Dipsticks: Efficient Ways to Check for Understanding by Todd Finley

2 01 2015

JULY 30, 2014

What strategy can double student learning gains? According to 250 empirical studies, the answer is formative assessment, defined by Bill Younglove as “the frequent, interactive checking of student progress and understanding in order to identify learning needs and adjust teaching appropriately.”

Unlike summative assessment, which evaluates student learning according to a benchmark, formative assessment monitors student understanding so that kids are always aware of their academic strengths and learning gaps. Meanwhile, teachers can improve the effectiveness of their instruction, re-teaching if necessary. “When the cook tastes the soup,” writes Robert E. Stake, “that’s formative; when the guests taste the soup, that’s summative.” Formative assessment can be administered as an exam. But if the assessment is not a traditional quiz, it falls within the category of alternative assessment.

Alternative formative assessment (AFA) strategies can be as simple (and important) as checking the oil in your car — hence the name “dipsticks.” They’re especially effective when students are given tactical feedback, immediately followed by time to practice the skill. My favorite techniques are those with simple directions, like The 60 Second Paper, which asks students to describe the most important thing they learned and identify any areas of confusion in under a minute. You can find another 53 ways to check for understanding toward the end of this post, also available as a downloadable document.

In the sections below, we’ll discuss things to consider when implementing AFAs.

Observation: A Key Practice in Alternative Formative Assessment

A fundamental element of most AFAs is observation. In her Edutopia post, Rebecca Alber says there is much to learn by taking observational notes as students work in groups. “However,” she clarifies, “if it is quiet during this talk time, and they are watching you watch them, they are most likely lost.” Another Edutopia blogger, Elena Aguilar witnessed “a fantastic first grade Sheltered English teacher” who directed his students to respond to a story by making hand gestures and holding up picture cards. “In this way, the teacher was able to immediately see who was struggling with the concepts and provide corrective feedback.”

By methodically watching and recording student performance with a focused observation form, you can learn a lot about students’ levels of understanding in just a few moments. For example, on the Teach Like a Champion blog, watch how math teacher Taryn Pritchard uses an observation sheet, and note her description of how she pre-plans to assess students’ mastery levels in only ten seconds. Pre-planning methodical observations allow instructors to efficiently and effectively intervene when it counts most — the instant students start down the wrong path.

New to Alternative Formative Assessment? Start Slow

The National Capital Language Resource Center recommends the following when introducing alternative assessment for the first time:

  • Integrate alternative assessments gradually, while still using the traditional assessments.
  • Walk students through the rubrics and discuss expectations when you introduce assignments.
  • Learn to score alternative assessments yourself, and then gradually introduce students to self-evaluation.
  • Teach students how to thoughtfully give each other feedback as you introduce them to peer-response.

A Simple Way to Gain Information from Your Students: Ask Them

When preservice teachers are confused as to why their students performed poorly on an assignment, I gently say, “Did you ask them why?” After all, having learners use their own vernacular to articulate why they are stuck can be profoundly useful for identifying where to target support.

According to the American Institute of Nondestructive Testing, the simplest tool to encourage student self-assessment is evaluative prompts:

  • How much time and effort did you put into this?
  • What do you think your strengths and weaknesses were in this assignment?
  • How could you improve your assignment?
  • What are the most valuable things you learned from this assignment?

Learners can respond to those prompts using Padlet, a virtual corkboard where many computer users can simultaneously post their responses, followed by a focused whole-class discussion of students’ answers. The instructor doesn’t always have to develop prompts — students can invent and submit one or more potential exam questions and answers on relevant content. Tell them that you’ll include the best contributions on a forthcoming quiz.

Portfolios are a more complex form of ongoing self-assessment that can be featured during student-led conferences. James Mule, principal of St. Amelia Elementary School in New York, describes how children benefit from the student-led conferences that occur at his institution: “With the student in charge and the teacher acting as a facilitator, the authentic assessment gives students practice in self-evaluation and boosts accountability, self-confidence, and self-esteem.” Pernille Ripp’s Blogging Through the Fourth Dimension provides all the handouts needed.

The biggest benefit of integrating AFAs into your practice is that students will internalize the habit of monitoring their understanding and adjusting accordingly.

We created the following list as a downloadable reminder to post by your computer. In the comments section of this post, tell us which of these 53 ways you’ve used for checking on students’ understanding — or recommend other AFAs we should know about.

