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Some teachers, however, feel compelled to deviate from what the "Investigations" program recommends, by teaching and drilling the familiar (to us) conventional algorithms. There seem to be three reasons why teachers feel a need to explicitly teach the conventional American algorithms:The first reason reflects that we live in a real world where, alas, not everyone appreciates the "Investigations" approach. In these situations, you may have to make concessions and explicitly teach the conventional algorithms to some degree. But be sure to ask yourself: is this a real need from parents and teachers or just a perceived one? Talk to parents and other teachers to find out the real nature of their concern. Perhaps they just want students to be able to solve computation problems accurately and quickly, and they are unaware that other algorithms can accomplish this too. As for standardized tests, your students should do well on these tests without knowing the conventional algorithms. Perhaps all they need is some familiarity with particular notation and vocabulary.
- in response to pressure from parents who only know one method or from teachers at the next grade level
- to prepare students for a standardized test that assesses for knowledge of specific algorithms
- because they feel it is important for students' education
We need to be able to sue schools for educational malpractice.
Also textbook authors.
14 comments:
Linking through to the original article, I found this:
"'Ours is not to reason why; just invert and multiply.' -- rhyme used to remember the algorithm for dividing fractions"
"Teachers see it all the time: otherwise intelligent and curious children who check their brain at the door as math time begins. 'Just give me the rule' they demand, when asked to solve a problem. These children have learned, from previous school experience or from home, that mathematics is just following a bunch of rules that don't often make sense."
(" or from home")??? If the problem is from home, then why do they need a new curriculum?
The justification of TERC is based on a strawman. I don't recall any teacher I had who said that all I needed was to remember a rule. As for the basic algorithims of arithmetic, I don't recall anyone having any problems, unless they failed to remember their times table or were unable to actually do the problems. Understanding of basic math was never an issue.
Besides, a real understanding of how basic algorithms work requires a certain amount of algebra. A lack of "understanding" and "inventing" of basic algorithms is not the problem.
The problems comes with fractions and the transition from fractions in terms of a slice of pie to fractions as rational terms in algebra. The question I have for those who have access to TERC is when and how do they explain dividing fractions. What mathematical understanding do they teach if they don't teach the justification for invert and multiply? My teachers didn't just teach us rules. This would be bad teaching.
In the article, they make a big deal about students inventing their own algorithms. I would like to know how they do this for dividing fractions. Does TERC do this for all basic mathematical identities? How would a child "invent" a rule for dividing fractions? What if a child could not figure out a rule?
Remember the Everyday Math addition of fractions examples I gave before? Students had to find the pattern for the following:
1/2 + 1/3 = 5/6
If I invent the following rule:
To get the numerator of the answer, you add the denominators. To get the denominator of the answer, you multiply the denominators.
Does this show that I understand anything? If I'm not able to invent anything, does that mean I can't understand it? If the teacher has to teach me the rule, how do they teach understanding?
Those who focus on just the basic traditional algorithms fall into a trap. They are arguing at the wrong level. They need to focus on fractions and the transition to rational expressions. They need to focus on mathematical understanding based on basic math identities rather than invented patterns.
What is it about a child "discovering" a pattern that is so appealing to the constructivists?
I feel like EM and TERC have decided that if a child can identify a pattern, then they must have understanding.
Steve, you'll love this.
EM introduces solid geometry in unit 10 and 11 of the 2nd journal. For some reason, despite limited work on finding area, the kids have to find volume and then jump right into Euler's Formula.
They are to fill in a box containing vertices, faces, and edges of various regular shapes and then they have to "see" a pattern.
From the example of five shapes, they are encouraged to extrapolate to every known object. My pattern worked for these five, it must work for every shape.
At no point did the teacher or EM explain the difference between a "theorem" and being able to prove anything. No discussion of inductive or deductive reasoning. No discussion on why seeing a pattern in a limited set is not alway sufficient to say you know it will work in every set.
Plus the whole solid geometry unit is deeply flawed in that they covered a tremendous number of topic in such a supericial way that I doubt anything will carry over to next year.
"EM introduces solid geometry in unit 10 and 11 of the 2nd journal."
No worries. He will never get there by the end of the year!
