Don't Worry About Later Bells. Rest Easy

In his first post, Hampton Tutors senior coach David Westwood discusses research on sleep in adolescents and the Seattle School District’s recent implementation of delayed school start times


Schoolchildren of all ages have been obliged to follow early schedules for decades, with the first bell usually ringing shortly after or shortly before 8:00 a.m. throughout the country. This was not the case, however, when public education became institutionalized in the United States in the 19th century. During that period, public schools began the day at 9 o’clock, coinciding with the start of the canonical work day. Based on research and, undoubtedly, countless incidences of bags under students’ eyes, there is growing support for a return to later starts of the school day. In 2016, Seattle led the way by pushing its public schools’ first bells from 7:50 a.m. to 8:45 a.m. and the results of this experimental policy are beginning to stream in.

Delaying school start times became a topic of national discussion in 2014, when the American Association of Pediatrics released a statement calling for “efforts of school districts to optimize sleep in students and [urging] high schools and middle schools to aim for start times that allow students the opportunity to achieve optimal levels of sleep (8.5–9.5 hours) and to improve physical (e.g., reduced obesity risk) and mental (e.g., lower rates of depression) health, safety (e.g., drowsy driving crashes), academic performance, and quality of life.” The AAP made these suggestions in light of studies on the effects of adolescence on circadian rhythms. They cite two critical biological changes that occur during these years: nighttime melatonin release is delayed by a couple of hours in adolescents compared to younger schoolchildren and adolescents experience less of a drive to sleep versus preadolescent individuals. Regarding the former factor, a 2008 paper by Frey et al. in PLoS ONE demonstrated that many children change from morning people to night owls while going through puberty, which they linked to melatonin—it is simply easier to fall asleep if there are heightened levels of melatonin coursing through one’s system. Teens also experience a slower onset of fatigue compared to younger individuals, as shown by increased “time to fall asleep after being awake for 14.5 to 18.5 hours in postpubertal versus prepubertal teenagers.” Nevertheless, the optimal number of hours of sleep, which ranges from 8.5 to 9.5 hours, does not change as children grow older, suggesting that standard bell schedules will result in a net decrease in resting hours.

After the Seattle School District heeded the AAP recommendations and delayed its first bells, scientists at the University of Washington and the Salk Institute analyzed the effects of the altered schedule on student health. Two groups of students were established: one drawn from first period biology classes at Roosevelt High School and Franklin High School in 2016 and another drawn from “the same grade, classes, and schools and during the same time of the year but in 2017, when the new school start time had been in effect for 7 months.” Students made daily entries in online diaries, took mood assessments, and recorded their grades over the course of the study, after which the data were effectively normalized for preexisting sleep disorders, physically-demanding extracurricular activities, sex, race, and commute time. The study found that the delayed first bell resulted in a median increase of 34 minutes of sleep and a 4.5% increase in students’ median grades. Attendance also improved.

As education research advances and as pedagogical techniques focus more on the needs of students as individuals, there is hope that school districts will implement later bell schedules for the benefit of their students. Longer periods of sleep result in more attentive students, better absorption of material, and fewer bags under eyes.


Struggling With Math? Practice Leads To Progress

In his latest blog post, Hampton Tutors Coach Tim Barnes discusses a fascinating study of math skill development in kids over time. The results suggest that, when it comes to learning (and therefore teaching) math, routine and consistency are key.


What kinds of brain changes occur when you learn how to solve a new math problem? A large group of researchers at Stanford have recently published their results of scanning kids’ brains over years of math development. They looked for large-scale structural differences that tease apart different aspects of learning such a complex set of skills.

This is one of the first longitudinal studies of brain connectivity in developing children relative to a task; these studies are especially difficult to set up because participants move away or drop out, and the experiment lasts longer than the typical length of both a university research position and a government grant funding cycle. The alternative of scanning kids of all ages at one point in time is much more tempting, but is almost useless because one cannot truly disentangle the general effects of age and the specific effects of each child’s unique development schedule.

