classroom learning

Finger tracing helps children doing geometry problems

  • Finger tracing key elements in worked problems seems to help some students better understand and apply mathematical concepts.

I've reported before on studies showing how gesturing can help children with mathematics and problem-solving. A new Australian study involving children aged 9-13 has found that finger-tracing has a similar effect.

Students who used their finger to trace over practice examples while simultaneously reading geometry or arithmetic material were able to complete the problems more quickly and correctly than those who didn't use the same technique.

In the first experiment, involving 52 students aged 11-13, some students were instructed to use their index fingers to trace elements of worked examples in triangle geometry, involving two angle relationships (Vertical angles are equal; Any exterior angle equals the sum of the two interior opposite angles.). Students were given two minutes to study a short instructional text on the relationships and how they can be used to solve particular problems. They were then given two minutes to study two worked examples. The tracing group were given additional instruction in how to use their index finger to trace out highlighted elements. The non-tracing group were told to keep their hands in their lap. Testing consisted of six questions, two of which were the same as the acquisition problems but with different numbers, and four of which were transfer questions, requiring more thoughtful responses.

A ceiling effect meant there was no difference between the two groups on the first two test questions. The tracing group answered significantly more transfer questions, although the difference wasn't great. There was no difference in how difficult the groups rated the test items.

In the second experiment, involving 54 Year 4 students, the instruction and problems concerned the fundamental order of operations. The tracing group were told to trace the operation symbols. The tracing group did significantly better, although again, the difference wasn't great, and again, there was no difference in assessment of problem difficulty.

In another experiment, involving 42 Year 5 students (10-11 years), students were given 5 minutes to study three angle relationships involving parallel lines (vertical angles are equal; corresponding angles are equal; the sum of co- interior angles is 180°). While answers to the 'basic' test questions failed to show significant differences, on the advanced transfer problems, the tracing group solved significantly more test questions than the non-tracing group, solved them more quickly, made fewer errors, and reported lower levels of test difficulty.

In the final experiment, involving 72 Year 5 students, on the advanced test problems, students who traced on the paper outperformed those who traced above the paper, who in turn outperformed those who simply read the material.

The researchers claim the findings support the view that tracing out elements of worked examples helps students construct good mental schemas, making it easier for them to solve new problems, and reducing cognitive demand.

As with gesturing, the benefits of tracing are not dramatic, but I believe the pattern of these results support the view that, when cognitive load is high (something that depends on the individual student as well as the task and its context), tracing key elements of worked examples might be a useful strategy.

Further research looking at individual differences would be helpful. I think greater benefits would be shown for students with low working memory capacity.

http://www.eurekalert.org/pub_releases/2016-01/uos-ftc012816.php

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[4046] Hu F-T, Ginns P, Bobis J. Getting the point: Tracing worked examples enhances learning. Learning and Instruction [Internet]. 2015 ;35:85 - 93. Available from: http://www.sciencedirect.com/science/article/pii/S0959475214000929

[4043] Ginns P, Hu F-T, Byrne E, Bobis J. Learning By Tracing Worked Examples. Applied Cognitive Psychology [Internet]. 2015 :n/a - n/a. Available from: http://onlinelibrary.wiley.com/doi/10.1002/acp.3171/abstract

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Seeing exemplary peer work can undermine student performance

  • Two experiments show that exposing students to exceptional examples of work by their peers is discouraging for many.

A natural experiment involving 5,740 participants in a MOOC ( massive open online course) has found that when students were asked to assess each other's work, and the examples were exceptional, a large proportion of students dropped the course.

In the MOOC, as is not uncommon practice, course participants were asked to write an essay and then to grade a random sample of their peers' essays. Those randomly assigned to evaluate exemplary peer essays were dramatically more likely to quit the course than those assigned to read more typical essays.

Specifically, around 68% of students who graded essays of average quality finished and passed the course, earning a certificate. Among those who graded slightly above average essays (more than one standard deviation above the class mean, 7.5/9), 64% earned a certificate. But among those who graded the best essays (those more than 1.6 SDs above the mean), only 45% earned a certificate.

These numbers can be compared to the fact that 75% of students who wrote an average essay earned a certificate, and 95% of those who wrote a 'perfect' essay, 9/9, earned a certificate. The difference between these numbers is about the same (in fact, slightly less) than the effect of grading average vs top essays.

