6 – learning most students rely on learning strategies that are ineffective or inefficient. which...
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6 – Learning
Most students rely on learning strategies that are ineffective or inefficient.
Which learning strategies are best?
Which should be avoided?
Repetition / Rehearsal
Experiment
Ss heard word list: grass gold
toy luck gift
After each word, Ss said last g word: grass gold gold
gold gift
Words arranged so that S repeated each g word 0 - 12 times.
Later, Ss received surprise recall test on g words.
Results
Implication: Rehearsing is not an effective learning strategy
(Craik & Watkins, 1973)
% recall
# of repetitions
Craik and Watkins (1973)
# of times word was repeated aloud
% recall
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Highlighting
Experiment
Ss read 8000-word article from Scientific American
Three groups:
active highlighters (Ss told to highlight “particularly important” text)
passive highlighters (text is pre-highlighted)
read
Ss return one week later, spend 10 minutes looking over article, take test
Results: no differences
(Fowler & Barker, 1974)
Numerous experiments have examined efficacy of highlighting (or underlining)
Usual finding: reading with highlighting = reading without highlighting
A few studies show small benefit if Ss select the text (and if they select the correct text)
But this benefit probably reflects Ss’ decision about what to highlight.
Highlighting is ineffective because it not require understanding.
Caveat: highlighting is fine if aim is to merely mark the location
(for a review, see Dunlosky et al., 2013)
Demo: Read passage
The procedure is actually quite simple. First you arrange things into different
groups depending on their makeup. Of course, one pile may be sufficient
depending on how much there is to do. If you have to go somewhere else due to
lack of facilities, that is the next step; otherwise you are pretty well set. It is
important not to overdo any particular endeavor. That is, it is better to do too few
things at once than too many. In the short run, this may not seem important, but
complications from doing too many can easily arise. A mistake can be expensive
as well. The manipulation of the appropriate mechanisms should be self-
explanatory, and we need not dwell on it here. At first the whole procedure will
seem complicated. Soon, however, it will become just another facet of life. It is
difficult to foresee any end to the necessity for this task in the immediate future, but
then one never can tell.
Reading
Experiment
Ss read passage with or without its title.
Passage was incomprehensible without its title.
Later, Ss were asked to recall passage
Results: Titled passages better recalled.
Implication
Reading helps only if text is understood
Advice is obvious but unheeded.
(Bransford & Johnson, 1972)
Demo- Read passage
A newspaper is better than a magazine. A seashore is a better place than the street.
At first it is better to run than to walk. You may have to try several times. It takes
some skill but it is easy to learn. Even young children can enjoy it. Once successful,
complications are minimal. Birds seldom get too close. Rain, however, soaks in very
fast. Too many people doing the same thing can also cause problems. One needs
lots of room. If there are no complications it can be very peaceful. A rock will serve
as an anchor. If things break loose from it, you will not get a second chance.
Most U.S. students lack basic skills.
Most high school students cannot
write grammatically
solve simple math problems*
recall basic facts from Science, History, Geography*
* When tested at the end of the course (before they’ve had years to forget).
Large representative survey of U.S. students (public and private)
11th grade history students asked to identify the half century for each event.
(multiple-choice)
Event Answer % Correct
US Constitution written 1750-1800 61%
World War I 1900-1950 57%
Civil War 1850-1900 32%
Large representative survey of U.S. students (public and private) 8th grade math questions (multiple choice)
Multiply -5 and -7
Answer = 35 50% correct (5 choices)
A turkey is put in the oven at 10:30 a.m. If the turkey takes 2¾ hours to cook, when should it be taken out of the oven?
Answer = 1:15 p.m. 55% correct (4 choices)
(U.S. Dept of Education)
Study comparing mathematics proficiency of students in different countries
Percentage of students who were “advanced”
Taiwan 45Singapore 40South Korea 40Hong Kong 31Japan 26...
United States 6
Serbia 5
(Trends in International Mathematics and Science Study, 2007)
Slovenia 4
Survey Question: “I get good marks in mathematics.”
Percentage who Agreed or Strongly Agreed
Program for International Student Assessment (2003)
The same study measured performance ...
