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.
Tuesday, October 19, 2010
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.
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