A follow-up study, involving 361 participants, simulated this setting, in order to delve into what the students thought. Participants, recruited via Amazon's Mechanical Turk, were asked to write a minimum of 500 characters in response to a quote and essay prompt. They were told the best responses would go into a lottery to win a bonus. They were then asked to assess two very short essays (about 200 words) supposedly written by peers. These were either both well-written, or both poorly-written. This was followed by some questions about what they felt and thought, and an opportunity to write a second essay.

Unsurprisingly, those who were given exceptional essays to grade felt significantly less able to write an essay as good as those. They also decided that the ability to write an excellent short answer to such philosophical questions was not very important or relevant to them, and were much more likely not to write another essay (43% of those who read the poor essays went on to try again, while only 27% of those who read the excellent essays did so).

Until now, research has mainly focused on how students respond when peer work is of a standard that the student is likely to see as “attainable”. This research shows how comparisons that are seen as unattainable may do more harm than good.

http://www.eurekalert.org/pub_releases/2016-02/afps-sep020216.php

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Parents' math anxiety can undermine children's math achievement

  • 1st- & 2nd-grade children learned less math and developed more math anxiety when math-anxious parents frequently helped with their math homework.
  • Children with math-anxious parents who rarely helped with their math homework were not affected.

A study of 438 first- and second-grade students and their primary caregivers has revealed that parents' math anxiety affects their children's math performance — but (and this is the surprising bit) only when they frequently help them with their math homework.

The study builds on previous research showing that students learn less math when their teachers are anxious about math. This is not particularly surprising, and it wouldn't have been surprising if this study had found that math-anxious parents had math-anxious children. But the story wasn't that simple.

Children were assessed in reading achievement, math achievement and math anxiety at both the beginning and end of the school year. Children of math-anxious parents learned significantly less math over the school year and had more math anxiety by the year end—but only if math-anxious parents reported providing help every day with math homework. When parents reported helping with math homework once a week or less often, children’s math achievement and attitudes were not related to parents’ math anxiety. Reading achievement (included as a control) was not related to parents' math anxiety.

Interestingly, the parents' level of math knowledge didn't change this effect (although this is less surprising when you consider the basic-level of math taught in the 1st and 2nd grade).

Sadly, the effect still held even when the teacher was strong in math.

It's suggested that math-anxious parents may be less effective in explaining math concepts, and may also respond less helpfully when children make a mistake or solve the problem in a non-standard way. People with high math anxiety tend to have poor attitudes toward math, and also a high fear of failing at math. It's also possible (likely even) that they will have inflexible attitudes to how a math problem “should” be done. All of these are likely to demotivate the child.

Analysis also indicated that it is not that parents induced math anxiety in their children, who thus did badly, but that their homework help caused the child to do poorly, thus increasing their math anxiety.

Information about parental anxiety and how often parents helped their children with math homework was collected by questionnaire. Math anxiety was assessed using the short (25-item) Math Anxiety Rating Scale. The question, “How often do you help your child with their math homework?” was answered on a 7-point scale (1 = never, 2 = once a month, 3 = less than once a week, 4 = once a week, 5 = 2–3 times a week, 6 = every day, 7 = more than once a day). The mean was 5.3.

The finding points to the need for interventions focused on both decreasing parents' math anxiety and scaffolding their skills in how to help with math homework. It also suggests that, in the absence of such support, math-anxious parents are better not to help!

http://www.eurekalert.org/pub_releases/2015-08/uoc-pma080715.php

http://www.futurity.org/parents-math-anxiety-979472/

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Short online ‘pep talks’ can boost students

A large study shows how a 45-minute online intervention can improve struggling high school students' attitude to schoolwork, and thus their academic performance.

There's been a lot of talk in recent years about the importance of mindset in learning, with those who have a “growth mindset” (ie believe that intelligence can be developed) being more academically successful than those who believe that intelligence is a fixed attribute. A new study shows that a 45-minute online intervention can help struggling high school students.

The study involved 1,594 students in 13 U.S. high schools. They were randomly allocated to one of three intervention groups or the control group. The intervention groups either experienced an online program designed to develop a growth mindset, or an online program designed to foster a sense of purpose, or both programs (2 weeks apart). All interventions were expected to improve academic performance, especially in struggling students.

The interventions had no significant benefits for students who were doing okay, but were of significant benefit for those who had an initial GPA of 2 or less, or had failed at least one core subject (this group contained 519 students; a third of the total participants). For this group, each of the interventions was of similar benefit; interestingly, the combined intervention was less beneficial than either single intervention. It's plausibly suggested that this might be because the different messages weren't integrated, and students may have had some trouble in taking on board two separate messages.

Overall, for this group of students, semester grade point averages improved in core academic courses and the rate at which students performed satisfactorily in core courses increased by 6.4%.

