A Teacher’s Perspective on Metaphors for Memory

I have found that improving my understanding of human cognition — how thinking happens, what learning looks like in the human brain — helps me to better understand the learning that is happening or not happening in my classroom. The brain is complicated, and cognitive science has far more questions than answers. At the same time, there are things that cognitive science does understand. In communicating those ideas, metaphors are useful in illuminating how a theory plays out in practice.

I’m going to explore four metaphors for memory. Each is imperfect: each illustrates some principles and comes up short with others. I think that, together, they get at some core elements of thinking were not intuitive for me and can help to paint a rich picture of what is happening in students’ minds during classroom instruction. The first and third of these metaphors come from Dan Willingham in this blog post, which is well worth a read.

This model sets up the distinction between working memory and long-term memory. Working memory is where thinking happens, and long term memory is where knowledge and skills are accumulated. For information to enter long-term memory, it needs to go through working memory.

There are several shortcomings to this approach. One essential feature of cognition is that working memory is finite — we can only think about a few things at a time — while long-term memory is, as far as psychological science knows, infinite. At the same time, long-term memory is not just a database of information. The organization of information in long-term memory is essential, and knowledge that is chunked together effectively increases the capacity of long-term memory. Finally, working memory influences long-term memory. Every time we think about something, we influence that knowledge, which is not captured in the flowchart.

Another way to think of memory is to imagine the brain as a giant library, and thinking as a few flashlights shining on a certain areas of the library.

All knowledge is not created equal in this model, as the better organized the library is, the more information can be captured by the flashlights’ beams. It’s possible to know something and not be able to think about it if you’re unsure of where to shine the flashlight. It can also illustrate the limitations of what our minds are capable of. There are only a few flashlights, and if too many of those flashlights are searching for information or busy shining on something at one time, the mind gets overloaded. A central metaphor here is that thinking and memory are inextricably linked; they don’t happen in different places through different processes but are systems that are interconnected and working in parallel.

But this metaphor also falls short in important ways. It still does not get at the idea that thinking changes memory; thinking is not a passive flashlight shining but actually creates new knowledge and consolidates old knowledge. The library metaphor also further perpetuates the idea of memory as a static, inert database.

Hill of Sand 

Think of a hill of sand—that’s your mind. You pour water on it—water is thought. The water coursing over the sand creates gullies and rivulets. That’s memory. It’s a representation of where the water (thought) has been in the past and if the water moves through those same channels they will become a little deeper. The next time you think (pour water) it will likely happen in the channels it’s followed before….but not necessarily.  The new water also has the potential to change the gullies on the hill.

Long-term memory is what has been left behind by working memory; memory is the residue of thought. Every time you think about something, you both consolidate that knowledge further and change the nature of that knowledge. This metaphor emphasizes the connections between thinking and memory, and I think it says something powerful about the influence long-term memory has on thought. Those gullies that have been created by long-term memory guide all of future thinking, and can be thought of as our collection of beliefs, habits, and biases.

But this metaphor comes up short as well. Working memory capacity is finite; thinking of it as a stream of water can miss that point. A metaphor of cognition as a stream can miss the nuance of the connected networks of knowledge that make up useful and transferable skills. This metaphor also does not emphasize the difference between storage and retrieval; it’s possible to know something but not remember it at the time.

Stones in a Forest 
A final metaphor I’d like to entertain is of memory as stones in a forest. As I walk around arranging and building memories with stones, I also wear paths from one place to another creating links between different memories or connecting new memories to old ones.

Memory as a forest captures a distinction between retrieval and storage that I think is important. Once I’ve created a memory — arranged some stones in the forest — that memory will be very slow to degrade. The path to and from the memory will become overgrown quickly if it is not used (that’s forgetting), but an old path will reappear quickly when it starts to be walked again. These paths also do something to emphasize how, if knowledge is not richly connected with broader ideas, it is unlikely to be useful, and it will be challenging to synthesize that knowledge with other, new ideas.

Of course this metaphor has weaknesses as well. Thinking influences memory. While this model captures the importance of revisiting memories over time, it does not get at the idea that memories themselves are changed through continued use. It also focuses more on the nature of long-term memory than the process of working memory and the interaction between them beyond wearing paths between memories.

