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A Mind for Numbers by Barbara Oakley
Erik Jahner, PhD
Erik Jahner, PhD
March 21, 2023

A Mind for Numbers by Barbara Oakley

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A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra) is an excellently constructed tour of the mind improving your approach to learning and problem-solving. While there are many learning strategy books out there, Barbara Oakley’s stands out due to its entertaining, educational, well-researched, and refreshed cognitive foundation. You are in for a real treat as each bite-sized chapter draws you in with engaging stories, enticing your curiosity with cognitive principles and historical tidbits asking you to constructively reflect on the machinery of your mind.

While this book says it is for Math and Science learning, the concepts addressed here can be applied to a wide array of subjects from language learning to time management, procrastination, and reading. Although it is filled with useful and updated information about how memory works, it is not simply a book about memory techniques. Throughout, there is a continual nod to social-emotional learning concepts and metacognitive awareness, including understanding how the ways you are learning may lead you to develop a false sense of confidence in your knowledge. Enabling you to understand your own learning profile, Oakley shares both what is effective and what is not effective, making it a great book for study skills classes or anyone who just wants to identify what learning practices are helpful and which ones are simply a waste of time.

Faithful to its inner teachings, the book is organized into very useful chunks of information that allow the reader to build their stores of knowledge in a systematic way. Each chapter is packed with great lessons followed by a “Pause and Recall” section and containing “Now you try” sections, encouraging us to pull away from the reading for a moment and relate the concepts to our lives and process them at more meaningful and deeper levels. We also get nice neat summaries pulling the chapters together integrating across chapters and allowing for a quick skim of some of the highlights. This structure naturally lends itself to classroom discussions. As an instructor, I have even used some of the “Now you try” sections with my college students who find them to be useful and revealing reflections.

Illuminating the intriguing history of psychology, we are treated to fascinating discussions of real people including arsenic eaters, a man who had an unnatural ability to remember details at some cost to other cognitive abilities, and an infamous neuroscientist who was put into jail for building a small cannon that destroyed a neighbor’s gate. These little bits of historical psychology are a gateway for the psychology novice to enter the field and engage students. These morsels from history led me to also jump on the internet and learn a bit more about these characters, demonstrating Oakley’s ability to open up new worlds.

I would be leaving out an important part of this book if I did not mention the memorable, fun, and useful visuals in this book. I’m particularly fond of the octopus representing attention mechanisms in the brain and pinball machines representing the semantic closeness of ideas. When discussing the removal of faint connections, we are offered illustrations of ‘metabolic vampires’ that suck the remaining life from neurons¬–images that really leave a lasting impression. The creative use of these and other metaphors throughout the book will help the novice student grasp the concept and act as useful teaching tools for the instructor to reframe the concept and make it accessible while staying true to the science. While the metaphors and illustrations are fun, they are not diminutive. The reader never feels talked down to, and the material is not oversimplified.

From mathematics to learning a new hobby and managing your life, Oakley enhances the learning experience and makes you the game-maker in your learning adventure. She makes learning fun and you will walk away with a growth mindset and new tools opening your mind to try or try again to learn concepts you thought were out of your reach–’even if you flunked algebra.’

Does MOVEMENT Help LEARNING?
Andrew Watson
Andrew Watson
February 02, 2021

Does MOVEMENT Help LEARNING?

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In the exaggerated stereotype of an obsessively traditional classroom, students sit perfectly silent and perfectly still. They listen, and watch, and do nothing else.

Few classrooms truly function that way.

But, how far should we go in the other direction? Can teachers — and should teachers — encourage noise and movement to help students learn?

In recent years, the field of embodied cognition has explored the ways that we think with our bodies.

That is: movement itself might help students learn.

Of course, this general observation needs to be explored and understood in very specific ways. Otherwise, we might get carried away. (About a year ago, for instance, one teacher inspired a Twitter explosion by having his students read while pedaling exercycles. I’ve spent some time looking at research on this topic, and concluded … we just don’t know if this strategy will help or not.)

So, let’s get specific.

Moving Triangles

An Australian research team worked with 60 ten- and eleven-year olds learning about triangles. (These students studied in the intermediate math track; they attended a private high school, with higher-than-usual SES. These “boundary conditions” might matter.)

