When I started teaching Younger Daughter the times tables, she protested that she already knew them -- at least for 2, 3, 4, and 5 -- because she had been taught them at school. It turns out that what she knew is "skip counting" -- that is, "2, 4, 6, 8, ..." (Back in my day, when we weren't dying of the bubonic plague, we called this "counting by twos.") But if you asked Younger Daughter what 2 x 3 is, she had to think about it for a while, and the answer she came up with might or might not be correct. (She's getting better now, after a great deal of work on both of our parts.)
There's nothing wrong with skip counting, and it's a reasonable first step toward learning multiplication. This is typical of the new "fuzzy math" curricula (Younger Daughter's school uses "TERC Investigations") -- they include some reasonable first steps, but they don't follow through and actually teach the skills and facts you need to know.
The first unrelated-to-me kid that I tutored had a similar problem with adding fractions. I was impressed when she knew that 1/2 + 1/4 is 3/4. When I asked her how she knew this, she drew a little pie illustration. Again, this is a reasonable first step. But when I asked her what 2/50 + 1/100 was, she was completely stumped. That's because Trailblazers doesn't follow through and actually teach the algorithm for adding fractions.
Feh! The bad news is that the district of Upper Tax Bracket only adopted TERC Investigations a couple of years ago -- it'll take time for any kind of momentum to build against it. In the meantime, I'm teaching Younger Daughter math at home.
Showing posts with label Trailblazers. Show all posts
Showing posts with label Trailblazers. Show all posts
Friday, July 26, 2013
Friday, December 9, 2011
Dream Schools
I want a dream lover
So I won't have to dream alone.
-- Bobby Darin, "Dream Lover"
TeacHer asked, in a recent comment:
Thanks for the question, TeacHer!
My dream school would combine the best of the progressive and traditional philosophies. From the best of the progressives, I would take an interest in the child as a complete human being, with physical and emotional needs as well as academic ones, and the goal of developing an independent thinker, with a continuing interest in learning. From the best of the traditionalists, I would take a true understanding and appreciation of content knowledge, including technical content like math and science, and a respect for linear, well-designed curricula.
So, with the progressives, I would throw out authoritarian classroom-management systems like PBIS and WBT, but with the traditionalists, I would throw out bad curricula like fuzzy math, non-phonics reading instruction, and meta-meta-meta "comprehension" questions that baffle and alienate small children.
My dream elementary school would assign no homework, or optional homework, and never restrict recess as a punishment, or for any reason. Teachers and administrators would work towards a genuine partnership with parents, not the current Jeeves-and-Wooster farce that passes for "partnership".
My dream teacher would be well-educated and genuinely interested in learning, as well as humane and caring (that's a surprisingly difficult combo to find!). She would be open to new ideas, and not ideologically wedded to certain techniques (and yes, I understand that a lot of important classroom decisions aren't under the teacher's control any more.)
And here's an impossible dream for you: just once, I would like to read a newsletter written by a teacher with no grammatical errors.
Readers, what would your dream school look like? How about your dream teacher?
So I won't have to dream alone.
-- Bobby Darin, "Dream Lover"
TeacHer asked, in a recent comment:
I've been wondering about this recently: if you were to design the perfect school, how would it look? What about the perfect teacher?
Thanks for the question, TeacHer!
My dream school would combine the best of the progressive and traditional philosophies. From the best of the progressives, I would take an interest in the child as a complete human being, with physical and emotional needs as well as academic ones, and the goal of developing an independent thinker, with a continuing interest in learning. From the best of the traditionalists, I would take a true understanding and appreciation of content knowledge, including technical content like math and science, and a respect for linear, well-designed curricula.
So, with the progressives, I would throw out authoritarian classroom-management systems like PBIS and WBT, but with the traditionalists, I would throw out bad curricula like fuzzy math, non-phonics reading instruction, and meta-meta-meta "comprehension" questions that baffle and alienate small children.
My dream elementary school would assign no homework, or optional homework, and never restrict recess as a punishment, or for any reason. Teachers and administrators would work towards a genuine partnership with parents, not the current Jeeves-and-Wooster farce that passes for "partnership".
My dream teacher would be well-educated and genuinely interested in learning, as well as humane and caring (that's a surprisingly difficult combo to find!). She would be open to new ideas, and not ideologically wedded to certain techniques (and yes, I understand that a lot of important classroom decisions aren't under the teacher's control any more.)
And here's an impossible dream for you: just once, I would like to read a newsletter written by a teacher with no grammatical errors.
Readers, what would your dream school look like? How about your dream teacher?
Wednesday, April 6, 2011
How Many Handshakes?
