Successful adaptive behavior presupposes having at your disposal a veritable array of tools for adaption and the mental knowhow to selectively apply or modify those tools. In math, these tools include a good conceptual understanding of the mathematical concepts being tested, as well as the ability to play around, combine, re-fashion these concepts in new/novel problems.
Renaming numbers or “borrowing” can in a sense be an adaptive practice. A student of mine added 2 6/11 and 3 7/11 by “borrowing” 4/11 from 6/11, added 4/11 to 7/11 to make a whole, and gave an answer of 6 2/11. He did not go thru the usual route of 5 13/11 = 6 2/11. When asked how he learned this approach, he pointed to the fraction tower that I extensively use with my students, and explained that it’s easier to regroup to make whole than to add improper fractions. This student has a better conceptual understanding of fractions than some Grade 12 students I know.
While routine expertise is prized by many especially come exam time, it is of limited value in some tests (GRE, GMAT or SAT) when an increasing number of questions tests the level of your adaptive expertise. Many questions force you to view the problem from a different perspective, perhaps less analytically and more holistically, perhaps less algorithmically and more relationally (among variables). That requires a great deal of flexibility to switch between viewpoints, and creativity to re-invent your take of the problem compared to similar ones encountered in the past.
Adaptive thinking can be a habit, an evolving process to greater adaptive expertise. It is not necessarily linked to the years of math schooling, as I’ve seen veteran math teachers and postgraduate math graduates unable to discern patterns and relationships in non-routine problems. It behooves us to develop adaptive habits in our students.
Tuesday, November 16, 2010
Tuesday, November 9, 2010
Best Experience in 2-week practicum
The best part is observing other non-Math classes and the insights they afforded me to be a better math teacher. I was thrilled to observe a Music 9 class when the fledgling musicians took up their individual instruments and played as a band a classical music, a upbeat dance and a moving sentimental piece. After that class, I was bent to discover the mathematical dimension of music, or even play some classic background music at appropriate stages of class time.
Then I observed a Foods 10 class, where measurement, proportions and even shapes (e.g. why are English pears roundish compared to Asian pears?) of ingredients rule. Then off to a PE 10 basketball class which had a jogging warm-up, then were led to the different mechanics of proper shooting. I intently watched how parabolic the students shots were, as well as the recommended leg and arm angles lectured by the instructor.
When asked what they felt were the most serious learning obstacle in learning math, a number of math teachers mentioned the language component. Upon learning this, the principal arranged for me to sit in an ESL class of mostly Chinese and Japanese students. Noting how serious the language gaps of some students are, I will try to re-phrase concepts when I'm actually teaching, or write key words and explanations in the board.
These invaluable insights will help me integrate other school subjects to math, and also find ways to link outside students' interests (like computer games, sports) to mymath teaching.
Then I observed a Foods 10 class, where measurement, proportions and even shapes (e.g. why are English pears roundish compared to Asian pears?) of ingredients rule. Then off to a PE 10 basketball class which had a jogging warm-up, then were led to the different mechanics of proper shooting. I intently watched how parabolic the students shots were, as well as the recommended leg and arm angles lectured by the instructor.
When asked what they felt were the most serious learning obstacle in learning math, a number of math teachers mentioned the language component. Upon learning this, the principal arranged for me to sit in an ESL class of mostly Chinese and Japanese students. Noting how serious the language gaps of some students are, I will try to re-phrase concepts when I'm actually teaching, or write key words and explanations in the board.
These invaluable insights will help me integrate other school subjects to math, and also find ways to link outside students' interests (like computer games, sports) to mymath teaching.
Word problems
Word problem from Mathpower 11 (1999), p. 221, #98.
#98. Scaffold Construction: two Scottish construction workers broke the world record for scaffold construction by building a 2-storey scaffold in 25 min 53 sec. The volume occupied by the scaffold was 100 m3. The length of the scaffold was 4 times the height. The width was 4 meters less than the height. Find the dimensions of the scaffold.
Comments:
1. The concept of a scaffold may not be accessible to most students. “Scaffold” will most likely not be in the working vocabulary of many Grade 11 students. From the problem, scaffold has obviously something to do with construction. It can be surmised that this may be a 3-dimensional solid because volume is given.
2. Transforming a verbose problem into a more visually pleasing and compact yet concise pictorial representation is one of the most powerful problem solving tools. Since a good drawing captures the essence of the problem, it strips the unessentials, and greatly clarifies in the resolution of the problem. Not to be able to draw a geometric –based problem (and rely only in algebraic manipulation) diminishes the holistic understanding of the problem. A student who does not have any idea of a scaffold will not be able to draw one.
3. This is a case of dressing up only to undress the problem. Upon seeing the words: volume, length, height, width and dimensions (in that order), the auto response will be to simply plug in numbers and variables into the volume formula. Chances are, students will get the right answer. This reinforces their previous experience that you do not really need to understand the scenario presented in the problem; you simply need to be creative to mix-match the numbers given.
#98. Scaffold Construction: two Scottish construction workers broke the world record for scaffold construction by building a 2-storey scaffold in 25 min 53 sec. The volume occupied by the scaffold was 100 m3. The length of the scaffold was 4 times the height. The width was 4 meters less than the height. Find the dimensions of the scaffold.
Comments:
1. The concept of a scaffold may not be accessible to most students. “Scaffold” will most likely not be in the working vocabulary of many Grade 11 students. From the problem, scaffold has obviously something to do with construction. It can be surmised that this may be a 3-dimensional solid because volume is given.
2. Transforming a verbose problem into a more visually pleasing and compact yet concise pictorial representation is one of the most powerful problem solving tools. Since a good drawing captures the essence of the problem, it strips the unessentials, and greatly clarifies in the resolution of the problem. Not to be able to draw a geometric –based problem (and rely only in algebraic manipulation) diminishes the holistic understanding of the problem. A student who does not have any idea of a scaffold will not be able to draw one.
3. This is a case of dressing up only to undress the problem. Upon seeing the words: volume, length, height, width and dimensions (in that order), the auto response will be to simply plug in numbers and variables into the volume formula. Chances are, students will get the right answer. This reinforces their previous experience that you do not really need to understand the scenario presented in the problem; you simply need to be creative to mix-match the numbers given.
