Jump to content

What's with the ads?


Photo

Samuel Blumenfeld's How to Tutor "New Math has smothered arithmetic"

how to tutor htt

  • Please log in to reply
63 replies to this topic

What's with the ads?

#1 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 26 August 2012 - 07:47 PM

Has anyone used the math in How to Tutor? What do you think of the author's warnings that "New Math has smothered arithmetic". The current wide and conceptual math curriculums are pretty much the same as the old "New Math" right?

"New Math" was designed to produce more scientists for the cold war, and there is the same goal to produce more scientists now due to global competition? Am I oversimplifying this or just wrong?

The author says that arithmetic is the "tool of economic man" and "practical everyday living" and that mathematics is more closely allied with the sciences and philosophy. He says that arithmetic is best taught in isolation and "in a very orderly way" with memorization before understanding.

There are lots of charts that are to be used for copywork. Emphasis is on seeing the patterns in arithmetic. "Unit-grouping exercises" to "strengthen the child's understanding" of the "position in sequence or relationship to other combinations" is emphasized.

The author, Samuel Blumenfeld, says that arithmetic "is one of the most useful tools a child can learn to master" and "is vital to an individual's economic survival and success" and "should be given top priority". "Once the child has mastered the arithmetic system, he'll be in a much better position to deal with the often confusing theories and concepts of New Math."

I learn well from charts, so this curriculum appeals to me. With no printer the copywork and flashcard emphasis is very doable.

Is there anything wrong with STARTING with a narrow focus on arithmetic for below average students that are not going to compete for STEM jobs? What about average and above average students?

#2 regentrude

regentrude

    Qualified Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 14575 posts

Posted 26 August 2012 - 08:01 PM

Has anyone used the math in How to Tutor? What do you think of the author's warnings that "New Math has smothered arithmetic". The current wide and conceptual math curriculums are pretty much the same as the old "New Math" right?


Depends on which curricula you have in mind. Everyday Math and Connected Math, yes, same nonsense. AoPS? Conceptual yes, but absolutely not New Math.

"New Math" was designed to produce more scientists for the cold war, and there is the same goal to produce more scientists now due to global competition? Am I oversimplifying this or just wrong?

No, that was the basic motivation. They just got scared of the success of the Russians.

The author says that arithmetic is the "tool of economic man" and "practical everyday living" and that mathematics is more closely allied with the sciences and philosophy. He says that arithmetic is best taught in isolation and "in a very orderly way" with memorization before understanding.

I agree that arithmetic should be the foundation and needs to be taught before higher math and conceptually more abstract topics. I utterly disagree, however, that it should be taught with memorization before understanding.
It is perfectly possible to teach arithmetic with conceptual understanding, and in fact, this will lead to better retention, because students who know why they do what they do do not have to rely on memory, but will be able to derive anything they have forgotten.
This conceptual understanding becomes more crucial the harder the math is. Students who simply memorize operations with fractions will inevitably make mistakes because they misremember, whereas students who understand the concepts behind the procedures can think critically, can spot mistakes, can evaluate their results and see if they are meaningful. (Typical subject: dividing fractions, The student who understands WHY the divisor fraction is flipped will know exactly when to do this. OTOH, I have seen students without this understanding remember vaguely that they "must flip" and do this for addition and multiplication. Ouch.)

The author, Samuel Blumenfeld, says that arithmetic "is one of the most useful tools a child can learn to master" and "is vital to an individual's economic survival and success" and "should be given top priority". "Once the child has mastered the arithmetic system, he'll be in a much better position to deal with the often confusing theories and concepts of New Math."

Absolutely. Without arithmetic, students can not function in daily life: they need it to budget, balance accounts, figure percentages...
But I would say the confusing New Math curricula should not be used at all in math education.

Is there anything wrong with STARTING with a narrow focus on arithmetic for below average students that are not going to compete for STEM jobs? What about average and above average students?

No. In fact, it is necessary to start like this. Typically, elementary education is almost entirely devoted to arithmetic with positive integers. Only after this has been thoroughly mastered, the student can be taught arithmetic with fractions and negative integers.
Aside from some geometry topics (which are interesting, useful, and fun for students), students study arithmetic until they are ready to move on to algebra. I would not advise skimping on arithmetic mastery for any student, be he below, at, or above, average. But I would encourage the teaching of the "why" alongside the teaching of the "how". This does not require "new math".

