Education Reviews · PhD Life · Science of Disney · Uncategorized

Mythbusting Mr. Incredible’s “New Math”

“Why would they change math?! Math is math! MATH is MATH!!” Bob Parr (aka Mr. Incredible) protested in Incredibles 2.

Most people who enjoyed or are heading out to buy the Incredibles 2 (released November 6th, 2018) will likely laugh at this joke. The joke works because so many parents today are struggling with the changes brought about by Common Core math. But is Common Core deserving of all of the jokes at its expense?

Although I sympathize with parents’ difficulties with helping their students with these “new” math problems, as a PhD student studying math education and math teacher education, I also want to share my knowledge about why Common Core math is a change for the better and also not much of a change at all.

Math Education Reform in the 1960s

The Incredibles and Incredibles 2 are set in the early 1960s, a time in which a different math reform called “New Math” was underway. New Math was a movement spurred by the launch of Sputnik and the United States feeling like they were falling behind the Soviet Union in a global competition for success and innovation (which had few tangible measures). The curriculum for new math wanted to enhance students’ conceptual understanding of math – knowing why standard algorithms work – rather than just procedural understanding – knowing how to do a standard algorithm.

Much like what we refer to as “inquiry learning” today, New Math wanted students to try and solve problems before being given the so-called rules, or standard algorithm (this practice is very common in math classes in countries like Japan whose students consistently earn higher scores than American students on international exams). Advocates of New Math argued for less repetitive drilling and instead for math that looked more like what mathematicians did – finding patterns and engaging with ideas like set theory and number theory. Even in this era, parents and teachers bemoaned the new content and strategies for teaching the content that involved more hands-on learning with objects rather than pen and paper arithmetic that they were used to.

Although New Math was effective in shifting the amount of time spent on arithmetic towards incorporating more advanced topics like geometry and calculus, teachers did not receive enough support in the mathematical content to achieve the goals of the movement in their classrooms besides using logic games.

After the end of the New Math movement, the goals of school math continued to swing back and forth between conceptual and procedural understanding, coming to a head in the so-called “math wars” of the 1990s, partly instigated by the Nation at Risk report which again highlighted how the United States was falling behind the rest of the world. Over the course of the decades between New Math and today, research in math education and the psychology of how we learn has made leaps and bounds and has been trying to impact what math looks like in schools.

Common Core in 2000s and 2010s

In the 2000s, the United States again felt like it was falling behind in several global economic indicators relative to countries like China, Japan, and South Korea as well as education leaders like Finland and Canada. The Common Core State Standards were seen as one important step towards preparing our students for the jobs of the future by again foregrounding conceptual understanding. Unfortunately, there are several myths about what the Common Core is and is not. I want to help dispel some of these so that parents and students alike can see how Common Core has not changed math in a negative way.

Myth 1: Common Core is a national curriculum.

This is false! First, the Common Core is not a curriculum.

The Common Core is a set of descriptions of what students are expected to be able to do in each year of school. They do not specify exactly what the content should be to help students master the set of skills nor do they specify exactly how teachers should teach it. To put in terms that the super-speedy Dash Parr would understand, the Common Core is determining where the finish line of the race is – not telling racers how they have to run, what equipment they have to wear, or whether or not they can use their superpowers.

Secondly, the Common Core is not federally mandated.

The Common Core was developed independently from the United States federal government by a bipartisan committee sponsored by a collection of governors and non-profit organizations. Before the Common Core, states were extremely varied in the race they were expecting their students to run. The developers were just trying to make sure the finish line was in the same spot for students in all 50 states so that a student who excelled at the 100-meter-dash wouldn’t be expected to run a mile if they moved from one state to another.  Furthermore, before the common core, it was difficult to compare proficiency levels across states. Continuing the analogy, a student who ran a 20-minute mile might be classified as “super” on one state’s assessment, but “not super” in another state because that state defined “super” as running a mile in less than 10 minutes.

After the standards were developed, states were able to choose whether they wanted to adopt these standards – and the standardized tests that accompany them – or not. Currently, only 43 states are using Common Core standards; several states declined to adopt them or have declined to use the standards-aligned tests. Instead, they’re developing their own standards and tests (that are often built on and look very similar to Common Core).

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The national government under Obama did have an effect on how many states decided to adopt Common Core though because they sponsored a competition called Race to the Top for states and school districts to receive bonus federal funding. To enter the competition and receive money, a district had to demonstrate how it was going to align with the Common Core standards, whether with curriculum or assessments. They had to provide some evidence that they were going to expect students to run the race with the finish line set by the Common Core.

Myth 2: Common Core is part of the Democrat agenda.

This is false also!