Tuesday, October 19, 2010
Response to Mason's Thinking Mathematically
Much like solving everyday problems, we normally approach textbook math problems rather informally, haphazardly, unscientifically, mostly a mental exercise, with a few tentative scribblings for a solution documented in whatever piece of paper is handy, with hardly any reflections on the actual process of problem solving. It is usually a one-go affair, usually unsuccessful, never to be visited again due to a resigned attitude of not being able to solve the problem.
Mason scrutinized the problem solving process in explicit detail, and suggested the twin approach of specializing and generalizing, plus the sub-strategies of entry, attack and review. The rubric that follows is helpful, for instance, how to introduce images and representations to represent the information you've already classified as needing a specialized skill, attempting a conjecture, framing a tentative resolution, checking and later, generalizing and even extending to accomodate various flavors of the original problem.
Mason gets very emphatic in recommending that you doggedly document your thinking and feelings, with all the false starts, detours, pitfalls, promising leads, frustration, the inevitable STUCK moments, and the delightful AHA moments. I thinks that's how great problems are solved by great mathematicians, as they religiously keep field notes of their thinking, and follow through their insights in a determined way.
Most definitely, we can use Mason's approach to our advantage, in Math classrooms and the larger room of life-long learning. Surely, I will adapt some of his ideas in my actual teaching.
Mason scrutinized the problem solving process in explicit detail, and suggested the twin approach of specializing and generalizing, plus the sub-strategies of entry, attack and review. The rubric that follows is helpful, for instance, how to introduce images and representations to represent the information you've already classified as needing a specialized skill, attempting a conjecture, framing a tentative resolution, checking and later, generalizing and even extending to accomodate various flavors of the original problem.
Mason gets very emphatic in recommending that you doggedly document your thinking and feelings, with all the false starts, detours, pitfalls, promising leads, frustration, the inevitable STUCK moments, and the delightful AHA moments. I thinks that's how great problems are solved by great mathematicians, as they religiously keep field notes of their thinking, and follow through their insights in a determined way.
Most definitely, we can use Mason's approach to our advantage, in Math classrooms and the larger room of life-long learning. Surely, I will adapt some of his ideas in my actual teaching.
Sunday, October 17, 2010
Simnt on Math and Citizenship
That mathematical literacy is crucial in today's world is a given. Even if we recognize it or not, there is an underlying mathematical concept in everything, without exception.To be able to quantify things, situations and occurences lead to a richer, fuller understanding of them , where mere verbal descriptors fail to give. (This is not saying that qualitative anaysis is unimportant). That said, it behooves us to understand as much as we can, or run the risk of others deceiving us with what at first glance are "hard" facts. And the deceipt can be blatant, deliberate, and pervasive. The mass media is full of statistics that are taken out of context, have only a grain of truth but for the most part untrue, presented in a way that leads to false conclusions, or are outright lies. We know that numbers can speak for themselves, and yet leave so much unsaid. Numbers can easily be manipulated to serve one's purpose. Discerning the real picture behind the numbers can only be accomplished by one mathematically literate.
It is here that most math teachers, the math curricula and schools fail. We tend to adopt a microscopic, mainly algorithmic, highly structured, and product-oriented paradigm that we fail to emphasize , the many paths to problem solving and intelligent discussion thereof, the value of students defining (in mathematical terms) their own problems, of invariably relating algorthmic skills to real-life scenarios, and the value of persistence, critical thinking, journaling your thoughts, thinking outside the box, and constantly updating on global research for new perspectives.
Math teaching and learning is not only about mastering the current textbook, as new knowledge grows exponentially. One estimate is that new technical knowledge is created every 2 years, so that by the time a student in a technical field is in her third year in university, the things she learned in first years are passe. For me, teaching high school math is really about planting the the seeds of love and awe about the beauty and the awesome power behind the numbers, in their evolving quest for mathematical literacy.
It is here that most math teachers, the math curricula and schools fail. We tend to adopt a microscopic, mainly algorithmic, highly structured, and product-oriented paradigm that we fail to emphasize , the many paths to problem solving and intelligent discussion thereof, the value of students defining (in mathematical terms) their own problems, of invariably relating algorthmic skills to real-life scenarios, and the value of persistence, critical thinking, journaling your thoughts, thinking outside the box, and constantly updating on global research for new perspectives.
Math teaching and learning is not only about mastering the current textbook, as new knowledge grows exponentially. One estimate is that new technical knowledge is created every 2 years, so that by the time a student in a technical field is in her third year in university, the things she learned in first years are passe. For me, teaching high school math is really about planting the the seeds of love and awe about the beauty and the awesome power behind the numbers, in their evolving quest for mathematical literacy.