5-Minute University (Comedian Don Novello)
People who went to college can tell you what they learned in just 5 minutes. So, at my college, students learn just the same stuff – all in 5 minutes.
www.youtube.com/watch?v=kO8x8eoU3L4
Think deeply
Experiment
Ss heard list of words.
After each word, Ss answered a question.
½ Ss: Does it include an “e” ? shallow processing
½ Ss: Is the word pleasant? deep processing (think about meaning)
Later, Ss took surprise test
Results
deep >> shallow
Implication
Students should think about what they hear and read (obviously)
(e.g., Craik & Lockhart, 1972)
Self-Explanation During lecture or reading, student periodically explains material.
Experiment
Ss read chapter in Biology text
½ Ss were occasionally interrupted and asked to explain what they just read
½ Ss read and reread chapter
Later, Ss tested on material
Results: Explainers outscored Rereaders
(Smith et al., 2010)
Data: reading with self-explanation > reading alone (even if total time equated)
There are several ways to incorporate self-explanation (or self-query)
Example
Student reads, “If interference is proactive, prior learning disrupts subsequent learning.”
Student writes, “PI = learn A, learn B, can’t recall B. Why? A interfered with B.”
Example
Student reads article describing experiment.
When finished, student explains procedure and results.
PowerPoint Effect
Lecture slides lead students to sit passively rather than write notes in their own words.
(For reviews, see Dunlosky et al., 2013; Roediger & Pyc, 2012)
Answer Feedback After making mistake, student sees correct answer
Data: Seeing correct answer after error improves scores on subsequent test.
This rather obvious advice is often ignored
Many teachers don’t correct spelling.
Many college students don’t bother to look at answer key.
Many K-12 tests are secure (so students cannot see answers).
Retrieval Practice Try to recall information (rather than reread or rehearse)
Example
Flashcards
Many disciplines inherently require retrieval practice
Examples
Writing requires recall of spelling and syntax
Math problems require recall of procedures
Answering Spanish teacher requires recall of vocabulary, grammar
But many courses do not (e.g., history, geography, cognitive psychology).
Should these courses include retrieval practice?
Experiment
Ss learned Swahili-English pairs
Ss repeatedly cycled through pairs using 1 of 2 strategies
Study-Only
MASHUA – BOAT
LESO – SCARF
etc.
Retrieval Practice (with feedback)
MASHUA - ? … BOAT
LESO - ? … SCARF
etc.
Total learning time equated.
Ss took final test.
Results: Retrieval Practice much better.
(Karpicke & Roediger, 2008)
Countless studies have shown benefit of retrieval practice.
Thus, tests improve learning (and not only assess learning)
Another benefit of retrieval practice: students learn how well they know material
Educational implications:
Students should test themselves
Teachers should give lots of practice questions and quizzes
Demowww.sheppardsoftware.com/canada_G0_Click.html
Caveats
Tests don’t help learning unless students try to recall information
Example
Good Who invented cotton gin?
Bad Who invented cotton gin? a. Edison b. Bell c. Whitney d.
Carver
Students must use retrieval practice to learn the relevant material or skill
Example
Flashcard: quadratic formula
Knowing formula doesn’t mean that student can use it to solve a
problem
Note: In this course, students should work practice sets, not memorize definitions
Spacing Distributing a certain amount of study effort over a longer period of time
Example
Which French course is more effective?
Semester Course 3 hours per week for 15 weeks = 45 hours
Intensive Course 15 hours per week for 3 weeks = 45 hours
Experiment
Ss learned trivia.
Result: Spacers recalled 200% more! This is called the spacing effect.
The spacing effect is amazing because
1) Spacing does not require more study time
2) Spacing effect is not due to recent review because test delay is fixed.
(Cepeda et al., 2009)
TestStudy for 1 hour
Study for 1 hour
mass
test delay = 24 weeks
gap = 4 weeks
TestStudy for 1 hour
Study for 1 hour
space
test delay = 24 weeks
Spacing effect grows over time.
Experiment
Ss learned trivia.
Results
(Cepeda et al., 2008)
space
mass
test delay
Test Score
TestStudy for 1 hour
Study for 1 hour
mass
test delay = 1, 5, 10, or 50 weeks
gap = 15 weeks
TestStudy for 1 hour
Study for 1 hour
space
In many courses, students already space.