Students learned about isosceles triangles, and the relationships between side-lengths and angles, and so forth.

20 of the students studied in a “traditional way“: reading from the book.

20 studied by watching a teacher use software to manipulate angles and lengths of sides.

And, 20 studied by using that software themselves. That is: they moved their own hands.

Researchers wanted to know:

Did these groups differ when tested on similar (nearly identical) triangle problems?

Did they differ when tested on somewhat different problems?

And, did they rate their mental effort differently?

In other words: did seeing movement help students learn better? Did performing the movement themselves help?

The Envelope, Please

The software clearly helped. The actual movement sort-of helped.

Students who interacted with the software themselves, and those who watched the teachers do so, did better on all the triangle problems. (Compared — that is — to students who learned the traditional way.)

And, they said it took less mental effort to answer the questions.

HOWEVER:

Students who used the software themselves did no better than the students who watched the teachers use it. (Well: they did better on the nearly identical problems, but not the newer problems that we care more about.)

In other words: movement helped these students learn this material — but it didn’t really matter if they moved themselves, or if they watched someone else move.

The Bigger Picture

Honestly: research into embodied cognition could someday prove to make a big difference in schools.

Once we’ve done enough of these studies — it might be dozens, it might be hundreds — we’ll have a clearer picture explaining which movements help which students learn what material.

For the time being, we should watch this space. And — fingers crossed — within the next 5 years we’ll have an Embodied Cognition conference at Learning and the Brain.

Until then: be wise and cautious, and use your instincts. Yes, sometimes movement might help. But don’t get carried away by dramatic promises. We need more facts before we draw strong conclusions.

Bokosmaty, S., Mavilidi, M. F., & Paas, F. (2017). Making versus observing manipulations of geometric properties of triangles to learn geometry using dynamic geometry software. Computers & Education, 113, 313-326.

“If I Want My Students to Learn Math, Should I Teach Them More Math?”
Andrew Watson
Andrew Watson
October 06, 2020

“If I Want My Students to Learn Math, Should I Teach Them More Math?”

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We all agree, I suspect, that students should learn math. And reading. They should learn history. And science. SO MANY other topics.

What’s the best way to meet these goals?

If I want my students to learn math, is math teaching the best way to go? If I want them to understand history, should I teach more history?

Or, instead, is there a handy shortcut?

If I could help students improve their reading by teaching something other than reading, that alternate approach just might be more efficient and motivating.

In fact, two candidates get lots of attention as “alternative approaches.”  If either or both pan out, they would offer us more choices. Maybe even a higher chance of success.

Music and Math

I don’t remember where I first heard that music education improves math learning. Specifically: learning to play the violin ultimately makes students better at learning calculus.

The explanation focused on “strengthened neural circuits” “repurposed” for “higher cognitive function.” Something like that. That string of words sounded quite impressive, and inclined me to believe.

Given the complexity of calculus, that would be really helpful!

But: is it true?

A recent meta-analysis looked at 54 relevant studies, including just under 7,000 participants.

Their findings? Let me quote key points from their summary:

Music training has repeatedly been claimed to positively impact children’s cognitive skills and academic achievement (literacy and mathematics).

This claim relies on the assumption that engaging in intellectually demanding activities fosters particular domain-general cognitive skills, or even general intelligence.

The present meta-analytic review shows that this belief is incorrect.

Once the quality of study design is controlled for, the overall effect of music training programs is null.

It gets worse:

Small statistically significant overall effects are obtained only in those studies implementing no random allocation of participants and employing non-active controls.

In other words: you get this result only if the study isn’t correctly designed.

And worse:

Interestingly, music training is ineffective regardless of the type of outcome measure (e.g., verbal, non-verbal, speed-related, etc.), participants’ age, and duration of training.

That is: no matter what you measure, the answer is still “no.”

Violin training sure strengthened some neural circuits. But that additional strength doesn’t get “repurposed for ‘higher’ cognitive function.”

If I want my students to learn math, I should teach them math.

Chess and Intelligence

If you watch The West Wing, you know that President Bartlet is smarter than everyone else because he won a Nobel Prize, and he plays chess frequently. He says things like “rook takes queen in five.” And then Leo nods appreciatively.

So smart.

It might be true that being smart makes you better at chess. (Although, Anders Ericsson says “no.”)