Friday, October 15, 2010
Micro-teaching: Factoring Quadratic Trinomials with Algebra Tiles
Lesson Plan: Factoring Quadratic Trinomials Using Algebra Tiles
Group: Howard, Maria, Raman
Intended Students: Grade 10 Fundamentals and Pre-calculus
Group: Howard, Maria, Raman
Intended Students: Grade 10 Fundamentals and Pre-calculus
WHAT | HOW LONG | MATERIALS | |
BRIDGE | Give everyone a small sheet of paper. In 5 seconds, write as many factors of 60. | 1 minute | Sheets of paper/pen |
LEARNING OBJECTIVES | Using the algebra tiles, students will be able to: 1. Factor quadratic trinomials, including perfect square trinomials 2. Relate the dimensions of a rectangular area with finding the factors of a trinomial 3. Experience three modes of factoring trinomials: algebraic method, concrete algebra tiles, and virtual manipulatives | ||
TEACHING OBJECTIVES | 1. Maximum engagement of all students 2. Individual hands-on-learning using math manipulatives (algebra tiles) 3. Demonstration of using virtual manipulatives in factoring trinomials | ||
PRETEST | Each student will be given a worksheet 1. Factor the trinomial: x2 + 5x + 6. Write answer in worksheet. Ask for answer. Show of hands who got the correct answer. Ask a student to briefly explain his/her answer. | 2 minutes | whiteboard |
PARTICIPATORY LEARNING | 1. State the learning objectives. Tie-up bridge and pre-test to objectives. 2. What are the factors of 6? (3 and 2) How can we illustrate this geometrically? (Draw a 3 by 2 rectangle, divided into 6 squares). How are factors related to dimensions (of length and width), and product related to area? (Finding the factors of a number is the same as finding the dimensions of a rectangle whose area is the number). Will this geometric representation work for finding factors of a trinomial? 3. Distribute/introduce the algebra tiles, as a geometric method of finding factors of trinomials. Each student will be given a complete set of tiles, with a transparent tile board. Walk the students through the 3 different tile sizes representing x2 (green), x (white) and 1 (red). Explain that x is a variable that can represent any positive number. 4. Assemble 2-green x2, 5-white x tiles and 2-red 1-tiles. If all the 9 pieces represent the area of a rectangle, what algebraic expression represents this area? (2x2 + 5x + 2) How can we get the dimensions of this rectangle? * In your worksheet, complete equation #2: 2x2 + 5x + 2 = (2x + 1)(x + 2) 5. Empty your tile board. For our second rectangle, assemble 1- green x2, 6-white x and 9-red 1-tiles into a rectangle. What expression represents the area of this rectangle? (x2 + 6x + 9). What are the factors? (x + 3) and (x + 3). What do you notice with our rectangle? (It is a square). Introduce the perfect square trinomial (PST). * In your worksheet, complete equation #3: x2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 6. Virtual Manipulatives: Reiterate that finding the linear factors of a quadratic trinomial is very much related to finding the dimensions of a rectangle that contain the trinomial. The internet is full of virtual manipulatives that offer fun, creative, and interactive ways of factoring trinomial, which may appeal to today’s technology-savvy students. Factor x2 + 7x + 12. (x + 4)(x+3). | 9 Minutes | whiteboard Algebra tiles Virtual manipulatives |
POSTTEST | Using your algebra tiles, find values of k, where x2 + kx + 6 factors into 2 binomials. (k = 5, 7). Write answer in #5 of your worksheet. | * Algebra tiles | |
SUMMARY & WRAP-UP | Ask students what they have learned today, which should touch the following points: 1. That to the concept of factoring is very much related to finding the dimensions of a rectangle of a given area. 2. That a quadratic trinomial factors only if one can arrange it into a rectangle. 3. That we know that a trinomial is a perfect square if the tiles neatly arranges into a square, with 2 equal dimensions. 4. Ask students to complete # 6 & 7 of their worksheet. Collect worksheets. | 3 minutes |
Student Worksheet
Name: __________________ Topic: Factoring Quadratic Trinomials
1. Factor: x2 + 5x + 6
2. 2x2 + 5x + 2 = ( )( )
3. x2 + 6x + 9 = ( )( ) = ( )2
4. x2 + 7x + 2 = ( )( )
5. x2 + kx + 6 = ( )( ) or ( )( )
6. One thing I learned today is _____________________________________.
7. Algebra tiles do/do not help in understanding factoring trinomials
Name: __________________ Topic: Factoring Quadratic Trinomials
1. Factor: x2 + 5x + 6
2. 2x2 + 5x + 2 = ( )( )
3. x2 + 6x + 9 = ( )( ) = ( )2
4. x2 + 7x + 2 = ( )( )
5. x2 + kx + 6 = ( )( ) or ( )( )
6. One thing I learned today is _____________________________________.
7. Algebra tiles do/do not help in understanding factoring trinomials
because_______________________________________________________.
Wednesday, October 13, 2010
Reflections: Algebra tiles microteaching
The major insights I gained from this 15-minute experience and how these might impact on my actual teaching are:
1. The time factor can be an all-consuming goal for teachers. We try to cover as much material as mandated by the curriculum by a target date, with little regard to whether the students have the prerequisite knowlege base. This can lead to impatience when getting a wrong answer or to subconsciously asking leading questions as a guise for discovery learning. We look for shortcuts (instrumental vs relational) to buy time.
2. The more planning time you spend on a lesson, the better it gets as you incorporate small adjustments as you review a previous plan. The first tendency for new teachers is to want to do just about everything they've learned in education school. You must temper your eagerness with the practicality of your actions.
3. Teaching is not about you impressing the students of what you know. Rather, the focus should always be on how best the students learn. It is sometimes hard to rein back on all the complex mathematical concepts that we know. But we should be clear on one thing; as math teachers, we naturally like doing math; it is the exact opposite for the great majority of our students (and non-math teachers as well). We need to go down to their level, talk their language, address their anxiety, and access their personal mathematical experience.
4. Judicious use of technology, without doubt, greatly enhances mathematical learning. Multi-media, multi-modal presenattions complement traditional teaching startegies, and build on a more wholistic, integral understanding of mathematical concepts. The dynamic representations of changes in variables permit connections that might otherwise be lost on traditonal paper-pencil algorithms.
5 Manipulatives are a great way to concretize abstarct math concepts. They appeal to visual/spatial and kinesthetic learners who want to move things around. They provide a solid link to the physical world and hence appeals to the practical student. Algebra, especially geometry should do well capitalizing on its benefits.
6. Working through your chosen examples for class discussion is important as errors might undermine students confidence in you.
7. Good questioning stimulates critical thinking and maximizes students' engagement. This is both an art and a science that should improve through time.
Overall feedback on our group's micro-teaching:
1. The overwhelming majority found the topic interesting as they they have no previous experience on algebra tiles. Further, many did not make the connection between product - area and factoring - dimensions of a rectangle prior to the presentation.
2. Majority commented on the well-planned presentation, as well as different approaches used.
3. Time manangement was the chief must-improve item, as they felt that I talked too fast, without giving students time to ponder on questions.
Self-assessment:
1. The decision to try algebra tiles as a micro-teaching topic stemmed from a comment of a veteran math teacher in one of the biggest schools in the lower mainland, that the use of manipulatives, although strongly recommended by the Education Ministry, might have little practical use in a classrom setting. I think a carefully planned lesson build around these tiles can bring math concepts more tangible, accessible and palatable to many students. Solving equations, even systems of equations with algebra tiles is another must-try activity.
2. Apart from the usual content questions (including pre and posttest), we included a learning log question and an evaluative question at the end of the students' worksheet. The former forces the student to state what he has learned today; the latter asks the student if he found "value" in the whole exercise. We found both to be a useful imput in assessing the overall effectiveness of the lesson.