Example
Kids receive spelling words on Monday and take test on Friday.
In math class, a unit is spread out over two weeks.
Are these gaps sufficient, or does size of spacing gap matter?
Experiment
Results
Longer gaps better
(Cepeda et al., 2009)
Study Session 1
test delay = 168 daysStudy Session 2
Test
gap0, 1, 7, 28, 84, 168 days
Longest Cognitive Experiment Ever
Ss learned French or German vocabulary
Ss completed 13 study sessions
Spacing Gap = 2, 4, or 8 weeks
Test Delay = 1, 2, 3, and 5 years
Results on Final Test: Longest Spacing Gap always best
(Of course, briefer test delays better than longer test delays.)
(Bahrick, Bahrick, Bahrick, & Bahrick, 1993)
Educational implications of spacing effect.
Teachers should give cumulative exams
Exposure to a skill or concept should be distributed over a long period of time.
Examples
Spelling
Typical Weekly list includes 15 new words
Better Weekly list includes 10 new words and 5 old words
Math
Typical Each assignment includes 15 problems of the same
kind
Better Each assignment includes 5 problems of new kind,
15 from past
Spaced practice is one of the largest and most robust findings in memory.
Yet only 8 of these 22 cognitive texts mention spacing effect.
Anderson (1995). Cognitive Psychology and Its Applications (4th Ed)Ashcraft (2002). Cognition (3rd Ed.)Barsalou (1992). Cognitive Psychology: An Overview for Cognitive Scientists.Benjafield (1992). Cognition. Best (1999). Cognitive Psychology (5th Ed.)Galotti (1999). Cognitive Psychology In and Out of the Laboratory (2nd Ed).Goldstein (2005). Cognitive Psychology: Connecting Mind, Research, …Guenther (1998). Human Cognition. Howard (1983). Cognitive Psychology.Hunt & Ellis (2004). Fundamentals of Cognitive Psychology (7th Ed.)Jahnke & Nowaczyk (1998). Cognition. Kellogg (2003). Cognitive Psychology (2nd Ed.)Matlin (2002). Cognition (6th Ed.)Medin, Ross, & Markman (2005). Cognitive Psychology (4th Ed.). Payne & Wenger (1998). Cognitive Psychology. Reed (1992). Cognition (3rd Ed.). Reisberg (2001). Cognition: Exploring the Science of Mind (2nd Ed.). Robinson-Riegler & Robinson-Riegler (2004) Cognitive Psychology …Solso, MacLin, & MacLin (2005). Cognitive Psychology (7th Ed.). Sternberg (1996). Cognitive Psychology.Wickelgren (1979). Cognitive Psychology.Willingham (2004). Cognition: The Thinking Animal (2nd Ed.).
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Learning Style
Students learn better if instructional method matches student’s learning style.
Example
First, students are classified by style (e.g., visual vs. verbal)
Then students receive instruction tailored to their style.
Researchers have written hundreds of articles that propose this approach.
Consequently, many teachers and schools offer customized instruction.
Assessments, customized instructions, teacher training cost many tax dollars.
But is there evidence for this?
What kind of experiment provides support for tailoring instruction to student’s style?
Hypothetical
But this does not happen.
Conclusions about Learning Style
Students have different preferences.
Combining multiple approaches can help.
Example
Text with diagram may be better than either alone.
Optimal approach varies by topic.
Examples
Geometry requires visual instruction.
Literature requires verbal instruction.
But there is no evidence for “customized instruction.”
(Pashler et al., 2008)
Parents Of Nasal Learners Demand Odor-Based Curriculum
March 15, 2000 – The OnionCOLUMBUS, OH–Backed by olfactory-education experts, parents of nasal learners are demanding that U.S. public schools provide odor-based curricula for their academically struggling children.
A nasal learner struggles with an odorless textbook.
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Take-Home Story
Educators should advocate for strategies supported by experiments.
Unfortunately, many “education experts” dismiss this idea.
Instead, experts espouse learning methods based on
anecdotal evidence armchair theories
politically palatable views (all kids have same potential)
By contrast, medical field relies on true experiments.