I will repeat that the The Common Core was developed by a bipartisan committee and underwent revision from people on both sides of the political divide. Several conservative organizations like the Fordham Institute (a research organization) and former Florida Governor Jeb Bush support the Common Core because they acknowledge the positive outcomes that are associated with raising our expectations of schools. And several liberal figures and organizations like the National Education Association oppose the Common Core because they do not support standardized tests in general or are worried about the impact the change will have on teachers.

Although I am not sure where Mr. Incredible stands on the role of the federal government or liberal vs conservative ideals, I think more of this complaints about math reform would fall under the umbrella of the next myth.

Myth 3: Common Core makes math harder and more complicated.

Many of parents’ complaints that can be found online or in everyday conversation are that the procedures encouraged by the Common Core math standards are extremely inefficient or ridiculous. Frustrated parents think it is more important to get the correct answer to a problem efficiently than to show their work using these complicated procedures.


The main mistake is that Jack subtracted 306 (3 groups of 100 and 6 groups of 1) rather than 316 (another group of 10).

To these parents I want to say: although some of the procedures may involve strange steps or seem silly, having students solve problems in these ways or in multiple ways helps them achieve conceptual understanding, which is better for them in the long-run (Richland, Zur & Holyoak, 2007; Rittle-Johnson, Star, Durken & Loehr, 2018). Common Core makes math different, not necessarily harder, and is supported by research in math education about how learning the reasons why the efficient algorithm works is better for retention and problem-solving. With the new standards, students are expected to be able to explain and critique (two skills which are more highly valued in the workplace) different methods for solving problems rather than just memorizing one way.

But what about the fact that test scores have gone down since Common Core was implemented?

Yes, scores on yearly assessments have dropped in the first few years of implementing Common Core and this can be interpreted as evidence that Common Core makes math harder. But the tests that have been developed to align with the Common Core standards are considered to be harder partly because they are unfamiliar and ask questions in different ways than the ways students are used to being tested and the way teachers are used to teaching.

This last piece about how teachers and their teaching has been affected by the Common Core is one of the main reasons that I am critical of the Common Core. In many places, teachers have not been provided with enough support for successful implementation. Teachers have to master the very skills that they are expected to teach their students because their math education did not teach them in these ways. Teachers also have to learn new ways of teaching because along with new skills come new misconceptions and mistakes that learners are likely to make. Teachers have spent a lot of time figuring out how to best instruct and help students with mistakes related to the standard algorithm. Now, teachers need help with identifying and remedying the kinds of mistakes that students make when asked to solve problems that align with the Common Core.

But what about parents?

Unfortunately, this does not help parents much because they, like Mr. Incredible, still want to help their kids but aren’t receiving the kind of training that teachers have access to.

Websites, guides and books have been popping up to offer parents assistance with Common Core. But parents don’t need to become experts in Common Core methods; sometimes it can just be better to ask kids about their thinking and discuss it with them. Making claims and defending them is one of the most important skills that the Common Core tests assess, regardless of age level or math content area. Furthermore, try not to pass on math anxiety and instead foster a positive math environment by playing math games or showing your child how you use math confidently in your everyday life. One of the most effective ways to help your kids with math is to promote a growth mindset – the idea that intelligence or success, especially in something like math, is not pre-determined but rather can be developed through effort and a positive attitude – by praising your child for not giving up and trying their best. In my own tutoring of high school students, I find that just expressing enthusiasm for math and how cool I think it is when I realize a new aspect of how math is interconnected can go a long way.

The Future of Common Core

Social scientists in academia and in policy are doing a lot of research on how these new standards are changing the lives of students. Some places are showing signs of success, like closing the achievement gap between more and less disadvantaged students. Research is also being done on improving teachers’ ability to teach Common Core, which doesn’t mean having them just take more math classes, but rather understand the connections among math content areas like students are expected to do. More work still needs to be done on how to effectively help parents become more capable and confident with Common Core. One of the first steps towards helping students is acknowledging the historical precedent for Common Core math and embracing the new ways of solving not as infuriating challenges to adults’ superpowers of patience but rather as incredible opportunities to learn together.

Additional reading

Vox has a great explanation of the Common Core

NPR has a Common Core FAQ as well

Disney Trips · PhD Life · Science of Disney · Uncategorized

How spectacular is Epcot’s SpectacuLAB? From a teaching and learning perspective

Only the magic of Disney could turn the demonstration of a few “high-tech” measurement tools into an entertaining, interactive story that might actually be good at teaching kids (and adults!) some simple science concepts.