GPA average in core subjects (math, English, science, social studies) was calculated at the end of the semester before the interventions, and at the end of the semester after the interventions. Brief questions before and after the interventions assessed the students' beliefs about intelligence, and their sense of meaningfulness about schoolwork.

GPA before intervention was positively associated with a growth mindset and a sense of purpose, explaining why the interventions had no effect on better students. Only the growth mindset intervention led to a more malleable view of intelligence; only the sense-of-purpose intervention led to a change in perception in the value of mundane academic tasks. Note that the combined intervention showed no such effects, suggesting that it had confused rather than enlightened!

In the growth mindset intervention, students read an article describing the brain’s ability to grow and reorganize itself as a consequence of hard work and good strategies. The message that difficulties don't indicate limited ability but rather provide learning opportunities, was reinforced in two writing exercises. The control group read similar materials, but with a focus on functional localization in the brain rather than its malleability.

In the sense-of-purpose interventions, students were asked to write about how they wished the world could be a better place. They read about the reasons why some students worked hard, such as “to make their families proud”; “to be a good example”; “to make a positive impact on the world”. They were then asked to think about their own goals and how school could help them achieve those objectives. The control group completed one of two modules that didn't differ in impact. In one, students described how their lives were different in high school compared to before. The other was much more similar to the intervention, except that the emphasis was on economic self-interest rather than social contribution.

The findings are interesting in showing that you can help poor learners with a simple intervention, but perhaps even more, for their indication that such interventions are best done in a more holistic and contextual way. A more integrated message would hopefully have been more effective, and surely ongoing reinforcement in the classroom would make an even bigger difference.

http://www.futurity.org/high-school-growth-mindset-910082/

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Classroom Learning

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Mathematics

Reading

Older news items (pre-2010) brought over from the old website

Effect of schooling on achievement gaps within racial groups

Analysis of data from a national sample (U.S.) of 8,060 students, collected at four points in time, starting in kindergarten and ending in the spring of fifth grade, has found evidence that education has an impact in closing the achievement gap for substantial numbers of children. High-performing groups in reading were found among all races. About 30% of European Americans, 26% of African Americans and 45% of Asian Americans were in high-achieving groups by the spring of fifth grade — these groups included approximately 23% of African American children and 36% of Asian children who caught up with the initial group of high achievers over time. Only around 4% of European American students were in catch-up groups, because a higher percentage of European Americans started kindergarten as high achievers in reading. The situation was different for Hispanic students, however.  By the end of fifth grade, just over 5% of Hispanic children were high achievers in reading, while the remainder tested in the middle range. There were no low achievers and no catch-up groups. A different pattern was found in math. Only 17% of European American students were high-achievers in math by the end of fifth grade, including 13% who started kindergarten at a lower achievement level and caught up over time.  About 18% of Asian Americans were high-achievers at the end of fifth grade (11% catch-up). Only 0.3% of African Americans were high achievers at the end of fifth grade, and 26% were medium-high achievers. But about 16% of Hispanics were high achievers in math. There were no catch-up groups for either the African Americans or the Hispanics. This suggests that current schooling doesn't have as strong an impact on math achievement as it does in reading.

The study was presented in Washington, D.C. at the annual meeting of the Society for Research on Educational Effectiveness.

http://www.physorg.com/news123859991.html

Children's under-achievement could be down to poor working memory

A survey of over three thousand children has found that 10% of school children across all age ranges suffer from poor working memory seriously affecting their learning. However, poor working memory is rarely identified by teachers, who often describe children with this problem as inattentive or as having lower levels of intelligence. The researchers have developed a new tool, a combination of a checklist and computer programme called the Working Memory Rating Scale, that enables teachers to identify and assess children's memory capacity in the classroom from as early as four years old. The tool has already been piloted successfully in 35 schools across the UK, and is now widely available. It has been translated into ten foreign languages.

http://www.physorg.com/news123404466.html 
http://www.eurekalert.org/pub_releases/2008-02/du-cuc022608.php

Priming the brain for learning

A new study has revealed that how successfully you form memories depends on your frame of mind beforehand. If your brain is primed to receive information, you will have less trouble recalling it later. Moreover, researchers could predict how likely the participant was to remember a word by observing brain activity immediately prior to presentation of the word.

