Math Confidence and Calculators

Do Calculators Increase or Decrease Math Confidence?

The students in Math for Elementary Teachers campaigned to use calculators on their exam. To get their feedback on this subject, we had a discussion board on line (where they could post anonymously if they wanted). All student posts were in favor of the calculator.

On Thursday, 2/12, they were allowed to use a calculator during the last 30 minutes of the 75 minute period. I noticed that some students who had supported calculator use finished without calculators.

Although some of the bias may be generational of the calculator vs. non-calculator, I posted New York State guidelines for Grades 3 - 8 (only Grades 7 and 8 are allowed to use them on the non-multiple choice sections). It will be interesting to get feedback from these learners next week.

On this exam, the average was above 90.

Metacognition -- Thinking About Thinking

You have to know what you don't know!

Asking learners thoughtful questions, providing guidance and giving feedback (rather than answers) raises awareness of thinking processes and helps students to categorize challenges into Easy / Medium / Hard.

For example, out of 10 questions, 5 may be easy, 2 medium and 3 hard -- when the student recognizes the type of problem and the difficulty level, effort can be expended strategically. Multiple choice questions are excellent workouts for metacognition as learners work on why the right answer is right, why the incorrect answers are incorrect and what could go wrong to make them choose a "good wrong answer". Some content areas especially suited for "good wrong answers" are exponent rules, fractions, pictographs, perimeter vs area.

Much of the research for this metholodology was conducted informally at Stuyvesant High School, the L.C. Smith College of Engineering at Syracuse University, later as an MBA student at the Stern School of Business at New York University, and over the last 8.5 years working one-on-one and in the classroom to help people build Math Confidence.



Assessments Based on Content but Built for Detail, Accuracy and Stamina

Multiple Choice Physics (points taken off for guessing) -- the question in mph, answer in meters/sec...
Matching Answers Statics and Dynamics -- 36 answers for 25 questions -- computed generated in 1982...
Circuit Theory 1 and 2 -- untimed exams at night starting at 8 PM and on into the night




This process enables the student to maximize potential for peak performance under intense conditions (exams) and contributes positively to their overall academic and emotional well-being.

New Year's Riddle courtesy of Math Notations

"What do you call solving an equation twice on Jan 1st?"We had three "first responders" so I will close the contest down now and announce our winners in the order in which I rec'd their email solutions. By the way the answer can be found "hidden" near the bottom of this post! And the winners are...
SEAN HENDERSON (and his wife!)
JUSTIN TOLENTINO
ROBIN SCHWARTZ











A New Year's Re-Solution

MathNotations: The Number Warrior and the Mysterious Minds of Students!

MathNotations: The Number Warrior and the Mysterious Minds of Students!

Robin Schwartz said...
Negative exponents, fractional exponents, the zero exponent, exponent rules, logs – these should turn on the ALERT indicator – be careful -- danger ahead!! Even armed with a calculator, students will get tricked by these types of problems…Jason, did your students use calculators for this exam?Studying multiple choice questions can help refine thinking skills (and analyze potential errors) through the strength of the comparative.
An example from “Using ‘Good Wrong Answers’ To Achieve Math Confidence and Success” is:

What’s the value of 3^-2?
A) -2/3
B) -9
C) 1/9
D) -6

If they are comfortable with the zero power, they can write 3^-2 = (3^0)/(3^2). Since exponents (and logs) are not logical, I work with learners on extra alertness with this type of content. Unlike some content where they know they are guessing, here they feel sure they have done it right. Another exercise is shown below:
16^1 =
16^.5 =
16^0 =
16^-.5 =
16^-1 =
By increasing awareness of their thinking process, students can build confidence and enjoyment of Math and even improve their scores.
Robin A. Schwartz
Founder, www.mathconfidence.com
www.blogspot.mathconfidence.com

December 24, 2008 11:32 PM

Re: The Opportunities of Test Prep – learning, studying, Life skills: response to Slatalla article 12/4/08