Edited by regentrude, 26 August 2012 - 08:05 PM.


#3 Rai B.

Rai B.

    Hive Mind Worker Bee

  • Members
  • Pip
  • 142 posts

Posted 26 August 2012 - 09:06 PM

The current wide and conceptual math curriculums are pretty much the same as the old "New Math" right?

"New Math" was designed to produce more scientists for the cold war, and there is the same goal to produce more scientists now due to global competition? Am I oversimplifying this or just wrong?

Is there anything wrong with STARTING with a narrow focus on arithmetic for below average students that are not going to compete for STEM jobs? What about average and above average students?

Not quite the same. The point of the original New Math was that it was the kind of math mathematicians actually use. It also teaches the math that computer programmers would need to know. The biggest problem with the old New Math programs was that the teacher has to understand that material to teach a New Math program, and most teachers do not. Everyday Math and its ilk are NCTM (National Council of Teachers of Mathematics) based programs (based on curriculum standards established in 1989 and 2000). They intend to teach conceptually, but were not written by mathematicians and the conceptual part is part of the teaching pedagogy (constuctivist) rather than part of the end goal.

Not really oversimplifying otherwise though, the motivations of many people using these programs are that simple.

Go ahead and start with arithmetic.:)

#4 Violet Crown

Violet Crown

    Educating for the Nineteenth Century

  • Members
  • PipPipPip
  • 2957 posts

Posted 26 August 2012 - 09:27 PM

What does it even mean to teach arithmetic without mathematical concepts? When a child is no longer repeating the words "one two three four five" as a parroted chant and is actually counting, he's grasped that each number is exactly one greater than the one before. Each number name maps onto one unique number of objects. One number, zero, counts no things. Numbers apply not just to apples but to oranges and bananas as well. These are all "conceptual mathematics."

When people talk about "New Math," they often seem to mean teaching that makes these concepts explicit at a level appropriate to the child's age. I don't see anything wrong with that. In the Seventies, "New Math" sometimes seemed to refer to a sort of cargo-cult pedagogy whereby the mere vocabulary of mathematics was trusted to make the concepts underlying them apparent: thus endless worksheets that asked students to "Find the union of the set of red fish and the set of blue fish" rather than just to add the red fish and the blue fish together. Silly, but not harmful.

What I do find harmful is deciding in advance which children are capable of understanding mathematics and which will simply be made to memorize math facts. OP, I'm assuming that's not what you meant. But I don't think mathematical comprehension requires extraordinary intellectual gifts, and I do think all children should be given the opportunity to try to understand. Not because we need more people in STEM careers, but because mathematics is the language of the universe, and we all deserve a chance to join in the conversation as we are able.

Edited by Sharon in Austin, 26 August 2012 - 09:30 PM.


#5 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 26 August 2012 - 09:35 PM

regentrude,

Thanks so much for your response to this authors quotes. I found him very interesting, but maybe overly reactionary. Even if overly reactionary, I really like the looks of these charts, and am wondering how much they can be used as a stand alone curriculum, and how much they NEED to be supplemented.

Have you ever seen HTT? How much can a student discover conceptually by copying charts like these? http://www.amazon.co.../ref=pd_sim_b_7 The main book has the answers filled in and the charts are given as copywork, but otherwise this workbook has the same basic charts. They look similar to some Waldorf lesson book pages I have seen.

The author approaches the math, like he does the phonics. Everything is presented as copywork in patterns. In the phonics the explicit information is presented as well. Not quite like Spalding phonographs but close to it. I was thinking I could add explicit instructions to supplement these math charts with what I have learned from Grube's method, the Eclectic Teacher's manual, The African Waldorf pdfs and Professor B.

I have failed with all those conceptual curricula, because there are few/no visuals, and the hands on and recitations instructions just don't let me SEE the organization at a glance. The brain damage I have sustained makes it harder for me to see the big picture. I get lost in the text that is not clearly divided and labeled and illustrated. I understand it while I'm reading it, but a week later it is impossible to apply, unless I reread it ALL again and teach soon after I read it.