U. S. Memory contest
VHS tape Scientific American Frontiers Segment 2 (about 15 min)
link not working (?)
www.pbs.org/saf/1102/video/watchonline.htm
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mnemonic learning strategy for a specific kind of material
ROY G. BIV
Stalactites on the ceiling, stalagmites on the ground
King Phillip Came Over For Great Spaghetti
Please Excuse My Dear Aunt Sally
SOH - CAH - TOA
Mnemonics are useful only in very specific cases.
peg-word mnemonic
Used to learn a list of items in a particular order (e.g., first 10 amendments)
Step 1. Learn peg-words. 1-bun 6-sticks
2-shoe 7-heaven
3-tree 8-gate
4-door 9-vine
5-hive 10-hen
Step 2. Form bizarre image of peg-word & study item
Demo: Close eyes and listen to words. For each words, form interactive image between word and peg word.
1. bun elephant2. shoe sailboat3. tree clock4. door foot 5. hive rainbow6. sticks telephone7. heaven radio8. gate corn9. vine brain 10. hen bottle
Now try to recall words in order
1. bun elephant2. shoe sailboat3. tree clock4. door foot 5. hive rainbow6. sticks telephone7. heaven radio8. gate corn9. vine brain 10. hen bottle
Bill of Rights (Amendments 1 through 10 of the Constitution)
1. freedom of speech, press, religion, assembly
2. bear arms
3. no soldier may stay in house without owner’s consent
4. reasonable search and seizure
5. grand jury; no self-incrimination; no double jeopardy
6. speedy public trial, may confront witness
7. trial by jury
8. no cruel or unusual punishment (or excessive bail)
9. there are other rights besides those listed here
10. states have powers not delegated or prohibited by constitution
Cranial Nerves
1 olfactory
2 optic
3 oculomotor
4 trochlear
5 trigeminal
6 abducens
7 facial
8 vestibulo/cochlear
9 glossopharyngeal
10 vagus
11 spinal accessory
12 hypoglossal
method of loci - like peg-word method, but cues are pre-learned locations
Experiment
Ss memorized path through campus with 40 salient locations
Ss heard 40 words (13 s each) and formed location-word image
Test given next day.
Mean Test Score = 34 of 40 in correct serial position
(Ross & Lawrence, 1968)
Method of Loci attributed to the following legend (recounted by Cicero):
The Greek poet Simonides was at a banquet. He stepped outside to talk to
messengers, and, while he was outside, the roof of the banquet hall collapsed, killing
everybody inside. The mangled corpses could not be identified, but Simonides was
able to identify every victim by visualizing where everyone had been sitting.
March 3, 2013 – ABC News
It was a solemn scene Thursday as U.S. Navy veteran Ron White stood at a black wall in downtown Fort Worth, and wrote in white the full name and rank of each of the 2,200 military members who had died in Afghanistan. White had no list. He had memorized the 2,200 names and ranks, more than 7,000 words.
White is a two-time USA Memory Champion who has made memorization his business over the past 22 years. He began working on memorizing the 2,200 names in May 2012, using the loci method.
"Essentially, what you do is you memorize a map of your city, your town where you live," White said. "So there's 2,200 service members who paid the ultimate sacrifice. So that means I memorized 2,200 locations in my hometown of Forth Worth, Texas.” The locations included places and sign posts like stop signs, trees, walls, pictures, restaurant booths and restaurant cash registers. He then took each name and turned it into a picture that reminded him of that person, and he visualized a name at each location.
"When I was at the wall, I simply stood at the wall and took an 11-hour mental walk around downtown Fort Worth and wrote out the names," White said.
Memory contests.
