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Evaluating the Best Classroom Practices for Teaching Math
Andrew Watson
Andrew Watson

What strategies work best for math teaching?

math teaching

And, crucially, how do we know?

To answer this question, we might rely on our teacherly instincts. Perhaps we might rely on various educational and scientific theories. Or, we might turn to data. Even big data.

Researchers in Sweden wondered if they could use the TIMSS test to answer this question.

(“TIMSS” stands for “Trends in International Mathematics and Science Study,” given every four years. In 2015, 57 countries participated, and 580,000 students. That’s A LOT of students, and a lot of data.)

3 Math Teaching Strategies

When students take these tests, they answer questions about their classroom experience.

In particular, they answer questions about 3 math teaching strategies. They are asked how often they…

Listen to the teacher give a lecture-style presentation.

Relate what they are learning in mathematics to they daily lives.

Memorize formulas and procedures.

Researchers want to know: do any of these teaching practices correlate with higher or lower TIMSS scores? In other words, can all these data help us evaluate the effectiveness of specific teaching practices?

2 Math Teaching Theories

Helpfully, the researchers outline theories why each of these practices might be good or bad.

As they summarize recent decades of math-teaching debate, they explain that “researchers with their roots in psychology and cognitive science” champion

formal mathematical notions,

explicit instruction where teachers show students how to solve math problems,

practicing and memorizing rules and worked examples.

On the other hand, “researchers with their roots in the reform movement” champion

connecting math to students’ daily lives,

a problem-solving approach,

understanding ideas and connections, rather than memorization.

Doubtless you’ve heard many heated debates championing both positions.

Predictions and Outcomes

These theories lead to clear predictions about TIMSS questions.

A cognitive science perspective predicts that “lecture-style presentations” and “memorizing formulas” should lead to higher TIMSS scores.

A reform-movement perspective predicts that “relating math to daily life” should lead to higher scores.

What did the data analysis show?

In fact, the cognitive science predictions came true, and the reform predictions did not.

In other words: students who listened to presentations of math information, and who memorized formulas did better on the test.

Likewise, students who applied math learning to daily life learned less.

An Essential Caveat

As these researchers repeatedly caution, their data show CORRELATION not causation.

It’s possible, for instance, that teachers whose students struggle with math resort to “daily life” strategies. Or that both variables are caused by a third.

Potential Explanations

“Connecting new math learning to real life situations” seems like such a plausible suggestion. Why doesn’t it help students learn?

These researchers offer two suggestions.

First, every math teaching strategy takes time. If direct instruction is highly effective, then anything that subtracts time from it will be less effective. In other words: perhaps this strategy isn’t harmful; it’s just less effective than the others.

Second, perhaps thinking about real-life examples limits transfer. If I use a formula to calculate the area of a table, I might initially think of it as a formula about tables. This fixed notion might make it harder for me to transfer my new knowledge to — say — rugby fields or floor plans.

At present, we can’t know for sure.

A final point. Although this research suggests that direct instruction helps students learn math, we should remember that bad direct instruction is still bad.

Lectures can be helpful, or they can be deadly tedious.

Students can memorize pertinent and useful information. Or, they can memorize absurd loads of information.

(A student recently told me she’d been required to memorize information about 60 chemical elements. Every science teacher I’ve spoken with since has told me that’s ridiculous.)

And so: if this research persuades to you adopt a direct-instruction approach, don’t stop there. We need to pick the right pedagogical strategy. And, we need to execute it well.

Cognitive science can help us do so..

Does Media Multitasking Really Interfere with Student Thinking?
Andrew Watson
Andrew Watson

media multitaskingTo many teachers, it just seems obvious: all that screen times MUST be bad for student brains.

To many other teachers, it just seems obvious: technology will unleash academic possibilities and revolutionize education.

So, which is it? Does media multitasking damage students’ cognitive capabilities? Or, does it allow them new avenues to creative possibilities?

Here’s What We Know

In a recent analysis, Uncapher and Wagner surveyed research into this topic.

Sure enough, they found some troubling evidence.

In half of the studies they examined, people who often use multiple streams of technology scored lower on working memory tests than those who don’t.

In two studies, they had a harder time recalling information from long-term memory.

Studies also showed problems with sustained attention.

Here’s a place where media multitasking might help: task switching. Given all the practice that multitaskers get diverting attention from one gizmo to another, they might well get better at doing so.

Alas, most of the research that U&W examined didn’t support that hypothesis.

Here’s What We Don’t Know: A LOT

Although all of the sentences above are true, they don’t answer most questions with any certainty.