Click to download a PDF of these 53 AFA strategies (435 KB).
  1. Summary Poem Activity
    • List ten key words from an assigned text.
    • Do a free verse poem with the words you highlighted.
    • Write a summary of the reading based on these words.
  2. Invent the Quiz
    • Write ten higher-order text questions related to the content. Pick two and answer one of them in half a page.
  3. The 411
    • Describe the author’s objective.
  4. Opinion Chart
    • List opinions about the content in the left column of a T-chart, and support your opinions in the right column.
  5. So What? Journal
    • Identify the main idea of the lesson. Why is it important?
  6. Rate Understanding
  7. Clickers (Response System)
  8. Teacher Observation Checklist
  9. Explaining
    • Explain the main idea using an analogy.
  10. Evaluate
    • What is the author’s main point? What are the arguments for and against this idea?
  11. Describe
    • What are the important characteristics or features of the main concept or idea of the reading?
  12. Define
    • Pick out an important word or phrase that the author of a text introduces. What does it mean?
  13. Compare and Contrast
    • Identify the theory or idea the author is advancing. Then identify an opposite theory. What are the similarities and differences between these ideas?
  14. Question Stems
    • I believe that ________ because _______.
    • I was most confused by _______.
  15. Mind Map
    • Create a mind map that represents a concept using a diagram-making tool (like Gliffy). Provide your teacher/classmates with the link to your mind map.
  16. Intrigue Journal
    • List the five most interesting, controversial, or resonant ideas you found in the readings. Include page numbers and a short rationale (100 words) for your selection.
  17. Advertisement
    • Create an ad, with visuals and text, for the newly learned concept.
  18. 5 Words
    • What five words would you use to describe ______? Explain and justify your choices.
  19. Muddy Moment
    • What frustrates and confuses you about the text? Why?
  20. Collage
    • Create a collage around the lesson’s themes. Explain your choices in one paragraph.
  21. Letter
    • Explain _______ in a letter to your best friend.
  22. Talk Show Panel
    • Have a cast of experts debate the finer points of _______.
  23. Study Guide
    • What are the main topics, supporting details, important person’s contributions, terms, and definitions?
  24. Illustration
    • Draw a picture that illustrates a relationship between terms in the text. Explain in one paragraph your visual representation.
  25. KWL Chart
    • What do you know, what do you want to know, and what have you learned?
  26. Sticky Notes Annotation
    • Use sticky notes to describe key passages that are notable or that you have questions about.
  27. 3-2-1
    • Three things you found out.
    • Two interesting things.
    • One question you still have.
  28. Outline
    • Represent the organization of _______ by outlining it.
  29. Anticipation Guide
    • Establish a purpose for reading and create post-reading reflections and discussion.
  30. Simile
    • What we learned today is like _______.
  31. The Minute Paper
    • In one minute, describe the most meaningful thing you’ve learned.
  32. Interview You
    • You’re the guest expert on 60 Minutes. Answer:
      1. What are component parts of _______?
      2. Why does this topic matter?
  33. Double Entry Notebook
    • Create a two-column table. Use the left column to write down 5-8 important quotations. Use the right column to record reactions to the quotations.
  34. Comic Book
    • Use a comic book creation tool like Bitstrips to represent understanding.
  35. Tagxedo
    • What are key words that express the main ideas? Be ready to discuss and explain.
  36. Classroom TED Talk
  37. Podcast
    • Play the part of a content expert and discuss content-related issues on a podcast, using the free Easypodcast.
  38. Create a Multimedia Poster with Glogster
  39. Twitter Post
    • Define _______ in under 140 characters.
  40. Explain Your Solution
    • Describe how you solved an academic problem, step by step.
  41. Dramatic Interpretation
    • Dramatize a critical scene from a complex narrative.
  42. Ballad
    • Summarize a narrative that employs a poem or song structure using short stanzas.
  43. Pamphlet
    • Describe the key features of _______ in a visually and textually compelling pamphlet.
  44. Study Guide
    • Create a study guide that outlines main ideas.
  45. Bio Poem
    • To describe a character or person, write a poem that includes:
      • (Line 1) First name
      • (Line 2) 3-4 adjectives that describe the person
      • (Line 3) Important relationship
      • (Line 4) 2-3 things, people, or ideas the person loved
      • (Line 5) Three feelings the person experienced
      • (Line 6) Three fears the person experienced
      • (Line 7) Accomplishments
      • (Line 8) 2-3 things the person wanted to see happen or wanted to experience
      • (Line 9) His or her residence
      • (Line 10) Last name
  46. Sketch
    • Visually represent new knowledge.
  47. Top Ten List
    • What are the most important takeaways, written with humor?
  48. Color Cards
    • Red = “Stop, I need help.”
    • Green = “Keep going, I understand.”
    • Yellow = “I’m a little confused.”
  49. Quickwrite
    • Without stopping, write what most confuses you.
  50. Conference
    • A short, focused discussion between the teacher and student.
  51. Debrief
    • Reflect immediately after an activity.
  52. Exit Slip
    • Have students reflect on lessons learned during class.
  53. Misconception Check
    • Given a common misconception about a topic, students explain why they agree or disagree with it.

http://www.edutopia.org/blog/dipsticks-to-check-for-understanding-todd-finley?utm_source=facebook&utm_medium=post&utm_campaign=blog-dipsticks-to-check-for-understanding-image-repost

Other Assessment Resources

In Edutopia’s The Power of Comprehensive Assessment, Bob Lenz describes how to create a balanced assessment system.

The American Federation of Teachers (AFT) describes dozens of Formative Assessment Strategies.

The Assessment and Rubrics page of Kathy Schrock’s Guide to Everything website hosts many excellent assessment rubrics.

More Rubrics for Assessment are provided by the University of Wisconsin-Stout.

Jon Mueller’s Authentic Tasks and Rubrics is a must see-resource in his Authentic Assessment Toolbox website.





Debunking the Genius Myth

21 12 2014

| August 30, 2013 |

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Picture a “genius” — you’ll probably conjure an image of an Einstein-like character, an older man in a rumpled suit, disorganized and distracted even as he, almost accidentally, stumbles upon his next “big idea.” In truth, the acclaimed scientist actually said, “It’s not that I’m so smart, it’s just that I stay with problems longer.” But the narrative around Einstein and a lot of accomplished geniuses — think Ben Franklin, the key and the bolt of lightning — tends to focus more on mind-blowing talent and less on the hard work behind the rise to success. A downside of the genius mythology results in many kids trudging through school believing that a great student is born, not made — lucky or unlucky, Einstein or Everyman.

