I love my 7th-graders! Today was the second time I've played games with them on a Friday afternoon, and they absolutely go nuts for it every time.
One concern I always have during a group game is that only the kids who are up for their turn are engaged. Today I improved that by letting TWO kids from each team go up together, and they can only collaborate AFTER they have both tried the problem independently. When they ring the bell, both people should already have written down the answer on their paper, and I'll randomly call on one of them to EXPLAIN their answer. (So, they had to confer before ringing the bell, and the stronger student was always eagerly explaining to the weaker student how to do it.)
The problems built up in difficulty, so I'd always make sure all students from the class understood the explanations to the previous question before starting another round. Students at their desks were also practicing because they didn't want to be the person to hold their partner up when it's their turn. Overall, the rotations went twice as fast, and everyone was constantly engaged.
By the end of the 80-minute period, even my weakest students were getting through the problems by themselves, and EVERYONE WAS SO EXCITED TO EXPLAIN THEIR ANSWERS! It was great!!
Grab the file here. This one was on filling in missing numbers into linear patterns, and some of it involved integers and decimals. The kids were brilliant! (We also had two exchange students from China playing with us. It was COOL; one of them explained to me in Chinese, and I translated for her in real time for the class, and high-fived her after that round! It was sweet.)
Friday, September 30, 2011
Sunday, September 25, 2011
Numerous Connections to Quadratic Functions
I think that as secondary math teachers, we ought to be aware of as many ways as possible to model quadratic patterns, so we can seamlessly integrate those techniques into our curricula. It's the first non-straight forward pattern that our kids encounter, and it seems to trouble them throughout their secondary careers. Thankfully, there are ties to quadratic modeling from MANY different topics!
To discuss this concretely, let's first have a look at a quadratic pattern:
(2, 7), (3, 9), (4, 12), (5, 16)
In this discussion, I am going to assume that you've already taught kids how to recognize that this is a quadratic pattern, since that discussion should absolutely precede any quadratic skills, methinks!
Neat, eh? I think it's pretty amazing how you can approach a simple quadratic pattern from so many different algebraic methods, and I am a true believer that the more of these integrated approaches we take, the better kids will understand each individual method and be able to transfer their knowledge.
By the way, an 8th-grader came up to me and made an insightful observation, the day following our quick class discussion about why we need 3 points in order to find a path of trajectory (quadratic function) using Dan Meyer's basketball pictures. She asked me, "We need 3 points to find a quadratic function, but we need at least 4 points to verify that the pattern IS quadratic, right?" and she drew me a table on the board with some first and second differences. Brilliant! Yes - 3 points is enough to find an equation, but that's like saying two points form a line. That's the minimal amount of info needed to get the equation, but as soon as you have redundant data points, you will know if your regression form is actually correct! How astute of her to make that observation!
To discuss this concretely, let's first have a look at a quadratic pattern:
(2, 7), (3, 9), (4, 12), (5, 16)
In this discussion, I am going to assume that you've already taught kids how to recognize that this is a quadratic pattern, since that discussion should absolutely precede any quadratic skills, methinks!
Method #1: Helping kids understand at a middle-school level that we find missing coefficients inside equations by plugging in given x's and y's.
For example, given our points above, y = ax^2 + bx + c becomes:
7 = 4a + 2b + c
9 = 9a + 3b + c
12 = 16a + 4b + c
This we can solve in MIDDLE SCHOOL using systems of equations/elimination method.Method #2: When kids get into high school and they start learning about matrices, we can set the same system up as a problem of 3-by-3 matrix, and help them use the calculator to solve it:
(It's a good time to make sure everyone understands why we're using inverse matrices, and what identity means in math, etc.)Method #3: Of course, now that they're in the realm of graphing calculators, we should also teach them how to use quadratic regression to get the values of the coefficients. Method #4: There is another sneakier way of finding the equation using pure quadratics algebra. It's called the Legrange Method, and I learned it this summer at PCMI.