Is it true that playing chess makes you smarter? If we want our students to learn math and reading and science, should we teach them more chess? Would some neural circuitry get repurposed?

Let’s go to the tape:

In contrast to much of the existing literature, we find no evidence of an effect of chess instruction upon children’s mathematics, reading or science test scores.

In this case, by the way, the “tape” is a randomized control trial with more than 4,000 students in it. So: that result seems impressively well established.

So far, it seems that if I want my students to be better at X, I should teach them X. Teaching them Y and hoping that Y makes them better at X hasn’t panned out well…

Social Studies and Reading

Reading might be an interesting exception to this rule. On the one hand, reading is a skill that students must acquire.

And, at the same time, they have to apply the skill of reading to the content being read. The more that students know about the content, maybe the better they’ll do at reading.

In any case, that’s a plausible hypothesis.

A recently released report from the Thomas Fordham Institute crunches the numbers, and finds that additional time devoted to social studies instruction ultimately improves reading scores.

Two key sentences from the executive summary:

Instead of devoting more class time to English language arts, we should be teaching elementary school children more social studies — as in, rich content about history, geography, and civics.

…

Literacy gains are more likely to materialize when students spend more time learning social studies.

In fact, they find that social studies instruction most benefits students from lower-income households, and from non-English speaking homes.

For a variety of reasons, this study looks at correlation, and so can’t demonstrate causation.

However, the underlying theory makes sense. If students can decode the sounds of the words “Berlin” and “Wall,” but don’t know the geography of Germany or cold-war history, they’re unlikely to make much sense of a reading passage about that in/famous border.

In Sum

Students improve at the skills they practice. Those skills — alas —  rarely transfer to distantly unrelated disciplines.

To help students learn math, teach them math. To help them read, teach them to read — and also about the scientific, historical, geographic, and philosophical concepts that make reading so important and so worthwhile.

Concrete + Abstract = Math Learning
Andrew Watson
Andrew Watson
December 07, 2019

Concrete + Abstract = Math Learning

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Early math instruction includes daunting complexities.

We need our students to understand several sophisticated concepts. And, we need them to learn a symbolic language with which to represent those concepts.

Take, for example, the concept of equivalence. As adults, you and I can readily solve this problem :   3+4 = 4 + __

Early math learners, however, can easily stumble. Often, they take the equals sign to mean “add up all the numbers,” and believe the correct answer to that question is “10.”

How can we help them through this stage of understanding?

Strategy #1: Switch from Abstract to Concrete

The first answer to the question seems quite straightforward. If the abstract, symbolic language of math (“3+4=___”) confuses students, let’s switch to a more concrete language.

For instance: “If my frog puppet has three oranges, and your monkey puppet has four oranges, how many oranges do they have together?”

It just seems logical: the switch from abstract to concrete ought to help.

Alas, those concrete examples have a hidden downside.

As Dan Willingham argues in Why Don’t Students Like School?, humans naturally focus on surface features of learning.

When children see monkeys and frogs and oranges, they associate the lesson with those specific entities–not with the underlying mathematical properties we want them to learn.

In edu-lingo, concrete examples can inhibit transfer. Students struggle to transfer a lesson about oranges and puppets to anything else.

Strategy #2: “Fade” from Concrete to Abstract

Taking their cue from Jerome Bruner, psychology researchers wondered if they could start with concrete examples and then, over time, switch to more abstract examples.

For instance, students might start learning about mathematical equivalence by using a balance. When they put an equal number of tokens on both sides, the balance is level.

In the second step, they do practice problems with pictures of a balance and tokens.

And, in the final step, they see abstract representations: 2 + 5 = 5 + __.

They describe this technique as “concreteness fading.”

And, sure enough, it worked. In this case, “worked” meant that students who learned equivalence though a concreteness fading method transferred their knowledge to different–and more difficult–problems.

They did so better than students who learned in a purely abstract way. And, better than students who learned in a purely concrete way. (And even, as a control condition, better than students who started with an abstract representation, and then switched to concrete.)

By the way: these researchers tested their hypothesis both with students who had a relatively low level of knowledge in this area, and those who had a high level of knowledge. They got (basically) the same results both times.