Metaphors are useful, but they are also limited. I’m skeptical there exists an ideal metaphor for memory,, and each of these metaphors serves different functions and emphasizes different ideas while missing others. Together, they can create a useful understanding of how cognition occurs, but there’s plenty more left out. I’m sure some psychologists would say I’m missing the distinction between decision-making, auditory memory, and visual memory in my model of working memory. Others would argue that I’m oversimplifying chunking, the organization of knowledge, and the complexity of long-term memory. Many teachers would say that I am getting lost in theory without connecting it classroom actions. Others would say that I am focusing on memory at the expense of numerous other influences on students’ everyday learning. There are plenty more potential avenues for criticism.

But these metaphors serve a very specific purpose for me. As I think of memory through metaphors, I consolidate my understanding of my students’ thinking in new ways and organize it so the implications of that understanding are readily available as I teach. I think that these metaphors do contain some scientific accuracy, but my primary goal is for my knowledge to be useful for me in my teaching. I am constantly interpreting what happens in my classroom — the relationship between the learning experiences I design for students and the students’ learning, or lack thereof. If these metaphors can serve as a lens to understand why one strategy works or another strategy doesn’t, they have helped me to better understand my teaching and my students’ learning.


Dan Willingham’s blog post was my primary influence on this post. I also found these sources useful:

Robert Bjork on storage and retrieval

Anna Sfard on metaphors

John Sweller on Cognitive Load Theory

Make It Stick

Why Don’t Students Like School

An Equity Perspective on Learning Styles

The claim that students will perform better when the teaching is matched to their preferred learning style is simply not supported by science.

Letter in The Guardian

The standard argument against learning styles is simple: experimental research seems to have definitively shown that matching instruction to students’ learning styles does not improve learning.

In my experience, this argument is also unconvincing for many teachers. I’d like to try on a different perspective.

Anecdotally, the most common learning style that people identify is being a visual learner. I’ve also met many people who identify “learning by doing” as their preferred style. Certainly other styles exist, but these are the two I have seen the most, drawn from the VAK (visual/auditory/kinesthetic) framework that is popular. These are people who are likely to speak up in learning situations by saying things like, “I’m a visual learner, I need to see a visual to understand this”, or “I really need a chance to do this to get it”.

One way to understand these requests is to acknowledge that effective visuals and active “learning by doing” strategies are underutilized, and all learners benefit from learning in a variety of ways rather than through a single modality. From this perspective, identifying with a certain learning style functions to influence instruction to improve learning. The learning style could be understood as a privilege — through prior experience, individuals have learned to advocate for better instruction through the lens of learning styles. Those who don’t identify with a certain learning style do not have these tools to advocate for themselves. Inevitably, this creates inequities between outspoken learners whose position allows them to advocate for more varied instruction to meet all learners’ needs, and those who, for whatever reason, do not.

While learning styles can create inequities that advantage those who advocate for themselves, they can also cement existing disparities by creating an avenue for learners to say, “well that’s not how I learn”. Labeling a student, or allowing a student to label themselves, as a certain type of learner is not very different from allowing them to label themselves a “math person” or “not a math person”. Whether or not the underlying label is valid, it functions to limit what the learner believes themselves capable of and steers them toward learning situations where they don’t need to push outside of their comfort zone or try new things.

Do These Land? 
The research evidence convinces me, but it doesn’t land for many other teachers. Is this perspective a useful one to change minds on the value of learning styles?

Realism and Idealism

Math Curmudgeon juxtaposes two arguments I’ve seen on Twitter several times in the last few weeks:

“If kids in your class are more engaged by a fidget spinner than they are by your lesson, the spinner isn’t the problem. Your lesson is.”


“Learning is hard. Kids fidget. Fads come, then go. Your lesson doesn’t suck simply because two kids out of 25 are fiddling with this thing.”

I find the tension between these two perspectives one of the most important interior monologues I have as a teacher. It’s a tension between the idealism of teaching engaging, meaningful, purposefully-structured lesson every day and the realism of the challenges and imperfections of classroom teaching and doing the best with what I have to work with. I need to be able to see both perspectives. I need to ask myself every day if I can be more engaging, ask better questions, be more responsive to student needs, build stronger relationships, and structure more meaningful curriculum. I also need to acknowledge that much of what happens in my classroom is at least partially outside of my control, and a lesson that doesn’t go well or a disengaged student doesn’t mean I’m a terrible teacher.

Honoring the cognitive dissonance of realism and idealism applies beyond the recent craze of fidget spinners.