The researchers measured three general classes of variables: they measured each kid’s general math and reading aptitude with a standardized test (Pearson WIAT-II) at each visit; they measured their speed at performing specific math problems in the brain scanner; and they measured brain activity with functional magnetic resonance imaging (fMRI) while the subject did math problems or lied passively in the scanner.

A statistical test showed that the subjects’ standardized test scores generally stayed the same, so they assume that the kids generally all developed their math skills at a similar rate. Because the subjects have to hold still in the scanner, the types of math problems that can be done in the scanner is limited; in this experiment, simple equations were shown (2 + 3 = 7) and the subject had to respond whether the equation was correct or incorrect. Since this was a fairly simple task for most kids aged 7 to 14, brain connectivity profiles were compared by how quickly they answered the question (reaction time / RT), from 2-3 seconds near 7 years old to about a half second at 14.

These efforts to analyze the “effective connectivity” between math-related brain areas over time are truly heroic; they’re able to piece together an interesting story about how kids’ brain connectivity changes with age and math ability, but results like this should always be held very loosely.

The researchers show that, while doing math, certain brain areas’ activity patterns become more or less similar as the subject gets older. They focus on connections with an area whose overall activity level is correlated not just to age, but also to math aptitude: the intraparietal sulcus (IPS). This area, otherwise known to process quantity, becomes more collaborative on this task with areas that respond to the forms of letters and figures (ventrotemporal occipital cortex / VTOC), while simultaneously collaborating less with prefrontal cortex (PFC), which is otherwise associated with high-level reasoning and cognitive control.

In other words, math skills may develop by making calculations more habitual and automatic. The authors believe that these changes are related to general math ability, because a similar study on a specific 8-week training didn’t show any changes like these; the changes are faster than those expected by general genetic-driven brain development from aging; and older kids with dyscalculia have connectivity profiles that resemble those of the younger kids in this study.

These results corroborate one of our earlier posts that suggest that math skills are related to making certain math forms and processes instinctive. As a tutor, one should not be afraid to incorporate a moderate level of rote practice into teaching math skills. At the same time, if math skills are dependent on quickly memorizing the visual form of a calculation, these results are another call to being as consistent as possible with math notation when learning a new concept.

[1] Battista C, Evans TM, Ngoon TJ, Chen T, Chen L, Kochalka J, & Vinod Menon V (2018). Mechanisms of interactive specialization and emergence of functional brain circuits supporting cognitive development in children. npj Science of Learning 3(1):1.

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Hey Teens — Research Shows, It Really Does Get Better

In her fourth article in this series, Hampton Tutors coach and UW Master of Social Work candidate Gina Nepa discusses how psychoeducation—learning about one's own mental health—helps teenagers understand that their social difficulties aren't necessarily permanent.


Due to their unique crux of psychological and biological development, teenagers tend to be “more reactive to stress” than individuals in other life phases (Yeager et al., 2016). Adolescence also occupies a time in which relationship-building is key, and those who struggle to attain relationships of “status” may subsequently internalize negative feelings about self. In a social context in which 11 percent of individuals over the age of 12 are prescribed antidepressants (Pratt et al., 2011), we need to evaluate protective factors (including personality!) that can hinder the longevity of self-sabotaging feelings.

Adolescents are evolutionarily trained to be highly-responsive to environmental cues, in order to help facilitate learning (Knox, 2010). Although this stimulus response can aid in acquiring academic content, it also ties into the “more robust habit-forming ability that adolescents have, compared to adults” (Knox, 2010). Teenagers are bombarded with messaging that their social difficulties may plague them forever, when science in fact demonstrates that “socially relevant traits are malleable” (Yeager et al., 2016). Social adaptation, or seeming lack thereof, is a fluid process that is certainly not solidified during adolescence (which I think most of us are grateful for).