"My pattern worked for these five, it must work for every shape."
What if a student doesn't see a pattern? What if the pattern is wrong? It's harder to un-learn wrong things. What if the child doesn't discover anything?
As I have said before, they put very mixed ability kids together in groups to discover or construct knowledge. Probably only one child will "discover" anything (have any sort of "light bulb"-type effect). Then again, no real discovery might happen due to pre-teaching by parents. This student (probably one from the top end of the mixed ability group) will then attempt to directly teach his/her idea to the rest of the group. This is supposed to be better than having a well-trained teacher do the job? Also, this mixed-ability approach virtually guarantees that none of the lowest-ability students will ever discover anything.
Everyday Math does does not do a lot of group discovery work (at least I don't see it with my son), but there is a big probability that the patterns kids have to discover at home will be directly taught by the parents - along with a lot of additional explanation and reteaching.
I'm getting tired of my son telling me that he has to solve something a particular way because that's the way they do it in class. I thought EM was about having students come up with their own rules. I thought EM was about multiple ways of doing things. I think he just doesn't like my confusing the issue by providing too much input. This happens because EM isn't doing the job properly. I have to fix things before they get worse.
If a student "discovers" the invert and multiply pattern for dividing fractions, does he/she understand anything? Finding patterns is extremely over-rated.
Patterns are clues, not understanding. In math, understanding comes from a mastery of the basic identities or "rules". But reform math does not like rules. They want to (somehow) create understanding without rules or explicit procedures. That's why kids have to look for patterns or apply guess and check to solve word problems.
The power of math lies in its rules - definitions, axioms, theorems, corollaries, identities, etc.
" 'Just give me the rule' they demand,"
Yes! Give them the rule. Make sure they understand it. Make sure they practice it. Make sure they know how all of the rules (and there are very few of them) work together. Give them the power of math, not patterns and guess and check.
"For some reason, despite limited work on finding area, the kids have to find volume and then jump right into Euler's Formula."
Do they actually talk about Euler's Formula?
Vertices - Edges + Faces = 2
Are they trying to get the kids to discover this? Do they do this so that they can claim that the kids learn something about topology? Do they do this because it's not outside the realm of possibility that someone might happen to discover this relationship. Wow! that's interesting! Duh! What do you use it for? Well, fifth graders, if you major in math in college, you might see it in a course on topology. You could get a job at a solid modeling company.
How many EM students could ever get there?
Sometimes the highest-ability kid teaches the group what they've "discovered" but sometimes it is the bossiest kid, or the loudest pushiest kid. If a group is to come up with the answer, a mild-mannered brain can be completely intimidated by other group members.
Finally, the calculator is promoted to ultimate decision-maker and is given more authority in an EM classroom than the teacher.
I don't believe in this but here is what I understand is the reason for cooperative groups:
1. Teach students to work together. When they grow up and work at GM as scientists and engineers, they will need to develop products in a group manner.
2. Teach social skills. Students, starting at a very early age (K?, 1st?) will need to learn dispute strategies. Putting them in cooperative groups helps a teacher teach students how to deal with bossiness, anger, etc.
3. The groups are generally not grouped according to abilities. This is so that the lower performer can learn from higher performers and higher performers can learn from lower performers. This one is dicey to me but I suspect higher performers learn patience and leadership skills?
Again, if I write one more paper on this I will scream. Oh wait, I think I have about 10 more to write on cooperative groups alone. Doh.
dickey45,
Did you see that study where they observed the differences between boys and girls in building a bridge (or something like that, I can't remember)? What was interesting was that the boys wanted to work alone at first. Once they saw the other's work that they liked they started asking about their ideas so that they could improve on their own project. The girls were totally different.
Now, I'm sure there are girls that are competitive loners and boys who collaborate more, but the idea that one way is superior might need to be questioned.
My son ends up in a lot of these groups and it drives him nuts because he always feels like he has to "teach" to the others when he would rather learn.