If you haven’t had a chance to see it before, the SpectacuLAB is a 30-minute play about an intern’s first day in a science lab learning about the lab’s cool technology through “experiments” that involve members of the audience! As of 2018, the SpectacuLAB is sponsored by Murata and most of the actors are real scientists from Science from Scientists, a non-profit that aims to get more 4th-8th grade kids interested in science. The show aims to teach force, acceleration, barometric pressure, and sound waves using Murata’s accelerometer, pressure sensors, and ultrasonic sensors.

Overall, the show is endearing and fun to watch at least once (I even saw the show as part of a second date!), but it wouldn’t hurt to incorporate more research-based evidence about learning science into the experience.


The Good

1. The show’s scientists are actually good role models for increasing diversity in STEM.

Research has shown that having a personal relationship with a scientist as well as seeing positive images of scientists increases students’ commitment to STEM and the SpectacuLAB can facilitate both of these. Both times I’ve seen the show, at least one performer has been a person of color or a woman (or both!) and kids have opportunities to start to build relationships with these scientists by participating in the show and asking them questions during Q&A. Although the majority of the audience is likely to be White and there are still many other institutional barriers hindering increased diversity in STEM, this show is a step in the right direction.

2. The show uses effective analogies, both to real-life experiences and across contexts.

To help explain pressure, the show makes analogies between ultrasonic sound waves making music to a car’s back-up monitoring, and to ears popping on airplanes.Incorporating students’ real-life experiences into science instruction is one of the most highly encouraged practices in the Next Generation Science Standards because it helps students see science all around them. There are also analogies between several applications of force – smashing one cup, smashing 1000 cups, applying weight to balloons, and laying on a bed of nails. This kind of careful control of what is being compared highlights what aspects are important for the scientific principle to hold – the distribution of force across a greater area – while downplaying more extraneous aspects – like the materials.

3. The show incorporates Jungle Cruise-esque humor and moments of suspense.

Although the empirical evidence on the effectiveness of humor and suspense on learning is weaker than for other techniques, there is some support that both are effective for creating memorable experiences. Humor has been shown to help memory (for punny jokes, in particular, see Summerfelt, Lippman, & Hyman Jr. 2010), and increases student motivation. On the flip side, the fear associated with moments of suspense may also be effective for learning. Usually suspense and fear are stressful and stress hinders learning, but the SpectacuLAB resolves the highly suspenseful moment of having someone lay on a bed of nails. Just like when music or horror movies resolve tension, the resolution of a suspenseful moment may help learning because it is a strong emotional experience. Both humor and suspense affect the dopamine reward system in the brain; linking the memory for science content with happiness from dopamine may help the learner retain the information longer.

The Less Good

1. The science doesn’t always work perfectly – especially the barometric pressure sensors.

The barometric pressure sensor game is very entertaining to watch but the sensors don’t always change color predictably, which can make the game frustrating for the players and confusing for trying to understand the science behind it. I think this demonstration could benefit from a screen showing a representation of air pressure in the room to explain what the different colors represent more than just a graph of where pressure is high or low. Even when the other experiments don’t go according to plan (like balloons popping unexpectedly or the ultrasonic drums being slow to respond), the scientists roll with the punches pretty well and this provides more opportunity for interrogation of the ideas after the show.

2. The show doesn’t allow for enough active learning opportunities.

Although holding a sensor is more active than sitting and watching someone else hold a sensor, this still isn’t the most effective way to learn. Active learning is when students (of any age!) are “engaged in” the learning process. Some examples that have been shown to be effective are self-explanation or even think-pair-share. Self-explanation – the process by which a student explains their own ideas to themselves through writing or speaking – has been shown to be effective for learning and retention, especially when dealing with multiple representations like in science. Think-pair-share is a common practice in classrooms at all grade levels in which students first think about their response, then tell a partner, and then share out to the whole class. These strategies are more effective for learning because they involve more construction of knowledge and questioning of relationships among ideas. Due to time constraints, these strategies are hard to incorporate in such a short show, but hopefully the show still inspires kids and adults alike to continue conversations about the content after they leave the theater to engage in more active learning.

3. The show could more strongly encourage more scientific practices.

Although the audience is exposed to a fair amount of scientific practices in the show – such as replication of results by repeating an experiment, manipulating independent variables (like stronger versus weaker applications of force), and interpreting data in graph forms – more opportunities could be built in for practices like making and revising hypotheses, and creating representations. When explaining force and pressure, there is some encouragement of making hypotheses but this could be incorporated more frequently and more broadly by getting the whole audience to share their ideas with even just a show of hands. Going along with the lack of active learning strategies, audience members don’t get the chance to demonstrate their understanding in scientist-like ways. Providing activity pages that encourage them to draw what pressure looks like, or make models about their predictions for additional scenarios could help them feel more like scientists. Outside of the theater, there are small interactive exhibits that allow visitors to practice more with the ideas from the show in authentic, scientist-like ways but these aren’t even mentioned during the show.