  • Teaching the last class of the day or the last day before vacation, I need to bring my best and believe that students can learn every minute while acknowledging that those classes are going to look different than mid-morning classes on a typical Tuesday.
  • Working with students with math anxiety, I need to believe that every student can learn and achieve at high levels while acknowledging that the most practical way forward may be to adjust my expectations in the short term.
  • When I look at a set of exit slips and see that my students all have the misconception I tried to address in class that day, I need to ask myself how to improve that lesson next time while acknowledging that human brains are mysterious, learning is hard, and even the best lesson plans often come up short.

Every moment of teaching is useful feedback on what works and what can be better next time; every moment of teaching is an imperfect teacher with imperfect students in an imperfect institution trying to do what’s best in that moment.

Transfer: The Low Road and the High Road

I often justify my existence as a math teacher by arguing that math is worth learning because it teaches humans to think clearly and reason abstractly. Or, in the words of Underwood Dudley:

What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better.

Is this true? Does a mathematical education teach transferable skills that can be applied beyond the classroom?

Thanks to Michael Pershan for sharing an informative research paper on this topic that gave me a new perspective on teaching for transfer: Are Cognitive Skills Context-Bound?, by D. N. Perkins and Gavriel Salomon.

Half a century ago, many psychologists would argue that effective thinking is a function of intelligence and general strategies for problem solving and critical thinking. Polya’s work in problem solving was particularly influential, as he identified a number of heuristics such as breaking a problem into subproblems or examining extreme cases that could be applied to a wide variety of problems in different domains. Many thought that expertise consisted largely of general strategies like these that helped humans to reason across a range of contexts.

In the following decades, a growing body of research provided evidence that seemed to contradict this hypothesis. Studies of experts in various fields showed that their knowledge did not transfer readily outside of the domain in which it had been learnt, suggesting they had developed a specialized skill rather than a broad array of general reasoning strategies. Chess players could excel at chess, but not broader strategic thinking. Doctors who were expert diagnosticians in one field were no better than chance in another. Research looking specifically for transfer found that it rarely happened spontaneously and seemed much more elusive than had been previously thought.

For many today, this is a dominant paradigm of psychology: knowledge gained in one domain is unlikely to support thinking in another domain. However, there are plenty of contradictory results suggesting there is more to the story. That transfer exists is self-evident; under some conditions humans are capable of solving problems they haven’t seen before. This has been replicated in some studies but not in others; the question is, what are the conditions to make transfer more likely?

The Low Road and the High Road to Transfer
The authors describe two conditions under which transfer seems more likely. The “low road” to transfer involves “much practice, in a large variety of situations, leading to a high level of mastery and near-automaticity” (22). Practice and fluency in a domain makes it more likely that those skills will be drawn upon in a novel situation. The “high road” to transfer “depends on learners’ deliberate mindful abstraction of a principle” (22). Knowledge that is contextualized and connected with other ideas or broader principles is better primed to apply to a new situation. In short, there are two ways I can teach students to increase the odds they will be able to flexibly apply their knowledge in the future: effective practice and mindful abstraction.

Neither of these “roads” is certain, but they also provide a blueprint for learning that is unlikely to transfer: if there is insufficient practice and learning only takes place in one context without explicit abstraction, transfer seems all but impossible.

Where To From Here?
I often get frustrated with the arguments between the inquiry-oriented “progressive” folks and the explicit instruction “traditionalist” crowd. From my perspective, they’re both right. Learning needs to focus on connections to different elements of prior knowledge, to examine the ways that mathematical content can apply in other domains, and to focus on depth and flexibility of knowledge, all arguments of progressive educators. At the same time, purposeful practice spaced over time leads to fluency and automaticity with key ideas, making it more likely that they can be applied and synthesized with other knowledge in the future, as emphasized by traditionalists.

I think both sides have a point. I’ve seen too many students struggle with a challenging problem because they lack prior knowledge I wish they had — whether that’s addition, fraction operations, integers, or reasoning about the structure of functions. With more practice and better fluency and automaticity, they could be more successful. And I’ve seen too many students struggle with a challenging problem because their prior knowledge is totally context-bound — they’ve only solved problems from one perspective, and are unable to see how their knowledge applies to a novel problem at hand. Prior mindful abstraction of the principles they need would support this thinking by making explicit connections they may be able to make use of in the future.

It’s not an either-or, it’s a both-and. I need to be teaching students so that they have access to both the low road and the high road to transfer. And doing that depends on the content, the students in the room, and where we are in the broader curriculum. There are no easy answers. But I think that considering these two pathways to transfer is a useful touch point for my pedagogical priorities in the classroom.