You split the function into smaller, easier quadratic functions by splitting the table and hiding two points at a time:
With a bit of simple algebra, you can figure out that
y1 = 3.5(x - 3)(x - 4)
y2 = -9(x - 2)(x - 4)
y3 = 6(x - 2)(x - 3)
Then, when you simplify each and combine them, you'd get your equation for y! This way is not the easiest, but it does reinforce the idea of roots, plugging in numbers to solve for parts of equation, and expansion/simplification of binomials.Method #5: Individual elements within a quadratic sequence are created as a summation of arithmetic/linear steps! For example, if you look at the example we have here: (2, 7), (3, 9), (4, 12), (5, 16)
f(2) = 7
f(3) = 7 + 2
f(4) = 7 + 2 + 3
f(5) = 7 + 2 + 3 + 4
Inductively,
f(x) = 7 + { 2 + 3 + 4 + ... +(x-1) }
Or, if we use an arithmetic series formula to represent the sum-of-steps:
f(x) = 7 + [2 + (x-1)][(x-2)/2]
Since my arithmetic series has first term = 2, last term = (x-1), and altogether (x-2)/2 pairs of elements.Method #6: There is a name for this trick, but I forget what it is. If we draw out the quadratic sequence as dots, we get:
Here I used the gray dots to represent the initial value.
If you double the added dots at each stage (see blue/black dots below), it now becomes instantly easier to find the pattern, because the length and the width of the resulting rectangle is a simple linear progression in terms of x:
f(x) = initial value + half of rectangular area
f(x) = 7 + 0.5(length times width of rectangle)
f(x) = 7 + 0.5(x + 1)(x - 2)
Neat, eh? I think it's pretty amazing how you can approach a simple quadratic pattern from so many different algebraic methods, and I am a true believer that the more of these integrated approaches we take, the better kids will understand each individual method and be able to transfer their knowledge.
By the way, an 8th-grader came up to me and made an insightful observation, the day following our quick class discussion about why we need 3 points in order to find a path of trajectory (quadratic function) using Dan Meyer's basketball pictures. She asked me, "We need 3 points to find a quadratic function, but we need at least 4 points to verify that the pattern IS quadratic, right?" and she drew me a table on the board with some first and second differences. Brilliant! Yes - 3 points is enough to find an equation, but that's like saying two points form a line. That's the minimal amount of info needed to get the equation, but as soon as you have redundant data points, you will know if your regression form is actually correct! How astute of her to make that observation!
Friday, September 23, 2011
Planned Time to Discuss Math
Today, I am happy because one of my real-life colleagues offered to sit down with me once a week to talk about math. I like the way she put it: "You are good at some things, and I am good at some different things. We can ask each other questions!"
I've never had a setup like this before! I can just talk about whatever I want to talk about, related to math and teaching, with someone else who is years more experienced than me!! We're not really planning partners, so it's not like we're bogged down by needing to discuss X number of lessons or Y issues each week, either.
This, my friends, is going to be sweet!
I've never had a setup like this before! I can just talk about whatever I want to talk about, related to math and teaching, with someone else who is years more experienced than me!! We're not really planning partners, so it's not like we're bogged down by needing to discuss X number of lessons or Y issues each week, either.
This, my friends, is going to be sweet!
Thursday, September 22, 2011
A Fun Pattern
I wanted to share a fun little pattern of the day. This is an extension of other visual patterns we've worked on, but this one right now stomps my 8th-graders, and it's kind of fun to see them stomped. Only about one or two kids have managed to figure out the second equation. A girl asked me if she can model the irregular shape by chopping it up into two regular rectangles, finding each quadratic equation separately, and then recombining them. YES! That's brilliant.
Questions to ask your students:
1. Can you conjecture what the next two stages will look like?
2. Which pattern is linear and which is quadratic? Which geometric property likely affects the type of function that the pattern will have?
3. For each pattern, find an equation that helps you predict the size (number of blocks) in stage x.
4. Can you figure out how many blocks will be in stage 50 of each pattern?
5. Can you figure out when (in which stage) there will be exactly _____ blocks?
Let expansion and factorization fall out as a matter of necessity. I think that's how we can help kids really sink their teeth into those skills, even before we get to "real world" problems.
Questions to ask your students:
1. Can you conjecture what the next two stages will look like?
2. Which pattern is linear and which is quadratic? Which geometric property likely affects the type of function that the pattern will have?
3. For each pattern, find an equation that helps you predict the size (number of blocks) in stage x.
4. Can you figure out how many blocks will be in stage 50 of each pattern?
5. Can you figure out when (in which stage) there will be exactly _____ blocks?
Let expansion and factorization fall out as a matter of necessity. I think that's how we can help kids really sink their teeth into those skills, even before we get to "real world" problems.