For example, if half of the studies showed that high multitaskers do worse on working memory tests, that means that half of the studies DON’T reach that conclusion.

(It’s important to note that NONE of the studies showed that high multitaskers were better at working memory tasks than their counterparts.)

Uncapher and Wagner repeatedly emphasize this point. We don’t have lots of studies — and those we do have don’t all point the same direction.

Another important question: causality. Perhaps multitasking reduces sustained attention. Or, perhaps people who have trouble sustaining attention multitask more often.

Firm Conclusions

At present, we can conclude with confidence that we don’t have enough evidence to conclude anything with confidence.

Overall, the evidence suggests heavy media multitasking might cause (or might result from) relative weaknesses in several cognitive functions.

We certainly don’t have evidence that encourages us to promote multi-gizmo use.

I myself try to stick to one device at a time. Until more evidence comes in, I’ll gently suggest my students do likewise.

(For thoughts on technology and attention, click here.)

Avoiding Extremes: Common Sense in the Middle
Andrew Watson
Andrew Watson

Teachers feel passionate about our work. As a result, we can advocate exuberantly — occasionally too exuberantly? — for a particular position.

Advocates for (or against) Social-Emotional Learning can make zealous claims for their beliefs. Same for PBL, or direct instruction. Or for flipped classrooms, or traditional ones.

Of course, given the variety of teachers, students, schools, curricula — and the variety of societies in which they all operate — we perhaps should hesitate to make absolute claims.

Today’s Shining Example

I recently rediscovered a marvelous example of comfort with the ambiguous middle ground.

In this EdSurge post, Art Markman explains how mindfulness can help. And: how it might not help.

He explains the benefits of a growth mindset. And: its potential detriments.

When asked “if schools teach the way students learn,” he doesn’t scream “OF COURSE!” Nor does he bellow “NEVER!”

Instead, he offers this answer: “Sometimes, but often not.”

In other words: we’re not all spectacular successes or hideous failures. Contrary to much of the rhetoric you hear, we live somewhere in between.

I hope you enjoy reading this interview. And, that Markman’s sensible example offers guidance on moderation and nuance.

I myself look forward to reading more of his work.

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Andrew Watson
Andrew Watson

Here on the blog, I write A LOT about the benefits of “retrieval practice.” (For example: here and here.)

retrieval practice limitations

In brief: our students often review by trying to put information into their brains. That is: they “go over” the material.

However, they learn more if — instead — they review by trying to pull information out of their brains. That is: they fill in blanks on Quizlet, or use flashcards, or outline the chapter from memory.

AT THE SAME TIME…

I also write about the importance of “boundary conditions.”

A particular research finding might be true for this group (say, college students learning chemistry) but not that group (say, 3rd graders learning spelling rules).

(For example: here and here.)

So, I really should ask myself: what are the boundary conditions for retrieval practice?

Retrieval Practice Limitations?

In the first place, retrieval practice has become so popular because it works so well in so many circumstances.

It helps 2nd graders and adult learners.

It helps with declarative knowledge and procedural knowledge.

And, it helps Red Sox fans and Dodgers fans. (I might have made that one up.)

However, I have recently seen research into two retrieval practice limitations, and I think they’re important for teachers to keep in mind.

“Narrow” vs. “Broad” Learning

Researcher Cindy Nebel (nee Wooldridge) wanted to know if retrieval practice helps students learn only the information they retrieve. That is, it might have a narrow, focused effect.

Or perhaps it helps students remember ideas related to the information they retrieve. Retrieval of one memory might broadly influence other memory networks.

In my geography class, for instance, students might learn that the capital of Egypt is Cairo, and that its main economic drivers are tourism and agriculture.

I encourage my students to make flashcards to help them remember capitals. When a student looks at her Egypt flashcard, will remembering its capital (“Egpyt!”) help her remember its main industries as well? Or, does it help consolidate only that specific memory network?

Alas, according to Nebel’s research, RP has a “narrow,” not a “broad” effect. It helps students remember the specific information they retrieved, but not related concepts.

Practically speaking, this finding suggests that we should be sure to tailor retrieval practice exercises quite precisely to the specific memory we want students to form. A question about triassic fossils won’t necessarily help them recall specifics about the end of the cretaceous era.

If we want them remember asteroid impacts, we should use RP to foster those memories.

Question Difficulty, Difficult Questions

A more recent study has looked at other retrieval practice limitations: fluid intelligence, and question difficulty. This research is still behind a paywall, and so I haven’t looked at the specifics.

The abstract, however, suggests that — especially on difficult items — students with relatively low fluid intelligence might benefit more from review than RP.