It’s Not How You Learn, It’s What You Do With It

One mistake we make in the school system is we emphasize understanding. But if you don’t build those neural circuits with practice, it’ll all slip away. You can understand out the wazoo, but it’ll just disappear if you’re not practicing with it.

-Barb Oakley, source

I stumbled across the above quote in a recent interview in the Wall Street Journal, and it struck me as a useful way to think about my teaching.

When I first started teaching, I spent most of my planning time thinking about how I wanted to introduce new topics to my students. I was always looking for clever ways of explaining ideas and interesting new perspectives and hooks relating content to prior knowledge or student interests. I designed inquiry lessons carefully leading students to the big mathematical ideas I wanted them to grapple with.

Now, I spend much more of my time thinking about practice. Not that how I introduce a topic is irrelevant, just less important than what students actually do with the knowledge they’ve gained. I think about how to space that practice and interleave different topics, how to build toward more rigorous applications, how to ensure students engage with a topic in multiple contexts and use multiple representations over time. I work to create collaborative structures that will support students in doing challenging math while still providing individual accountability. I design sequences of activities that move between whiteboarding, technological manipulatives, and pencil-and-paper to keep students engaged for a full class.

The core principle of my teaching is that students are active in their learning. Students learn math by doing math. Practice can have a negative connotation among teachers, and research suggests repetitive practice on low-level tasks is ineffective for learning. But focused, purposeful practice that pushes students outside their comfort zone, is designed to move toward meaningful goals, and involves useful feedback is absolutely necessary for deep, durable learning.

There’s a constant balance here. John Sweller’s Cognitive Load Theory suggests that if the demands of problem solving are too great, students may not retain what we want them to learn even if they are successful in solving the problem. I am partial to Ben Blum-Smith’s summary: “any thoughtful teacher with any experience has seen students get overwhelmed by the demands of a problem and lose the forest for the trees”. At the same time, Robert Bjork’s work on desirable difficulties suggests that if students don’t experience any difficulties in the learning process, what they learn is unlikely to be retained in long term memory or transfer to new contexts. Meaningful learning is hard; if it feels easy it’s likely a missed opportunity.

I’m uninterested in arguing about whether discovery or direct instruction is better. From my perspective, those terms have been overused and caricatured to become meaningless pejoratives. As Dan Willingham says, memory is the residue of thought. What are students thinking about? What does that thinking look like? Those are the key questions I’m interested in, and I think they lead conversations past surface features to the substance that has a real influence on learning.

So students learn math by doing math, and my job is to constantly monitor what that experience is like for students. To what extent are they challenged and thinking deeply about mathematics? To what extent are they overwhelmed and struggling to connect the dots? If I can find a balance between these two poles while keeping students doing substantive math that builds toward ambitious goals, it’s a good day for me.

Research to Practice: Feedback

The purpose of this post is to digest the research on feedback and explore results that I wish I had learned about earlier in my teaching career. This is not a formal research review; I am cherry-picking topics I find useful and ignoring areas where, from my perspective, researchers have had trouble agreeing or results have tenuous links to classroom practice. I’m also including my own extrapolations to how these ideas apply to my teaching. I’m not an expert in this field, so take things with a grain of salt.

By far the most useful source has been Valerie Shute’s review Focus on Formative Feedback. It is very readable and I highly recommend it. I’ve also learned a great deal from Kluger & DeNisi’s review, Dylan Wiliam’s chapter on feedback that moves learning forward in Embedding Formative Assessment, and John Hattie’s book Visible Learning. Collectively, these authors cite several thousand sources. My goal is not to provide an exhaustive bibliography, but to explore a small number of key ideas I find useful.

Use Feedback Sparingly 
One commonly cited result on feedback is that, in Kluger & DeNisi’s review of nearly a century of research, 38% of feedback interventions resulted in negative effects; that is, in 38% of experiments, feedback resulted in less learning than a control condition with no feedback. There’s reason to be skeptical of this number. It does not imply that 38% of teacher feedback is preventing student learning. Rather, it suggests that many things that a teacher might intuitively think could be useful feedback may be less useful than they think. My corollary here is that, in most cases, giving feedback takes a great deal of teacher time. There are lots of other things I can do in a day that I am fairly confident support student learning. I should consider alternatives before giving individual feedback and carefully evaluate what offers me the most value.