1. The time factor can be an all-consuming goal for teachers. We try to cover as much material as mandated by the curriculum by a target date, with little regard to whether the students have the prerequisite knowlege base. This can lead to impatience when getting a wrong answer or to subconsciously asking leading questions as a guise for discovery learning. We look for shortcuts (instrumental vs relational) to buy time.
2. The more planning time you spend on a lesson, the better it gets as you incorporate small adjustments as you review a previous plan. The first tendency for new teachers is to want to do just about everything they've learned in education school. You must temper your eagerness with the practicality of your actions.
3. Teaching is not about you impressing the students of what you know. Rather, the focus should always be on how best the students learn. It is sometimes hard to rein back on all the complex mathematical concepts that we know. But we should be clear on one thing; as math teachers, we naturally like doing math; it is the exact opposite for the great majority of our students (and non-math teachers as well). We need to go down to their level, talk their language, address their anxiety, and access their personal mathematical experience.
4. Judicious use of technology, without doubt, greatly enhances mathematical learning. Multi-media, multi-modal presenattions complement traditional teaching startegies, and build on a more wholistic, integral understanding of mathematical concepts. The dynamic representations of changes in variables permit connections that might otherwise be lost on traditonal paper-pencil algorithms.
5 Manipulatives are a great way to concretize abstarct math concepts. They appeal to visual/spatial and kinesthetic learners who want to move things around. They provide a solid link to the physical world and hence appeals to the practical student. Algebra, especially geometry should do well capitalizing on its benefits.
6. Working through your chosen examples for class discussion is important as errors might undermine students confidence in you.
7. Good questioning stimulates critical thinking and maximizes students' engagement. This is both an art and a science that should improve through time.
Overall feedback on our group's micro-teaching:
1. The overwhelming majority found the topic interesting as they they have no previous experience on algebra tiles. Further, many did not make the connection between product - area and factoring - dimensions of a rectangle prior to the presentation.
2. Majority commented on the well-planned presentation, as well as different approaches used.
3. Time manangement was the chief must-improve item, as they felt that I talked too fast, without giving students time to ponder on questions.
Self-assessment:
1. The decision to try algebra tiles as a micro-teaching topic stemmed from a comment of a veteran math teacher in one of the biggest schools in the lower mainland, that the use of manipulatives, although strongly recommended by the Education Ministry, might have little practical use in a classrom setting. I think a carefully planned lesson build around these tiles can bring math concepts more tangible, accessible and palatable to many students. Solving equations, even systems of equations with algebra tiles is another must-try activity.
2. Apart from the usual content questions (including pre and posttest), we included a learning log question and an evaluative question at the end of the students' worksheet. The former forces the student to state what he has learned today; the latter asks the student if he found "value" in the whole exercise. We found both to be a useful imput in assessing the overall effectiveness of the lesson.
quadratic factoring with algebra tiles
Topic: Factoring Quadratic Trinomials Using Algebra Tiles
Group: Howard, Maria, Raman
Intended Students: Grade 10 Fundamentals and Pre-calculus
Intended Students: Grade 10 Fundamentals and Pre-calculus
I. BRIDGE - (1 minute)
Give everyone a small sheet of paper. In 5 seconds, write as many factors of 60.
II. LEARNING OBJECTIVES: Using the algebra tiles, students will be able to:
2.1 factor quadratic trinomials, including perfect square trinomials
2.2 relate the dimensions of a rectangular area with finding the factors of a trinomial
2.3 experience three modes of factoring trinomials: algebraic method, concrete algebra
tiles, and virtual manipulatives
III. TEACHING OBJECTIVES
3.1 maximum engagement of all students
3.2 individual hands-on-learning using math manipulatives (algebra tiles)
3.3 demonstration of using virtual manipulatives in factoring trinomials
III. TEACHING OBJECTIVES
3.1 maximum engagement of all students
3.2 individual hands-on-learning using math manipulatives (algebra tiles)
3.3 demonstration of using virtual manipulatives in factoring trinomials
IV. PRE-TEST (Each student will be given a worksheet sheet (2 minutes)
4.1 Factor the trinomial: x2 + 5x + 6. Write answer in worksheet.
Show of hands who got the correct answer. Ask a student to briefly explain her answer.
V. PARTICIPATORY LEARNING (9 minutes)
5.1 State the learning objectives. Tie-up bridge and pre-test to objectives.
5.2 What are the factors of 6? (3 and 2) How can we illustrate this geometrically? (Draw a 3 by 2 rectangle, divided into 6 squares). How are factors related to dimensions (of length and width), and product related to area? (Finding the factors of a number is the same as finding the dimensions of a rectangle whose area is the number) Will this geometric representation work for finding factors of a trinomial?
5.3 Distribute/introduce the algebra tiles, as a geometric method of finding factors of trinomials. Each student will be given a complete set of tiles, with a transparent tile board. Walk the students through the 3 different tile sizes representing x2 (green), x (white) and 1 (red) Explain that x is a variable that can represent any positive number.
5.4 Assemble 2-green x2, 5-white x tiles and 2-red 1-tiles. If all the 9 pieces represent the area of a rectangle, what algebraic expression represents this area? (2x2 + 5x + 2) How can we get the dimensions of this rectangle? In your worksheet, complete the equation #2: 2x2 + 5x + 2 = ( ) ( )
5.5 Empty your tile board. For our second rectangle, assemble 1- green x2, 6-white x and 9-red 1-tiles into a rectangle. What expression represents the area of this rectangle? (x2 + 6x + 9). What are the factors? ( x + 3) and (x + 3). What do you notice with our rectangle? (It is a square). Introduce the perfect square trinomial (PST). In your worksheet, complete equation #3: x2 + 6x + 9 = ( ) ( ) = ( )2
5.6 Virtual Manipulatives: Reiterate that finding the linear factors of a quadratic trinomial is very much related to finding the dimensions of a rectangle that contain the trinomial. The internet is full of virtual manipulatives that offer fun, creative, and interactive ways of factoring trinomial, which may appeal to today’s technology-savvy students. Factor x2 + 7x + 12. = ( )( ).
VI. Posttest: Using your algebra tiles, find values of k, where x2 +kx + 6 factors into 2 binomials. (k=5,7). Write answer in #5 of your worksheet.