This research finding raises several questions: how, precisely, do we measure question difficulty?

And: how much stock do we want to put into measures of fluid intelligence?

Classroom Decisions

As always, the question comes down to this: “what should I, as the classroom teacher, actually do?

Based on this research, I think we can reach a few clear conclusions:

In many circumstances, retrieval practice helps students remember more than simple review.

As much as possible, we should ensure that we have students retrieve the precise information (or process) we want them to remember. Nearby questions might not help enough.

When working with difficult material, or with students who really struggle in school, we should keep an open mind. Try different learning strategies, and see which ones prove most effective with this student right here.

I’ll keep you posted as I read more about boundary conditions for retrieval practice.

Life Without Memory: Your Hippocampus and You
Andrew Watson
Andrew Watson

Who are you without your memory?

In neurobiological lingo: who are you without your hippocampus?

The Best-Known Answer

No doubt you’ve heard of Henry Molaison, aka H. M., whose hippocampi were removed in order to cure debilitating epilepsy.

The good news: the operation (more-or-less) fixed the epilepsy.

The (very) bad news: without his hippocampi, Henry couldn’t form new long-term memories. In fact, he struggled to recall prior memories as well.

So much of our knowledge about memory formation comes from Henry’s life.

We understand the brevity of working memory because of H. M.

We distinguish between declarative memory (“knowing what”) and procedural memory (“knowing how”) better because of H. M.

As Suzanne Corkin describes in Permanent Present Tense, research into Henry’s very rare brain tells us more about each of our brains.

Today’s News: A New Henry

On December 29 of 2007, artist Lonni Sue Johnson came down with a bad case of viral encephalitis. As a result, she ended up with severe damage to both her hippocampi. This damage, in fact, resembles H.M.’s surgical lesions.

You can read about her case in a remarkable book by Michael D. Lemonick, The Perpetual Now: A Story of Amnesia, Memory, and Love.

Lonni Sue’s situation resembles Henry’s in many ways — they both live in a “perpetual now” — but their stories differ as well.

First: Henry was relatively young at the time of his surgery, and so he hadn’t yet developed professional skills. (Because his epilepsy also proved quite debilitating, he didn’t get very far in school.)

Lonni Sue, however, was an accomplished artist and musician — even an amateur pilot.

For example: she drew several covers for the New Yorker magazine. You might recognize her whimsical style if you google her art.

Second: Her family decided soon after her illness that they would be as public as Henry’s family had been private. They want her remarkable condition — as much as possible — to benefit science, and the public’s understanding of the brain.

For that reason, when Lonni Sue’s sister Aline ran into Lemonick on the street, she asked if he wanted to write about her life without memory.

Third: Lonnie Sue brought a remarkable good cheer to a life that might seem so depressing, even terrifying, to others.

When Lemonick first met her, she brightly introduced herself and showed him her drawings. Then, she introduced him to a word game she often played: “singing the alphabet.”

She sang a list of words that grew in alphabetical order. Here’s what she sang that first time (and, notice how cheerful the words are!):

“Artists beautifully creating delightful exquisite finery giving hospitable inspiration joining keen laughter’s monthly necessities openly preparing quiet refreshment sweetly turning under violet weathervane xylophones yearning zestfully”

Life Without Memory: Research Findings

For the same reasons that Aline invited Lemonick to write about her sister, she has also invited researchers to learn what they can from Lonnie Sue’s brain.

Lemonick does a wonderful job of explaining these research findings. He does go into the methodological details. But he maintains a big-picture emphasis on the history and meaning of the research.

For instance, we saw that research on Henry helped solidify a distinction between procedural and declarative memory. Further research with Lonni Sue suggests that these categories often overlap.

Her knowledge of music, for example, acts like both declarative and procedural knowledge at the same time.

For teachers, this finding just makes sense.

So many of the skills students learn require them to know facts AND procedures. A chemistry lab, a historical investigation, a business plan: all these school accomplishments ask students to know stuff, and to do things with that knowledge.

The Perpetual Now won’t necessarily help classroom teachers design better lesson plans. But, it does help us understand the rich complexity of human memory.

And, it tells the story of an extra-ordinary life: one where “xylophone weathervanes yearn zestfully.”

I recommend the book enthusiastically.

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Andrew Watson
Andrew Watson

Neil Selwin argues, dramatically, that “EdTech is Killing Us All.”

His point is not that technology is bad for learning, but that it’s bad for the environment. As we think about the educational work we do, we should keep this perspective in mind.