Harvard-educated tutors Hunter Maats and Katie O’Brien began to notice that this belief about being born smart was creating a lot of frustration for the kids they tutored, and sometimes unwittingly reinforced by their parents. “We had sessions working with a student where the mom would walk by and say, ‘Oh, he didn’t get the math gene!’” said O’Brien. “And I’d think, Gee, give the kid a reason to never even try.”

“Try,” it seems, is the magical and operative word that has the possibility to transform how well a student does in school — once they understand a little about how to try, and a little about how learning and the brain works. How students think about learning makes a difference in what they’re able to achieve. Groundbreaking research conducted by Stanford psychologist Carol Dweck has shown that when students take on a growth mindset – one in which they believe that the brain is malleable, and they can improve at a task with effort – they handle setbacks better and improve academically.

“Kids are sent to school with no manual on how to use their brains. Not what to learn buthow to learn.”

Maats and O’Brien knew about all the research, and began sharing information about learning and the brain with their students. They turned their one-on-ones into a book, The Straight-A Conspiracy, to show teenagers that they had control, for a large part, over how they did in school, and that believing certain kids were born talented was a grand conspiracy to keep them down and stressed out (with tongue planted firmly in cheek). The authors use the latest research in psychology and neuroscience to try and convince teens, with lots of pop culture references and humor thrown in, that understanding how their brain learns can help them “totally rule the world.”

Maats explained that often students he tutored had watched another kid in class blow through an assignment and assumed they were just naturally good at it, that they didn’t even have to try. But he began clarifying the real reason they worked so fast; the student knew the answer right away because “it had become automatic,” he said. “They looked effortless, but they only became effortless through hard work.” Unlike sports or music, where students can see others practicing, much of schoolwork practice happens at home, builds slowly over time, and goes unseen. “You don’t see the work others are doing, so it looks like it never happened,” Maats said.

[RELATED: Eight Ways of Looking at Intelligence]

O’Brien said that “geniuses” also know how to focus their attention, and that’s why they may appear calm. “That overwhelmed feeling is coming from attention being focused on too many things that are not automated at once,” she said. “You can’t focus on two things you don’t know – but neither could Einstein!” Explaining one of the largest conspiracies they face with students, and parents’ biggest complaint, student multitasking, they disseminate the research for the teenage brain:

“Your attention can only deal with one unautomated task at a time. The idea that your attention can multitask is a major myth… When you’re trying to do all four of these tasks – walking, chewing gum, talking to your friend and reading Huckleberry Finn – the first two won’t be affected, because you’ve automated them. You can keep walking and chewing gum without even noticing they’re happening. But each of the new activities – holding a new conversation and reading a new book – requires your full attention in order to go well… But the more important point is that you just don’t want to put yourself through that! It’s totally manic!… The more stuff you pile on at once, the more time pressure you add to the situation, the more you start to feel really overwhelmed.”

By using concrete research in a way that speaks directly to teenagers, Maats and O’Brien hope to dispel the image of the rumpled genius, being brilliant in spite of himself. Instead, they want students to know that there are proven techniques that can improve their school performance and get parents and teachers off their back (a particular favorite is “Go Cyborg on Your Mistakes,” an extended “Terminator” metaphor that relates the idea of focused practice). And they seem to relish explaining that the straight-A student is working harder than kids think.

[RELATED: Can Everyone Be Smart At Everything?]

“You would never put a child into the driver’s seat of a car, with no license and no drivers’ ed, and expect him to be able to cruise down the highway successfully, with no fear or hesitation,” said O’Brien. “And yet kids are sent to school with no manual on how to use their brains. Not what to learn but how to learn. The result is that everyone spends their days in school guessing what might be the best approach, the most effective technique…and the questioning about the how takes a lot of time and attention away from what needs to be learned.”





The Delicate Balance

8 12 2014

By Jill Jenkins

With school turning out more runners, jumpers, racers, tinkerers, grabbers, snatchers, fliers, and swimmers instead of examiners, critics, knowers, and imaginative creators, the word ‘intellectual,’ of course, became the swear word it deserved to be.”
Ray Bradbury, Fahrenheit 451

                What is important for students to know?  What should our schools be teaching? If you listen to media, all the schools should be focused on is STEM: Science, Technology, Engineering, and Mathematics. Just like in the 1950’s our society is demanding that education provide more STEM education to provide a technological suave population who can produce a profit for our corporations. Are schools created to serve our corporations or the individual needs of our students?  Society certainly rewards students who perform well in science, technology, engineering and mathematics, but not every student has the desire or the aptitude to do well in those areas.  Are we doing those students a disservice? Since girls have stronger verbal skills and brains wired for an education in communications is this a subtle form of prejudice?  Before we write our curriculum, it is important to determine what is important to know to help our students become both productive citizens and principled people.  We need a more balanced approach to serve all of the needs of all of the varied students in our classes?