What are your thoughts on the SpectacuLAB? Which learning or physics concept would you want to know more about?

Additional Reading

Tsui, L. (2007) Literature review on strategies to increase the diversity of STEM fields

Allen-Ramdial, S. A. & Campbell, A. G. (2014) Reimagining the pipeline: Advancing STEM diversity, persistence, and success

Humor and learning popular press article

Humor and learning research article

Suspense and learning

Prince, M. (2004) Active learning

Berthold, K., Eysink, T. H. S., & Renkl, A. (2009) Self-explanation

Gentner, D. & Toupin, C. (1986) Surface similarity hinders analogy

Next Generation Science Standards

Listicles · PhD Life · Princess Life · Uncategorized

Pixar Women, PhD Part 2

EVE – Ecology.

EVE’s sole mission was to collect data about life on Earth. She’ll likely need to work on her data analysis skills but armed with a PhD, she will be invaluable to the re-colonization of the planet. Specializing in urban ecology or sustainability, she will have a lot to contribute to establishing a healthier relationship between humans and their new ecosystems.


Ellie – Latin American Studies.

Ellie’s one dream in life was to go to Paradise Falls in South America. Based off of Iguazu Falls in Argentina, Ellie’s dream might have come true sooner if she studied Latin American Studies in graduate school. She could have done field research near the falls or at least traveled to conferences hosted in the region – and conference travel is a great way to see new parts of the world using university (or external scholarship) funding! Focusing on ecotourism or the relationship with animals across different Latin American cultures would be perfectly suited for her experience as a zookeeper!

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Joy – Positive Psychology.

Eternally optimistic, Joy would want to figure out how to bring that same happiness to others. The positive psychology branch of psychology focuses on human flourishing rather than floundering, on what makes a good life worth living, with several studies on techniques for promoting feelings such as self-confidence, compassion and gratitude. Joy might even build on work by Fowler & Christakis about how contagious happiness is. I’m sure she’d be looking out for Riley too by investigating how this phenomenon holds in young adult populations. I wonder if this might even explain the popularity of boy bands.


Sadness – Social Psychology.

On the flip side of Joy, Sadness would likely study depression, dabbling a little bit in its relationship to memory formation. I would be worried about Sadness facing impostor syndrome while in graduate school but with Joy by her side and studying depression herself, I think she’s well-equipped to take on any emotions that come with the hard work required.


Cruz Ramirez – Mechanical Engineering.

Cruz is apparently quite a whippersnapper with the latest technology but she’s also mastered motivational speaking. She’ll put those smarts to work studying ways to improve engines (and other machines to train her fellow cars). Her personality will really shine through when she leads her own lab because she will likely be a very engaged mentor and well-regarded member of her department.


Mama Imelda – Musicology.

Mama Imelda always wanted to do music but repressed that part of her when Hector left her; pursuing a PhD in Musicology would allow her to fully appreciate music. While she would not be performing as much, she would be able to study music and its cultural contexts. I think she might be particularly interested in exploring several case studies of familial relationships with music in Mexican culture.

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Coco – Marriage and Family Therapy.

If only Coco had earned her PhD earlier, maybe Hector could have visited his daughter on Día de los Muertos a few more times. Coco would definitely use her doctoral training to help her own family and dozens of others resolve their interpersonal conflicts. She could even specialize in Latin American therapy and would be a valuable resource in the era of family separation at the U.S. border.

Voyd – Astrophysics.

I can already see it: Violet and Voyd earning their PhDs in physics simultaneously even if they are across the country or world from one another. With the ability to create wormholes, Voyd would likely be fascinated with the next best thing: black holes. Thus, a PhD in astrophysics would give her the training on the underlying mechanisms of her own powers and she’s already got a get-up suited for space travel.

Edna Mode – Textiles.

Although most fashion designers do not have their PhDs, Edna is something else. These programs are found more outside of the U.S. but would be well-suited for her passions for materials that need to perform in extreme conditions. With her reputation, she would likely be a superstar at conferences and would no doubt be able to find a great deal of funding. She might not enjoy the longer-term projects typical of PhD research but she seems well-suited for pulling all-nighters to prepare for presentations the next day.


Listicles · PhD Life · Princess Life · Uncategorized

Pixar Women, PhD Part 1

Bo Peep – Animal Sciences.

Bo Peep has conspicuously been missing in the last two Toy Story movies and I think the possibility of her disappearing to go earn her PhD in Animal Sciences is somewhat likely (although I don’t know how she would be able to manage without her support system of the rest of Andy’s toys). This shepherdess would be overly qualified to conduct research on sheep behavior and agricultural practices that can improve the lives of her baaaa-es and other ruminants.