VII. SUMMARY/CLOSURE: Ask students what they have learned today, which should touch the following points: (3 minutes)
1. That to the concept of factoring is very much related to finding the dimensions of a rectangle
V. PARTICIPATORY LEARNING (9 minutes)
5.1 State the learning objectives. Tie-up bridge and pre-test to objectives.
5.2 What are the factors of 6? (3 and 2) How can we illustrate this geometrically? (Draw a 3 by 2 rectangle, divided into 6 squares). How are factors related to dimensions (of length and width), and product related to area? (Finding the factors of a number is the same as finding the dimensions of a rectangle whose area is the number) Will this geometric representation work for finding factors of a trinomial?
5.3 Distribute/introduce the algebra tiles, as a geometric method of finding factors of trinomials. Each student will be given a complete set of tiles, with a transparent tile board. Walk the students through the 3 different tile sizes representing x2 (green), x (white) and 1 (red) Explain that x is a variable that can represent any positive number.
5.4 Assemble 2-green x2, 5-white x tiles and 2-red 1-tiles. If all the 9 pieces represent the area of a rectangle, what algebraic expression represents this area? (2x2 + 5x + 2) How can we get the dimensions of this rectangle? In your worksheet, complete the equation #2: 2x2 + 5x + 2 = ( ) ( )
5.5 Empty your tile board. For our second rectangle, assemble 1- green x2, 6-white x and 9-red 1-tiles into a rectangle. What expression represents the area of this rectangle? (x2 + 6x + 9). What are the factors? ( x + 3) and (x + 3). What do you notice with our rectangle? (It is a square). Introduce the perfect square trinomial (PST). In your worksheet, complete equation #3: x2 + 6x + 9 = ( ) ( ) = ( )2
5.6 Virtual Manipulatives: Reiterate that finding the linear factors of a quadratic trinomial is very much related to finding the dimensions of a rectangle that contain the trinomial. The internet is full of virtual manipulatives that offer fun, creative, and interactive ways of factoring trinomial, which may appeal to today’s technology-savvy students. Factor x2 + 7x + 12. = ( )( ).
VI. Posttest: Using your algebra tiles, find values of k, where x2 +kx + 6 factors into 2 binomials. (k=5,7). Write answer in #5 of your worksheet.
VII. SUMMARY/CLOSURE: Ask students what they have learned today, which should touch the following points: (3 minutes)
1. That to the concept of factoring is very much related to finding the dimensions of a rectangle
of a given area.
2. That a quadratic trinomial factors only if one can arrange it into a rectangle.
3. That we know that a trinomial is a perfect square if the tiles neatly arranges into a square,
2. That a quadratic trinomial factors only if one can arrange it into a rectangle.
3. That we know that a trinomial is a perfect square if the tiles neatly arranges into a square,
with 2 equal dimensions.
4. Ask students to complete # 6 & 7 of their worksheet. Collect worksheets.
4. Ask students to complete # 6 & 7 of their worksheet. Collect worksheets.
Student Worksheet :
Name: __________________ Topic: Factoring Quadratic Trinomials
1. Factor: x2 + 5x + 6
2. 2x2 + 5x + 2 = ( ) ( )
3. x2 + 6x + 9 = ( ) ( ) = ( )2
4. x2 + 7x + 12. = ( ) ( )
5. x2 + kx + 6 = ( )( ) or ( )( )
6. One thing I learned today is _____________________________________.
7. Algebra tiles do/do not help in understanding factoring trinomials because_______________________________________________________.
Name: __________________ Topic: Factoring Quadratic Trinomials
1. Factor: x2 + 5x + 6
2. 2x2 + 5x + 2 = ( ) ( )
3. x2 + 6x + 9 = ( ) ( ) = ( )2
4. x2 + 7x + 12. = ( ) ( )
5. x2 + kx + 6 = ( )( ) or ( )( )
6. One thing I learned today is _____________________________________.
7. Algebra tiles do/do not help in understanding factoring trinomials because_______________________________________________________.
Friday, October 8, 2010
Undecipherable Zero
For something that means nothing
Zero can be so intriguing.
You can surely see an apple, even 10 raised to the negative one of an apple
But nor zero apple (or is it apples?)
A ten, a billion, even google and infinity
Comes before zero in the alphabet parade.
Zero is the great equalizer
The biggest or the smallest of numbers
The most mundane or exotic of numbers
Are all equally easy when added to zero.
It transforms into a black hole in multiplication
And swallows everything beside it indiscriminately
It can be choosy and temperemental
Dividing into zero is your ticket to the black hole
Or you quickly become a pariah when zero is divided into you.
How else can we explain...
Poor men who have nothing, yet feel so richly blessed
And rich men who have everything, yet feel utterly empty.
Or a child giving out her last candy to comfort a friend
Flooded with goodness and happiness in her simple gift.
Or parents who have lost a child
Choose to transform their grief into concrete actions
That will save many children.
Something out of nothing
That's zero in the mathematics of life.
Zero can be so intriguing.
You can surely see an apple, even 10 raised to the negative one of an apple
But nor zero apple (or is it apples?)
A ten, a billion, even google and infinity
Comes before zero in the alphabet parade.
Zero is the great equalizer
The biggest or the smallest of numbers
The most mundane or exotic of numbers
Are all equally easy when added to zero.
It transforms into a black hole in multiplication
And swallows everything beside it indiscriminately
It can be choosy and temperemental
Dividing into zero is your ticket to the black hole
Or you quickly become a pariah when zero is divided into you.
How else can we explain...
Poor men who have nothing, yet feel so richly blessed
And rich men who have everything, yet feel utterly empty.
Or a child giving out her last candy to comfort a friend
Flooded with goodness and happiness in her simple gift.
Or parents who have lost a child
Choose to transform their grief into concrete actions
That will save many children.
Something out of nothing
That's zero in the mathematics of life.
Wednesday, October 6, 2010
Divide/Zero Musings
Divide means to take groups of things and partition them to equal sets. Or it could mean deciding how many persons or things can fit into a given whole. For example, we want to know how many buses are needed for 40 students if a bus can take 10 at a time. Alternatively, if we know that a bus can seat only 10 students, how many buses are needed for 40 students.