Yeager et al. (2016) tracked 205 ninth-graders over the course of a year; half of the teens were informed of the malleability of socially relevant traits, whereas the other half were provided with no psychoeducation. The results were stark. “The teens who were exposed to the idea that people can change coped better on days when they reported more stressors … exhibited higher GPAs 7 months later compared with their peers … [and] reported lower rates of depressive symptoms at the end of the school year” (Yeager et al., 2016).

This research indubitably demonstrates that psychoeducation, and evidence-based psychological interventions, among adolescents is impactful on a far-reaching scale. As educators, we have an obligation to provide teens with the information that physiologically, their internalized social defeat is not static.

Knox, R. (Host). (2010, March 1). The Teen Brain: It’s Just Not Grown Up Yet [Radio broadcast episode].

Pratt, L., Brody, Debra J, Gu, Qiuping, & National Center for Health Statistics. (2011). Antidepressant use in persons aged 12 and over: United States, 2005-2008 (NCHS data brief (Series); no. 76). Hyattsville, MD: U.S. Dept. of Health and Human Services, Centers for Disease Control and Prevention, National Center for Health Statistics.

Yeager, D., Lee, H., & Jamieson, J. (2016). How to Improve Adolescent Stress Responses. Psychological Science, 27(8), 1078-1091.

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Can Video Games Actually Make Students Better Multitaskers?

In this article, Hampton Tutors coach and resident cognitive sciences expert Tim Barnes discusses a recent study that aimed to uncover the puzzling link between video gaming and improved multitasking skills. 


Video game nerds are having their heyday in the psychological sciences: a number of experiments have shown that playing first-person action video games significantly improves one’s ability to visually attend to multiple objects, and even to increase the number of ideas one can hold in their head at once.

Few people feel comfortable with the idea of assigning violent video games to students as homework, however, so the race is on to find out exactly what aspects of these games improve players’ executive control and multitasking ability.

One idea is that, unlike puzzle or strategy games, action video games require the ability to solve problems while constantly ‘moving’ in the game’s virtual space; Bender and colleagues created a stripped-down lab version of such a game to investigate whether cognitive training while managing constant movement would increase cognitive capacity [1].

The study

Subjects in this experiment were trained in two tasks, either simultaneously or in separate sessions. One of these tasks was to use a computer mouse to keep a cursor centered within a circle that would move on the screen like a billiard ball; this task is supposed to have continuous movement similar to an action video game. The other task was to watch the center of the screen as various shapes were presented about once per second, and to press a key when the shape matched a target shape indicated before each trial.

Sessions of these tasks were sandwiched between a series of evaluations. Each subject was given a set of cognitive tests before task training, then evaluated on how good they were at the two tasks before training. After being trained for about an hour a day for five days on the tasks, they were re-evaluated, first on their performance on the specific tasks, then on the general cognitive tests.

All the data was then grouped by whether a subject spent half an hour on the mouse-tracking task and then half an hour on the shape-matching task, or whether each training day contained one session where the two tasks were happening simultaneously. The authors also made sure afterward that one group wasn’t naturally better at any of the tasks coming into the experiment, and that one group didn’t enjoy playing the games more than the other group.

The results

In the end, the multitasking-trained group was better at moving the mouse and recognizing the shape at the same time, but they didn’t show any improvement on the general cognitive tests. To the extent that action video games really do confer multitasking skills on players, continuous engagement and movement doesn’t seem to be the key variable.

Until that key is found, tutors should continue to rely on teaching organizational skills to help students multitask and manage multiple deadlines, tasks, and priorities.

[1] Bender AD, Filmer HL, Naughtin CK, & Due, PE (2017). Dynamic, continuous multitasking training leads to task-specific improvements but does not transfer across action selection tasks. npj Science of Learning 2(1):14.

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Cognitive Behavioral Therapy and School Avoidance

In the third article in her series on mental health and tutoring, Hampton Tutors coach and UW Master of Social Work candidate Gina Nepa talks about how cogntitive behavorial therapy is helping students improve school attendance. 