Records (as of Feb 2013)
Event U.S. World
5 minutes to learn list of random digits 303 333
15 minutes to learn list of random words 120 214
Memorize order of shuffled deck of cards 63 sec 24 sec
15 minutes to memorize poem (1 pt per word) 234 305
Poem for U.S. 2006 Championships When I am old and gray, and life has run her course through my veins, I shall stand before a beveled glass of reckoning ~ and I shall weep. Like rain to a parched and arid soul, my tears shall fall into the ravine of my heart ~ and then shall I survey the serpentine’d path of my soul’s Sadhana. I shall bow my head beneath the weighted glory of mem’ry, and pray to know that all I have done, I have done well. I shall reach, with hands withered and veined, for the finer silk of love’s gentle thread, which interwove itself through the fabric of who I was and who I became ~ in those yesterdays, which are now my todays. As striated lights against the lens of my perception, the colors of my being shall explode across my vision ~ rendering me speechless with wonder at vast potential explored; destiny pondered and embraced. When I am old and gray, the awe in which I have lived my life shall wear as a mantle, woven upon legacy’s loom: stretched taut across the framework of purpose, and driven hard by courage, and tenacity, and the indefatigable determination to draw all mediocrity out of the thread on which I toiled, and over which I prayed. And you shall be there, weaver’s teacher: strong and steady stream of light which guided the spindles, threaded the bobbin ~ and somehow kept tapestry’s perfection in mind; forever blueprinting the pattern into my heart. But let me not wait for tomorrow, to touch the fabric of this life; to hold close the breathing aliveness of a storm’s invisible power to electrify, and to set my spirit ablaze. Let me not wait another moment to peer through the beveled glass, and find perfection’s refraction, reflected. Let me enjoy, today, the magic of the journey ~ Sadhana’s tapestry rich of texture and vibrant of color! Let me breathe in the aromatherapeutic essence of choice, and a destiny fulfilled, with every step I take upon a journey into the heart of me! Let me find the Holy Grail today ~ while youth still dances in my spirit, and grace aligns my feet to a path firm and true ~ and set before me as surely as a dance hall ticket, already purchased. Let my tears be of joy, which dry upon the loom ~ bleeding their way into every careful stitch and turn. And when I am done, and the tapestry is complete ~ I shall know only peace, at having been blessed to sit beside you, all these years, to share the soul and heart which have bonded us together since the beginning.
To attain rank of grand master, you must
1) memorize 1,000 digits in an hour
2) memorize order of 10 shuffled decks of cards in an hour
3) memorize one shuffled deck in less than two minutes.
As of 2005, there were 36 grand masters in the world.
(Foer, 2005)
The 16th USA Memory Championship will be held on Saturday, March 16,
2013 at Con Edison at 4 Irving Place in New York City.
Another popular memory contest:
memorizing digits of pi
Rajan
1981 - Recited 31,811 digits of
2005 - Akira Haraguchi recited 83,431 digits
But RM averaged 4 digits / sec – AH just 1 digit / sec.
Conway & Gathercole: married couple, both memory researchers
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Summary of Learning Strategies
Strategy Efficacy
repetition / rehearsing poor
highlighting poor
reading good if material is
understood
answer feedback good
self-explanation good
retrieval practice good
spacing good
interleaving good for math
learning style no evidence
mnemonics useful in rare scenarios
Demo
1
2
3
4
5
6
7
8
9
10
11
12
Demo
In a moment, I will quickly read a list of words related to the same topic. Please close your eyes and listen carefully.
School Supplies
notebookpenscissorsfolderstaplereraserbinderpenciltaperulerglueprotractor
Next Step
Your page is numbered 1 – 12. Write as many of the words as you can remember.Order does NOT matter.
Next Step
Next to each word, write your confidence that the word was on list. 1 = not sure 5 = certain
Answer Key:
School Supplies
notebookpenscissorsfolderstaplereraserbinderpaperpenciltaperulerglueprotractor
The End
Experiment
Ss rated words as pleasant or not.
Before seeing the list,
Ss were told they would later be tested (intentional learning)
or not (incidental learning)
Test results: No difference!
Note: effect probably depends on materials
Examples
finding would replicate with plot of TV show but not face-name pairs.
(Hyde & Jenkins, 1973)
Recalling name is hard; no foolproof method, but again, visual imagery helps.
keyword method
Link name to object, then form image of object and face
Examples
Conrad con rat - visualize him in prison garb with rat on his face
Diana Princess Diana - visualize her wearing tiara
Keith key – visualize key stuck in his face
Similar approach – focus on link between name and salient feature
name face easier than face name
Preceding surveys concerned material that students had studied recently.
