reflection. She is hugely skilled, and two stand-out features are apparent: she has an excellent relationship with her students so can push and probe without them feeling uncomfortable, and she doesn’t appear to need to be the “expert”, or finish every conversation herself. She happily empowers her students in this area.”

Given that this task comes at the end of a unit, it may be the case that students had mastered the required content and therefore gained something from this process. It is hard to tell. However, the way this is written is strongly progressivist. We have a teacher as guide on the side, offering no explanations, only probing questions. We have students working collaboratively and setting each other problems. We have students being ’empowered’.

Despite seemingly being the main point of the article, exactly how the students are developing their ‘metacognitive thinking’ is something that is left to us to imagine. It seems unlikely that they would be debating the relative merits of blocked versus distributed practice, even if knowledge of the advantages of spaced practice is one of the disaparate elements grouped together as ‘metacognition’ that has some evidence to support it.

I think I can see where this is all going. Watch out for old progressivist meatloaf, reheated and served up as metacognition, coming soon to a school near you.

Possibly the worst argument I have ever read

My Twitter feed lit up at the start of last week. There is a blogger in Geneva or somewhere who is big into identity politics, writes blogs that he calls ‘essays’ and complains about ’emotional labour’ if challenged. I blocked him a while back because he decided he was going to ‘hound’ researchED Canada. ResearchED is a fine organisation that I am proud to be associated with.

Anyway, it seems that some educational progressivists have decided that he is their new brain. One of these has got a gig with the UK trade magazine, Schools Week, writing blog reviews, and she decided to share one of these essays.

People on Twitter alerted me to the fact that the contents of this essay are pretty offensive. It accuses me of being a ‘neoconservative reactionary’ and claims that other bloggers have directly appropriated the language of the alt-right. In other words, it deploys a fancy-sounding, politicised form of ad hominem. I took this up with Schools Week but they essentially couldn’t care less and took no responsibility for the contents of the blog review. I don’t know what I expected, to be honest.

However, I now found myself reading the whole of the essay and it contains possibly the worst argument I have ever read.

The subject is school shaming. This is the process where a tabloid newspaper, or latterly a education consultant on Twitter, picks on an aspect of a named school’s approach and whips up a social media mob. Maybe the school has introduced silent lesson transitions or has insisted on students adhering to its uniform policy.

I am against school shaming because it invariably descends into claims that members of the school are Nazis and various other forms of personal attack. Often, schools will be contacted for days and weeks by trolls. @oldandrewuk has documented examples of this. One aspect that makes these claims particularly unfair is that they often involve allegations made by parents or students that the school cannot respond to for reasons of confidentiality. The school’s version of events is left unheard.

There is a valid argument in favour of school shaming that points to schools being in receipt of public money and suggests they should therefore be accountable in this way. I don’t think the negatives are worth this kind of capricious, mob-driven accountability, but if I were on the other side of the debate then that is the argument I would be advancing.

However, this is not the argument in the essay. Instead, we are informed that schools cannot be shamed because a school is a building and buildings can’t feel shame.

Yes, that’s right. That’s the argument.

If this were true, then a child who says, “I love my school” or “I hate my school” would be passing comment on the state of the corridors and playground facilities and nothing else. Clearly, a school is a community made-up of people and their myriad interactions. Headteachers and senior leaders endure vicious personal attacks in school shaming incidents, and it is likely that teachers and students feel an associated stigma. An attack on a school is an attack on a community set-up for the advancement of young people. The bricks and cement have little to do with it.

The progressivist school shamers are going to need a new brain.

The four issues that define this blog

I am currently in a reflective mood and I have been thinking about is what it is that I stand for. I have argued before that you should be suspicious of those who apparently stand for nothing and only ever question. They have an agenda too, but for some reason they are hiding it from you. In contrast, I think my agenda is pretty clear. Nonetheless, from time to time, it is worth rethinking and restating our positions.