An Essential Detail

When we teachers try to incorporate psychology research into our teaching, we can sometimes find that it conflicts with actual experience.

In this case, we might find that our young math learners just “get it” faster when we use frog puppets. Given that experience, we might hesitate to fade over to abstract teaching.

This research shows an intriguing pattern.

Sure enough, students who began with concrete examples made fewer mistakes on early practice problems. And, that finding was true for both the “concrete only” group and the “concreteness fading” groups.

In other words, the “abstract only” group did worse on the early practice problems than did those groups.

But…and this is a CRUCIAL “but”…the “concrete only” group didn’t do very well on the transfer test. Their raw scores were the lowest of the bunch.

However, the “concreteness fading” group did well on the early problems AND on the transfer test.

It seems that, as the researchers feared, too much concrete instruction reduced transfer.

 

In sum: “concreteness fading” gives young math learners both a helpfully clear introduction to math concepts and the abstract understanding that allows transfer.

Fyfe, E. R., McNeil, N. M., & Borjas, S. (2015). Benefits of “concreteness fading” for children’s mathematics understanding. Learning and Instruction, 35, 104-120.

When Parents Teach Reading, Do They Also Promote Math Skills?
Andrew Watson
Andrew Watson
July 28, 2019

When Parents Teach Reading, Do They Also Promote Math Skills?

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Parents begin teaching children well before schooling starts. Obviously.

In fact, parents often teach children topics that we might consider “academic”: say, how to read, or, how to count.

Researchers might investigate this parental pre-school teaching with some reasonable hypotheses.

For instance:

Presumably, the way that parents teach reading influences the reading skills that their children develop.

Likewise, presumably, the way parents teach numbers and counting influences the math skills that their children develop.

Let’s ask a more counter-intuitive question:

Does the way that parents teach reading influence their children’s math skills?

In other words, does early teaching in one discipline influence understand in a different discipline?

That question might raise skeptical eyebrows, for a number of reasons. In particular, most research that asks this kind of transfer question comes back with a negative answer.

That is: learning in one discipline (say: playing piano) doesn’t usually make you better at another discipline (say: doing calculus).

Today’s Study

Researchers in England wanted to explore this surprising hypothesis. They had hundreds of parents fill out questionnaires. Some questions focused on parental approaches to reading:

How often does the child discuss the meaning of a story with an adult?

or

How often is the child encouraged to name letters or sound out words?

Other questions focused on parental approaches to numbers:

How often is the child encouraged to identify numbers in books or the environment?

They then tested the children on a variety of number and math skills.

Can you put two ducks in the pond?

Can you point to the number 5?

If two horses are on the path, and another joins them, how many horses are on the path?

So, what did they find? Did either of the reading approaches predict number and math skill? Did they predict those skills better than the parents’ direct focus on numbers and math?

The Results

Yes, and yes.

The parents’ approach to reading predicted math success better than the parents’ focus on numbers.

And, when comparing the two approaches to reading,

A focus on letters and sounds led to better math performance than did a focus on the meaning of the story.

In the dry language of research:

Only letter-sound interactions could predict statistically significant unique variance in counting, number transcoding and calculation.

What Should Parents Do?

This research pool is deep and complicated, and — as far as I can see — we’re not yet able to offer definitive parenting advice.

So, this study found that parental focus on letter-sound interactions improved later math skills.

But:

Self-reports aren’t always reliable (although they’re very common in this field), and

The differences weren’t all that great, and

We have many different goals when we teach children to read.

That is: if our only goal were to help students understand numbers, then this study would encourage parents to focus substantially on letter-sound relationships.

But, of course, we want our children to think about the meaning of stories too. That’s one way they learn important developmental lessons. That’s how they think about meaning in their own lives.

This study — especially if it’s confirmed by later research — encourages us to use several strategies to teach our children about words and reading.

And, it gives us reason to think that those multiple approaches will help them with books, and with numbers too.

Website of the Day…
Andrew Watson
Andrew Watson
March 09, 2017

Website of the Day…

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AdobeStock_81917513_Credit

You’d like to understand probability and statistics more richly? Perhaps you’d like to visualize some of the more abstract concepts?