Sooner Isn’t Always Better
“Feedback should happen as soon as possible” is often treated as a truism in teaching. However, in a number of studies analyzed by Shute, research suggests that delayed feedback may be more effective than immediate feedback. It is difficult to sort through the various interactions here. One element is that feedback should not interrupt the learner during a task; this is one example of feedback that can be harmful for learning. A second is that, in some situations, immediate feedback causes the learner to rely on the feedback rather than their own thinking. Third, delayed feedback seems to be useful for simpler tasks. One possible mechanism is that, by revisiting a topic later through feedback and spacing learning, the student has an opportunity for more useful thinking than finishing a session with that feedback or relying on it rather than doing additional thinking. Immediate feedback seems more important for complex tasks. This is a tricky one, and there are no clear-cut answers, but it’s worth hesitating to consider the interaction between the learner and the feedback before giving it during or immediately after a task.

No matter how well feedback is articulated, it is only useful if the learner engages with it. Wiliam explores three triggers that may cause a learner to reject feedback. Each of these triggers is dependent on the learner’s perception of the feedback, rather than the intention of the teacher. In other words, no matter how thoughtful feedback is, and even if that same feedback was successful for another student, if a student perceives it in certain ways they are unlikely to learn. Truth triggers occur when a learner gets feedback that they perceive as incorrect or unfair. Relationship triggers occur when a learner gets feedback from an individual they don’t trust or don’t think has their best interests in mind. Identity triggers occur when a learner interprets feedback as saying something about who they are as a person rather than communicating concrete ideas about their thinking or their work. To consider the flip side of each of these triggers, feedback needs to consider the student’s perspective, needs to be built on authentic relationships, and needs to communicate concrete ideas about the student’s work rather than giving grades that are value-laden for many students. Building off of the last idea, research suggests that comments are much more useful than grades for promoting learning. What is more surprising is that, when comments and grades are given together, there is little difference for learning than if the grade was given alone; the grade acts as an identity trigger that causes the learner to focus on themselves rather than making use of the feedback in the comments.

Verification and Elaboration 
Shute writes that effective feedback often combines two elements: verification — communicating the extent to which student ideas are right or wrong; and elaboration — explanatory information communicating analysis of the work or areas for further thinking. Verification should avoid potential triggers by communicating about specific features of the task rather than using grades or similar value-laden information that can distract from specific features of student work.

Feedback Should Cause Thinking 
Dan Willingham writes that memory is the residue of thought. Wiliam explores this idea as well, writing that if feedback does not cause the learner to do additional thinking, it is unlikely to lead to learning. The central idea of this principle is that feedback should be concretely connected to some student action that involves future learning. This can happen in a variety of ways, for instance when the feedback is presented in a way that requires thoughtful interpretation by the student, leads to revision or reassessment, or sets up a classroom activity engaging with the ideas presented in the feedback. If the feedback is more work for the donor than the recipient, or if the feedback focuses more on the past than the future, it may present a missed opportunity for additional learning.

Shute notes that many of the experiments informing the research results above were conducted in laboratory settings where motivation was largely controlled for. This should cause classroom teachers to take research on feedback with healthy skepticism. While a certain feedback strategy might be research-based and well-intended, if students are unmotivated it is likely to be ineffective. This leads me to two ideas. I need to communicate to students that I care about the quality of their work. This can happen through feedback; it can also happen through other means like student work analysis or targeted review. But one useful function of feedback is to improve motivation by communicating that student work matters. Whether through feedback or other means, I need to find ways to let students know that their work matters regularly. Second, I need to clearly articulate to students the feedback strategies I am using and why I am using them, so they understand how feedback connects to their learning and the purpose behind my classroom decisions.

Looking Back 
Considering these principles of effective feedback, I think they are fairly limited in their utility. I don’t think there’s enough to go on to really design research-based classroom feedback strategies that will work across a variety of contexts. That said, I spent the first few years of teaching without any benchmarks for what effective feedback looked like; I did what seemed right at the time and leaned heavily on what teachers around me were doing and what I had experienced as a student. I think that the ideas above offer a useful lens to move beyond those anecdotal experiences to more purposeful strategies focused on maximizing student learning.

Problem Solving and Creativity

I try not to put too much stock in the endless hole of people talking about Silicon Valley and startups and the tech world on the internet. But this quote caught my eye:

You need to know the things that you need to know to solve the problem. And you need to not believe things that will get in the way of solving the problem.

Sourced here, which credits the quote to Scott Klemmer, though I can’t find the original anywhere. He was talking about design and what research says about creativity, but I think it applies well to problem solving in math. Leads me to the question: is there anything else to problem solving?