Gypsy – Quantum Physics.

Armed with the knowledge that her moth (and other butterfly) wings are composed of nanostructures with incredible potential for new technologies, Gypsy would want to better understand the physics underlying her own biology. Because the structures are so small that they need to be studied with electron microscopes, Gypsy would have to pursue knowledge in the realm of quantum physics. Then, she would be able to explain how light interacts with these materials for various applications even if most people think it’s just magic.


Rosie – Theater and Performance Studies.

Rosie was quite theatrical in her role as rhinoceros beetle tamer in P.T. Flea’s circus. She would combine her affinity for the performing arts with her curiosity about gender studies. With most of her life having been defined by her relationship with her husbands, Rosie would spend the years of her PhD examining how women were portrayed in various plays, musicals, and other genres.


(For the princesses of Bug’s Life, check out this post)

Jessie – Conservation Biology.

Jessie would turn her passion for taking care of various critters into a career. Although she might have had a mildly traumatic experience with museum conservation, biological conservation is much more up her ally. She’d be able to save species on the brink of extinction by collecting data on ecosystems – her focus of course would be on the deserts and mountain areas of the American West and their animal inhabitants.


Boo – Mythological and Religious Studies.

Boo’s formative experience in the Monster World would shape her desire to understand why other people develop a fear of monsters when she clearly was not afraid. This field would give her the tools to explain why people believe what they believe as well as document how societies cultivate shared beliefs. If she really has time-traveling powers as the Pixar theory says she does, she could even do comparative analyses of medieval Scottish beliefs with ideologies of Monstropolis.


Roz – Criminal Justice.

I’m pretty sure that Roz already has her PhD in Criminal Justice, but even if she doesn’t, now that her mission at Monsters, Inc., she has plenty of time to go back to school. Her dedication to finding the bad guy and maintaining secret identities is the glamorous side of studying criminological theory, public policy, statistics, and forensic science, all of which she would be trained in as a PhD student.


Dory – Cognitive Psychology.

Motivated by wanting to better understand the relationships between her own memory loss (specifically, anterograde amnesia or the inability to form new memories) and her seemingly more proficient place-based memory, Dory would pursue a PhD in Cognitive Psychology. She would likely study various techniques to assist in coping with anterograde amnesia while using those techniques herself to become a successful academic. She might also specialize in spatial and place-based memory and spend a few years on a project investigating the neuroscience of hippocampal place cells.


Helen Parr/Elastigirl/Mrs. Incredible – Material Sciences and Engineering.

Although the origin of supers’ powers in the Pixar universe is still unknown (whether it is due to genetic mutation or some sort of government experimentation), Elastigirl would want to leverage her expertise in flexibility in her PhD program. She would research the stretchiness of various materials like hydrogels and conditions that affect this property to improve its durability and functionality in military and civilian applications.


Violet Parr – Physics.

Violet’s powers are both a function of visible light, so she would study physics, and more specifically optics, to earn her doctorate. Because invisibility requires modifying the frequency of light reflected off of objects to be in a range that is not visible to the human eye, Violet would build on the research that Edna had to have known about to develop the teen’s supersuit so that the phenomenon can be more deeply understood. While her shield generation also seems to rely upon ultraviolet light, it defies Newtonian laws of physics so she would definitely have to do a few experiments to be able to explain this ability.

Sally Carrera – Law.

Sally says that she used to be an attorney so she has earned her doctorate in law already. Although she did not very much enjoy her legal career, she used her knowledge to help out Lightning McQueen when he got into trouble in Radiator Springs. She could do plenty more pro-bono work for other troubled cars but also seems to be well-suited to consulting for people fighting to preserve historical landmarks and fighting against business owners who want to destroy such things to turn a profit.

Colette Tatou – Organizational Psychology.

Colette’s experiences in Chef Skinner’s kitchen would make her interested in improving the working conditions of several other kitchens, especially for women in the industry. Earning her PhD in Organizational Psychology would give her the theories and tools to be a consultant to numerous restaurants to improve their productivity and employee satisfaction. Maybe Disney Parks and Resorts would even hire her!


Which Pixar PhD woman are you most like?

PhD Life · Uncategorized

PhD Friday Week 1 Synthesis

For the past few months, I’ve been in a rut regarding my research topic. On one hand, I feel like I am just scratching the surface of an immensely large body of literature on analogy and comparison for learning in laboratory and classroom settings, which makes is both overwhelming and motivating. But on the other hand, I also feel like pursuing my current line of research is too theoretical and not preparing me for the kind of career I want to impact learning beyond the classroom by designing educational technology.