                Schools need to prepare students to be productive citizens, but to be honest with as rapidly as technology is changing that is not an easy task.  As a child, I remember laughing at Maxwell Smart and his shoe telephone.  Now, all of us carry telephones around in our pockets that are not only communication devices, but small computers.  The truth is there will be careers that we can’t even imagine, so we have to give students skills to be life-long learners.  To achieve they must be willing to learn new skills through-out their lives. We need to prepare students to adapt to world that we cannot conceive existing. 

                Research shows that females learn differently than males. According to the article, “How Boys and Girls Learn Differently” by Dr. Gail  Gross from the Huffington Post,boys have less serotonin and oxytocin which makes girls more sensitive to other’s  feeling subtly communicated through body language and they can sit still for longer periods of time.  Girls have larger hippocampus, where memory and language is stored.  This means they develop language skills, reading skills and vocabulary much sooner than boys. On the other hand, boys have a larger cerebral cortex which means they learn visually and have better spatial relationships.  This could improve their ability in engineering and technology.  These differences become less dramatic as the child grows older.  Perhaps schools need to focus on presenting a broad spectrum of disciplines in a variety of ways to serve all of students.

Even though our society does not value careers where communications rather than subjects like science, technology, engineering and mathematics are the primary focus, they may still be important careers for our society.  For example, teachers are essential if we want to continue to produce an educated workforce, but if pay is the measurement of value, they are not valued by society.  In the state where I taught science, engineering, technology and math teachers were all paid $5000.00 a year more than any other kind of teacher.  Still, if we want to be realistic students’ need a balance of both to be successful.  For example, my daughter is a journalist; however, she also needs to know how to write computer coding because the magazine that employs her is on-line. Most scientists must document whatever they do which means they need writing and reading skills. Furthermore, who is to say who will be the next poet laureate .   The arts, history and language arts are all equally important skills for students to master as math, science and technological based skills.

Even more important, the humanities:  literature, history and the arts force people to ask “why.”  Certainly, we can’t think about Nazi Germany without realizing, there was a reason that Hitler banned books.  We can’t read a Michael Critchton book without discussing ethics in science and medicine.  We can’t read Charles Dickens’ Oliver Twistwithout questioning the social problems caused by poverty and homelessness.  Reading, writing, history, the arts are all connected to science, math, technology and engineering. A quality education is a balance.  All of it is equally important.  Teachers should be compensated equally and students should be provided with an equal balance.  Teachers should help students develop their own individual talents, so they can become all that they can be.  Schools should prepare each student to become “all that they can be,” not a product to serve the needs of industry.





Bestowing the Gift of Self-Confidence to Students

7 12 2014

The Gift of Self-Confidence

The moment you doubt whether you can fly, you cease for ever to be able to do it.”
J.M. Barrie, Peter Pan

            One of the most important gifts that a teacher can impart to a student is the gift of self-confidence.  To succeed at anything, a person must believe that success is possible.  Many students lack the belief that they could possibly be successful in school or anywhere else; as a result, these same people have difficulty succeeding in life. Students who doubt their abilities often lack any motivation to try.  If a person does not try, they have no possibility of succeeding.  As a result, an educator must first impart the ability to believe in oneself before the student can begin to succeed. Educators must become Peter Pan to help students fly.

When I was teaching in an Alternative Education, I was amazed at the students who had no desire to do well in school or even to attempt to do well in school.  After getting to know these students, I discovered that most of them had suffered so many humiliating failures at school that they believed that they were not capable of learning. They found it was less painful to do nothing, than to attempt anything and fail.  To continually have the belief that they could not succeed reinforced was just too painful for them.  Some of them used outrageous behavior as a way of avoiding this failure. I remember one particular student, Juan, who would not stay in his seat, sang loudly and yelled obscenities across the room to avoid a writing assignment. To reach students like Juan, I had to break down their barriers, get to know them as individuals, persuade them that I was their advocate and I was going to show them how to be successful by celebrating even their smallest achievement.  Being successful can be  rewarding, but to convince these students of that, the teachers needs to break successful behavior into its smallest components and reward for the successful completion of each small step.  For example, I began by rewarding students for coming to class prepared.  Each student who had a pencil and paper was rewarded with a small piece of candy.  Next I created a chart on the board showing the relationship of how a student would feel if he brought this parents a report cards with all “A’s” on it compared to how he would feel if he brought his parents a report card with all “F’s” on it.  Helping a student understand that happiness is directly connected to their success in school is an important step to motivating them to want to succeed.

Students who feel socially inept are often unhappy at school.  Girls, especially, suffer from social bullying that goes unnoticed by educators.  Our society puts so much emphasis on physical beauty and social position in school that students who do not fit the norm are often isolated.  Girls often exclude these girls from social situations and do not include them even in conversations.  Shunning can be cruel treatment that can cause scars that last a lifetime.  Some of this bullying takes the form of cruel comments in social media or scathing remarks made in a classroom or a hallway.  Students who suffer from these vicious assaults lose their self-esteem and as a result, do poorly academically or feel badly about continuing their education because it is too painful.   As an educator, protecting and supporting students’ self-esteem should be one of our goals. Helping students learn to accept and embrace people who are different from them should be another. For students to do well, all students must feel safe and appreciated.

When teachers are writing goals for their classrooms, academic goals are only one dimension of education.  Helping a student feel safe and good about his ability to succeed should be high on the list of objectives. Helping a student accept that others may differ from him, but should still included  in the community without ridicule or attack.   School should prepare students to succeed in life.  If a student has doubts or is not empowered with self-confidence, he cannot succeed.  Like Peter Pan, teachers must bestow the gift of self-confidence.