PS--for all you EM parents, take a look at this Edspresso post (if you haven't already seen it) from this parent. She clearly feels your pain.
http://www.edspresso.com/2007/05/im_failing_first_grade_elena_b.htm
1. Engineers work in groups. They don't do group work.
2. Kids get lots of practice with social skills without a steady diet of group work.
3. Not only do they have very mixed ability kids in the same classroom, they expect them to work together on the same material. It's one thing to develop an appreciation for all levels of learners, but quite another to learn together.
These goals require some very fuzzy ideas of academics. They have to un-link thinking and understanding from content and mastery.
I loved the edspresso article, especially how counting money initiated her smoking habit. I think it increased my gray hair and made me seriously consider nightly drinking, though my current first grader figured it out right away. My oldest has never recovered from 1st grade EM and still has counting money IEP goals in 6th grade. I have flashbacks when my current 1st grader brings home some of those home links and can remember lots of tears and door slamming.
What profession does group work? Can't think of any.
I've been part of many a "team" but typically members have discrete areas of responsibility for a larger project. No one teaches the rest of the group. If you can't make it on your own merits, you might find yourself looking for a new profession.
I like those homeschooling studies that show homeschoolers are as well, or better, adjusted socially than their peers. Hmm.
If we could get those homeschoolers to do more group work, then maybe the public schools could narrown that gap.
"As for standardized tests, your students should do well on these tests without knowing the conventional algorithms. Perhaps all they need is some familiarity with particular notation and vocabulary."
Last night my husband was reviewing math concepts for the SOL and noticed he something.
There are still small gaps in his math knowledge. He had no concept of overestimate and underestimate...rounding off.
He became so bogged down in trying to "solve" the problem that he missed what the question was asking him to do. He has learned, however, to become a great guesser. He has adapated.
It was obvious from the math review last night that there was no familiarity with a "particular notation and vocabulary" for my son.
He will do fine on the state test, but these basic skill gaps are worrisome. TERC is not good for him.
I would like to add that when I talked with my third grader's teacher about TERC and the SOL test, she said this sort of math would not benefit students on the test. She would have to supplement.
"No worries. He will never get there by the end of the year!"
Wait! My son started counting boxes to find volume last night. I think it was in unit 9. One of the issues I have with EM is that the kids just bring tear-out sheets home for homework in their folder. We have the reference book at home (not helpful), but I want to see the two workbooks. I want to look ahead and behind. I also want to see which lessons the teacher is skipping!!!
If you look at EM worksheets out of context, they never look too bad (except for ones like finding patterns for adding fractions). The problems are the rapid spiraling, important topics are only briefly covered, and the lack of practice. It all looks like math, but it won't get the job done. I know math and what is needed.
Then I think about spelling, vocabulary, reading comprehension, and writing. I don't know what is needed. I look at what the school does and it looks fine on a day-to-day basis, but I can't tell if they are getting the job done. (Actually, I don't even know what they are doing and I think they want it that way.) I don't think I could say what the job is. If they can't get math right, then that makes me concerned about what they're doing in other areas.
Outside of spelling and vocabulary, it seems like they are doing ... well, I don't know what they are doing. They read some books that few fifth-grade boys would like. They have reading circles, write in journals, and do peer reviews. What I don't see is an effort to really work on reading comprehension and writing. I don't see ANY of his writing edited or reviewed by his teacher. They do stuff without any specific expectations.
I guess I need to do my homework.
Susan, that sounds like a great study. I'll try to find it. Reminds me of an NPR story a few weeks back on the # of boys entering college. They are going down, below girls. Then I thought, huh, usually strong math skills are an indicator of college ability (or something like that). Well if you change the WAY in which kids learn math at an early age, might you be messing with the pathway to college for boys? So you stick boys in math groups to "talk" about how to do math and "write" about it. Both are NOT their strength. Are we setting them up for failure?
Steve, that you for your points. I will take them to heart. Remember, I'm only regurgitating and I might put my own spin or understanding on it. I happen to work in groups where we design a database structure on a white board or web forms on a white board. But once that's done, we divy up the work and go back to our cubicles.
I think some of you guys definitely have the motivation (maybe not the time) to design a good curriculum that is openly available to the public. I you are all in CA, it would be easy to write it against those well defined CA standards. And it would be fun. Wiki anyone? I have some curriculum you guys could look at (I haven't seen Saxon or Singapore).
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