As internal and external pressures escalate within schools, more and more children are opting to avoid school, and often isolate instead. This pattern of avoidance in turn affects appropriate developmental socialization, and delays the onset of effective coping mechanisms and executive functioning skills. A four-week implementation of cognitive behavioral therapy (CBT) among children who refused school attendance due to anxiety was found to not only produce instantaneous attendance improvements among 88 percent of students, but also illustrated long-term attendance consistency for 81 percent of students (King et al., 2001).

Cognitive behavioral therapy focuses on “correcting” the interactions between thoughts, feelings, and behaviors. The treatment is rooted in adapting negative cognitive beliefs that result in counterproductive behaviors; in order to modify unhelpful patterns, core beliefs about self must be changed (Benjamin et al., 2011). For school avoidance particularly, several CBT techniques can aid in reducing a child’s negative associations with school.

One clinically significant technique is Aversive Conditioning. This strategy relies on the use of “dissuasion” to provoke a child to lose interest in pursuing a counterproductive behavior (such as skipping a particular class). Instead of associating skipping Algebra class with a positive stimulus of avoiding a stressful exam, the child may be conditioned to associate skipping class with a negative stimulus, such as loss of time with friends.

The importance of incorporating CBT techniques with school-averse children early cannot be understated. The amygdala within the brain regulates vigilance, and many children are hypervigilant in their quests to confirm that school is a terrible, anxiety-ridden place. Neuroimaging studies post-CBT treatment concluded that the amygdala was thus “significant[ly] deactivated” when changing the associations children have to historically negative stimuli (such as school), mimicking results of individuals who had positive responses to antidepressants (Almeida et al., 2013).

An abundance of evidence to date thus confirms that school avoidance does not need to be a lifelong battle.

Almeida, A. G. D., Araujo Filho, G. M. D., Berberian, A. D. A., Trezsniak, C., Nery-Fernandes, F., Araujo Neto, C. A. & Oliveira, I. R. D. (2013). The impacts of cognitive-behavioral therapy on the treatment of phobic disorders measured by functional neuroimaging techniques: a systematic review. Revista Brasileira de Psiquiatria, 35(3), 279-283.

Benjamin, Puleo, Settipani, Brodman, Edmunds, Cummings, & Kendall. (2011). History of Cognitive-Behavioral Therapy in Youth. Child and Adolescent Psychiatric Clinics of North America, 20(2), 179-189.

King, N., Tonge, B., Heyne, D., Turner, S., Pritchard, M., Young, D., Rollings, S., Myerson, N., Ollendick, T. (2001). Cognitive-Behavioural Treatment of School-Refusing Children: Maintenance of improvement at 3- to 5-year follow-up. Scandinavian Journal of Behaviour Therapy,30(2), 85-89.

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Coaching Anxiety Workshop with Dr. Jennifer Tininenko of EBTCS

In this latest blog installment, Hampton Tutors founder Andy Williamson discusses a recent anxiety workshop his team participated in and the reason why this type of training is increasingly important for today's educators.


Anxiety is the great underlying cause of the majority of school-based problems. Today's high school (and increasingly middle school) students face unrelenting pressure from their school work, the omnipresence of social media, and the arms race of college-attracting extracurriculars.

While social media usually takes the brunt of the criticism for rising anxiety, schools and colleges are increasingly competitive environments (there are roughly 50% more college applications in 2017 than in 2010), meaning educational institutions must admit their role in an emerging mental health crisis. In effect, high schoolers are constantly operating at a level of extreme stress, to a degree unsustainable for developing brains. 

Dr. Jennifer Tininenko is the co-director of the Child Anxiety Center at Evidence Based Treatment Centers of Seattle (EBTCS). On Friday 12 January, she came to Hampton Tutors to advise our coaches on working with students with anxiety, and how EBTCS' research can help us in coaching students on a weekly basis with their school work. 