When I look at these charts, I feel more confident. and I think gradually I could pencil in conceptual lesson pages numbers from the other curricula, to support these copywork charts. And I'm thinking when I'm feeling most confused I could just resort to copying the charts as good enough, till I get clearer, and still keep everyone on track.

I can sometimes make mental jumps and discoveries from charts. I personally could have learned a LOT of arithmetic from these charts alone as a child, and even now feel like they are strengthening my skills, just by looking at them. It's hard for me to know how helpful these charts are to average and below average students, because they are so powerful for me.

I'm thinking maybe I need to mark up my Professor B books in some way to make them more organized looking. Maybe that will help. I once color coded a text only writing curriculum and it was much easier to teach from after that. But HTT looks effortless for me to teach from. There are so many things I adore about PB, but I feel like I wade through it with my eyes closed. HTT feels the opposite of that, because it's so visual.

#6 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 26 August 2012 - 09:51 PM

What I do find harmful is deciding in advance which children are capable of understanding mathematics and which will simply be made to memorize math facts. OP, I'm assuming that's not what you meant. But I don't think mathematical comprehension requires extraordinary intellectual gifts, and I do think all children should be given the opportunity to try to understand. Not because we need more people in STEM careers, but because mathematics is the language of the universe, and we all deserve a chance to join in the conversation as we are able.


Right now I am tutoring primarily adults, and always remedial for math right now. I asked about clear cases of no chance of STEM, and average, in different sentences, to discuss these groups differently. My 50 year old student that has taken 2 days to fully understand grade 1 science, that penguins do NOT eat bananas, will clearly NOT be going into a STEM career. So I did mean sometimes closing the door to STEM with the hopes of just possibly managing to help a student function better in "practical everyday living", especially when they are clearly already so "behind". Also if arithmetic first in general is harmful, if the wider topics are going to be added LATER. I'd kind of like to talk about different difficult scenarios. HTT is written as a tutoring book, to be used preventively and remedially, but I am also curious about it's broader potential use.

For severely behind students I wonder what would be the most efficient supplement to help them join in the conversation. I do think that is important!!

Edited by Hunter, 26 August 2012 - 09:57 PM.


#7 Violet Crown

Violet Crown

    Educating for the Nineteenth Century

  • Members
  • PipPipPip
  • 2957 posts

Posted 26 August 2012 - 09:56 PM

Ah. I may well have just misunderstood you. I missed that you were talking about older adults.

#8 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 26 August 2012 - 10:14 PM

When students begin to fall further and further "behind" the 8th grade Algebra 1 plan, how much of the math curriculum should be devoted to arithmetic vs mathematics in general?

How much of Blumenfeld's general ideas on math should be separated from the potential use of the math curriculum he wrote?

Is "joining in the conversation" conceptual? Is it exclusive of arithmetic? According to Blumenfeld, "Euclid used no arithmetic or logistic at all and exercised a strict taboo against using it." and "Mathematics was considered the purist form of philosophy, to be studied apart from any consideration for it's practical use."

There is a $9.99 Kindle version of this book.

#9 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 26 August 2012 - 10:18 PM

Ah. I may well have just misunderstood you. I missed that you were talking about older adults.


I'm also curious about larger populations, but there is no immediate worries of me neglecting a child :-) I'm all over the place with my questions, I know. All minors are safe for the MOMENT though :lol: I'm just fascinated with this book. I hope someone can download the book and post some quotes that they like or are driven mad by.

Edited by Hunter, 26 August 2012 - 10:21 PM.


#10 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 26 August 2012 - 10:22 PM

pdf versions of the book can be downloaded here

https://howtotut.ipo...om/paradigm.htm

#11 regentrude

regentrude

    Qualified Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 14575 posts

Posted 26 August 2012 - 10:48 PM

regentrude,

Thanks so much for your response to this authors quotes. I found him very interesting, but maybe overly reactionary. Even if overly reactionary, I really like the looks of these charts, and am wondering how much they can be used as a stand alone curriculum, and how much they NEED to be supplemented.