Zero- Zero can mean nothing. Or if you initially have something and someone takes all of that something, then you have nothing, or zero. Zero can also mean a reference point, a starting b=point, much like the zero in the number line. Zero can also mean adding 2 opposite numbers that cancel each other. As a reference point, zero can mean elevation at sea level (ground0, or zero temperature.
Zero- Zero can mean nothing. Or if you initially have something and someone takes all of that something, then you have nothing, or zero. Zero can also mean a reference point, a starting b=point, much like the zero in the number line. Zero can also mean adding 2 opposite numbers that cancel each other. As a reference point, zero can mean elevation at sea level (ground0, or zero temperature.
Thursday, September 30, 2010
Interview Questions to Students and Teachers
Questions to students:
1. Name three things that will greatly improve your math learning. Explain why.
2. Is math among your most or least favorite subjects? Explain further. (Here, the student many be comparing the characteristics of other subjects to math)
3. Apart from being a university requirement, is a good math education important? (Is math relevant in real-life?)
4. Does the use of technology (graphing calculators, smartboards, virtual manipulatives) enhance your learning? Why or why not?
Questions to teachers:
1. Name three things that you find most challenging about teaching mathematics to secondary students.
2. How do you motivate disinterested, underperforming, don't-care-if-I-pass students to love math?
3. What for you is authentic assessment? How do you practice this?
4. How do you balance the demands of the curriculum with teaching for understanding?
5. How do you ensure judicious use of technology to maximize learning?
1. Name three things that will greatly improve your math learning. Explain why.
2. Is math among your most or least favorite subjects? Explain further. (Here, the student many be comparing the characteristics of other subjects to math)
3. Apart from being a university requirement, is a good math education important? (Is math relevant in real-life?)
4. Does the use of technology (graphing calculators, smartboards, virtual manipulatives) enhance your learning? Why or why not?
Questions to teachers:
1. Name three things that you find most challenging about teaching mathematics to secondary students.
2. How do you motivate disinterested, underperforming, don't-care-if-I-pass students to love math?
3. What for you is authentic assessment? How do you practice this?
4. How do you balance the demands of the curriculum with teaching for understanding?
5. How do you ensure judicious use of technology to maximize learning?
Wednesday, September 29, 2010
Mr. Not-so-great Fictional letter
Hi Mr. Not-so-great teacher,
Just a short hi from a former Grade 10 Math student who is now a Math teacher in Bahrain. I bet you still remember me as I my always spell trouble- homework/seatwork.worksheest are either incomplete, obviously copied hurriedly, or non-existent for all I care. It's amazing how I can come up with creative ways to tune out of your class for almost the entire year, while managing a semblance of deep thought or profound reflection. I've honed my astral travel skills that year, my thoughts in some far-flung exotic resort, while my body pinned down in my desk. I have not learned an iota of useful information from you, with all the mumbo-jumbo about projectile paths, exponential decay or inconsistent systems. To be fair, I've not learned anything at all from my other teachers as well. One thing I can credit you though, you generally leave me in peace.Perhaps you've divined the futility of any effort. Oh, and that brings me to why I'm writing after all these years. For some odd reason, my Grade 10 report card survived the Great Bonfire. That was when I burned all my school stuff in a nice bonfire when I freaked out in the early part of Grade 12 coz I was so pressured to graduate that year.
Except my Grade 10 card, which is now right in front of me. At the end of the term, you said and I quote: Bryan has a lot of potential in excelling in math as he is can be focused if he wants to. I don't know if that potential stuff lay buried in my unconscious mind all this time. What I know is that I'm now a math teacher, and a pretty good one at that. So what are the odds of that, eh?
Incorrigible student
Greatest hopes: That my students will appreciate the inherent beauty of Math and gain profound satisfaction in doing math the rest of their lives.
Greatest fears: That despite best intentions, my Math class was so traumatic or boring or irrelevant that a student would want to erase all memories of it in her long-term memory.
Just a short hi from a former Grade 10 Math student who is now a Math teacher in Bahrain. I bet you still remember me as I my always spell trouble- homework/seatwork.worksheest are either incomplete, obviously copied hurriedly, or non-existent for all I care. It's amazing how I can come up with creative ways to tune out of your class for almost the entire year, while managing a semblance of deep thought or profound reflection. I've honed my astral travel skills that year, my thoughts in some far-flung exotic resort, while my body pinned down in my desk. I have not learned an iota of useful information from you, with all the mumbo-jumbo about projectile paths, exponential decay or inconsistent systems. To be fair, I've not learned anything at all from my other teachers as well. One thing I can credit you though, you generally leave me in peace.Perhaps you've divined the futility of any effort. Oh, and that brings me to why I'm writing after all these years. For some odd reason, my Grade 10 report card survived the Great Bonfire. That was when I burned all my school stuff in a nice bonfire when I freaked out in the early part of Grade 12 coz I was so pressured to graduate that year.
Except my Grade 10 card, which is now right in front of me. At the end of the term, you said and I quote: Bryan has a lot of potential in excelling in math as he is can be focused if he wants to. I don't know if that potential stuff lay buried in my unconscious mind all this time. What I know is that I'm now a math teacher, and a pretty good one at that. So what are the odds of that, eh?
Incorrigible student
Greatest hopes: That my students will appreciate the inherent beauty of Math and gain profound satisfaction in doing math the rest of their lives.
Greatest fears: That despite best intentions, my Math class was so traumatic or boring or irrelevant that a student would want to erase all memories of it in her long-term memory.
Fictional letter
Dear Mr. Favorite Teacher,
Hello, still remember me from your Math 12 Foundations & Pre-Calc class? Perhaps not, with so many students milling around you, asking endless questions, seeking help in your assigned homework or even former students asking math questions assigned by OTHER teachers! You seem not to have time for yourself in school, and even after school as you go online with the class blog every single night, responding to our endless queries, comments, complaints about Math in particular and life in general. I know you once said that you're a face person and not a name person so I"m sending you a before (was it really 10 years ago?) and after (with my husband & 5-year old daughter) picture of me. (Download attached file first before proceeding any further).