Wednesday, September 21, 2011
Progression of Middle School Skills
Teaching so many grades at once has made me re-think (because I have to!) the sequence of topics that we teach in school. This is a work in progress, but here is a rough list of topics in sequence, and what I personally think is ABSOLUTELY NECESSARY for the kids to understand regarding each topic:
* We absolutely need to introduce the idea of linear patterns as soon as possible. Something like:
_____, ______, 3, _______, _______, ________, 47, ________
The kids should be able to do this starting in early 7th grade (or before), well before they even consider what x's and y's are. They should be able to reason it out like this: "In 4 steps, the value goes up 44, so in each step, how much does it go up by??"
This proportional/linear reasoning underlies so much algebra but can be introduced completely without!! (And notice that you can use it to reinforce integer operations and also fractional understanding.)
_______, 0, ______, _______, 1, _______, _______, _______
* Instead of grade 7 being a re-teach of grade 6 skills, we need to focus on estimation skills in grade 7. Can kids plot points on a graph when scale is 3 or bigger?? Can kids plot points on a graph when they have mixed number types? Can kids mentally estimate 5% and 20% and 15% and 90% QUICKLY because they understand how to benchmark against 10%? Can kids figure out which 2 integers 23/9 falls between, AND which of those integers it is closer to??
Estimation needs to be the focus in Grade 7, to help kids see the "bigger picture" of values and less of the rote algorithms.
* Work in Grade 7 on helping kids set up equations. Help them figure out that "x" is the quantity that's being asked. Help them list out all the givens. Help them figure out that most equations are written like "part + part + part = TOTAL" or "part * part = TOTAL" so that they can always try to look for the total in a question. Help them figure out that "each" and "every" represent rates, which represent multiplications somewhere.
* Let kids figure out how to go backwards to find a Mystery Number. Don't try to teach rote algebra. Instead, draw arrow diagrams that represent the steps of evaluation, and kids'll figure out on their own how to work backwards.
* When kids are flying through easy equation setups, give them a twist. Make them figure out that if two quantities add to 20, then they are x and 20-x. Help them figure out why it's not x and x-20. Help them figure out that if two quantities multiply to be 70, then they are x and 70/x, not x and x/70. Because they'll need that flexible understanding down the road.
* Then, proportions. Can kids figure out WHICH situations are proportions and which are NOT?! How about proportions in the coordinate plane - what do they tell us? What about part-to-part ratios vs. part-to-whole ratios?
It sounds silly, but making tables works wonders in reinforcing proportional relationships. Kids will naturally observe that proportional quantities always start at (0, 0) and grow "proportionately" (in lack of another word), and they'll want to develop "shortcuts" like figuring out the scale factor immediately.
* Introduce bivariate linear patterns and introduce non-linearity alongside it. Before we teach any algebra, help kids learn that: x is the cause and y is the effect. Almost ALWAYS. For goodness' sake, don't tell them which variable is x and which is y, and DON'T always put the x on the left side of the table. Let them figure it out.
* Visually introduce linear patterns. Let them play with the idea of building equations and then making predictions, forwards and backwards, with the simple equation. Emphasize the utter importance of thinking about what X and Y each represents in the problem, in order to figure out where to plug in the given starting values. They'll need this understanding down the road in higher algebra, to keep track of the meaning of their computed results.
* Tie slope to UNIT rates. Yes, rise is a rate. But rise/run gives us the UNIT rate, which is more useful more often. Let them graph while THINKING about the meaning of unit rate and initial value. Don't feed them graphing algorithms!
* Then, introduce quadratic patterns and the idea of second differences. DON'T introduce quadratic skills before kids learn to visually recognize quadratic patterns from a table!!
* Help kids learn to find quadratic equations. Even if it's not the easiest way. (I do the Legrange Method and I think my kids hate me for it. Down the road I'll teach them to solve for a, b, c, using a system of equations.)
* Now, can the kid make predictions with a quadratic equation? Forwards and BACKWARDS. Don't let them start on factoring until they see the value of going backwards. If you need to, give them a graphing calculator for now, and tell them that soon we're going to figure out how to "not cheat" with the calculator and how to do that backwards solving-for-x by hand.
* Factor with the box method, and MAKE SURE THAT KIDS UNDERSTAND WHY factoring helps us solve the problem. (Here, insert joke about product of hair counts at NY Yankee Stadium being 0. Why?? It only takes 1 bald man. Start referring to your "zeroed factor" as a bald person.) Then introduce geometric problems that force kids to set up quadratic equations with x's on both sides, to solve them, and then to eliminate nonsensible solutions. The whole shebang. Focus on the idea that not all algebraic solutions always make sense.