Have you ever seen HTT? How much can a student discover conceptually by copying charts like these? http://www.amazon.co.../ref=pd_sim_b_7 The main book has the answers filled in and the charts are given as copywork, but otherwise this workbook has the same basic charts. They look similar to some Waldorf lesson book pages I have seen.


I am sorry, but I can not answer your question whether this is going to work. I have never taught elementary math (started homeschooling in 5th grade). Both my kids are gifted, so I have no recollection of their process to learn arithmetic since it was so effortless.
What I think the worksheets need, if they can be used, is the supplementation with visuals - c-rodes, for example. Especially for students with the problems you describe, this seems to be absolutely necessary, in order to translate something they know from experience (one block plus one block makes two blocks) into the language of numerals.
I have never worked with learning disabled students or brain damaged students; this may or may not work. I am also not sure how severe the students' limitations are and what is possible; I have a brother who is unable to do single digit addition.
I hope you get better insights from people with more experience in that area.

#12 regentrude

regentrude

    Qualified Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 14575 posts

Posted 26 August 2012 - 10:58 PM

When students begin to fall further and further "behind" the 8th grade Algebra 1 plan, how much of the math curriculum should be devoted to arithmetic vs mathematics in general?


I do not understand this question. What "mathematics in general" are you talking about? Geometry?
I would say arithmetic should be heavily stressed, since many fields of math can not be pursued without (number theory or probability for example). Geometry, however, should be included. It can help the student develop spatial and abstract concepts. I am not a neurologist, but I believe that studying geometry may be helpful in developing abstract thinking which, in turn, will help in other areas of math. So, I would definitely study geometry alongside arithmetic.

Is "joining in the conversation" conceptual? Is it exclusive of arithmetic? According to Blumenfeld, "Euclid used no arithmetic or logistic at all and exercised a strict taboo against using it." and "Mathematics was considered the purist form of philosophy, to be studied apart from any consideration for it's practical use."

What is "joining in the conversation"? Sorry, I do not understand what it is you are asking.

#13 NASDAQ

NASDAQ

    Hive Mind Royal Larvae

  • Members
  • PipPip
  • 1569 posts

Posted 27 August 2012 - 01:18 AM

I know from other threads that you are not generally a fan of "Asian math," but hear me out here.

We in North America tend to use Singapore Math alone, or almost alone. In Singapore, or Japan, children use it along with a supplemental "cram school" program that parents provide. In math, these programs involve tons and tons and tons of drill. Kumon is one -- the actual program, not the workbook. So those books are used as conceptual _in addition_ to a heavy dose of memorisation. This surely contributes to the time crunch for children in those countries, of course.

I have not read Mr. Blumenfeld's book. I've read his articles and gather he is quite the opinionated fellow -- a man after my own heart!

As to conceptual vs. memorisation, I have a few thoughts, so I'll number them:
1. While "New Math" is a form of concept-first math, I don't think that any conceptual program right now is a perfect reflection of a New Math program.
2. Memorisation allows for more flexibility in presentation. It is much more important to introduce conceptual math at the 'correct' stage. A child can memorise "2+2=4" whether or not he is capable of understanding it, and he can use that information when he _is_ capable of understanding it. It's generally preferable to memorise material one understands, because it's easier, but it's not a lost cause. I think that this would be a big consideration for a teacher, though. It would be exceedingly difficult to catch an entire class of children at the right time for a given concept.
3. Conceptual math is sometimes sold as making math more meaningful to students who are poor in math, but I don't think this works out. The students who gain the most from conceptual math are the ones who have the most math talent.
4. Conceptual math is harder to teach and requires more teacher training.
5. Some-but-not-all conceptual math systems expect a lot of explanations and therefore put weak writers and readers at a further disadvantage. IME this isn't the case with Singapore Math, but is the case with some of the systems that have been used in schools.
6. Conceptual math taught properly to a child who is ready for it can be a lot of fun, whereas memorisation rarely is.
7. Some children so fully understand the concepts that they don't need separate memorisation help.
8. If a child hasn't over-learned the math facts from the previous unit of conceptual math, he or she can easily become stalled during the next unit.