Here’s a website that will make your day. The landing page alone is worth the click…

Research Morsel: Gender Differences in Math (Again)
Andrew Watson
Andrew Watson
December 12, 2016

Research Morsel: Gender Differences in Math (Again)

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AdobeStock_41992792_Credit

The journal Intelligence recently published an interesting study [1] analyzing gender differences in cognitive abilities in the US and India.

The question hovering in the background is—as it so often is—“are there innate gender differences in cognitive abilities?”

That is: we have lots of data showing gender differences in various measures of academic success; are those differences inherent in genders, or are they socially created? Or, a combination of both?

To answer this question, you might look at the very best performers in—say—math. If there are substantially more boys in the top 5% of math scorers, and if that substantial difference persists over time, then you might think that–all other things being equal–boys are innately better at math.

This study, however, shows that the difference has shrunk in the last twenty years—in both the US and India. If gender differences in math are innate, then these results are a shocker.

Of course, other readers might see this study reinforcing a theory of innate gender differences.

  1. Although there is a smaller difference in math success between genders, that difference does persist. That is: there are still more boys than girls at the very highest end of math performance.
  2. The gender difference at the high end of verbal performance has not changed. Girls still score higher than boys do on such tests.

In my view, this study tends to confirm the hypothesis that social forces exaggerate—and perhaps create—gender differences in academic performance.

  1. I suspect that gender differences in verbal performance haven’t changed because we haven’t focused on them with the same energy and creativity that we’ve used to promote girls’ self-perception in math.
  2. While math gender differences persist in the US and India, they vary quite widely by country [2]: an odd finding indeed if boys are naturally mathier than girls.

Reasons to be cautious when interpreting this article—and this blog post:

  1. This research looks at gender differences in one very specific way: math and verbal performance at the very high end (“the extreme right tail” of the bell curve). There are MANY other ways to consider these complex questions, and we shouldn’t let any one way determine our answer.
  2. We have only recently begun to understand that gender isn’t always binary. I don’t think many researchers in this field have found ways to analyze math performance of transgender students.
  3. The article is still behind a paywall, so I haven’t seen the numbers. You might want to look at the underlying data to see if you find it persuasive.
  4. I, of course, have my own biases:
    1. I think that gender differences in academic performance are much more likely to be socially created than innate [3, 4]. And so, it’s not surprising that I interpret this article as I have. (It’s also not surprising that I’ve decided to write about it for the blog.)
    2. More broadly, I think the “innate differences” hypothesis just isn’t helpful to teachers. My job is to help this student learn academic material—these facts, these procedures, these moral habits, these life lessons. If I clutter my brain with the belief that “girls can’t do math,” I do my students a deep disservice because I make it harder for them to learn. That is: my potentially false belief turns into a self-fulfilling prophecy. All of Carol Dweck’s research [5, 6] and all Claude Steele’s research [7, 8], highlights this point.

If you’re especially interested in this topic, Lisa Damour—Director of Laurel School’s invaluable Center For Research on Girls—has produced many evidence-based summaries that can be helpful to your thinking.

  1. Makel, M. C., Wai, J., Peairs, K., & Putallaz, M. (2016). Sex differences in the right tail of cognitive abilities: An update and cross cultural extension. Intelligence, 59, 8-15.
  2. Else-Quest, N. M., Hyde, J. S., & Linn, M. C. (2010). Cross-national patterns of gender differences in mathematics: a meta-analysis. Psychological Bulletin, 136(1), 103.
  3. Hyde, J. S. (2005). The gender similarities hypothesis. American Psychologist, 60(6), 581.
  4. Eliot, L. (2009). Pink brain, blue brain: How small differences grow into troublesome gaps – and what we can do about it. Boston: Houghton Mifflin Harcourt.
  5. Dweck, C. (2008). Mindset: The new psychology of success. New York: Ballantine Books.
  6. Rattan, A., Good, C., & Dweck, C. S. (2012). “It’s ok—Not everyone can be good at math”: Instructors with an entity theory comfort (and demotivate) students. Journal of Experimental Social Psychology, 48(3), 731-737.
  7. Steele, C. (2010). Whistling Vivaldi: How stereotypes affect us and what we can do. New York: W. W. Norton & Company.
  8. Murphy, M. C., Steele, C. M., & Gross, J. J. (2007). Signaling threat how situational cues affect women in math, science, and engineering settings. Psychological Science, 18(10), 879-885.