Can Quiet Cognitive Breaks Help You Learn?
Andrew Watson
Andrew Watson

We write a lot on the blog about “desirable difficulties” (for example, here and here). Extra cognitive work during early learning makes memories more robust.

cognitive breaks

Retrieval practice takes more brain power than simple review — that is, it’s harder. But, it helps students remember much more.

Wouldn’t it be great if some easy things helped too?

How about: doing nothing at all?

Cognitive Breaks: The Theory

When a memory begins to form, several thousand neurons begin connecting together. The synapses linking them get stronger.

Everything we do to help strengthen those synapses, by definition, helps us remember.

We know that sleep really helps in this process. In fact, researchers can see various brain regions working together during sleep. It seems that they’re “rehearsing” those memories.

If sleep allows the brain to rehearse, then perhaps a short cognitive break would produce the same result.

Cognitive Breaks: The Research

Michaela Dewar and colleagues have been looking into this question.

They had study participants listen to two stories. After one story, participants had to do a distracting mental task. (They compared pictures for subtle differences.)

After the other, they “rest[ed] quietly with their eyes closed in the darkened testing room for ten minutes.”

Sure enough, a week later, the quiet rest led to better memory. As a rough calculation, they remember 10% more than without the quiet rest.

10% more learning with essentially 0% extra cognitive effort: that’s an impressive accomplishment!

Classroom Questions

A finding like this raises LOTS of practical questions.

Dewar’s study didn’t focus on K-12 learners. (In fact, in this study, the average age was over 70.) Do these findings apply to our students?

Does this technique work for information other than stories? For instance: mathematical procedures? Dance steps? Vocabulary definitions?

Does this finding explain the benefits of mindfulness? That is: perhaps students can get these memory benefits without specific mindfulness techniques. (To be clear: some mindfulness researchers claim benefits above and beyond memory formation.)

Can this finding work as a classroom technique? Can we really stop in the middle of class, turn out the lights, tell students to “rest quietly for 10 minutes,” and have them remember more?

Would they instead remember more if we tried a fun fill-in-the-blank review exercise?

I’ll be looking into this research pool, and getting back to you with the answers I find.

Cognitive Breaks: The Neuroscience

If you’d like to understand the brain details of this research even further, check out the video at this website. (Scroll down just a bit.) [Edit 11/4/19: This link no longer works; alas, I can’t find the video.]

The researchers explain a lot of science very quickly, so you’ll want to get settled before you watch. But: it covers this exact question with precision and clarity.

(By the way: you’ll hear the researchers talk about “consolidation.” That’s the process of a memory getting stronger.)

If you do watch the video, you might consider resting quietly after you do. No need to strain yourself: just let your mind wander…

hat tip: Michael Wirtz

T/F: Timed Tests Cause Math Anxiety?
Andrew Watson
Andrew Watson

Questions about math and anxiety have been on the uptick recently.

Over at Filling the Pail, Greg Ashman offers his typically direct analysis. You might disagree with his opinion, but he’s always worth a mental debate.

By the way, a casual aside in his post deserves attention of its own. Here’s how Ashman frames his tests: “I’m just checking in to see how well I’ve taught you.”

That simple sentence accomplishes many useful goals — it’s one I might use myself. It’s hard to imagine an easier way to reduce test stress…

How to Stop Cheating: An Awkward Debate
Andrew Watson
Andrew Watson

We would, of course, LOVE to prevent cheating.

prevent cheatingIt does moral damage to the cheater. It undermines classroom trust. And: it makes it hard for us to know how much our students are actually learning.

So: what techniques might help us do so?

How To Prevent Cheating: “Moral Reminders”

For some time now, Dan Ariely has made this his field. (Check out his book:  The (Honest) Truth about Dishonesty: How We Lie to Everyone — Especially Ourselves.)

Over the years, he developed a clever research paradigm to see how much people cheat. With that in place, he tested various strategies to prevent cheating.

(He can also promote cheating, but that’s not exactly what we’re looking for.)

One strategy that has gotten a lot of attention over the years: moral reminders.

Ariely asked some students to write down ten books they had read in high school. He asked the others to write down the 10 Commandments.

That is: he made them think about foundational moral standards in our culture.

Sure enough, once reminded about moral standards, students cheated less. (The Cohen’s d was 0.48, which is an impressive effect for such an easy intervention.)

Then Again, Maybe Not

In a study published just a month ago, Bruno Verschuere (and many others) retested Ariely’s hypothesis. Whereas the original study included 209 students, this meta-analysis included almost 4700. That is … [checks math] … more than 20 times as many students.

Studying much more data, they found that “moral reminders” made no difference.