* Embed linearity and quadratics in visuals, applications, word problems. Take your time. Make sure kids can viscerally connect to the idea of a slope being a rate and a y-intercept being an initial value; make them go backwards from equations to filling in blanks in a story. Make sure kids graph EVERYTHING and they understand always how the graph and table and equation are all interchangeable.
* Secretly introduce the idea of linear systems through applications like a savings race. Help kids approach the idea of break-even points from a graphical perspective, and use visual puzzles to get them thinking about equivalent quantities and substitution. Finally, wrap it up in a neat algebra bow by tying those methods to traditional algebra symbols and operations.
Make sure your kids can explain how to properly check a system!
* Get back to quadratics and now ask the kids to find quadratic equations fitting tables, using a systems of equations approach. Integrate different concepts into one seamless application for them. Don't let them walk away from this unit not knowing how to find a quadratic equation given ANY quadratic table. Re-emphasize now why we need exactly 3 points to find a quadratic equation. Use Dan Meyer's basketball pictures to let it sink in.
* Introduce square roots and combining square roots through Tangrams and other composite shapes made of right triangles. Tie this into combining like terms in general, and get them to think about decomposing shapes into smaller parts...
* Make sure kids understand dimensions. THEN introduce scaling in the coordinate plane.
* Lastly, exponents. I don't have good ideas for reinforcing exponent rules, except making kids write out all the terms being multiplied. Do you??
I feel like in the end, I'm still missing some topics, but I would be VERY HAPPY if every middle schooler I have the pleasure of teaching can leave with all of these concepts integrated together under their mathematical belt. That'd give us so much to build off of in high school, when the algebra gets trickier. What is your list of middle school must-knows??
* We absolutely need to introduce the idea of linear patterns as soon as possible. Something like:
_____, ______, 3, _______, _______, ________, 47, ________
The kids should be able to do this starting in early 7th grade (or before), well before they even consider what x's and y's are. They should be able to reason it out like this: "In 4 steps, the value goes up 44, so in each step, how much does it go up by??"
This proportional/linear reasoning underlies so much algebra but can be introduced completely without!! (And notice that you can use it to reinforce integer operations and also fractional understanding.)
_______, 0, ______, _______, 1, _______, _______, _______
* Instead of grade 7 being a re-teach of grade 6 skills, we need to focus on estimation skills in grade 7. Can kids plot points on a graph when scale is 3 or bigger?? Can kids plot points on a graph when they have mixed number types? Can kids mentally estimate 5% and 20% and 15% and 90% QUICKLY because they understand how to benchmark against 10%? Can kids figure out which 2 integers 23/9 falls between, AND which of those integers it is closer to??
Estimation needs to be the focus in Grade 7, to help kids see the "bigger picture" of values and less of the rote algorithms.
* Work in Grade 7 on helping kids set up equations. Help them figure out that "x" is the quantity that's being asked. Help them list out all the givens. Help them figure out that most equations are written like "part + part + part = TOTAL" or "part * part = TOTAL" so that they can always try to look for the total in a question. Help them figure out that "each" and "every" represent rates, which represent multiplications somewhere.
* Let kids figure out how to go backwards to find a Mystery Number. Don't try to teach rote algebra. Instead, draw arrow diagrams that represent the steps of evaluation, and kids'll figure out on their own how to work backwards.
* When kids are flying through easy equation setups, give them a twist. Make them figure out that if two quantities add to 20, then they are x and 20-x. Help them figure out why it's not x and x-20. Help them figure out that if two quantities multiply to be 70, then they are x and 70/x, not x and x/70. Because they'll need that flexible understanding down the road.
* Then, proportions. Can kids figure out WHICH situations are proportions and which are NOT?! How about proportions in the coordinate plane - what do they tell us? What about part-to-part ratios vs. part-to-whole ratios?
It sounds silly, but making tables works wonders in reinforcing proportional relationships. Kids will naturally observe that proportional quantities always start at (0, 0) and grow "proportionately" (in lack of another word), and they'll want to develop "shortcuts" like figuring out the scale factor immediately.
* Introduce bivariate linear patterns and introduce non-linearity alongside it. Before we teach any algebra, help kids learn that: x is the cause and y is the effect. Almost ALWAYS. For goodness' sake, don't tell them which variable is x and which is y, and DON'T always put the x on the left side of the table. Let them figure it out.