To the Editor:

Thank you for the article on the College Board’s new program (Michelle Slatalla’s “My Child’s Fate, All Laid Out by 13”, 12/4/08). While there can be an overemphasis on testing, as a parent and Math educator, I try to focus on the positive opportunities that studying and learning can offer.
While some see multiple choice tests as counterproductive, the “good wrong answers” (the tempting ones that test takers may choose due to being tricked or not reading carefully) can help students see their mistakes and learn from them. Identifying potential errors leads to more metacognition (thinking about thinking), stronger comprehension, better grades, improved problem-solving skills and more enjoyment of the process.
In addition to content, the study of Math has other benefits:
· instilling values of discipline and excellence
· improving memory and focus
· preparing students for success in the ‘knowledge economy’
This positive perspective can help students, parents, teachers, and administrators meet the challenges of ‘teaching to the test’ by viewing it as an opportunity to address knowledge gaps and common errors while sharpening critical thinking and gaining confidence.
The testing of 8th graders can lead to strengthening of knowledge and skills to build a strong foundation so learners can realize their potential. This may alleviate remediation at the college level while inspiring lifelong learning in the next generation.

Robin A. Schwartz

link to NYT article
http://www.nytimes.com/2008/12/04/fashion/04spy.html

Invert and multiply...which one????

Oh, just flip it upside down and then multiply (“invert and multiply”).
But they are not sure which one to flip!!

Students learn this in late elementary, but when asked “Why?”,
they will say “Oh, it’s the rule”.

An example such as “How much is 12 divided by ½?” can illustrate the concept and learners can follow this example rather than a rule!!

How many half dollars are in $12?
How many half inches in 1 foot?

12 x 2/1 = 24!! ;)

Subject: Trig homework can be pleasant ;)

Dear Professor Fisman:

Your Slate (NYT, http://www.nytimes.com/2008/06/15/weekinreview/15read.html?_r=1&scp=5&sq=trigonometry&st=cse) article from June about One Laptop Per Child has some terrific points, however,

“Perhaps not surprisingly, the lesson from Romania’s voucher experiment is not that computers aren’t useful learning tools, but that their usefulness relies on parents being around to assure they don’t simply become a very tempting distraction from the unpleasantness of trigonometry homework.”

The role of a Math peak performance coach goes beyond the algebra and trig knowledge and focuses on the benefits of brain fitness, ‘flow’, persistence and critical thinking – assets for academic, professional and personal success.

Math could sure use some good PR – last month, the headline for a New York Times article read: "Video Game Helps Math Students Vanquish an Archfiend: Algebra"

Trig homework can be pleasant! Many learners find that there is enjoyment in the engagement or ‘flow’ of studying and that brain fitness can be challenging and sometimes even fun!

Robin A. Schwartz, MBA, BSEE
CEO, Math Confidence

Thanks. I loved this message. I was a math guy until I got to graduate school, but I know I’m not normal. Anything you can do on this front is all to the greater good.

Best, ray fisman, Lambert Family Professor of Social Enterprise, Columbia Business School

Algebra as an Archfiend?

Education is about stretching knowledge, skills and attitude.
Last month, the headline of a New York Times article about a Math video game for middle schoolers designed by Dimension M:

"Video Game Helps Math Students Vanquish an Archfiend: Algebra"speaks volumes about attitudes about Math! In the print verson, the word "foe" was used.

How about "ally", "comrade", "challenge", "opportunity"?

"Students at Intermediate School 30 in Brooklyn played a video game on Monday, and learned a little algebra at the same time. " was the caption on a photo.
The juxtaposition of this sentence would be:
"Students at Intermediate School 30 in Brooklyn learned algebra on Monday and played a little bit of video game at the same time. "

http://www.nytimes.com/2008/10/08/nyregion/08video.html

Linked In Question:
How did you succeed in a difficult course while in high school or college?
Students all over the country are getting ready for major tests and final examinations. It seems that students always have one difficult course or instructor. I would like to hear about strategies that you used to get through a difficult course. What are some of the best study habits? I would like to share your strategies with students. Please confirm that I can share your suggestions.