The previous discussion reminds me of a radio show that the Head of School at Natural Friends did last summer. As an example of the kind of deep problem that kids at Natural Friends might work on, he offered, "there are three people at a party. How many handshakes?"
I was completely baffled by this the first time I heard it. How the heck would I know how many handshakes? There are too many possibilities for a divergent thinker like myself.
There might be zero handshakes, because everyone in the room is germ-phobic.
There might be hundreds of handshakes, because it's a meeting of the Obsessive-Compulsive Handshakers Support Group.
If two of the three people are married to each other, they wouldn't need to shake hands, so there would be two handshakes (the husband and wife each shaking the third person's hand.)
If two of the three people are Orthodox Jewish men (not allowed to touch a woman other than their own wife), and the third is an unrelated woman, the only legal handshake would be between the two men.
Or it could be that each person shakes every other person's hand exactly once, in which case you have three handshakes.
That last possibility is actually the preferred one; the handshake question turns out to be a classic problem. For the problem to work as intended, it must be assumed (or better yet, stated) that each person shakes every other person's hand exactly once.
The problem is fairly sophisticated; as you add people to the party, you wind up taking the sum of an arithmetic series. I think it would go right over the head of a kid brought up on the thin gruel of Trailblazers Math.
I was completely baffled by this the first time I heard it. How the heck would I know how many handshakes? There are too many possibilities for a divergent thinker like myself.
There might be zero handshakes, because everyone in the room is germ-phobic.
There might be hundreds of handshakes, because it's a meeting of the Obsessive-Compulsive Handshakers Support Group.
If two of the three people are married to each other, they wouldn't need to shake hands, so there would be two handshakes (the husband and wife each shaking the third person's hand.)
If two of the three people are Orthodox Jewish men (not allowed to touch a woman other than their own wife), and the third is an unrelated woman, the only legal handshake would be between the two men.
Or it could be that each person shakes every other person's hand exactly once, in which case you have three handshakes.
That last possibility is actually the preferred one; the handshake question turns out to be a classic problem. For the problem to work as intended, it must be assumed (or better yet, stated) that each person shakes every other person's hand exactly once.
The problem is fairly sophisticated; as you add people to the party, you wind up taking the sum of an arithmetic series. I think it would go right over the head of a kid brought up on the thin gruel of Trailblazers Math.
Monday, April 4, 2011
Neither Fish, nor Flesh, nor Good Red Herring
Every school my kids have attended so far is neither truly progressive, because kids have very little choice in what they do, nor truly traditional, because the academic content is so weak.
Even curricula that are marketed as "progressive", like Trailblazers Math, contain all the worst aspects of traditional ed: bad homework, scripted lessons that kids are relentlessly marched through, and authoritarian pronouncements on how problems may be solved (no standard algorithms!)
It seems to me that the biggest stumbling block that prevents schools from achieving true progressive ed is simply this: control. To be truly progressive, to allow students as much freedom and autonomy as possible, just goes against the grain for educators. School is a place where kids are told what to do.
The biggest stumbling block that prevents schools from achieving true traditional ed is a pervasive lack of interest in real content. Professional-class parents are often more concerned about college admissions than learning. Too many educators don't know what real content looks like, and mistake activity for engagement.
Even curricula that are marketed as "progressive", like Trailblazers Math, contain all the worst aspects of traditional ed: bad homework, scripted lessons that kids are relentlessly marched through, and authoritarian pronouncements on how problems may be solved (no standard algorithms!)
It seems to me that the biggest stumbling block that prevents schools from achieving true progressive ed is simply this: control. To be truly progressive, to allow students as much freedom and autonomy as possible, just goes against the grain for educators. School is a place where kids are told what to do.
The biggest stumbling block that prevents schools from achieving true traditional ed is a pervasive lack of interest in real content. Professional-class parents are often more concerned about college admissions than learning. Too many educators don't know what real content looks like, and mistake activity for engagement.
Friday, March 25, 2011
Thursday, February 24, 2011
What is Student Engagement?
 | |||
| (from Trailblazers Math, grade 2.) |
Unfortunately, life is not that simple. It is possible to be mentally engaged while sitting quite still (e.g., while watching a movie), and it is also possible to be mentally absent while moving around (e.g., daydreaming while folding laundry.) You can make kids move around, but that doesn't necessarily mean that they're interested in what they're doing, or that they're learning anything in particular. A bad project is less engaging than a good lecture.
Tuesday, February 22, 2011
I Have Measured Out My Life with Coffee Spoons
For I have known them all already, known them all:—
Have known the evenings, mornings, afternoons,
I have measured out my life with coffee spoons;
I know the voices dying with a dying fall
Beneath the music from a farther room.