After reading the papers I summarize below, I feel like I’ve finally had a breakthrough that can combine both my theoretical and applied interests. In addition to helping me think of some new research ideas, I think the learning principles discussed are relevant to teachers and learners outside of academia. I look forward to discussing any ideas that the following article summaries make you think about!

Gamification of Cognitive Training 

Two different protocols of working memory training using an N-back task led to mostly similar improvements and performance on transfer tasks. As expected, participants reported being more engaged and expending more effort in the gamified condition of the N-back task which was designed to have more motivational elements than the non-gamified condition. Participants in the gamified condition showed greater improvements in working memory after four training sessions than did participants in the non-gamified condition.

My Two Cents: This paper harkens back to an early stage of my research career when I was completing an honors thesis on the psychological construct of executive function, which includes working memory, inhibitory control, and switching ability (which is kind of a combination of the other two). Although I focus less on these psychological constructs now, the principles for what makes a good game are relevant citations for me to follow up on when I design my own educational software.

Variation Theory

The authors outlined some tenets of variation theory that inform the selection of examples for instructional purposes. Variation theory recommends a progression of examples that first enable contrast, then generalization and then fusion. What this means is that students need to see non-examples as well as examples to illustrate the important point that a teacher wants them to learn. For example, it is hard to know what defines a triangle until a triangle is contrasted with a square or a circle. Then after seeing several examples of triangles being contrasted with other things, you learn each of the properties that makes a triangle special. Combining all of these triangle properties helps you understand the concept of a triangle. In the rest of the paper, the authors detail what is made possible to be learned (different from what is actually taught and what from students learn from a lesson) through a teachers’ selection of examples and tasks in two different lessons on solving equations.


My Two Cents: This is where my breakthrough started to materialize even though the paper was still very much focused on in-classroom learning. Variation theory is basically what all of my ideas about why comparisons are important boils down to and I have yet to see it explicitly implemented in educational technology.

On top of giving a very applied explanation of how variation theory might be used to plan a lesson, this article happened to be in a special issue that includes other articles I was excited about: applying principles of cognitive psychology to teaching. Even though the other articles in the special issue barely mentioned variation theory, they all seemed to fundamentally depend on these ideas.

Example-based learning, self-explanation and comparison

Example-based learning is the idea of providing students with a set of examples that helps them develop the common procedural or conceptual thread between them. Different from traditional math instruction in which a teacher demonstrates solving several math problems in the same way, example-based instruction would be more akin to a student looking over the worked out examples and the explanations for each step in a textbook. In his article on example-based learning, Renkl made the case that this technique is most effective when students engage in self-explanation while they are studying the worked examples, especially when students are new to the material.

Self-explanation was a separate technique further outlined in another article by Rittle-Johnson, Loehr & Durkin (2017). In contrast to a common practice in several math classrooms in which the teacher has the main responsibility for explaining the reason for a problem-solving step or the principle that two problems have in common, self-explanation is when students do this work themselves. The greater mental effort expended by the student when self-explaining helps improve retention over hearing someone else explain. However, students need to be trained in how to do high-quality self-explanations. Unfortunately, not much detail was provided on how to do this besides

These two techniques were combined in a chapter on comparisons more broadly. Durkin, Star & Rittle-Johnson (2017) acknowledge that there are several types of comparisons that are beneficial for learning depending on what is being compared and what question is asked. Comparing an incorrect solution with a correct solution (an example of example-based learning) is beneficial but is more beneficial for students with higher prior knowledge who are better able to distinguish the important differences and figure out the reasons why one is correct (self-explanation). For students who are less experienced with material to be taught, it is more helpful to compare the same solution method for multiple problems in order for them to gain confidence with one method before introducing them to alternative methods for solving similar problems.

My Two Cents: I see each of these techniques relying on variation theory. In order to have effective example-based learning, examples that highlight different and similar features need to be carefully selected. Prompting students to make comparisons helps draw attention to these differences and similarities as well as encourages them to explain their reasoning for why these differences and similarities exist.

Which principle is most relevant to your teaching and learning experiences?


Durkin, K., Star, J. R., & Rittle-Johnson, B. (2017). Using comparison of multiple strategies in the mathematics classroom: Lessons learned and next steps. ZDM, 1-13.

Kullberg, A., Kempe, U. R., & Marton, F. (2017). What is made possible to learn when using the variation theory of learning in teaching mathematics?. ZDM, 1-11.

Mohammed, S., Flores, L., Deveau, J., Hoffing, R. C., Phung, C., Parlett, C. M., … & Zordan, V. (2017). The benefits and challenges of implementing motivational features to boost cognitive training outcome. Journal of Cognitive Enhancement, 1-17.