Dr. Tininenko's workshop demonstrated that anxiety issues amongst students manifest in a variety of ways. The body/brain/behavior nexus is crucial in identifying anxiety in students, since different individuals may have different symptomatic responses to anxiety. Some may experience physical symptoms such as stomach ache or headaches (body), some may 'catastrophize' situations (brain), while others may show avoidant tendencies (behavior). Strategies for helping these students in an academic context depend very much on which responses to anxiety they demonstrate. 

That said, Dr. Tininenko outlined some clear strategies for mitigating (and avoiding) anxiety that can be integrated into our coaching. For example, coaches can develop metrics with students to gauge a work's 'completion' - particularly with open-ended projects like essays and creative writing. Furthermore, the use of 'chunking' as a tactic is a means of diminishing the formidable size of a project. 

Overall, what the workshop demonstrated was that anxiety disorders are often the product of poor feedback mechanisms, which perversely often make issues worse over time. Each "I can't" begets further "I can'ts". For an academic coach, taking steps to address this vicious circle can demonstrate to students that they have the agency to succeed. By developing clear, structured tactics for work, it can become a virtuous circle; "I can" begets "I can". Additionally, tactics drawn from mindfulness practice can help students to confront challenges on an everyday basis. Ultimately, the goal is to help students to develop their own toolkit for dealing with anxiety in their schoolwork.   

Hampton Tutors would like to thank Dr. Tininenko for her workshop. For our coaches, it provided clear, concrete steps for helping students who have anxiety disorders. By integrating these steps into our coaching, we can benefit students of all anxiety levels. Anxiety may be a growing issue, but we feel more equipped to deal with the range of challenge it presents.

Building Resilience and Responding to Defeat

In the second article in her series on mental health and tutoring, Master of Social Work candidate at UW, Gina Nepa talks about how to respond to setbacks in education. 


Maintaining the role of student for twelve years brings with it numerous opportunities for feelings of self-defeat, discomfort, and failure to thrive. Many of us have witnessed our peers or our children wither in response to unforeseen circumstances, whether those circumstances look like an unanticipated low grade on a test or a change within the family. Facing hardship is an inevitable, uncomfortable part of the life span; however, building up a “toolbox” of skills to strengthen resolve in times of distress can aid in promoting academic and socioemotional health. Starting this process of socioemotional tool accumulation at a young age can indubitably generate stronger long-term outcomes for success and well-being. So, which elements separate the students who are able to adapt and, in fact, grow when faced with discomfiting circumstances from the students who sink into self-loathing?

Wolpow et al. (2011) recommend utilizing “high expectations, reasonable limits, and consistent routines” (p. 16) as markers to build resilience, recommending that regardless of a student’s trajectory, conceptions of expectations should never be changed. Lowering markers for achievement was demonstrated to negatively impact student psyche, as students were inadvertently indoctrinated with familiar notions that they were not “good enough.”

Additionally, public schools tend to incorporate a substantial amount of group work into instruction, as reinforced by state Common Core standards. Although many students balk in response to having to work alongside strangers of varying work ethics and communication styles, the qualities that mark the students that thrive in group instruction are worth noting. A 30-year longitudinal study of resiliency[1] concluded that the children who fostered connections with “pro-social person[s]” (positive peer influences) had greater health outcomes decades later than children who were not able to build those strong connections. Werner’s (1989) study coincides with Wolpow et al.’s research in that both teams emphasize the ability of students to help other students as integral to building resilience.

Being amenable to the group process as an opportunity to build skills of teaching and explanation will help students generate their own feelings of competency and autonomy, in turn providing them with better academic outcomes.

1 Werner, E. (1989). High-Risk Children in Young Adulthood: A Longitudinal Study from Birth to 32 Years. American Journal of Orthopsychiatry, 59(1), 72. Wolpow, R., Johnson, Mona M, Hertel, Ron, Kincaid, Susan, & Washington. Superintendent of Public Instruction. (2009). The heart of learning and teaching: Compassion, resiliency, and academic success. Olympia, WA: Office of Superintendent of Public Instruction (OSPI) Compassionate Schools.