Posted by Jill Jenkins 





SOME COOL FACTS ABOUT MATHEMATICS.

3 11 2014
brain
  1. The word ‘mathematics’ comes from the Greek “máthēma”, which means learning, study, science.
  1. Do you know a word known as Dyscalculia? Dyscalculia means difficulty in learning arithmetic, such as difficulty in understanding numbers, and learning math facts!
  1. In America, mathematics is known as ‘math’, they say that ‘mathematics’ functions as a singular nounso as per them ‘math’ should be singular too.
  1. ‘Mathematics’ is an anagram of ‘me asthmatic’. (An Anagram is word or phrase made by transposing or rearranging letter of other words or phrase.)
  1. Notches (cuts or indentation) on animal bones prove that humans have been doing mathematics since around 30,000 BC.
  1. The word ‘hundrath’ in Old Norse (old language from where English language originated), from which word ‘hundred’ derives, meant not 100 but 120.
  1. What comes after a million, billion and trillion? A quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion and undecillion.
  1. The number 5 is pronounced as ‘Ha’ in Thai language. 555 is also used by some as slang for ‘HaHaHa’.
  1. Different names for the number 0 include zero, nought, naught, nil, zilch and zip.
  1. Zero ( 0 ) is the only number which can not be represented by Roman numerals.
  1. The name ‘zero’ derives from the Arabic word “sifr” which also gave us the English word ‘cipher’ meaning ‘a secret way of writing’ .
  1. What is the magic of No. Nine (9)? Multiply any number with nine (9 ) and then sum all individual digits of the result (product) to make it single digit, the sum of all these individual digits would always be nine (9).
  1. Here is an interesting trick to check divisibility of any number by number 3. A number is divisible by three if the sum of its digits is divisible by three (3).
  1. The = sign (“equals sign”) was invented by 16th Century Welsh mathematician Robert Recorde, who was fed up with writing “is equal to” in his equations.
  1. Googol (meaning & origin of Google brand ) is the term used for a number 1 followed by 100 zeros and that it was used by a nine-year old, Milton Sirotta, in 1940.
  2. The name of the popular search engine ‘Google’ came from a misspelling of the word ‘googol’.
  1. Abacus is considered the origin of the calculator.
  1. Have you ever noticed that the opposite sides a die always add up to seven (7).
  1. 12,345,678,987,654,321 is the product of 111,111,111 x 111,111,111. Notice the sequence of the numbers 1 to 9 and back to 1.
  1. Plus (+) and Minus (-) sign symbols were used as early as 1489 A.D.
  1. An icosagon is a shape with 20 sides.
  1. Trigonometry is the study of the relationship between the angles of triangles and their sides.
  1. If you add up the numbers 1-100 consecutively (1+2+3+4+5…) the total is 5050.
  1. 2 and 5 are the only primes that end in 2 or 5.
  1. From 0 to 1,000, the letter “A” only appears in 1,000 (“one thousand”).
  1. A ‘jiffy’ is an actual unit of time for 1/100th of a second.
  1. ‘FOUR’ is the only number in the English language that is spelt with the same number of letters as the number itself
  1. 40 when written “forty” is the only number with letters in alphabetical order, while “one” is the only one with letters in reverse order.
  1. In a group of 23 people, at least two have the same birthday with the probability greater than 1/2 .
  1. If there are 50 students in a class then it’s virtually certain that two will share the same birthday..
  1. Among all shapes with the same perimeter a circle has the largest area.
  1. Among all shapes with the same area circle has the shortest perimeter .
  1. In 1995 in Taipei, citizens were allowed to remove ‘4’ from street numbers because it sounded like ‘death’ in Chinese. Many Chinese hospitals do not have a 4th floor.
  2. The word “FRACTION” derives from the Latin ” fractio – to break”.
  1. In working out mathematical equations, the Greek mathematician, Pythagoreans used little rocks to represent numbers. Hence the name of Calculus was born which means pebbles in Greek.
  1. In many cultures no 13 is considered unlucky, well, there are many myths around it .One is that In some ancient European religions, there were 12 good gods and one evil god; the evil god was called the 13th god. Other is superstition goes back to the Last Supper. There were 13 people at the meal, including Jesus Christ, and Judas was thought to be the 13th guest.
  1. Have you heard about Fibonacci? It is the sequence of numbers wherein a number is the result of adding the two numbers before it! Here is an example: 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.
  1. Want to remember the value of Pi (3.1415926) in easy way ? You can do it by counting each word’s letters in ‘May I have a large container of coffee?’
  1. Have you heard about a Palindrome Number? It is a number that reads the same backwards and forward, e.g. 12421.




How to turn every child into a “math person”

13 08 2014

How to turn every child into a “math person”

Last month, the US Math Team took second place in the International Math Olympiad—for high school students—held in Cape Town, South Africa. Since 1989, China has won 20 out of 27 times (including this year), and in the entire history of the Olympiad, the US Math Team has won only 4 out of 55 times, so second place is a good showing. According to the American Mathematical Association website: “team leader Loh noted that the US squad matched China in the individual medal count and missed first place by only eight points.”

