I just can never do Math! My brain just doesn’t get Math. Mrs. Jones said that I’d never be good in Math. I memorized how to do it, but now I can’t remember. None are true!
Overcome Math Anxiety 1. Q uality teaching requires a positive attitude . 2. Teach for u nderstanding! 3. A ssure students that mistakes are normal. 4. Share your l ove of math with enthusiasm. 5. I nitiate student practice with ongoing assessment. 6. Differentiate your t eaching. 7. Y earn for student success! Quality
1. Q uality teaching requires a positive attitude! Overcome Math Anxiety Top 6 Keys to Being a Successful Teacher by Melissa Kelly ( http://712educators.about.com/od/teachingstrategies/tp/sixkeys.htm ) All teachers must have a positive attitude and the belief that you have the ability to achieve it.
Connecticut Bricklayer 1968 in Yankee magazine Your attitude is everything!
2. Teach for u nderstanding! Overcome Math Anxiety Classic example: Teaching about More later
3. A ssure students that making mistakes are the best way to learn. Overcome Math Anxiety Honor your mistakes! You learn more from failure than success: in business, in athletics, and in math!
4. Share your l ove of math with enthusiasm & a sense of humor. Overcome Math Anxiety 3 in. 4 in. x Find x
#4 - continued <ul><li>Humor in the Mathematics Classroom? But Seriously . </li></ul><ul><li>- Chuck Nicewonder </li></ul><ul><li>Math Forum @ Drexel - Humor ( http://mathforum.org/library/resource_types/humor/ ) </li></ul><ul><li>Math Jokes for Teachers ( http://www.sonoma.edu/math/faculty/falbo/jokes.html ) </li></ul><ul><li>Mathematical Humor ( http://www.math.utah.edu/~cherk/mathjokes.html ) </li></ul>^ ^ (@) (@) | |____| Now, Abbott & Costello 13 x 7 = 28
5. I nitiate student practice, with ongoing assessment Overcome Math Anxiety 9 Principles of Good Practive for Assessing Student Learning ( http://ultibase.rmit.edu.au/Articles/june97/ameri1.htm#9 ) “ Assessment works best when it is ongoing”!
6. Differentiate your t eaching. Overcome Math Anxiety Differentiated teaching is a process which includes planning curriculum aims, teaching strategies, resources, teaching methods and ways of interacting with students while giving special consideration to: 1) differentiation between students according to ability; 2) the different ways students learn; and 3) the different speeds at which students learn. (http://www.doctorgus.net/ClassroomDifferentiated.htm) Internet4Classrooms - http://www.internet4classrooms.com/di.htm How to differentiate Instruction - http://www.teachnology.com/tutorials/teaching/differentiate/planning/
7. Y earn for student success! Overcome Math Anxiety ...even with word problems!
Problem Solving How NOT to introduce word problems.
Yes, word problem can be used to foster an understanding of various mathematical topics.
Pedagogical Implications for Problem-Solving <ul><li>No other decision that teachers make has a greater impact on students' opportunity to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages the students in studying mathematics . (Lappan, Michigan State, http://darwin.nap.edu/html/hs_math/ch13.html ) </li></ul><ul><li>The activity in which knowledge is developed and deployed . . . is not separate from or ancillary to learning and cognition. Nor is it neutral. Rather, it is an integral part of what is learned. Situations might be said to co-produce knowledge through activity . (Brown, Collins, & Duguid, 1989) </li></ul><ul><li>IOW, HOW you teach is the most important part of the learning process </li></ul><ul><li>So engage the student and co-produce knowledge </li></ul>
Overcome Math Anxiety 1. Q uality teaching requires a positive attitude . 2. Teach for u nderstanding! 3. A ssure students that mistakes are normal. 4. Share your l ove of math with enthusiasm. 5. I nitiate student practice with ongoing assessment. 6. Differentiate your t eaching. 7. Y earn for student success! Quality Now, a little more about this.
Without using any numbers and also without using the words “multiply”, “divide”, or “ratio”, describe exactly what the symbol means. An example of teaching without understanding
<ul><li>“ 3.14”, they answered </li></ul><ul><li>“ NO!…not exactly”, I replied. </li></ul><ul><li>“ Oh, 22/7 is exact”, they said. </li></ul><ul><li>“ NO!…not exactly”, I replied. </li></ul><ul><li>“ Oh, it’s 3.14 with a lot of other digits after that!”, they said. </li></ul><ul><li>“ No!… not exactly”, I replied. </li></ul><ul><li>“ What? ”… they said with looks of exasperation & bewilderment </li></ul>What is ∏...exactly?
Even when I tried to coach them... <ul><li>“ Who discovered it?”, I asked. </li></ul><ul><li>“ Some really smart mathematician”, they replied. </li></ul><ul><li>“ If the world lost the value of Pi, how could we rediscover it?”. </li></ul><ul><li>“ Don’t know”, they replied. </li></ul><ul><li>“ What’s the equation for the circumference of a circle?” </li></ul><ul><li>Some pondered, “ π r 2 or is it 2∏r ?” </li></ul>
And now at the university level… The one correct answer came from a PE major… he had a high school teacher who was passionate about his students understanding
Our friend... … is simply how many times the diameter a circle will fit around it’s circumference.