Tests of long-term retention are more discouraging.
Sample of studies of student retention showing dramatic, quick loss.
algebra (Bahrick & Hall, 1991)
foreign language (Bahrick, 1984)
587 Ss give Spanish quiz 1- 50 years later
Recall of vocabulary dropped quickly
cognitive psychology (Conway, Cohen & Stanhope, 1991)
Instructor tested his students from last 12 years
Recall of concepts: 3 mo. = 60% 3 y, 3 mo. = 30%
“A few years ago, someone wrote a proposal for a book called How to Study
in College, offering [this] advice: Students should postpone their studying
until the last possible moment, because if they studied earlier, they would
forget it all. If you have ever waited until the night before a test to start your
study, you know this advice is bad. So how could its author, a college
professor who had read some research on memory, have come to such a
faulty conclusion?
- James W. Kalat
p. 239, Introduction to Psychology (6th Ed.)
Gap0 months 3 months 6 months
Test
Score
0%
40%
But students already space. Should their spacing gaps be longer?
Experiment
(Cepeda et al., 2009)
Study Session 1
test delay = 168 daysStudy Session 2
Test
gap0, 1, 7, 28, 84, 168 days
best gap = 28 days
gap
TestScore
Spacing effect is large, and spacing does not require additional study time.
No boundary conditions are known.
Different topics
text comprehension (e.g., Rawson & Kintsch, 2005)
biology (Reynolds & Glaser, 1964)
mathematics (Rohrer & Taylor, 2006; 2007)
foreign language vocabulary (e.g., Bahrick & Phelps, 1987)
Complex tasks
surgery (Moulton et al., 2006)
subtle syntax rules (Bird, 2010)
Kids (e.g., Metcalfe et al., 2007; Seabrook et al., 2005)
Does optimal gap depend on test delay?
Experiment
1354 subjects randomly assigned to 26 conditions
Ss learned trivia facts.
(continued)
Study Session
1
test delay 7 days 35 days70 days350 days
Study Session
2
Test
gap0, 1, 2, 7, 21, or 35 days0, 1, 4, 7, 11, 21, or 105 days0, 1, 7, 14, 21, or 105 days0, 1, 7, 21, 35, 70, or 105 days
Results
(Cepeda et al. , 2008)
TestScore
Optimal gap got longer as the test delay got longer.Optimal gap was about 20% of test delay.
test delay7 days
35 days
70 days
350 days
100%
TestScore
0%110 gap (days) 0
350 test d
elay (days)
0
21ln111scoretest dgcbt
agepp
Strategies that optimize test scores often produce inferior practice scores.
Examples
spacing (rather than massing)
interleaving (rather than blocking)
Such strategies are called desirable difficulties. (Bjork and colleagues)
Common misconception
If a strategy optimizes practice performance, it also optimizes test score.
This partly explains why students and teachers prefer massing and blocking.
[in left column, paintings by same artist]
[in right column, paintings by different artists]
Experiment
Ss saw landscapes by artists with similar style.
Ss saw 6 paintings per artist, each with artist’s name.
For half the artists, paintings were blocked by artist
6 paintings by Artist A, followed by 6 paintings by Artist B, etc.
For other half, paintings were interleaved. A B C D etc.
Test: Ss saw previously unseen paintings by same artistsS tried to select artist’s name from a list.
Test Results: Blocked < < Interleaved
Thus, interleaving improved category learning.
(Kornell & Bjork, 2008)
In almost all math texts, most assignments provide blocked practice.
Example
With blocked practice, students know the strategy before they read problem.
Example Problem #10 would be hard if it appeared alone on test.
But #10 is easy if it follows 9 problems requiring same strategy.
Multiply. Write in simplest form.
1. (3/5) ∙ (5/7) 2. (4/5) ∙ (3/8) 3. (6/7) ∙ (7/6)
4. (- 1/8) ∙ (4/9) 5. (- 2/9) ∙ (3/8) 6. (- 12/13) ∙ (- 2/3)
7. (1 1/3) ∙ (5 1/2) 8. (2 1/2) ∙ (1 2/5) 9. (- 6 3/4) ∙ (1 7/ 9 )
10. Rhode Island is the smallest state in the United States. Its area is about 1/6 the area of New Hampshire. If the area of New Hampshire is about 9,270 square miles, what is the approximate area of Rhode Island?