Ring any bells? I can almost hear your gasp and hear your OMG! Yes I'm the quiet student always sitting in the back row where I prefer to work alone. But unlike other teachers, you always found a way to reach me and check on me every single class! I really like it when you give us time to work on our own worksheets and you move about checking on ALL students, and commenting on how well we're progressing in our worksheets. I'm eternally afraid to volunteer my answers in class, moreso, to ask questions if something is not clear. Looking down on my work, you seem to know exactly what part of the process I'm having trouble with, and nudge me gently towards the correct path. For that, I'm really grateful as it took the stress of learning math.
Hello, still remember me from your Math 12 Foundations & Pre-Calc class? Perhaps not, with so many students milling around you, asking endless questions, seeking help in your assigned homework or even former students asking math questions assigned by OTHER teachers! You seem not to have time for yourself in school, and even after school as you go online with the class blog every single night, responding to our endless queries, comments, complaints about Math in particular and life in general. I know you once said that you're a face person and not a name person so I"m sending you a before (was it really 10 years ago?) and after (with my husband & 5-year old daughter) picture of me. (Download attached file first before proceeding any further).
Ring any bells? I can almost hear your gasp and hear your OMG! Yes I'm the quiet student always sitting in the back row where I prefer to work alone. But unlike other teachers, you always found a way to reach me and check on me every single class! I really like it when you give us time to work on our own worksheets and you move about checking on ALL students, and commenting on how well we're progressing in our worksheets. I'm eternally afraid to volunteer my answers in class, moreso, to ask questions if something is not clear. Looking down on my work, you seem to know exactly what part of the process I'm having trouble with, and nudge me gently towards the correct path. For that, I'm really grateful as it took the stress of learning math.
Tuesday, September 28, 2010
Summary: Battleground Schools
The article describes two equally strong but polarized views of math education in 20th century North America, particularly the US. The prevailing view at any given point in history depends on the current political and economic state of the US and the global scenario. Hence the terrors of fascism in the 1940s together with increasing migration, urbanization and industrialization, rendered the static, rigid curriculum wholly inadequate, spawned a distrust for the infallible teacher and called for a paradigm shift to a more meaningful, practical education by "doing math." John Dewey figured prominently in the progressivist reform of early to mid-20th c.
Then a second major shift came in the post-war 60s when the US dominance in the global arena is threatened by Russia's seeming superior technological capabilities. Heavily influenced by the ultra-conservative, highly elitist Bourbaki group, math sentiment swung to the conservative end, where abstraction reign, and university-level math is injected in K-12 curriculum to bring out the scientist in every student. Its tentacles reached far and wide to Europe and developing countries, where suitability to local conditions invariably posed a problem, as both the teachers, parents and students were ill-prepared to handle the the high-level abstraction, perhaps appreciated (and understood) only by those headed for university physics or math.
Discontent with the impracticability of the New Math to most students, a third seismic shift began in the late 1970s, spilling into the next 2 decades. Back-to-basics, mandated national standards, and teacher and school accountability were the new buzz words. NCTM played a stellar role in the US when it developed its own standards program after innumerable consultations with education stakeholders: math teachers, parents, administrators, education specialists. NCTM's huge impact is such that it served as a model to develop curricular standards for other subject areas, as well as serving a focal reference for many of the states in the US, in Canada, as well as developing countries. NCTM is heavily leaned on the progressive end where sense-making, original thinking, conceptual understanding, exploratory process and meaningful problem solving are the emphasized more than teacher-constructed algorithm, fluency, end-answers.
As good as this sounds, discontent will always be there, which is expected as we adapt to changing national and global scenarios. Such was the case when the TIMSS results showed dismal performance of the US and the traditional superpowers, compared to the top performance of recent world players and emerging economies of Singapore, Taiwan and South Korea. Some western economies are closely monitoring Asian classrooms, and how they can possibly import some lessons in their quest for a better math education.
Then a second major shift came in the post-war 60s when the US dominance in the global arena is threatened by Russia's seeming superior technological capabilities. Heavily influenced by the ultra-conservative, highly elitist Bourbaki group, math sentiment swung to the conservative end, where abstraction reign, and university-level math is injected in K-12 curriculum to bring out the scientist in every student. Its tentacles reached far and wide to Europe and developing countries, where suitability to local conditions invariably posed a problem, as both the teachers, parents and students were ill-prepared to handle the the high-level abstraction, perhaps appreciated (and understood) only by those headed for university physics or math.
Discontent with the impracticability of the New Math to most students, a third seismic shift began in the late 1970s, spilling into the next 2 decades. Back-to-basics, mandated national standards, and teacher and school accountability were the new buzz words. NCTM played a stellar role in the US when it developed its own standards program after innumerable consultations with education stakeholders: math teachers, parents, administrators, education specialists. NCTM's huge impact is such that it served as a model to develop curricular standards for other subject areas, as well as serving a focal reference for many of the states in the US, in Canada, as well as developing countries. NCTM is heavily leaned on the progressive end where sense-making, original thinking, conceptual understanding, exploratory process and meaningful problem solving are the emphasized more than teacher-constructed algorithm, fluency, end-answers.
As good as this sounds, discontent will always be there, which is expected as we adapt to changing national and global scenarios. Such was the case when the TIMSS results showed dismal performance of the US and the traditional superpowers, compared to the top performance of recent world players and emerging economies of Singapore, Taiwan and South Korea. Some western economies are closely monitoring Asian classrooms, and how they can possibly import some lessons in their quest for a better math education.