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Remembering to learn by learning to remember.

In the second of our series, Tim Barnes PhD talks about contemporary neuroscience's impact on the world of academic coaching. 


We often think of the learning process as coming before the remembering process. After all, what are you remembering, if not something you've already learned? However, what if we could flip the system around, so that memory was a tool for learning? Evidence shows that we can harness this power to become smarter, more efficient learners. 

Education researchers have recently documented what they call a testing effect, where repeatedly quizzing a student on a recently learned fact helps them to solidify that fact better than repeated study sessions. [1] Hoping to reveal a brain mechanism or correlate for this effect, Wiklund-Hörnqvist and colleagues recorded brain activity while running a memory exercise for participants. [2] In effect, they wanted to see how remembering something could help you learn it. 

Subjects (mainly in their twenties) were given a “lesson” in a subject they (presumably) have no previous experience of: translating words between Swedish and Swahili. Each subject stared at a screen for a half hour, where each of sixty translation pairs was shown on the screen for five seconds, all repeated five times. Then each subject was tested on the word pairs while in an MRI scanner. After being shown one of the words, they were asked to push a button declaring how sure they were about their knowledge. Then they had to answer a multiple choice question about its translation into the other language: what’s the second letter in the other word? The key experimental condition was that sometimes they were given instant feedback on what the correct translation word was afterward, and sometimes there was no feedback. Words with feedback were presented three times in the testing period.

After collecting all the data, the authors analyzed only those responses for which the subjects said they were sure about their answers and then either got the question right every time (+++) or got the question wrong the first time and then learned from the corrective feedback (-++). The more active brain areas during presentation of effective corrective feedback were those that correspond to learning new rules or strategies for accomplishing a task. In other words, re-testing may help students consolidate learning by practicing remembering the relevant fact — reorganizing one’s mental filing cabinet in order to streamline fact retrieval.

Practically, this study reinforces what many tutors and teachers already know — practice-based learning is more effective than listening to a lecture. The testing effect, however, has only been verified when the student believes they grasp the content. So, for learners, testing oneself on the material as a first step may be a major tool for committing items to memory. So test yourself first, learn the material second, and there's a good chance it'll stay learned!

[1] Karpicke JD and Roediger HL (2008). The critical importance of retrieval for learning. Science 319(5865):966–968.

[2] Wiklund-Hörnqvist C, Andersson M, Jonsson B, and Nyberg L (2017). Neural activations associated with feedback and retrieval success. npj Science of Learning 2(1):12.

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Instant Gratification and the ADHD Brain: How to Navigate a Culture of Immediacy

In the first of a series exploring how Social Work research impacts our coaching, Hampton Tutors coach and candidate on UW's Master of Social Work Program, Gina Nepa, explains how an ADHD student can manage their learning. 

  Photo by Alice Moore on Unsplash

 Photo by Alice Moore on Unsplash


We live in a culture in which we are repeatedly indoctrinated with expectations of instantaneous delivery. Because we’re able to send and receive messages with a click, our brains remain constantly activated and on “alert.” That nagging feeling of never quite feeling caught up inadvertently seeps into school performance and, ironically, prevents students from accessing the emotional regulation necessary to efficiently complete schoolwork.

We have come to associate many personality markers with ADHD: executive function challenges, easy distractibility, a harried response to a highly-detailed assignment. Although ADHD clearly has a physiological basis, the diagnostic markers are also a somewhat normal response to a culture of immediacy and unrelenting social and academic pressures. Jackson and MacKillop (2016) found that individuals with ADHD tended to face more challenges delaying gratification than their non-ADHD peers. This ADHD sample, in fact, derived significant pleasure from short-term rewards, and subsequently funneled more energy into obtaining short-term rather than long-term benefits. We see these results come into play when our students neglect planning for a project or keeping up with textbook notes, but can always be relied on to answer that text message from a friend.