Reading about the US Math Team’s performance in the Olympiad this year takes me back to my senior year of high school in 1977 when, having taken 9th place in the US Math Olympiad, I was invited to travel to the International Math Olympiad in Belgrade as an alternate to the 8-member US Math Team. I chose not to go to Belgrade because the Olympiad conflicted with the National Speech Tournament, where my team couldn’t have tied on points for first place without me—while the US Math Team won without needing my help. This profoundly shaped my perception of myself as a “math person.”

Left: an article from 1976 where Miles placed 23rd in the US Math Olympiad; top: in 1977 Miles placed 9th in the competition; bottom: questions from the 1977 USA Math Olympiad.

More than 36 years later, I have come to the view that almost everyone should think of herself or himself as a “math person.” In our column “There’s one key difference between kids who excel at math and those who don’t,” Noah Smith and I wrote this about the often-heard statement: “I’m just not a math person.”

We hear it all the time. But the truth is, you probably are a math person, and by thinking otherwise, you are possibly hamstringing your own career. Worse, you may be helping to perpetuate a pernicious myth—that of inborn genetic math ability.

Not everyone agrees with us. Noah and I got some pushback for our rejection of the idea that inborn math ability is the dominant factor in determining math skill. So I did some more reading in the psychology literature on nature vs. nurture for IQ and for math in particular. The truth is even more interesting than the simple story that Noah and I told.

Math ability is not fixed at birth 

Three facts run contrary to the idea that inborn mathematical ability is a dominantfactor in determining whether or not someone is good at math compared to others of the same age.

First, it is a reasonable reading of the very inconsistent evidence from twin studies to think that genes account for only about half of the variation in mathematical skill among kids. For example, this 2007 National Institutes of Health Public Access twin study, using relatively transparent methods, estimates that genes account for somewhere in the range from 32% to 45% of mathematical skill at age 10. That leaves 55% to 68% of mathematical skill to be accounted for by other things—including differences in individual effort. (Other estimates of the percentage of variation of mathematical skill in kids due to genes range all the way from 19% to 90%. )

Second, a remarkable fact about IQ tests, including the mathematical components of IQ tests, is that every generation looks a lot smarter than the previous generation. This steady increase in performance on IQ tests is known as “the Flynn effect” after the political philosopher James Flynn, who discovered this remarkable fact. The American Psychological Association’s official report “Intelligence: Knowns and Unknowns” says:

… performance has been going up ever since testing began. The “Flynn effect” is now very well documented, not only in the United States but in many other technologically advanced countries. The average gain is about 3 IQ points per decade.

At that rate, an IQ test from 100 years ago would put an average American today at an IQ of 130—in the top 5% of everyone back then.  The American Psychological Association’s report goes on to say:

The consistent IQ gains documented by Flynn seem much too large to result from simple increases in test sophistication. Their cause is presently unknown, but three interpretations deserve our consideration. Perhaps the most plausible of these is based on the striking cultural differences between successive generations. Daily life and occupational experience both seem more “complex” (Kohn & Schooler, 1973) today than in the time of our parents and grandparents. The population is increasingly urbanized; television exposes us to more information and more perspectives on more topics than ever before; children stay in school longer; and almost everyone seems to be encountering new forms of experience. These changes in the complexity of life may have produced corresponding changes in complexity of mind.

In other words, although people a century ago were good at many things, many of them would have struggled with the kinds of abstract problems IQ tests focus on.

(As a simple example of how math standards have risen, my father tells me that when he was in high school, people thought calculus was too advanced for high school students. Nowadays, about one of every six high school students takes calculus in the US.)

Third (and I wish the research were clearer about this for math specifically), thefraction of differences in IQ that seem genetically linked increases dramatically with age. For children, about 45% of differences in IQ appear to be genetic, while for adults, about 75% of differences in IQ appear to be genetic. Think about that. How could it be that genes matter more and more as people get older—even though the older you get, the more environmental things have happened to you? What I think is the most plausible answer, is that the genes are influencing what people do and what they do in turn affects their IQ.

The “love it and learn it” hypothesis

No one yet knows exactly how genes, environment, and effort interact to determine mathematical skill. In light of the evidence above, let me propose what I call the “love it and learn it” hypothesisThis hypothesis has three elements:

  1. For anyone, the more time spent thinking about and working on math, the higher the level of mathematical skill achieved.
  2. Those who love math spend more time thinking about and working on math.
  3. There is a genetic component to how much someone loves math.

Despite emphasizing time spent on math as the driver of math skill, this can explain why identical twins look more alike on math skills than fraternal twins. Since time spent dealing with math matters, it allows plenty of room for the average person to be better at math now than a hundred years ago. And the effect of loving math on math experience and therefore math skill is likely to only grow with time.

To get better at math, act like someone who loves math

If the “love it and learn it” hypothesis is true, it gives a simple recommendation for someone who wants to get better at math: spend more time thinking about and working on math. Best of all: spend time doing math in the kinds of ways people who love math spend time doing math. Think of math like reading. Not everyone loves reading. But all kids are encouraged to spend time reading, not just for school assignments, but on their own. Just so, not everyone loves math, but everyone should be encouraged to spend time doing math on their own, not just for school assignments. If a kid has a bad experience with trying to learn to read in school, or is bored with the particular books the teacher assigned, few parents would say “Well, maybe you just aren’t a reader.” Instead, they would try hard to find some other way to help their kid with reading and to find books that would be exciting for their particular kid. Similarly, if a kid has a bad experience trying to learn math in school, or is bored with some bits of math, the answer isn’t to say “Well maybe you just aren’t a math person.” Instead, it is to find some other way to help that kid with math and to find other bits of math that would be exciting for their particular kid to help build her or his interest and confidence.