#14 kiwik

kiwik

    Hive Mind Level 4 Worker: Builder Bee

  • Members
  • PipPipPip
  • 3147 posts

Posted 27 August 2012 - 02:56 AM

I did new maths as a child. I was one of the best students in the class and started off liking maths but that didn't last long. IMHO new maths was an experiment in legal mental torture.

#15 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 27 August 2012 - 05:33 AM

Regentrude,

I was referring to Sharon's quote below.

What I do find harmful is deciding in advance which children are capable of understanding mathematics and which will simply be made to memorize math facts. OP, I'm assuming that's not what you meant. But I don't think mathematical comprehension requires extraordinary intellectual gifts, and I do think all children should be given the opportunity to try to understand. Not because we need more people in STEM careers, but because mathematics is the language of the universe, and we all deserve a chance to join in the conversation as we are able.


I was first introduce to this phrase back in the late 90's, when it was advocated that all students, especially marginalized ones, have access to the "Great Books". I'm thinking that many at this board are carrying some aspect of the "Great Books Conversation" over to math, and I'm wanting to tease out which parts of mathematics, that are not covered in arithmetic/logistics, are most essential to "joining in the conversation". I understand this somewhat, and agree with this is theory, but struggle with practical application.

#16 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 27 August 2012 - 05:47 AM

I have never taught elementary math (started homeschooling in 5th grade). Both my kids are gifted, so I have no recollection of their process to learn arithmetic since it was so effortless.


I started afterschooling my gifted rising 4th grader, and pulled him out, a little more than a year later in December of his 5th grade.

Remediating my own brain damage as it pops up, and tutoring ESL and LD adults is NEW to me too, but fascinating. I'm not really equipped to do it, but people stalk me to do it anyway, saying I'm just being stingy about sharing my knowledge, when I try to send themselves elsewhere :001_huh: And I have people who rely on my home library to help their siblings and nieces and nephews with homework.

Because all this fascinates me, and I'm being stalked, I just dive in despite my pitiful qualifications. And then I come here and torment you all with my questions. I'm throwing out quotes and asking questions, but I know I don't really know what I'm talking about. The book fascinated me though. I think if someone else reads it and quotes it, better conversation about it can take place.

Edited by Hunter, 27 August 2012 - 05:50 AM.


#17 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 27 August 2012 - 06:01 AM

I know from other threads that you are not generally a fan of "Asian math," but hear me out here.

We in North America tend to use Singapore Math alone, or almost alone. In Singapore, or Japan, children use it along with a supplemental "cram school" program that parents provide. In math, these programs involve tons and tons and tons of drill. Kumon is one -- the actual program, not the workbook. So those books are used as conceptual _in addition_ to a heavy dose of memorisation. This surely contributes to the time crunch for children in those countries, of course.

I have not read Mr. Blumenfeld's book. I've read his articles and gather he is quite the opinionated fellow -- a man after my own heart!

As to conceptual vs. memorisation, I have a few thoughts, so I'll number them:
1. While "New Math" is a form of concept-first math, I don't think that any conceptual program right now is a perfect reflection of a New Math program.
2. Memorisation allows for more flexibility in presentation. It is much more important to introduce conceptual math at the 'correct' stage. A child can memorise "2+2=4" whether or not he is capable of understanding it, and he can use that information when he _is_ capable of understanding it. It's generally preferable to memorise material one understands, because it's easier, but it's not a lost cause. I think that this would be a big consideration for a teacher, though. It would be exceedingly difficult to catch an entire class of children at the right time for a given concept.
3. Conceptual math is sometimes sold as making math more meaningful to students who are poor in math, but I don't think this works out. The students who gain the most from conceptual math are the ones who have the most math talent.
4. Conceptual math is harder to teach and requires more teacher training.
5. Some-but-not-all conceptual math systems expect a lot of explanations and therefore put weak writers and readers at a further disadvantage. IME this isn't the case with Singapore Math, but is the case with some of the systems that have been used in schools.
6. Conceptual math taught properly to a child who is ready for it can be a lot of fun, whereas memorisation rarely is.
7. Some children so fully understand the concepts that they don't need separate memorisation help.
8. If a child hasn't over-learned the math facts from the previous unit of conceptual math, he or she can easily become stalled during the next unit.