* Visually introduce linear patterns. Let them play with the idea of building equations and then making predictions, forwards and backwards, with the simple equation. Emphasize the utter importance of thinking about what X and Y each represents in the problem, in order to figure out where to plug in the given starting values. They'll need this understanding down the road in higher algebra, to keep track of the meaning of their computed results.
* Tie slope to UNIT rates. Yes, rise is a rate. But rise/run gives us the UNIT rate, which is more useful more often. Let them graph while THINKING about the meaning of unit rate and initial value. Don't feed them graphing algorithms!
* Then, introduce quadratic patterns and the idea of second differences. DON'T introduce quadratic skills before kids learn to visually recognize quadratic patterns from a table!!
* Help kids learn to find quadratic equations. Even if it's not the easiest way. (I do the Legrange Method and I think my kids hate me for it. Down the road I'll teach them to solve for a, b, c, using a system of equations.)
* Now, can the kid make predictions with a quadratic equation? Forwards and BACKWARDS. Don't let them start on factoring until they see the value of going backwards. If you need to, give them a graphing calculator for now, and tell them that soon we're going to figure out how to "not cheat" with the calculator and how to do that backwards solving-for-x by hand.
* Factor with the box method, and MAKE SURE THAT KIDS UNDERSTAND WHY factoring helps us solve the problem. (Here, insert joke about product of hair counts at NY Yankee Stadium being 0. Why?? It only takes 1 bald man. Start referring to your "zeroed factor" as a bald person.) Then introduce geometric problems that force kids to set up quadratic equations with x's on both sides, to solve them, and then to eliminate nonsensible solutions. The whole shebang. Focus on the idea that not all algebraic solutions always make sense.
* Embed linearity and quadratics in visuals, applications, word problems. Take your time. Make sure kids can viscerally connect to the idea of a slope being a rate and a y-intercept being an initial value; make them go backwards from equations to filling in blanks in a story. Make sure kids graph EVERYTHING and they understand always how the graph and table and equation are all interchangeable.
* Secretly introduce the idea of linear systems through applications like a savings race. Help kids approach the idea of break-even points from a graphical perspective, and use visual puzzles to get them thinking about equivalent quantities and substitution. Finally, wrap it up in a neat algebra bow by tying those methods to traditional algebra symbols and operations.
Make sure your kids can explain how to properly check a system!
* Get back to quadratics and now ask the kids to find quadratic equations fitting tables, using a systems of equations approach. Integrate different concepts into one seamless application for them. Don't let them walk away from this unit not knowing how to find a quadratic equation given ANY quadratic table. Re-emphasize now why we need exactly 3 points to find a quadratic equation. Use Dan Meyer's basketball pictures to let it sink in.
* Introduce square roots and combining square roots through Tangrams and other composite shapes made of right triangles. Tie this into combining like terms in general, and get them to think about decomposing shapes into smaller parts...
* Make sure kids understand dimensions. THEN introduce scaling in the coordinate plane.
* Lastly, exponents. I don't have good ideas for reinforcing exponent rules, except making kids write out all the terms being multiplied. Do you??
I feel like in the end, I'm still missing some topics, but I would be VERY HAPPY if every middle schooler I have the pleasure of teaching can leave with all of these concepts integrated together under their mathematical belt. That'd give us so much to build off of in high school, when the algebra gets trickier. What is your list of middle school must-knows??
Saturday, September 17, 2011
MYP and Grading
I realized when I started teaching in the IB system that a lot of changes were going to have to take place, but what I hadn't realized was that my grading scheme would involuntarily shift towards the mode of Standards-Based Grading.
My understanding is that in MYP (Middle Year Programme, which runs from 6th through 10th grade), students are graded based on four assessment criteria: Knowledge and Understanding (out of 8 points), Communication in Mathematics (out of 6 points), Reflections in Mathematics (out of 6 points), and Mathematical Modeling (out of 8 points). A sort of weighted average of all four criteria of assessment results in their final grade in the class, which is out of 7 possible points.
One clear advantage of this system is that it makes it very clear for these MYP students, which of the four categories they need to focus on and improve. (Hence, I say it's more SBG-style.)