Otten, L.J., Quayle, A.H., Akram, S., Ditewig, T.A. & Rugg, M.D. 2006. Brain activity before an event predicts later recollection. Nature, published online ahead of print 26February2006

http://www.nature.com/news/2006/060220/full/060220-19.html
http://www.eurekalert.org/pub_releases/2006-02/uoc--uri022806.php
http://www.eurekalert.org/pub_releases/2006-02/ucl-ywr022206.php

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Meditating leads to better grades

Three classroom experiments have found that students who meditated before a psychology lecture scored better on a quiz that followed than students who did not meditate. Mood, relaxation, and class interest were not affected by the meditation training.

The noteworthy thing is that the meditation was very very basic — six minutes of written meditation exercises.

The effect was stronger in classes where more freshmen students were enrolled, suggesting that the greatest benefit is to those students who have most difficulty in concentrating (who are more likely to drop out).

05/2013

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More evidence of the value of gesture in teaching math

A new study claims to provide ‘some of the strongest evidence yet’ for the benefits of gesturing to help students learn.

The study involved 184 children aged 7-10, of whom half were shown videos of an instructor teaching math problems using only speech, while the rest were shown videos of the instructor teaching the same problems using both speech and gestures. The problem involved mathematical equivalence (i.e., 4+5+7=__+7), which is known to be critical to later algebraic learning.

04/2013

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Immediate reward improves low-performing students’ test scores

July, 2012

A large study involving Chicago public school students has found conditions in which rewards offered just before a test significantly improve test performance.

In contradiction of some other recent research, a large new study has found that offering students rewards just before standardized testing can improve test performance dramatically. One important factor in this finding might be the immediate pay-off — students received their rewards right after the test. Another might be in the participants, who were attending low-performing schools.

The study involved 7,000 students in Chicago public schools and school districts in south-suburban Chicago Heights. Older students were given financial rewards, while younger students were offered non-financial rewards such as trophies.

Students took relatively short, standardized diagnostic tests three times a year to determine their grasp of mathematics and English skills. Unusually for this type of research, the students were not told ahead of time of the rewards — the idea was not to see how reward improved study habits, but to assess its direct impact on test performance.

Consistent with other behavioral economics research, the prospect of losing a reward was more motivating than the possibility of receiving a reward — those given money or a trophy to look at while they were tested performed better.

The most important finding was that the rewards only ‘worked’ if they were going to be given immediately after the test. If students were told instead that they would be given the reward sometime later, test performance did not improve.

Follow-up tests showed no negative impact of removing the rewards in successive tests.

Age and type of reward mattered. Elementary school students (who were given nonfinancial rewards) responded more to incentives than high-school students. Younger students have been found to be more responsive to non-monetary rewards than older students. Among high school students, the amount of money involved mattered.

It’s important to note that the students tested had low initial motivation to do well. I would speculate that the timing issue is so critical for these students because distant rewards are meaningless to them. Successful students tend to be more motivated by the prospect of distant rewards (e.g., a good college, a good job).

The finding does demonstrate that a significant factor in a student’s poor performance on tests may simply come from not caring to try.

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Support for link between physical activity & academic success

March, 2012

A review supports the benefits of physical activity for children’s and adolescent’s scholastic performance, but points to the need for better studies. A recent study looks at the effects on attention of different types of physical activity.

A review of 10 observational and four intervention studies as said to provide strong evidence for a positive relationship between physical activity and academic performance in young people (6-18). While only three of the four intervention studies and three of the 10 observational studies found a positive correlation, that included the two studies (one intervention and one observational) that researchers described as “high-quality”.

An important feature of the high-quality studies was that they used objective measures of physical activity, rather than students' or teachers' reports. More high-quality studies are clearly needed. Note that the quality score of the 14 studies ranged from 22%! to 75%.

Interestingly, a recent media report (NOT, I hasten to add, a peer-reviewed study appearing in an academic journal) spoke of data from public schools in Lincoln, Nebraska, which apparently has a district-wide physical-fitness test, which found that those were passed the fitness test were significantly more likely to also pass state reading and math tests.

Specifically, data from the last two years apparently shows that 80% of the students who passed the fitness test either met or exceeded state standards in math, compared to 66% of those who didn't pass the fitness test, and 84% of those who passed the fitness test met or exceeded state standards in reading, compared to 71% of those who failed the fitness test.

Another recent study looks at a different aspect of this association between physical exercise and academic performance.

The Italian study involved138 normally-developing children aged 8-11, whose attention was tested before and after three different types of class: a normal academic class; a PE class focused on cardiovascular endurance and involving continuous aerobic circuit training followed by a shuttle run exercise; a PE class combining both physical and mental activity by involving novel use of basketballs in varying mini-games that were designed to develop coordination and movement-based problem-solving. These two types of physical activity offered the same exercise intensity, but very different skill demands.