(from The Love Song of J. Alfred Prufrock, by T. S. Eliot.)
We find that in doing measurement activities, children learn about number. It’s a two-way street. Our approach is not, “Well, we’ll teach the kids about number, and once they understand number, then we can teach them about measurement, because measurement is based on number.” We find it works the other way, too: by doing various measurement activities, which are very engaging for the students, they’re building their number ideas.
(from Math Trailblazers, quoting one of the developers.)
I think the program is pretty good with measurement. Kids like to measure things, and the program builds on that; it builds on their interests.
(from Math Trailblazers, quoting a 3d grade teacher.)
Have known the evenings, mornings, afternoons,
I have measured out my life with coffee spoons;
I know the voices dying with a dying fall
Beneath the music from a farther room.
(from The Love Song of J. Alfred Prufrock, by T. S. Eliot.)
We find that in doing measurement activities, children learn about number. It’s a two-way street. Our approach is not, “Well, we’ll teach the kids about number, and once they understand number, then we can teach them about measurement, because measurement is based on number.” We find it works the other way, too: by doing various measurement activities, which are very engaging for the students, they’re building their number ideas.
(from Math Trailblazers, quoting one of the developers.)
I think the program is pretty good with measurement. Kids like to measure things, and the program builds on that; it builds on their interests.
(from Math Trailblazers, quoting a 3d grade teacher.)
"Kids like to measure things?" Really? I've never noticed my kids show any particular interest in measuring. I have noticed, however, that Trailblazers spends an unbelievable amount of time and effort on measurement, from the Bouncing Ball lab to the above worksheet, part of an entire section that has the kids measuring random objects in hands and cubits (forearm lengths). Why? In order to demonstrate what a bad idea it is to use non-standard units!
Wednesday, February 9, 2011
Bouncing Ball Math Lab
So far, the Head of School at Natural Friends seems to have decided that what their math program really needs is better PR. To this end, he included a message from the math specialist in the school's bulletin. The math specialist has worked at Natural Friends for many years while the kids were graduating right into remedial math at their next school. She is a lot more likely to be part of the problem than part of the solution, IMHO. Here's the specialist's account of a 4th-grade "math lab":
Teacher 4 holds out a tennis ball a little above his waist height and asks his class, "Do you think you could predict how high this ball would bounce if I dropped it from 1 meter? "
Students suggest, "60 cm." "1 ½ meters.' 'To the ceiling."
"What could we do to make reasonable predictions?"
An animated discussion follows. The students know from past experience that they will have to experiment by dropping their ball a number of times from various heights to be able to make a good prediction. They decide to work in groups of three . Each group gets a ball and agrees to drop it several times from 40 cm, 60 cm and 120cm. They purposely omit dropping it from 100 cm (1 m) because they are intent on predicting the answer to Teacher 4's question before actually measuring it. From the many labs they have done in earlier grades they know that they will need to take at least three trial drops from each height and then find the average in order to get a more accurate result.
And so another Math Lab begins. By now the students are familiar with the process of planning an experiment, collecting data, graphing their data, and finally analyzing their results and testing their predictions.
These fourth graders have just learned how to calculate the mean and they will use this new skill to find the average of the three drops from each height. (In previous labs they usually took the median [the middle number of the three trials] as the average.) This lab will reinforce skills such as measuring, graphing, making 'best fit lines', and working cooperatively. It will also enable students to practice their new ability to find the mean.
There is a lively hum in the class as they students spread around the room and begin to plan their group work. They tape their meter sticks to the wall and practice dropping the ball and reading the results while the ball is in motion. Then they are ready to carry out the experiment, record the data and graph their results..
Later they find a 'best fit line' for their data points, so that they can predict how high the ball would probably bounce if it were dropped from exactly 1 meter. They then test their hypothesis by dropping the ball from 1 meter and compare this with their predictions.
This math lab is one of eight labs fourth graders will do during this academic year. For each lab, students sketch their plan, gather their materials, collect data in a data chart, then graph and analyze the data.
We have found these labs to be an excellent way for students to use and extend the mathematics that they are learning in a meaningful and highly motivated way.
and my response, sent to the Head of School and the math specialist:
I read the description of the 4th grade math lab in today's newsletter, and I have a question. When the kids take an average of three measurements, how do they perform the calculation? Do they add the numbers using the standard addition algorithm, then divide using long division? Or do they just use a calculator? If they're using calculators, they've missed an opportunity to practice their math skills. Over-reliance on calculators is one of the problems that lead to kids graduating from NF without the skills they need. (I see a calculator next to a student's elbow in one of the photos.)