Sometimes we do tell them that ∏ is the ratio of C/d… why then don’t they remember? <ul><li>Remember the adage from teacher training…? </li></ul><ul><li>Tell me & I’ll forget </li></ul><ul><li>Show me & I’ll understand </li></ul><ul><li>Let me do it & I’ll remember </li></ul>The problem may be just that... we tell them & frequently, that’s all.
A few techniques to understand <ul><li>Group work with various size circles… three dimensional objects are easiest for measuring circumference </li></ul><ul><li>Get class average… always got very close </li></ul><ul><li>Whole class activity in gym </li></ul><ul><li>With chalk and large circular objects draw on outside parking areas </li></ul>
Now about the area... If they can’t remember if the area is r 2 or 2 r, ask them to think about the area of a rug…. Does it have dimensions of sq. ft. or just ft? Discuss that areas always involve squared dimensions... A r 2 Show why the area must be less than 4r 2 ? As always, best to allow each student to discover the formula for himself or herself!
Method to discover the area... <ul><li>Cut a circle into fourths or eighths or more if feasible. </li></ul><ul><li>Spread the fourths out… </li></ul><ul><li>Do the same for a second circle </li></ul><ul><li>Fit the two together </li></ul>
Now the grouped figure looks like this... Which closely approximates the appearance of a parallelogram of the same area…. The more sections originally cut from the circle, the closer is the approximation.
The dimensions of this parallelogram... 2 ∏ r r r 2 ∏ r
First and foremost, students should understand the relationships between radius, diameter, & circumference... <ul><li>The diameter is twice the radius. </li></ul><ul><li>2. The circumference is a slightly more than 3 times the diameter. </li></ul>IOW, double the radius, you’ll get the diameter. Triple the diameter, you’ll approximately get the circumference!
<ul><li>Unfortunately, textbooks frequently get in the way of understanding. </li></ul><ul><li>Never rely exclusively on the text! </li></ul>Confront Math phobia immediately! Never let them say this!
Engage students first…then open the text! Math phobic nightmare!!!
Charts can be good, but would the average student understand the trend here… especially without being involved in the calculations? Number of sides of polygon Perimeter Area 4 5.66 r 2.00 r 2 6 6.00 r 2.60 r 2 8 6.12 r 2.83 r 2 10 6.18 r 2.93 r 2 20 6.26 r 3.09 r 2 30 6.27 r 3.12 r 2 100 6.28 r 3.14 r 2
Don’t just read through this intro… instead have students do the measurements! Addison-Wesley Mathematics , 1987 What’s wrong on this page? 53589
Texts always seem to put the applied sections at the end! Why? ~ Try starting with it instead! Why not do this 1st?
The text reinforces that problems are hard by putting them at the end. #20 & 21 Are basic to understanding
<ul><li>First do hands-on activities for discovery </li></ul><ul><li>Then check for basic understanding </li></ul><ul><li>Then do some easy word problems </li></ul><ul><li>Then and only then do your typical math problems for reinforcement. </li></ul>So try this!
<ul><li>Suppose this was a multiple choice exam: </li></ul><ul><li>19.4 cm b) 383.1 cm c) 191.54 cm d) 1319.6 cm </li></ul>Teach them to find the answer without any calculations! IOW, what is the only one that closely approximates three times the diameter? a: 61/3.14 … C=d/ b: (2 * 61) * 3.14… mistakes 61 for the radius c: Correct 3.14 * 61… C= d d: 3.14 * (61) 2 … uses C= d 2 The answers :
Approximation 400/3 = 133 405.06/3.14 = 129 Actual C ~ d ~
π may not be rational, but it is understandable!!!
From Research <ul><li>As modern civilization requires relentless quantification and critical evaluation of information in daily transactions, it becomes necessary to develop new ways of thinking and reasoning that can be used to learn and do mathematical activities. </li></ul><ul><li>Through problem solving for instance, we acquire a functional understanding of mathematics needed to cope with the demands of society. </li></ul>Limjap, De La Salle University
<ul><li>Word problems are much more likely to have real world applications </li></ul><ul><li>Often can be use in a lab setting </li></ul><ul><li>Look especially for problems that have applications to sports, music, cars, etc. </li></ul>When will we ever use this stuff?
Simple charts can show trends How many gallons of 50% antifreeze must be mixed with 80 gallons of 20% antifreeze to get a mixture that is 40% antifreeze? Gallons of antifreeze Concentra'n x .50 80 .20 x + 80 .40 .50*x + .20*80 = .40 * (x+80)
Moral of the story… do all you can to enhance student understanding! Work forward Work backwards Estimation “ The philosophical and theoretical view of knowledge and learning embodied in constructivism offers hope that educational processes will be discovered that enable students to acquire deep understanding rather than superficial skills .” (Blais, 1988)