(In fact, they found that students who recalled the 10 commandments were just a smidge likelier to cheat; but, the difference was tiny — not even approaching statistical significance.)

As we’ve seen in other cases of the “replication crisis,” seemingly settled results are back in question.

What’s a Teacher to Do?

Of course, Ariely had other suggestions as well. Signing  pledges not to cheat reduces cheating.  And, of course, teachers who supervise students closely reduce their opportunities to cheat.

As far as I know, these strategies have not been retested (although the second one seems too obvious to need much retesting).

For the time being, sadly, we should rely less on indirect moral reminders, and more on direct pledges — and direct supervision.

The Fear Factor: How One Emotion Connects Altruists, Psychopaths, and Everyone In-between by Abigail Marsh
Rebecca Gotlieb
Rebecca Gotlieb

Abigail Marsh’s 2017 book , reviews research by her and others showing that extraordinary altruists and psychopaths may be two extremes of a bell-curve of human caring with altruists and psychopaths distinguished by how sensitive they are to feelings of fear. She employs an evolutionary perspective to argue that having evolved to care for vulnerable young has equipped us with the neural architecture to care for other people more generally. She concludes by arguing that we both can and should strive to be more altruistic.

Marsh, associate professor of psychology at Georgetown University, has herself been both the beneficiary of extraordinary altruism and the victim of assault. At age 19 she was in a serious car accident that might have left her dead had an anonymous stranger not intervened, at great risk to himself, to rescue her. A few years later a man she did not know grabbed her in a sexually inappropriate manner. When she reacted with outrage, he proceeded to punch her in the face, knocking her to the ground. Marsh has devoted her career to understanding the roots of and individual variability in these extremes of behavior, how we can help people with psychopathic tendencies, and how we can behave more like altruists.

Early in her career, Marsh uncovered a peculiar relationship; the ability to recognize fear in faces predicted altruism. This effect held across different types of altruistic actions and was more robust than many factors that had traditionally been expected to be closely related to altruism. Intrigued by this, she went on to study responses to fearful faces in children and adolescents known for their lack of altruism and lack of remorse for causing other people pain—youth with callous-unemotional traits (or psychopathic tendencies). Marsh conducted functional MRI scans of the brains of these youth as they were exposed to faces displaying different emotions. Previous research had suggested that the amygdala—a brain structure located deep in the middle of the brain—responds strongly when an individual is exposed to fearful expressions. In children with psychopathic traits, however, this brain pattern was not observed—a fact consistent with these children stating that they rarely experienced fear. It may be that psychopathic individuals’ impaired ability to feel fear impairs their ability to empathize with others’ fear, and this makes it difficult for them to understand why it can be wrong to make people feel afraid.

Given that many human traits can be modelled with a bell-curve shape and that the distribution of psychopathy in the population looks like a bell-curve cut in half, Marsh hypothesized that perhaps a normal curve could be constructed to represent the full distribution of caring in the population with extraordinary altruism balancing out psychopathy. Although many people engage in altruistic actions frequently, extraordinary altruists do so for people they do not know at great cost to themselves, even when there is no expectation to do so.

Humans can function normally with one kidney but are born with two. Anonymous kidney donors—people who voluntarily undergo kidney surgery, incurring financial costs and medical risks, in order to give one of their kidneys to person they do not know—served as an ideal case study of extraordinary altruists. Marsh recruited these extraordinary altruists to undergo similar functional MRI scans to the ones she had had youth with callous-unemotional tendencies undergo. Interestingly, she identified complementary findings. Extreme altruists’ amygdalae were especially responsive to fearful faces and their right amygdalae were about 8% larger than those of people in the general population. Marsh concludes that, although there is a common trope that extraordinary altruists are fearless to be able to help others in the way they do, on the contrary these individuals are actually hypersensitive to the fear of other people and motivated to act because of it.
That hypersensitivity to fear may stem from a co-opting of our parental instincts. We have evolved to love our small, underdeveloped babies, enjoy physical contact with them, and produce nutritious mother’s milk to feed them. We are built to parent, and are drawn to provide care for other beings, even those who are not our children.

Marsh argues that people are more compassionate than we typically recognize. Better quality of life is associated with greater caring. Thus, as quality of life continues to improve, more people may become increasingly compassionate. To bring about greater altruism we should cultivate a humble understanding that strangers’ welfare is worth as much as our own. Most simply, Marsh argues that there is a virtuous cycle of giving, so, “if you want to be more altruistic, just start!” (P. 50).

Marsh, A. (2017). The Fear Factor: How One Emotion Connects Altruists, Psychopaths, and Everyone In-between. New York: Basic Books.