The way a teacher presents a mathematical principle or method in class may not work for you—or, as Elizabeth Green suggested in the New York Times, the whole American pattern of K-12 math instruction may be fatally flawed. If you loved math, you would think about that principle or method from many different angles and look up and search out different mathematical resources, until you found the angle that made most sense to you. Even if you don’t love math, that would be a good way to approach things.

Many people think that because they can’t understand what their math teacher is telling them, it means they can’t understand math. What about the possibility that your teacher doesn’t understand math? Some people are inspired to a life-long love of math by a great math teacher; others are inspired to a life-long hatred of math by an awful math teacher. If you are unlucky enough to have an awful math teacher, don’t blame math for your teacher’s failings.

Cathy O’Neil—who blogs at mathbabe.org—describes well what I like to call “slow-cooked math”:

There’s always someone faster than you. And it feels bad, especially when you feel slow, and especially when that person cares about being fast, because all of a sudden, in your confusion about all sort of things, speed seems important. But it’s not a race. Mathematics is patient and doesn’t mind.

Being good at math is really about how much you want to spend your time doing math. And I guess it’s true that if you’re slower you have to want to spend more time doing math, but if you love doing math then that’s totally fine.

I was lucky to have a dad and older brother who showed me a bit of math early on, in a way that was unconnected to school. Then in school, I spent at least as much time on math when I wasn’t supposed to be doing math as when I was. It was a lot more fun doing math when I wasn’t supposed to be doing math than when I was.

For one thing, when I did it on my own, I could do it my own way. But also, there were no time limits. It didn’t matter if it took me a long time. And nothing seemed like a failure.

I spent a lot of time doing math. And very little of that math was done under the gun of a deadline. I spent some time on literal tangents in geometry and trigonometry. But I spent a lot more time on figurative tangents, running into mathematical dead ends. When Euclid told King Ptolemy “there is no Royal Road to geometry,” it had at least two meanings:

  1. Everyone—even a king or queen—has to work hard if he or she wants to learn geometry or any other bit of higher math.
  2. The path to learning geometry, or math in general, is not always a straight line. You may have to circle around a problem for a long time before you finally figure out the answer.

What can be done

I feel acutely my own lack of expertise in math education for students younger than the college students I teach. Fortunately, there are a wealth of practical suggestions for teaching and learning math by others who know more than I do, or have a different perspective from their own experience.

Noah and I received many comments in response to our post but the comments I learned the most from were from these people, who let me turn their comments into guest posts on my blog:

In Green’s article “Why Americans Stink at Math,” she talks about how differently math is taught in Japanese classrooms, and how we should hope that we might someday get that kind of math instruction in the US. The key difference is that in Japan, the students are led by very carefully designed lessons to figure out the key math principles themselves. That kind of teaching can’t easily be done without the right kind of teacher training—teacher training that is not easy to come by in the United States.

But some teachers at least encourage their students to follow a “slow-cooked math” approach where they can dig in and wrap their heads around what is going on in the math, without feeling judged for not understanding instantly. Elizabeth Cleland gives a good description here of how she does it.

Even when a student is lucky enough to have good teachers at school, a little extra math on the side can help a lot. Kids who arrive at school knowing even a tiny bit of math will have more confidence in their math ability and will probably start out liking math more. Even quite young kids will be interested in a Mobius strip made out of paper where a special twist makes what looks like two sides into just one side.  And putting blocks of different lengths next to each other as in aMontessori addition strip board is exactly how I have always pictured addition in my head.

A Montessori addition strip board.Image via jsmontessori.com

Extra math doesn’t all have to come from parents. In some towns, enough Little League soccer coaches are found for almost every kid to be on a soccer team. And even I was once drafted as a Cub Scout Den Leader. If people realized the need, many more adult leaders for math clubs for elementary and middle school kids could be found. In addition to showing kids some things themselves, math club leaders can do a lot of good just by checking out and sorting through the growing number of great math videos and articles online, as well as old-style paper-and-ink books.

I use Wikipedia regularly as a math reference. (There is no reason to think Wikipedia is any less reliable than the typical math textbook; textbooks are not 100% error-free either.)  I have a post on logarithms and percent changes that is one of the most popular posts on my blog. (Maybe it is the evocation of piano keyboards and slide rules, or the before and after pictures of Ronald Reagan.) And Susan Athey, the first woman to win the John Bates Clark Medal for best American economist under forty, highly recommends Glenn Ellison’s Hard Math for Elementary School as a resource for math clubs. All of that just scratches the surface of the resources that are out there.

The obvious issue raised by the “love it and learn it” hypothesis is that some people may not start out loving math, and some may never love math. Acting as if you love math when you don’t may work, but it can be painful. So it is important to figure out what can be done to instill a love of math. Even if they only know a little math themselves, people who can get kids who don’t start out loving math to come to love it are a national treasure. As the brilliant business guru Clay Christensen (among others) has pointed out, in an age when lectures from the best lecturers in the world can be posted online, the kind of help students need on the spot is the help of a coach.