If the kids use calculators for anything more challenging than 2+2, how much math is really being taught here? Kids need to achieve fluency with numbers, which comes -- wait for it -- from working with numbers. I think the calculators are a crutch used by the teacher to avoid confronting the reality of how weak the kids' math skills really are.
I didn't mention it in my e-mail, but I am also deeply skeptical of the picture painted in this description. Were the kids really engaged by the question of how high the ball would bounce? I doubt that either of my daughters would be.
Teacher 4 holds out a tennis ball a little above his waist height and asks his class, "Do you think you could predict how high this ball would bounce if I dropped it from 1 meter? "
Students suggest, "60 cm." "1 ½ meters.' 'To the ceiling."
"What could we do to make reasonable predictions?"
An animated discussion follows. The students know from past experience that they will have to experiment by dropping their ball a number of times from various heights to be able to make a good prediction. They decide to work in groups of three . Each group gets a ball and agrees to drop it several times from 40 cm, 60 cm and 120cm. They purposely omit dropping it from 100 cm (1 m) because they are intent on predicting the answer to Teacher 4's question before actually measuring it. From the many labs they have done in earlier grades they know that they will need to take at least three trial drops from each height and then find the average in order to get a more accurate result.
And so another Math Lab begins. By now the students are familiar with the process of planning an experiment, collecting data, graphing their data, and finally analyzing their results and testing their predictions.
These fourth graders have just learned how to calculate the mean and they will use this new skill to find the average of the three drops from each height. (In previous labs they usually took the median [the middle number of the three trials] as the average.) This lab will reinforce skills such as measuring, graphing, making 'best fit lines', and working cooperatively. It will also enable students to practice their new ability to find the mean.
There is a lively hum in the class as they students spread around the room and begin to plan their group work. They tape their meter sticks to the wall and practice dropping the ball and reading the results while the ball is in motion. Then they are ready to carry out the experiment, record the data and graph their results..
Later they find a 'best fit line' for their data points, so that they can predict how high the ball would probably bounce if it were dropped from exactly 1 meter. They then test their hypothesis by dropping the ball from 1 meter and compare this with their predictions.
This math lab is one of eight labs fourth graders will do during this academic year. For each lab, students sketch their plan, gather their materials, collect data in a data chart, then graph and analyze the data.
We have found these labs to be an excellent way for students to use and extend the mathematics that they are learning in a meaningful and highly motivated way.
and my response, sent to the Head of School and the math specialist:
I read the description of the 4th grade math lab in today's newsletter, and I have a question. When the kids take an average of three measurements, how do they perform the calculation? Do they add the numbers using the standard addition algorithm, then divide using long division? Or do they just use a calculator? If they're using calculators, they've missed an opportunity to practice their math skills. Over-reliance on calculators is one of the problems that lead to kids graduating from NF without the skills they need. (I see a calculator next to a student's elbow in one of the photos.)
If the kids use calculators for anything more challenging than 2+2, how much math is really being taught here? Kids need to achieve fluency with numbers, which comes -- wait for it -- from working with numbers. I think the calculators are a crutch used by the teacher to avoid confronting the reality of how weak the kids' math skills really are.
I didn't mention it in my e-mail, but I am also deeply skeptical of the picture painted in this description. Were the kids really engaged by the question of how high the ball would bounce? I doubt that either of my daughters would be.
Tuesday, February 1, 2011
Reforming Math 5
(Continuing from Reforming Math 1, Reforming Math 2, Reforming Math 3, and Reforming Math 4.)
Here's the latest from the Head of Natural Friends:
I have read your e-mail several times and the more I have thought about it the more I am convinced that the distance between your position and mine is small.
I share the conviction that computation is an important component of mathematics. I share the conviction that this vital curricular thread (computation) has developmentally appropriate expectations associated with it. For example, multiplication tables 0 to 12 known at the end of fourth grade, and addition, subtraction, multiplication, and division of parts of wholes (fractions, decimals, and percents) mastered at the end of fifth grade.
You seem to object to constructivist, mathematical inquiry only in so far is it is not the whole of the math curriculum (and not primary) or insofar as it is tiring, tedious, time-wasting (and alliterative).
I want you to know that our mathematics specialist shares the conviction that both computation and interesting, constructivist, mathematical inquiry are important elements of a strong mathematics curriculum. I want you to know by way of example that our fourth grade teacher has identified the aforementioned multiplication tables expectation (in writing and at curriculum night) for his class parents this year.
I can not speak to the history of the successful implementation of this point of view at Natural Friends but I can tell you that I am determined that we as a faculty will successfully implement a mathematics curriculum that contains both elements and reflects clearly both rote and more creative expectations going forward.
and my reply:
I expect we agree on goals more than we do on methods.