What is very interesting to me is that a typical quiz that I give may only cover two assessment criteria (Knowledge and Understanding, and Communication in Mathematics). I look at their overall performance on the quiz, and if they understood just over half of the material, for example, I would assign a 5 out of 8. I also look at the amount of work shown, plus the one or two explanation questions, and assign a Communications in Mathematics grade out of a total of 6 points. (If a kid shows a lot of appropriate work throughout the quiz but fails to complete the explanation question, for example, I see that as a 3 out of 6.) This means that a kid could have missed some concepts on the quiz, but if they are really good about explaining what they did and showing every step of the work, they are earning credit toward a different assessment criterion. This I like, because it shows an emphasis toward the process used, as well as the final result.
As from last year, I continue to offer opportunities for students to revise their graded work and to re-submit for points. This year, I'm even more generous - they can revise even their project writeups, since my classes are smaller and I can handle the extra grading/feedback workload. And my new perspective is that the quiz process is truly three parts: 1. pre-quiz practice (ie. practice quizzes, done in class), 2. the quiz itself, returned with copious written feedback but not a score, 3. the kid revises the quiz, comes to explain every question to me. I used to think that the last step was optional, but this year my opinions on that have changed, under the new MYP grading system. Under the new grading system, I expect my MYP kids to do this revision step, and it counts towards their "Reflections in Mathematics" grade (as well as bumps up their original quiz score). I think of Reflections in Mathematics as a grade I am assigning towards meta-cognition in mathematics, if you will. I will take a similar approach with projects that are returned, by asking kids to take a close look at a rubric and to pick out specific things they need to work on the next time in order to show improved mastery.
Another thing that I am trying is to introduce formal structure for mathematical lab writeup as soon as possible. I don't quite think my seventh graders are ready for it yet, but by 8th and 9th grade, the students should be doing at LEAST 2 or 3 formal lab writeups a year, complete with intro, plan, data analysis and prediction, and final error analysis. This is because at the end of Grade 11, as part of their IB curriculum, they are supposed to do an "internal assessment" of this form, and I think if we leave it until 11th grade for the teacher to tackle this idea of a formal writeup in math, that is way too late.
So, to that end, I took the formal Grade 11 lab writeup rubric from the IB program and I modified it into a more kid-friendly language for use with younger students. You are welcome to take a look! The first trial run of this lab writeup I am doing is with the cup-stacking activity from Dan Meyer. My (low) Grade 9 group has been working on the rough draft of this writeup in class so that they can ask me for help. Some of them were pretty disorganized and had already lost their data from before, so I let them re-measure the data and explain the steps to me to ensure that they understand what they're doing. It's a very worthwhile process, and I think that forcing them to do this lab writeup has truly enhanced every kid's understanding of the activity. Even from my English language-learners in class, I have seen much more understanding be articulated than I could have hoped for, even within the rough drafts. It's great!
Here are some files that might be useful, if you're thinking about doing this kind of thing: lab writeup checklist and the modeling rubric. They are works in progress, but I figured they're a good starting point.
My understanding is that in MYP (Middle Year Programme, which runs from 6th through 10th grade), students are graded based on four assessment criteria: Knowledge and Understanding (out of 8 points), Communication in Mathematics (out of 6 points), Reflections in Mathematics (out of 6 points), and Mathematical Modeling (out of 8 points). A sort of weighted average of all four criteria of assessment results in their final grade in the class, which is out of 7 possible points.
One clear advantage of this system is that it makes it very clear for these MYP students, which of the four categories they need to focus on and improve. (Hence, I say it's more SBG-style.)
What is very interesting to me is that a typical quiz that I give may only cover two assessment criteria (Knowledge and Understanding, and Communication in Mathematics). I look at their overall performance on the quiz, and if they understood just over half of the material, for example, I would assign a 5 out of 8. I also look at the amount of work shown, plus the one or two explanation questions, and assign a Communications in Mathematics grade out of a total of 6 points. (If a kid shows a lot of appropriate work throughout the quiz but fails to complete the explanation question, for example, I see that as a 3 out of 6.) This means that a kid could have missed some concepts on the quiz, but if they are really good about explaining what they did and showing every step of the work, they are earning credit toward a different assessment criterion. This I like, because it shows an emphasis toward the process used, as well as the final result.