So what can be done? There are several methods of appeasing instant gratification urges while simultaneously orienting the brain to long-term needs. Creating a homework space that incorporates “background” rewards, such as music[1] or snacks, can aid students in satisfying immediate pleasures while staying task-directed. Maintaining a rewards or “points” system with your student can also incentivize the brain to reorient. Just as stimulants produce a seemingly opposite effect on those with ADHD, so too can the intentional implementation of “distractors” in your student’s space.

Jackson, & Mackillop. (2016). Attention-Deficit/Hyperactivity Disorder and Monetary Delay Discounting: A Meta-Analysis of Case-Control Studies. Biological Psychiatry: Cognitive Neuroscience and Neuroimaging, 1(4), 316-325.

Pelham, William E., Waschbusch, Daniel A., Hoza, Betsy, Gnagy, Elizabeth M., Greiner, Andrew R., Sams, Susan E., Carter, Randy L. (2011). Music and Video as Distractors for Boys with ADHD in the Classroom: Comparison with Controls, Individual Differences, and Medication Effects. Journal of Abnormal Child Psychology,39(8), 1085-1098.

[1] Pelham et al., “Music and Video as Distractors,” p. 1085.

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Fractions speak louder than words

In the first of a series exploring how Neuroscience research impacts teaching, Hampton Tutors coach and PhD in Neuroscience, Tim Barnes, explains how research is shining light on the way students understand Math. 


Answer as quickly as you can: which of the following numbers is closest to the correct answer to the question below?

13/11 + 7/8

A) 1  B) 2  C) 19  D) 20

[The correct answer is given at the bottom of this post]

In a recent study, the 4th to 8th graders who were recently asked this question tended to give an answer that was midway between the two fractions; they were, in effect, taking the average of the fractions rather than adding them [1]. The authors of the study concluded that this was a conceptual problem, as students instinctively know that adding two numbers should result in a higher number.

Research conducted by Professor Bob Siegler at Carnegie Mellon has attempted to use fractions to answer the fundamental question of why some people learn mathematical concepts more easily than others [2]. To answer this question, scientists need to determine what stage of the process causes differences between students. Common theories include differences in how students mentally model the size of numbers; how they relate those sizes to written digits; their ability to hold and manipulate concepts in working memory; their ability to hear and understand math-related words; and socioeconomic factors [3,4]. The 'fraction experiment' checked that the subjects’ errors were more likely to be conceptual than to be related to their mental estimates of fraction magnitudes, their ability to do whole number arithmetic, or their ability to do the same estimation problem for whole numbers.

Practically, this means that tutors and coaches should remember that students rely on informal thinking to check their math work, something that may actually increase, rather than diminish, as they age [5]. Students can benefit, therefore, from improving their 'informal learning' along with their formal arithmetic methods. At its simplest, working on mental math (i.e. math without a calculator) reinforces math learning even at the highest levels. In some ways, there is an elegance that even the highest mathematical concepts are held together by the simplest mental techniques.  

[The answer was B) 2. The actual answer is 2 5/88, or 2.056]

[1] Braithwaite DW, Tian J, and Siegler RS (2017). Do children understand fraction addition? Developmental Science:e12601.

[2] Siegler RS, Fazio LK, Bailey DH, and Zhou X (2013). Fractions: The new frontier for theories of numerical development. Trends in Cognitive Sciences 17(1):13–99.

[3] Vanbinst K and De Smedt B (2016). Individual differences in children's mathematics achievement: The roles of symbolic numerical magnitude processing and domain-general cognitive functions. Progress in Brain Research 227:105–130.

[4] Siegler RS and Lortie-Forgues H (2017). Hard lessons: Why rational number arithmetic is so difficult for so many people. Current Directions in Psychological Science 26(4):346–351.

[5] Braithwaite DW, Goldstone RL, van der Maas HLJ, and Landy DH (2016). Non-formal mechanisms in mathematical cognitive development: The case of arithmetic. Cognition 149:40–55.

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