Glencoe McGraw-Hill (Grade 8)
Another example:
Identify the common factors of each set of numbers. 1. 45, 75 2. 12, 21, 30
Find the greatest common factor of each set of numbers. 3. 8,32 4. 24, 60 5. 3, 12, 18 6. 4, 10, 14
Oliver has 14 chocolate cookies and 21 iced cookies.7. If Oliver gives each friend an equal number of each type of cookie, what is the greatest number of
friends with whom we can share his cookies? 8. How many cookies did each friend receive?
Identify the common factors of each set of numbers. 9. 45, 75 10. 36, 90 11. 6, 21, 30 12. 16, 24, 40
Find the greatest common factor of each set of numbers. 13. 12, 18 14. 18, 42 15. 48, 60 16. 30, 72 17. 14, 35, 8418. 9, 18, 42 19. 16, 52, 76 20. 12, 30, 72 21. 37, 64, 72 22. 35, 63, 84
Anika is placing photos in a scrapbook. She has 8 large photos, 12 medium photos, and 16 small photos. Each page will have only one size of photo. Anika wants to place the same number of photos on each page.
23. What is the greatest number of photos that could be on each page? 24. How many pages will she use in all?
Glencoe McGraw-HillMath Connects 1 (Grade 6)
(pp. 199-200; Day et al., 2011)
This is an excerpt from an assignment with interleaved practice.
(But it includes a small block of problems, 9 – 11, on the most recent lesson.)
Each problem includes the corresponding lesson number.
Only 3 problems relate to the immediately preceding lesson.
Practice Session 1 Practice Session 2 Test
Proportion
Correct
0
1
Interleaved
Blocked
.63
.20
.87
.43
.91
.78
Experiment
Ss learned to find volume of 4 kinds of obscure solids
Practice problems were blocked : aaaa bbbb cccc dddd
or interleaved : abcd bdac cadb dcba
Conclusion: Interleaving required Ss to choose strategy, just like the test.
(Rohrer & Taylor, 2007)
In almost all math texts, each assignment is devoted to one skill or concept
Example
With blocked practice, students know the strategy before they read problem.
Example
#10 is easy if it follows 9 problems requiring same strategy.
But #10 by itself if it appears alone on a test.
Thus, math problem has two steps: 1) choose strategy 2) solve problem
Multiply. Write in simplest form.
1. (3/5) ∙ (5/7) 2. (4/5) ∙ (3/8) 3. (6/7) ∙ (7/6)
4. (- 1/8) ∙ (4/9) 5. (- 2/9) ∙ (3/8) 6. (- 12/13) ∙ (- 2/3)
7. (1 1/3) ∙ (5 1/2) 8. (2 1/2) ∙ (1 2/5) 9. (- 6 3/4) ∙ (1 7/ 9 )
10. Rhode Island is the smallest state in the United States. Its area is about 1/6 the area of New Hampshire. If the area of New Hampshire is about 9,270 square miles, what is the approximate area of Rhode Island?
Glencoe McGraw-Hill (Grade 8)
Experiment
Grade 7 kids worked math assignments for 10 weeks
Every kid saw the same problems but problem order varied
Two kinds of assignment:
blocked Every problem was the same kind (e.g., all slope problems)
interleaved Problems of different kinds were intermixed
Two weeks after last assignment, Ss received surprise math test
Test scores: blocked 38%
interleaved 72%
Here are some possible outcomes
Which findings supports learning style idea? A, B, C.
Why? Because, for those findings, optimal method is not the same for all Ss.
A
TestScore Method 2
Method 1
A Style B StyleLearners Learners
B
Method 2Method 1
C
A Style B StyleLearners Learners
A Style B StyleLearners Learners
Method 2
Method 1
D
TestScore
Method 1
Method 2
E
Method 1
Method 2
F
Method 1
Method 2
A Style B StyleLearners Learners
A Style B StyleLearners Learners
A Style B StyleLearners Learners