I told you about NF's current reputation among the surrounding schools, not because I'm asking you to fix the past, (although if you can, be my guest), but because I'm trying to convince you that the status quo should not be allowed to continue.
If I could wave a magic wand and get anything I wanted, I would have you throw out Trailblazers/Connected Math and bring in Singapore Math. Singapore Math is clear, it's concise, and it teaches real math. TB/CM, by contrast, just doesn't teach a long list of important algorithms and concepts (I hope you've had a chance to peruse my list -- the e-mail is titled "Missing Math"), and it wastes a great deal of time and energy on substandard algorithms (like "partial sums" and "partial quotients"). There is also way too much time spent on discussion and writing in English, and not enough time spent learning the symbolic language of mathematics. TB/CM is very weak on the abstract qualities of math; it doesn't go near logic and inference.
Now, I'm not against constructivist inquiry if the kids enjoy it. Of course, kids should be encouraged to ask questions and make sense of what they're learning. If they can exercise their math skills while engaged in an interesting project or game, I'm all for it. But they also need a coherent, complete curriculum. Clearly stated goals (as in your example of the 4th grade) are an important first step.
I'm not a constructivist or traditionalist or drill-and-killer. I'm a pragmatist. I want to know that if Younger Daughter stays at NF through the 6th grade, she will go on to her next school able to handle their math program, without remediation or extra tutoring. I want her to know math, like math, and have justified confidence with it. Can NF deliver all this?
Sincerely,
FedUpMom.
P.S. Speaking of games, here's one that Older Daughter likes. It's about probability.
Can't Stop
And here's a list of math games:
Math Games
Here's the latest from the Head of Natural Friends:
I have read your e-mail several times and the more I have thought about it the more I am convinced that the distance between your position and mine is small.
I share the conviction that computation is an important component of mathematics. I share the conviction that this vital curricular thread (computation) has developmentally appropriate expectations associated with it. For example, multiplication tables 0 to 12 known at the end of fourth grade, and addition, subtraction, multiplication, and division of parts of wholes (fractions, decimals, and percents) mastered at the end of fifth grade.
You seem to object to constructivist, mathematical inquiry only in so far is it is not the whole of the math curriculum (and not primary) or insofar as it is tiring, tedious, time-wasting (and alliterative).
I want you to know that our mathematics specialist shares the conviction that both computation and interesting, constructivist, mathematical inquiry are important elements of a strong mathematics curriculum. I want you to know by way of example that our fourth grade teacher has identified the aforementioned multiplication tables expectation (in writing and at curriculum night) for his class parents this year.
I can not speak to the history of the successful implementation of this point of view at Natural Friends but I can tell you that I am determined that we as a faculty will successfully implement a mathematics curriculum that contains both elements and reflects clearly both rote and more creative expectations going forward.
and my reply:
I expect we agree on goals more than we do on methods.
I told you about NF's current reputation among the surrounding schools, not because I'm asking you to fix the past, (although if you can, be my guest), but because I'm trying to convince you that the status quo should not be allowed to continue.
If I could wave a magic wand and get anything I wanted, I would have you throw out Trailblazers/Connected Math and bring in Singapore Math. Singapore Math is clear, it's concise, and it teaches real math. TB/CM, by contrast, just doesn't teach a long list of important algorithms and concepts (I hope you've had a chance to peruse my list -- the e-mail is titled "Missing Math"), and it wastes a great deal of time and energy on substandard algorithms (like "partial sums" and "partial quotients"). There is also way too much time spent on discussion and writing in English, and not enough time spent learning the symbolic language of mathematics. TB/CM is very weak on the abstract qualities of math; it doesn't go near logic and inference.
Now, I'm not against constructivist inquiry if the kids enjoy it. Of course, kids should be encouraged to ask questions and make sense of what they're learning. If they can exercise their math skills while engaged in an interesting project or game, I'm all for it. But they also need a coherent, complete curriculum. Clearly stated goals (as in your example of the 4th grade) are an important first step.
I'm not a constructivist or traditionalist or drill-and-killer. I'm a pragmatist. I want to know that if Younger Daughter stays at NF through the 6th grade, she will go on to her next school able to handle their math program, without remediation or extra tutoring. I want her to know math, like math, and have justified confidence with it. Can NF deliver all this?
Sincerely,
FedUpMom.
P.S. Speaking of games, here's one that Older Daughter likes. It's about probability.
Can't Stop
And here's a list of math games:
Math Games
Subscribe to:
Posts (Atom)