Saturday, September 25, 2010
How to Make a Hole in Postcard-sized Paper Big Enough to Pass Through
| What | Time | Materials |
| Greet students. Today we'll explore one of the most pefect shapes in nature. Can you guess what | 5 s | |
| this may be? (Circle) | ||
| At the end of the lesson, students will be able to: | ||
| 1. have a deeper appreciation of the area of a circles | ||
| 2. problem solve creatively | ||
| 3. think outside the box | ||
| 1. to motivate students as to be 100% engaged | ||
| 2. to elicit maximum participation involving the 3 senses (visual, auditory, touch) | ||
| 1. Can everyone construct a circle with their 2 fingers? 3 fingers? 4 fingers? 5 fingers? 10 fingers? | ||
| 2. Was making a good circle easy? Compared to say makng/drawing a sqaure or a rectangle? Why? | ||
| 3. Compare the circles formed by someone with shorter fingers with a person with longer fingers. | ||
| What do you call the area inside the circle/hole? | ||
| 1. Let's explore the biggest circle/hole we can make out of a piece of paper. Who has some idea of | 7 m | whiteboard, |
| this is done? Show in the whiteboard. | markers | |
| 2. Discuss: areas of a unit circle vs . 2x2 circle vs. 3x3 circle. | ||
| 3. Draw conclusions (The bigger the paper, the bigger the circle that can be cut in it. | 10 postcard-size | |
| 4. Show a piece of postcard-size paper. What is the biggest circle/hole you can make out of it. | cardboard | |
| Is the hole large enough to put your fist through? Your head through? For the whole length of you | with pre-drawn | |
| pass through? | line | |
| 5. What if I tell you a secret that it is possible to cut a hole through this cardboard big enough for you | markings | |
| to pass through? What woud the pattern of the cutting be like? (zigzag/irregular) Why? | ||
| 6. Give each student a postcard with pre-drawn line markings. Ask them to describe the pattern | scissors | |
| of dotted lines. Ask them to visualize the resulting hole this pattern will produce. | ||
| 7. Give general instructions on how to cut the paper. Guide each student as they do the actual cuts. | ||
| 8. End result: a huge hole big enough for a person to pass through. | ||
| 1. How did you use the concept of area to cut a super-sized hole out of a small paper? | 1 m | |
| 1. Summarize activity. | 2 m |
Teacher/Student Interviews
TEACHING AND LEARNING MATHEMATICS
It is well known that good teachers love the subject they are teaching. On the other hand, if you as the teacher feel negative towards mathematics, it may show up when you are teaching your students and can affect them similarly. Little children usually like numbers and math - yet many kids in schools develop 'math anxiety or phobia' or end up disliking math. A major factor in that is the way math is taught and the way the teachers feel about math.
We have been to a Senior Secondary School, and asked several questions to both math’s teachers and students. The results were not that surprising. After interviewing several students, we found that most of them do not like mathematics and find it boring. By analyzing different students, we realized that they want to be taught by instrumental way. However, they want to understand purpose of math they are studying. They could not relate the purpose of learning math other than measuring, estimating the bills. Even when they are asked that what they want to change in curriculum, their answer was that want math which can be used in daily life. Therefore, the students might get more motivated if she/he knows where all maths is needed. So many times kids question the needfulness of things they study. Emphasizing and pointing out the everyday applications of math may help them. Even when they are asked that what they want to change in curriculum, their answer was that want math which can be used in daily life.
Also, students prefer hands on learning rather than lecture method. They like teachers who involve activities in their lessons. One student said that she wants math more hands on which means that they want more activity oriented curriculum. Thus, by including different activities in class we can motivate students.
We had a chance to interview and observe secondary mathematics teacher in a class. Students were taking interest and paying attention to what they were taught. We talked about her teaching strategies. She told us that one very important factor in motivating students to study math is that you yourself, as the teacher, stay positive about math - if possible, enthusiastic! .
Secondly, we need to get the student involved! One of the reasons for math anxiety is the way math is often taught as "There is only ONE way to do this, and you need to do learn it and do it right." Math is presented as 'given from above'. Students can be much more motivated if they are asked open questions, involved in the development of concepts, given very open-ended exercises. Granted, this kind of teaching style may require a lot of planning from the teacher, probably a good understanding in math, and good materials.
Thirdly, the teacher should not put a wrong answer down. Instead, say, "Please can you explain how you came up with that?" In a classroom, a teacher can ask, "Did someone else get the same result as you? OK. Did somebody get a different result? OK, we have two (or three) different answers here. Let's figure them out." Wrong answers are valuable. You get insight into student's thinking and where he went wrong, and what needs a rethought. Students and kids need to be treated as humans and not feel put down or stupid for their answers.
Last but not least,take the emphasis off from tests. Tests are a part of school but they don't need to be the ultimate goal. She told us that, she returns the quizzes back to students and ask them to do the corrections. The goal is to learn math so the child can use it in her life.
IT’S THE ATTITUDE NOT THE APTITUDE THAT DECIDE THE ALTITUDE OF YOUR SUCCESS.
IT’S THE ATTITUDE NOT THE APTITUDE THAT DECIDE THE ALTITUDE OF YOUR SUCCESS.
Wednesday, September 22, 2010
Non-math microteaching
Strength's of the group's presentation: 1. Variety on interesting topics 2. Engaging personalities of presenters 3. Uncomplicated activity suitable for a 10-minute time frame 4. a plan B in case there's spare time
My presentation: Wasn't too happy as I messed up with my main material. Also, poor time management.
Lessons learned: Be sure to pre-try your planned activity at least twice. Choose a suitable topic within the given time frame.
My presentation: Wasn't too happy as I messed up with my main material. Also, poor time management.
Lessons learned: Be sure to pre-try your planned activity at least twice. Choose a suitable topic within the given time frame.
Dave Hewitt: Model Math Teacher #2
Dave Hewitt's totally unconventional approach to teaching blew my mind away, something akin to a 20-minute medium-sized twister. He defied a lot of hallowed (read traditional) conventions of teaching. First, he ignored the primary function of the whiteboard for the better part of the lesson and used it instead as a rhythmic background sound for his imaginary number line. One can doze off copying notes but not with a regular, very audible tapping in the board. Besides, Dave hardly takes his eyes off his students, as if instantaneously processing & learning from each moment. Second, he circled the room with his stick, tapping back & forth, to teach skills with increasing complexity. One can never be sure what's next; anticipation, hence engagement is high. Third, the lessons progressed seamlessly to the physical placement of integers in the number line, to addition/subtraction of integer(s), to equivalent expressions, to the sneaky way of introducing variables, to solving equations using properties of equalities & inverse operations, & order of operations. The beauty is that the increasing complexity of the lesson appeared to be the most natural way an average person can think for himself; there were no explicit definitions, rules, or algorithms given as a reference starting point. It takes a teacher great courage, insight & dedication to want to do untested things. But we owe it to our students and ourselves to be in a perpetual state of creative dissatisfaction to continually find better ways to teach and learn. Bill Gates recently said that if you want your child to be educated well, it is not as important to send him to a good school than to a great teacher.
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