For too long, we have depended too heavily on overburdened math teachers whohave remarkably little time in school to actually teach math, and whom the system has deprived of the kind of training they need to teach math as well as it can be taught. It is time for all of us to take the responsibility for learning math and doing what we can to help others learn math–just as we all take responsibility for learning to read and doing what we can to help others learn to read.

Most of us who participated as kids in a sport or other competitive pursuit remember a coach who got us to put in a lot more effort than we ever thought we would. Math holds out the hope of victory not just in a human competition, but in understanding both the visible universe and the invisible Platonic universe. There is no impossibility theorem saying there can’t be math coaches in every neighborhood who make the average kid want to gain that victory.





Math- When Am I Ever Gonna Use This?

11 08 2014

by 

 When Am I Ever Gonna Use This?
Posted: 10/21/2012 11:40 am
 “When am I ever gonna use this?” As an eighth-grade algebra teacher, I hear this refrain at least once a week. It’s a difficult question to answer. I mean, when is the last time that your employer asked you to factor a polynomial or prove two polygons congruent? The truth is that most of us will never use the myriad of math facts and algorithms in our post-school lives. However, that does not mean that math does not have some valuable lessons for us. The following are lessons that can be learned in an algebra classroom and applied in your life. No calculator required.

Mistakes

Pencils come with erasers for a reason. Mistakes in a math classroom are inevitable. In fact, they are beneficial, as they often uncover misconceptions and maladaptive beliefs. We often believe that we must write our lives in ink and that mistakes permanently mar us. We try to hide them, turn our heads in shame. There is always a lesson in every error. Rather than crossing out the mistake, examine it and learn from it.

Growth at the Edge

I always want my students to be just slightly uncomfortable; I push them a little beyond their comfort zone because this is where the learning occurs. We don’t all have algebra teachers following us around in life, but we can still push ourselves past our self-imposed boundaries. Growth occurs at the edge of comfort and panic. Find that space and embrace it.

Take Risks

I can do anything with a student who is willing to try. I have respect for those who will volunteer even when they are unsure of their answer. They have learned that taking risks can bring reward in the form of a correct answer, a deeper understanding, or the respect of their peers. Life is no different. If you never risk anything, you will never gain anything either. Don’t be afraid to raise your hand.

Break It Down

In algebra, students are presented with complex problems. One of the first skills they learn is to break a large problem into a series of simpler ones, focusing on one step at a time. When you feel overwhelmed in life by what seems to be an insurmountable obstacle, try breaking it into manageable tasks. You might just be amazed at what you can accomplish.

Eliminate the Unnecessary

“Six-year-old Suzy has four apples and 5-year-old Bobby has two. How many apples do they have together?” It’s pretty clear that this example has extraneous information, clutter that can be ignored without detracting from the answer. Examine your life. Do have unnecessary clutter that you can eliminate? Get rid of it and you will find clarity in what remains.

“See” the Effect

I train students to anticipate the effect of a step in a problem before they make it. I want their minds to always be slightly ahead of their pencil as they “see” the impact of a decision prior to carrying it out. This is beneficial in the world at large as well as it encourages thoughtful and deliberate decisions. If you understand effect, you can change the cause.

Think Logically

Every year I have students that attempt to “prove” lines parallel by stating that they look parallel. As tempting as that reasoning may be, it is simply not valid. Our minds are sometimes lazy and try to make assumptions without proof. Watch yourself and check to see if the data supports your conclusions. It’s good to listen to your gut, but don’t divorce it from your brain.

Communication

Every year I have students that can execute the mathematics perfectly, yet cannot communicate to anyone else how or why they made the decisions they did. Their work is then almost useless, as no else can understand or build upon their results. Our thoughts and ideas are only as good as our ability to communicate them to another. Learn to be clear in your words so that they may be understood.

Balance

“Whatever you do to one side, you have to do to another.” Algebra students across the country recite this line as they learn how to balance equations. Perhaps we should all be uttering this line as a reminder to create balance in our own lives. Remember that when you add something to one area of your life, you will need to make a change in another so that balance is restored.

Work Backwards

Sometimes my students face problems that feel impossible. They can’t even figure out the first step. I teach them to start from the goal and work backwards, teasing out the steps as they go along. In your life, start with your goals and figure out what you need to do to achieve them. By seeing the steps in reverse, even the loftiest dreams can be made possible. Try it.

Persevere

Math can be hard. So can life. In both cases, it takes perseverance and tenacity to see difficulties through until the end. The rewards that come from determination and dogged spirit are so much sweeter than those that come without the sweat. The only way that failure is certain is if you do not try.

Basics Matter

It’s difficult to understand algebra if you don’t know multiplication. Likewise, it’s difficult to find fulfillment if you’re not meeting your basic needs. Start at the beginning and make sure you have a strong foundation upon which to build.

Confidence

So many of my students enter my room in August convinced that they cannot “do” math. In my 11 years of teaching, I have yet to meet a student that was correct in this belief. My first job with these uncertain pupils is to convince them that they can. Until they believe in themselves, they will continue to fail. The biggest lie we tell ourselves is, “I can’t.” Stop lying. It may be scary to try, but just think of the possibilities.

Participate

One of my favorite quotes hangs right above the board in my classroom:

“Math is not a spectator sport.” — Jerry Mortensen

Neither is life. Don’t stand on the sidelines watching it unfold in front of you. Get out there and play!