Math Confidence Response:
As a peak performance Math coach (and a former Engineering and Finance student who made it through Electromagnetics and Corporate Finance), there are key questions for test takers that raise awareness of their thinking process:
How did you get your answer?
Was it Easy? Medium? Hard?
Is that your final answer? (courtesy of Who Wants to Be a Millionaire?)

Other Success Factors:
Study Groups turn up the volume of thinking through discussion
Read over the entire test at the beginning
Do not leave anything blank -- write something :)
Know examples rather than formulas (12 divided by 1/2 = 24 is better than invert and multiply) Keep review sheets clean -- do not write the answers on the review sheet! This will force reworking of the problems --increasing the chance of success on the test
View the test as a learning opportunity -- test takers are making new connections while they are in the exam

While studying for finals may not have an exact parallel in the workplace, gaining math (and other academic) confidence can help with the following Life Skills:
Problem-Solving
Critical Thinking
Optimizing Your Potential
Escaping the Perfectionism Trap
Appreciating Effort vs. Obsessing about Ability
Financial / Medical Information Fluency
Expanding your self-teaching Skills
Finding a (new) career
Lifelong Learning
Brain Fitness


Posted by
Robin A. Schwartz, 11/26/08
http://www.mathconfidence.com/
mathconfidence@aol.com

Math Confidence's Robin Schwartz's Letter to the Editor
The New York Times

LETTERS; Ups, Downs and America's Mind-Set

Published: September 29, 2008

To the Editor:
Positive thinking is not about always getting what you want; it is making the best of what you have while planting seeds to create opportunities in the future.
My role as an educator and parent is to empower the learner with independence, knowledge, skills and attitude to achieve personal, professional and academic success.
Motivational speakers like Tony Robbins, Jim Rohn and Stephen R. Covey have helped millions of people stay on course, some of whom might have turned to other coping mechanisms, like alcohol. Their messages may not resonate with all audiences, but they help to motivate and inspire millions of people to persevere through life's (and the financial markets') inevitable ups and downs.
Robin A. Schwartz Bronx, Sept. 25, 2008
The writer is the founder of mathconfidence.com.

Trainer As Consultant 12/08

This certificate at the NYU School of Continuing and Professional Studies has expanded my mindset regarding learning and teaching and has been a positive experience professionally and personally.

The next certificate -- Coaching -- will be completed in 2009 and has helped to me to better understand and support learners in the classroom and one-on-one.

Permanent Weight Loss is a Metaphor for Life

While this may seem off topic for a Math blog, the methodology and mindset for permanent weight loss is very similar to that of success in Math. In 2003, I lost 40 pounds off of a 5'2" frame after a health issue forced me to give up sugar!

I have kept the weight off for 3.5 years and since it was always my #1 New Year's Resolution, I have had to reorient my goals and find new ones to keep myself going!

Looking long-term, making a commitment, planning in advance, staying on course!

Motivation is a Key to Success

More and more I realize that what I do is not really about the Math...much of it is about motivation and coaching. How do you build a student who is willing and eager to learn? How does one instill the idea of doing one's best? That phrase is overused so the one I like to use is living up to one's potential. I know that when I took History is high school I did not do very well -- my interest level was low. I got less and less enthusiastic with the less work that I put in. In hindsight, if I had put in a little more I would have gotten a lot more out.

It is not necessary or even desirable to do everything excellently, just to know that you put forth the effort makes a big positive difference!

Excellent Opportunities

Math for companies, Math for educators, Math for everyone ;)

Mental Math

Students can sometimes procedurally solve arithmetic problems but have to use different skills when asked to compute 85 x 99 without a pencil.
The idea of 99 x 85 being equivalent to 100 x 85 - 1 x 85 is a different non-mechanical way of thinking for many students. The mechanics of Mathematics are important -- however, when used in conjunction with mental Math and clever but easier procedures, students are empowered and become more confident and self-reliant.