Renkl, A. (2017). Learning from worked-examples in mathematics: students relate procedures to principles. ZDM, 1-14.

Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM, 1-13.


Listicles · PhD Life · Princess Life · Uncategorized

If Princesses were PhD Students Part 2

Some of the best and most interesting Disney characters to me are the unofficial Disney princesses. What would these nine fearless females choose to study in graduate school?

Megara – Media and Communication.

She seemed to be quite confident in giving Hercules advice about how to adjust his public image. Becoming an expert in how rhetoric manifests in various mediums both ancient and modern would be quite lucrative for this dame; she could use these skills to develop her own media consulting company for all of the big stars or start her own magazine that would rival Cosmo with bitingly witty pieces on sandals, weak ankles, togas and how to become equals with any man despite how godly they might seem to the general populace.

Jane – Evolutionary Anthropology.

This likely descendant of Belle shares the bookworm’s affinity for beastly creatures as demonstrated by her fondness for gorillas, chimpanzees and of course Tarzan. Living in the jungle for so many years would make her one of the most qualified candidates to be a leading researcher of apes, so I expect that she would apply to the Max Planck Institute for Evolutionary Anthropology. I’m sure she’d leverage her strong bonds with the bands of gorillas to study their behavior and communication very intimately and become a well-published expert as long as the animals don’t trash her camp repeatedly.

Anna – Art History.

During the time that Anna spent in the halls of Arendelle’s castle growing up, she seemed to deeply appreciate (and talk extensively to) the paintings lining the walls. Studying art history would give Anna the skills to preserve the works of art as well as learn more about the customs of Arendelle that she can no longer remember. As a result of her studies, she likely would not stop talking about the rich artistic traditions of Scandinavia, which Kristoff would happily endure and perhaps even contribute to himself.

Elsa (yes, she’s technically a queen) – Civil Engineering.

She already seems to know a lot about building with ice so she’d probably revolutionize the field of civil engineering by using her expertise of ice’s properties as well its advantages and disadvantages to design structures in regions other than Arendelle. Elsa would also shatter some glass ceilings as she rise in the ranks of the male-dominated field. I would just feel really sorry for Elsa networking at conferences because everyone would come up to her “wanting to break the ice” which would probably get old very quickly. On a positive note, her several years in solitude have prepared her well for hours writing papers alone.

Nala – Marriage and Family Therapy.

Where would the animal kingdom be if not for Nala’s amazing skills at helping Simba confront his past and his own greatest fears? Once the famine of Scar’s years as king subsides, Nala would be better able to serve the kingdom if she could receive additional training to hone her skills at negotiation. I’m sure there’s way more drama in the pride than is shown in the movie and she would be just the lion to help resolve any conflicts.

Kida – Religious Studies.

Like Moana, the mythology of her people would spark a need to know more about why people believe what they believe. Learning about Mesoamerican and Southeast Asian religious practices would only be the launching point for writing a thesis on Atlantean beliefs and customs. Several of her people would follow in her footsteps to share and develop more expertise about the unique Atlantean architectural styles and functioning of crystals.

Vanellope von Schweetz – Computer Science.

Having been erased from the code of her own game, this spunky gal would go to school to ensure that none of the citizens of Sugar Rush face the same fate as she did. Understanding the intricacies of several programming languages and how videogames store their data could make her the heroine of the whole arcade if it were ever in danger again as well as make her much more adept at navigating the Internet (Wreck-It Ralph 2 *hint hint*). Furthermore, gaining expertise in computer science could also help her make faster and more environmentally friendly electric racing cars; who’s to say that even a gingerbread car couldn’t run on electricity?

Atta – Biology.

As such a broad field, biology would be best suited for Atta’s need to know about ecosystems, agriculture, animal (or insect) behavior as well as environmental policy in order to be an informed leader of her colony. Knowing more about photosynthesis and the nutrients and farming practices that would best support high, continuous crop yield would ensure that she would rarely have difficulty feeding all of her “subjects” and maybe even feed some grasshoppers in the mean time. She might have to improve upon her skills at standing up for herself in the field but standing up to Hopper would definitely make for a good personal statement.

Dot – Mechanical Engineering.

With Flik and Atta as role models, Dot will have witnessed dramatic revolutions in how her colony harvests food by the time she would be prepared for a PhD program. Her experiences with birds (both real and constructed) would likely have piqued her fascination with machines, especially flying ones. Thus, studying how things move would enable this princess to improve the harvesting machines and potentially lead the growing colony off the island either by riding the streams from rainstorms or via air travel.

Read Part 1 here!

Which characters would you like me to do next?


Listicles · PhD Life · Princess Life · Uncategorized

If Disney Princesses were PhD Students

Each and every Disney Princess is capable of putting in the dedication and effort required to earn her PhD. So if their dreams were to finish grad school, what would their theses be on?