The attention test was a short (5-minute) paper-and-pencil task in which the children had to mark each occurrence of “d” with double quotation marks either above or below in 14 lines of randomly mixed p and d letters with one to four single and/or double quotation marks either over and/or under each letter.

Processing speed increased 9% after mental exercise (normal academic class) and 10% after physical exercise. These were both significantly better than the increase of 4% found after the combined physical and mental exertion.

Similarly, scores on the test improved 13% after the academic class, 10% after the standard physical exercise, and only 2% after the class combining physical and mental exertion.

Now it’s important to note is that this is of course an investigation of the immediate arousal benefits of exercise, rather than an investigation of the long-term benefits of being fit, which is a completely different question.

But the findings do bear on the use of PE classes in the school setting, and the different effects that different types of exercise might have.

First of all, there’s the somewhat surprising finding that attention was at least as great, if not better, after an academic class than the PE class. It would not have been surprising if attention had flagged. It seems likely that what we are seeing here is a reflection of being in the right head-space — that is, the advantage of continuing with the same sort of activity.

But the main finding is the, also somewhat unexpected, relative drop in attention after the PE class that combined mental and physical exertion.

It seems plausible that the reason for this lies in the cognitive demands of the novel activity, which is, I think, the main message we should take away from this study, rather than any comparison between physical and mental activity. However, it would not be surprising if novel activities that combine physical and mental skills tend to be more demanding than skills that are “purely” (few things are truly pure I know) one or the other.

Of course, it shouldn’t be overlooked that attention wasn’t hampered by any of these activities!

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The problem in correcting false knowledge

February, 2012

Whether corrections to students’ misconceptions ‘stick’ depends on the strength of the memory of the correction.

Students come into classrooms filled with inaccurate knowledge they are confident is correct, and overcoming these misconceptions is notoriously difficult. In recent years, research has shown that such false knowledge can be corrected with feedback. The hypercorrection effect, as it has been termed, expresses the finding that when students are more confident of a wrong answer, they are more likely to remember the right answer if corrected.

This is somewhat against intuition and experience, which would suggest that it is harder to correct more confidently held misconceptions.

A new study tells us how to reconcile experimental evidence and belief: false knowledge is more likely to be corrected in the short-term, but also more likely to return once the correction is forgotten.

In the study, 50 undergraduate students were tested on basic science facts. After rating their confidence in each answer, they were told the correct answer. Half the students were then retested almost immediately (after a 6 minute filler task), while the other half were retested a week later.

There were 120 questions in the test. Examples include: What is stored in a camel's hump? How many chromosomes do humans have? What is the driest area on Earth? The average percentage of correct responses on the initial test was 38%, and as expected, for the second test, performance was significantly better on the immediate compared to the delayed (90% vs 71%).

Students who were retested immediately gave the correct answer on 86% of their previous errors, and they were more likely to correct their high-confidence errors than those made with little confidence (the hypercorrection effect). Those retested a week later also showed the hypercorrection effect, albeit at a much lower level: they only corrected 56% of their previous errors. (More precisely, on the immediate test, corrected answers rose from 79% for the lowest confidence level to 92% for the highest confidence. On the delayed test, corrected answers rose from 43% to 70% on the second highest confidence level, 64% for the highest.)

In those instances where students had forgotten the correct answer, they were much more likely to reproduce the original error if their confidence had been high. Indeed, on the immediate test, the same error was rarely repeated, regardless of confidence level (the proportion of repeated errors hovered at 3-4% pretty much across the board). On the delayed test, on the other hand, there was a linear increase, with repeated errors steadily increasing from 14% to 23% as confidence level rose (with the same odd exception — at the second highest confidence level, proportion of repeated errors suddenly fell).

Overall, students were more likely to correct their errors if they remembered their error than if they didn’t (72% vs 65%). Unsurprisingly, those in the immediate group were much more likely to remember their initial errors than those in the delayed group (85% vs 61%).

In other words, it’s all about relative strength of the memories. While high-confidence errors are more likely to be corrected if the correct answer is readily accessible, they are also more likely to be repeated once the correct answer becomes less accessible. The trick to replacing false knowledge, then, is to improve the strength of the correct information.

Thus, as recency fades, you need to engage frequency to make the new memory stronger. So the finding points to the special need for multiple repetition, if you are hoping to correct entrenched false knowledge. The success of immediate testing indicates that properly spaced retrieval practice is probably the best way of replacing incorrect knowledge.

Of course, these findings apply well beyond the classroom!

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