As from last year, I continue to offer opportunities for students to revise their graded work and to re-submit for points. This year, I'm even more generous - they can revise even their project writeups, since my classes are smaller and I can handle the extra grading/feedback workload. And my new perspective is that the quiz process is truly three parts: 1. pre-quiz practice (ie. practice quizzes, done in class), 2. the quiz itself, returned with copious written feedback but not a score, 3. the kid revises the quiz, comes to explain every question to me. I used to think that the last step was optional, but this year my opinions on that have changed, under the new MYP grading system. Under the new grading system, I expect my MYP kids to do this revision step, and it counts towards their "Reflections in Mathematics" grade (as well as bumps up their original quiz score). I think of Reflections in Mathematics as a grade I am assigning towards meta-cognition in mathematics, if you will. I will take a similar approach with projects that are returned, by asking kids to take a close look at a rubric and to pick out specific things they need to work on the next time in order to show improved mastery.
Another thing that I am trying is to introduce formal structure for mathematical lab writeup as soon as possible. I don't quite think my seventh graders are ready for it yet, but by 8th and 9th grade, the students should be doing at LEAST 2 or 3 formal lab writeups a year, complete with intro, plan, data analysis and prediction, and final error analysis. This is because at the end of Grade 11, as part of their IB curriculum, they are supposed to do an "internal assessment" of this form, and I think if we leave it until 11th grade for the teacher to tackle this idea of a formal writeup in math, that is way too late.
So, to that end, I took the formal Grade 11 lab writeup rubric from the IB program and I modified it into a more kid-friendly language for use with younger students. You are welcome to take a look! The first trial run of this lab writeup I am doing is with the cup-stacking activity from Dan Meyer. My (low) Grade 9 group has been working on the rough draft of this writeup in class so that they can ask me for help. Some of them were pretty disorganized and had already lost their data from before, so I let them re-measure the data and explain the steps to me to ensure that they understand what they're doing. It's a very worthwhile process, and I think that forcing them to do this lab writeup has truly enhanced every kid's understanding of the activity. Even from my English language-learners in class, I have seen much more understanding be articulated than I could have hoped for, even within the rough drafts. It's great!
Here are some files that might be useful, if you're thinking about doing this kind of thing: lab writeup checklist and the modeling rubric. They are works in progress, but I figured they're a good starting point.
Friday, September 16, 2011
Connect the dots
I am introducing my 7th-graders to graphing points, in anticipation of the upcoming discussions about algebraic/graphical linearity and similarity ratios.
To this end, I went ahead and enhanced an assignment I have already used in the past. They're going to do Connect the Dots individually on one picture I've designed, and then make big posters in groups on other designs while still practicing plotting points! (I'm very excited about this. I need some wall decorations very soon, and I think these posters are going to be perfect.)
I've come up with three designs besides the original individual one. The kids are not going to be told what the designs are until they graph them out. The feedback for me (and for them) is instantaneous, because I can visually see when their picture doesn't look quite right. We haven't done this yet, but I think doing it on the gridded chart paper has an added benefit of forcing them to use scales bigger than 1 (since there are fewer grids on the big paper) -- which will in turn force them to consider how to graph points that are not neatly on or halfway between two labels on the axis. Go approximation!
Doing it on big poster also will have other benefits, methinks: 1. encouraging group work (they will be encouraged to divide up the points to work in parallel), 2. extra neatness (using pencils to plot points and rulers to outline before outlining with markers).
Here are the files, in case you want to adopt this activity: Design #1 (individual work, in notebook) and #2 (group work/poster) and Design #3 and #4 (group work/poster).
I'm pretty excited. I'll even take pictures this time of their final products!!
To this end, I went ahead and enhanced an assignment I have already used in the past. They're going to do Connect the Dots individually on one picture I've designed, and then make big posters in groups on other designs while still practicing plotting points! (I'm very excited about this. I need some wall decorations very soon, and I think these posters are going to be perfect.)
I've come up with three designs besides the original individual one. The kids are not going to be told what the designs are until they graph them out. The feedback for me (and for them) is instantaneous, because I can visually see when their picture doesn't look quite right. We haven't done this yet, but I think doing it on the gridded chart paper has an added benefit of forcing them to use scales bigger than 1 (since there are fewer grids on the big paper) -- which will in turn force them to consider how to graph points that are not neatly on or halfway between two labels on the axis. Go approximation!
Doing it on big poster also will have other benefits, methinks: 1. encouraging group work (they will be encouraged to divide up the points to work in parallel), 2. extra neatness (using pencils to plot points and rulers to outline before outlining with markers).
Here are the files, in case you want to adopt this activity: Design #1 (individual work, in notebook) and #2 (group work/poster) and Design #3 and #4 (group work/poster).
I'm pretty excited. I'll even take pictures this time of their final products!!
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