Snow White – Materials Science and Engineering.

Based on her life at 14, which leaves a lot of time to grow and have her interests change before the typical age of entering graduate school (looking at you Princess Leia), Snow would likely get sick of cleaning up after seven little men pretty quickly. But channeling the inventiveness of real-life women who were looking to improve the efficiency of the tasks they were expected to do (and came up with a few handy things like windshield wipers, Kevlar, and solar-heated homes), Snow White would set out to study materials and their various cleaning properties. Who knows, she might even patent a robot besides the Roomba (also invented by a woman) that whistles while it works!

Cinderella – Clinical Psychology.

Her kindness and her own childhood experiences would prepare Cindy well for researching and providing therapy for her royal subjects, particularly those in foster care. Taking classes would give her knowledge of forms of care to prescribe other than befriending mice or expecting a fairy godmother. I just hope she’s more punctual with her clinical appointments by the time she graduates than she was with her the stroke of midnight.

Aurora – Neuroscience.

Specifically, she would study the still unanswered questions of why humans need sleep and how consciousness works during slumber. This would allow her to better defend her everyday arguments for why cursing the entire kingdom with 100 years of slumber was actually GOOD for everyone. She could also run studies on the neuronal activation patterns of true love’s kiss which I’m sure Prince Philip would happily volunteer for.

Ariel – Anthropology.

This little mermaid has always been fascinated by humans and would have a hard time settling in on ONE question or even just a few to answer with her dissertation. But she did seem particularly keen on fire and all its causes and consequences, although incorporating that into a likely career in museum curation (she’s already got gadgets and gizmos aplenty) might be a little difficult. Nevertheless, I’d love to see her take on an interactive exhibit about man’s red flower.

Belle – Comparative Literature.

This beauty’s obsession with stories and faraway places isn’t focused on any one time period or even language (the tale of Aladdin is rumored to be one of her favorites though). Reading through the entire library in the castle she shares with Beast would give her a voluminous head start on the adventure of earning her doctorate.

Jasmine – Economics.

She wanted to know how the people outside of her palace walls lived so she’d have to pick a program that allows for in-depth field work. This, coupled with Jasmine’s responsibilities as a sultana (I just learned a new word), makes studying economic theories a good fit. She’d learn how to determine how her policy decisions would affect the economy. This would help her devise better welfare programs than stealing bread and apples.

Pocahontas – Environmental Policy.

Protecting the natural environment is of utmost importance to this daughter of a village chief. Researching the effects of deforestation on ecosystems around the globe and the effectiveness of various grassroots organizations would probably be her choice of dissertation topic. I can see the title now – We are All Connected To Each Other: In a Circle, in a Hoop that Never Ends – because all good titles have colons.

Mulan – Political Science.

Excellent army generals are often appointed as higher-ups in government, and Mulan would definitely be my choice for Secretary of Defense. After years of researching women’s roles in government and military decisions, maybe picking up a law degree along the way, she’d have a few more diplomatic tricks up her sleeve than using fireworks as her weapon of choice. And she could pave the way for proving that women aren’t “too temperamental” to lead an army.

Tiana – Biochemistry.

Besides earning an MBA, Tiana would realize that gaining a better understanding of the chemistry of food and how humans enjoy it would be the next best thing for growing the success of her restaurants. With a working knowledge of the science of digestion, I wonder if she’d ever put frog legs on the menu.

Merida – Aerospace Engineering.

Stemming from her passion for archery, Merida would likely have an interest in other flying objects like planes and spaceships. Plus, it’s a field that has a lot of barriers to break – for women, the speed of sound, the speed of light, etc. She’d be in the thick of developing military technology as well, which would pair well with her experiences dealing with the other clans.

Rapunzel – Physics and Astronomy.

Despite the wealth of knowledge and skills that she accumulated in her years in her tower (somehow with only a few books and a lot of craft supplies at her disposal), what fascinated Rapunzel the most were the stars. Naturally, her new dream would be to study the forces that control everything from her hair’s incredible strength to the creation of “balls of gas, burning billions of miles away”, to quote one of Disney’s most philosophical characters. She might even win a few scientific images contests with her artistic inclinations.

Moana – History.

All she knows is that she is descended from voyagers and I’d bet ten coconuts that she wants to learn a little bit more about her people. Her persistence and curiosity would serve her well in navigating through the literature on Polynesian History and Culture to write an impressive dissertation. She would probably have to find a new advisor besides Maui though – who knows what the department policy on having a demi-god as a committee member would be?

What discipline do you think these twelve leading ladies would choose? Comment below with which ones you agree with most!