To this end, there are four issues that define my writing and the purpose of this blog.

1. Teacher professionalism

Teaching is the Cinderella profession. We are not a true profession because we do not regulate ourselves and we are not the ones who are in charge of creating and defining teachers’ professional knowledge. Everyone has been to school and everyone has an opinion, a phenomenon that often characterises education stories in the mainstream press, even if there are some great education journalists out there. If anything, we are even more badly served by our trade magazines which often seem driven by the desire to sell edtech or to platform consultants.

However, things are changing. There is a grassroots movement on social media, through Twitter and blogs, that involves teachers speaking to other teachers directly, avoiding the traditional gatekeepers. With the advent of researchED, we have seen a global awakening focused on connecting teachers to the best education research. I know that many teachers have felt empowered by a better understanding of research because they often write to me to tell me this. Which leads to the second issue…

2. Quality research

A great deal of education research is not worth the time of the person who conducted it, particularly much of the research that follows current trends in sociology. It is of no use at all to teachers who are seeking to improve their practice and it has no wish to be.

Beyond the genuinely pointless stuff, there is a vast swathe of literature that attempts to answer important questions that matter to teachers, but that uses flawed approaches to doing so. Look for the confounds. Look for the research that could only ever deliver a result in one direction. Look for research that extends the definitions of concepts into new territory in order to deliver forgone conclusions. These problems don’t disappear if you mush all this bad research together in a dodgy meta-analysis.

However, there is again reason for optimism. Although I have been critical of the UK’s Education Endowment Foundation and, by extension, its Australian offshoot, Evidence for Learning, at least we now have well-funded organisations committed to running randomised controlled trials in education. I am also hopeful that as the process of teachers professionalising themselves gathers pace, more will be persuaded to conduct their own serious-minded research.

3. Explicit teaching

My own research journey began when I read Hattie’s Visible Learning through which I chased down a reference to Kirschner, Sweller and Clark’s 2006 paperWhy Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. Up until this point, I knew the evidence suggested that inquiry-style forms of teaching were more effective, but I had never been able to make them work. Once I read this paper, I realised that this was not what the evidence suggested and I wondered how I had been led to think that is was. Now, instead of it being a guilty secret, I could actively seek ways of making my explicit teaching style more effective.

Significantly, explicit teaching is a whole system, not a part of a lesson. It is about the gradual release of control from the teacher to the students. It is the process of I do, we do, you do and all the complicated details involved in working out the most effective way to manage that process for different subject matter and different students. Rosenshine sums it up well.

4. Knowledge

The first post on my now defunct, Webs of Substance, blog was titled ’21st century knowledge’ and was a call to refocus the curriculum on knowledge. Knowledge has therefore been a driving force for me from day one. Academic skills such as reading and calculation are important, but they originate in knowledge and essentially represent the fluent application of knowledge held in long-term memory. Many other constructs that commentators proffer as generic skills are either not skills at all, or the generic component is trivial compared with the subject knowledge needed to deploy the skill.

There is a large potential opportunity cost in attempts to place an ever greater focus on (non)skills such as critical thinking, creativity, entrepreneurship and so on, and yet this seems to be the direction of travel for Australia’s curriculum reforms.

Plenty of work to be done

I think we are making progress on teacher professionalism and the quality of education research, even if we are not even past the end of the beginning when it comes to addressing these issues. I also get a sense that explicit teaching has become less of a bogeyman in recent years. The term is even quite popular in Australia, but I wonder whether this is because people think it is some discrete thing they can dip in and out of on occasion, or something that suits a narrow range of objectives. The education establishment is quite capable of absorbing words and making them mean the opposite of what they actually mean (e.g. ‘fluency’ and mathematics). The knowledge agenda is an interesting one. It is not something that even registers in Australia, where critical thinking is all the rage, but the English school system seems to be charging off in this direction at a pace. Perhaps that will allow us to draw useful contrasts in time.