Is a function increasing or decreasing?
The advent of the graphing calculator has helped students and teachers understand and explain this material much more easily and clearly. The best example is to use a right side up parabola and ask the student verbally. When shown the Power Rule of Calculus at the same time, the student can then support his/her answer (or change it accordingly!!). The next task is show the contrast between the parabolic and the cubic so that the student can see that y = x^3 is ALWAYS increasing!!
Then we can revisit the idea of a line and how it has a constant rate of change -- all of these concepts can be well demonstrated by using the xy table on the graphing calculator.

When working with a tenth grader on imaginary numbers, we used the graphing calculator to study the contrast between y = x^2 - 4 (that has 2 zeros -- at 2 and -2) and y = x^2 + 4 that hovers up above the x-axis and has no real roots only imaginary. When students are able to visualize this difference and then tie it together with the quadratic formula (seeing that the discriminant b^2 - 4ac will result in a negative number for x^2 + 4 -- and that there is no sqrt of a negative number), they can then understand and process the concepts and procedures that are key to success in working with imaginary numbers.

Imaginary numbers are best studied and remembered by sequentially (and boringly) rewriting each component of the equation. For example,
sqrt(-4) x sqrt(-36)
sqrt 4 sqrt (-1) sqrt 36 sqrt(-1)
2 i 6 i
12 i^2
Then from up above drop in a -1 for i^2
12 (-1) =

An interesting Letter to the Editor regarding Math homework in the New York Times today.
Here is the text:

To the Editor:
I wonder if the educators cited in the article have ever tried to teach the New York City-mandated math curriculum to students who enter high school at a sixth-grade level. Perhaps gifted students can “chill” during vacations.
But the 98 percent of students who don’t attend Stuyvesant and the other specialized high schools need to catch up — and homework is one of the few tools we teachers control that can help us provide individualized instruction.
We either raise the level of homework for these kids or lower our expectations of them — the choice is obvious.
Mitch Kurz
New York, April 4, 2007
The writer is a math teacher at the Bronx Center for Science and Math.

The Calculus in a freshman core terminal class!!

The idea of slope extended beyond the straight line

Contrast a line with a parabola graphically and with a table on the TI-84 and equation wise (algebraically) so that students can see that the rate of change for a nonlinear function changes in a nonconstant manner)

Show the slope of line is the "m"

Write the slope formula using dy/dx or as delta y / delta x (using the Greek letter delta triangle thingie)

Sixth grade homework can sometimes look like this: 5/12 divided by 5/9. I wish fractions problems would use easier numbers so that the student can feel if their answer makes sense. I like to use 12 divided by 1/2 or perhaps the more difficult problem used by Liping Ma and/or Deborah Ball : 1 3/4 divided by 1/2.

Students can build intuition by asking themselves if the answer would be more than 1 or less than 1 by using 3/12 or 12/3 and seeing which of these examples their problem resembles.


These fraction problems are usually also related to attention to detail and reading. One of the other problems was you have 12 yards of fabric and will use 2/3 of a yard for each item -- how many items can you make? The student will often perform the multiplication of12 x 2/3 without asking themselves the question: Can I make more than 12 items or less? Once they have answered that questions, then they have a more clear sense of what to do. 1/2 vs. 2 can be used here.

1/2 x 1/2 is a great example as students often add instead of multiply and do not feel that multiplication should make their answer smaller!!

For division of fractions, why do we invert and multiply? If the student just memorizes the procedure, they will not know which procedure to use. It might (definite maybe) be ok if students were able to memorize these concepts and procedures but they get all jumbled!! Students will better comprehend the material and have the ability to more accurately retrieve it if they rely upon examples rather than memorizing procedures.