It's not so much that I'm against Asian maths, as that I think there are PROBABLY better SIMILAR methods I want to explore FIRST, especially with my CURRENT student population. I listen to all of you WAY more than you think when you talk about them.

Your thought are some things that I've been discovering, and I don't think I've done a good job articulating them, or figuring out how to address them. I'm going to purposely NOT respond to them for now and I hope others will, so I can just LISTEN.

Edited by Hunter, 27 August 2012 - 06:04 AM.


#18 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 27 August 2012 - 04:06 PM

3. Conceptual math is sometimes sold as making math more meaningful to students who are poor in math, but I don't think this works out. The students who gain the most from conceptual math are the ones who have the most math talent.


Have others observed this too?

My observations have been that struggling students often cannot understand conceptual work until AFTER they have done some cookbook math. It's kind of like being taught all about the science of yeast reproduction, without ever having seen bread, never mind handled raw dough.

#19 lorisuewho

lorisuewho

    Luke 12:48b

  • Members
  • PipPip
  • 1540 posts

Posted 09 September 2012 - 01:24 PM

We in North America tend to use Singapore Math alone, or almost alone. In Singapore, or Japan, children use it along with a supplemental "cram school" program that parents provide. In math, these programs involve tons and tons and tons of drill. Kumon is one -- the actual program, not the workbook. So those books are used as conceptual _in addition_ to a heavy dose of memorisation. This surely contributes to the time crunch for children in those countries, of course.


I know this is an older thread, but I was only recently linked to it today. I missed it the first time around.

I did not know the above information, but I think that is what draws some of us homeschoolers to using more than one math curriculum. I know this is exactly why we have found happiness in the combination of a conceptual math program and an Amish math program. They both bring strengths to the table.

#20 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 09 September 2012 - 03:28 PM

Lori, I really hope you buy this book. The controversial math section fascinates me.

I've started teaching arithmetic and mathematics as 2 different subjects, and so far...I think it might work.

#21 lorisuewho

lorisuewho

    Luke 12:48b

  • Members
  • PipPip
  • 1540 posts

Posted 09 September 2012 - 03:46 PM

Lori, I really hope you buy this book. The controversial math section fascinates me.

I've started teaching arithmetic and mathematics as 2 different subjects, and so far...I think it might work.


I can't wait to read it because I *think* that is what I am doing this year but I never had a word for it. I just felt there were two streams feeding the river we call math.

I'll let you know as soon as it gets here. With used books you never know how long it will take.

#22 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 09 September 2012 - 04:43 PM

:hurray:

It's so exciting to know someone was already on the path, without even reading the book yet. See if your library has it, if you think it'll take awhile. Most have it. My used books have been taking FOREVER to arrive, lately. The library has been great though :001_huh:

Even if someone doesn't USE the book to TEACH from, this is a must read homeschooling book.

#23 Amy Jo

Amy Jo

    Hive Mind Queen Bee

  • Members
  • PipPip
  • 2110 posts

Posted 10 September 2012 - 12:20 AM

If you go to the amazon link and "look inside" the book, you can search "unit-grouping" and see the charts Hunter is referring to. The charts look great - I'm going to try them out on my oldest - writing seems to help him. I might have to get this book. Which I can't afford lol - y'all are bad for my budget! :lol: (And no, my library doesn't have it - [sarcasm]what a shock![/sarcasm])

Sadly, I can't answer the main question as I'm not sure what "new math" is, exactly. But I do agree with Hunter that memorization has a place. My neighbor is struggling to get her GED (some LDs & difficult family situations led to her dropping out of school many years ago). She has a horrible time with math. And it's not like she doesn't try! Should she memorize first, or should she go back to concepts first? I don't know. But I think it's clear there is no one-size solution.

I've also been thinking of separating math into 'arithmetic' (Practical Arithmetic, drill & maybe some math copywork) and 'mathematics' (MEP, games, things like Hands-on Equations & Calculus for Young People, living math books).

#24 lorisuewho

lorisuewho

    Luke 12:48b

  • Members
  • PipPip
  • 1540 posts

Posted 10 September 2012 - 05:29 AM

I've also been thinking of separating math into 'arithmetic' (Practical Arithmetic, drill & maybe some math copywork) and 'mathematics' (MEP, games, things like Hands-on Equations & Calculus for Young People, living math books).


Hunter, does the book give terminology to this? Would we call it "arithmetic" and "mathematics"? Because this sounds like what I'm doing.

Edited by lorisuewho, 10 September 2012 - 06:15 AM.


#25 54879525

54879525

    Beekeeping Professor

  • Members
  • PipPipPipPipPipPip
  • 55102 posts

Posted 10 September 2012 - 06:12 AM

What does it even mean to teach arithmetic without mathematical concepts?


It means you end up like me in the end. Sucky in math.

I was taught only memorization of math facts and then thrown into algebra. It felt like being thrown into the middle of the Bearing Sea and told "ok start swimming". Awful!

I've managed so far to relearn a lot of what I didn't along with my son. It's like for the first time I understand what the heck is going.

I don't know what "new math" is so I can't comment on that part.

#26 NilaGroveSchool

NilaGroveSchool

    Hive Mind Larvae

  • Members
  • 28 posts

Posted 10 September 2012 - 08:17 AM

Hunter,

I have the How to Tutor book. It was initially published in 1973 and in my opinion is a real classic. What a practical book! One of the subtitles is The Art of Tutoring. He discusses the one-to-one relationship involved in tutoring and gives many considerably helpful tips.

The math section is refreshingly logical! I used some of the "charts" from How to Tutor to aid my child in memorization and found that it helped her to cement the math facts.

My dd, age 7, "strongly dislikes" games (the kind you win or lose, or if they are timed :001_smile:) to teach math facts, especially those on the computer. The charts and tips from the How to Tutor math section assist in getting the repetitive and therefore boring part of math memorized. My ds who is now 23, on the other hand happily learned all his math facts when he was in Kindergarten to 3rd grade on Math Blaster, a computer game which may no longer be around.

I used to treat our main curriculum, MEP maths, as the most important part of math, however this year my intuition is saying that the memorization of facts is the most important part, so she does exercises in "arithmetic" before we do the MEP lesson. When the math facts are etched in the brain, in black and white, then we have a "fun" MEP math lesson.

We are in year 3 of MEP and I would classify my dd as average to above average in math? My husband and I have degrees in engineering and we understand and enjoy math...

My ds, 23 graduated from an exemplary (or whatever they call it), public school, I was rarely impressed with the math he was being taught and "trusted" the system, even though my intuition said something was missing (his schooling was easier than my own and more confusing all at the same time, plus it lacked any "meat")

I think it is important for all to learn "arithmetic".

#27 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 10 September 2012 - 08:40 AM

Hunter, does the book give terminology to this? Would we call it "arithmetic" and "mathematics"? Because this sounds like what I'm doing.


HTT says the Greeks called arithmetic logistics. HTT says that arithmetic is the art of counting and calculation, and the tool of commerce, and outside the domain of science and philosophy.

HTT says that mathematics deals with abstractions of numbers, relationships of geometric forms, most of which can be discussed rhetorically without the use of arithmetic. The Greeks considered it the highest form of philosophy. "Euclid exercised a strict taboo against using it."

In Language Arts, it's standard for many of us to teach explicit phonics separate from literature. I'm wondering why it's not common to do something similar in math.

Edited by Hunter, 10 September 2012 - 08:43 AM.


#28 wapiti

wapiti

    Apprentice Bee Keeper

  • Members
  • PipPipPipPipPip
  • 8405 posts

Posted 10 September 2012 - 08:43 AM

On terminology, FWIW, "Fuzzy Math" and "New Math" are completely different things.

"New Math" = 1960s/early 1970s, e.g. Dolciani

"New New Math" = "Fuzzy Math" = circa 1989 NCTM, e.g. Everyday Math (I think this was in development in the mid-80s?), TERC, Connected Math, etc.

I'm not completely certain, but I'd guess that emphasis on memorization and procedures would fall under "Traditional Math", pre-New Math

On the OP, I would suggest teaching to the person's strengths, without ultimately neglecting the other side of the equation (concepts vs memorization/procedures). Some people are strong with rote memorization and sequences, but weaker on understanding mathematical concepts. Some people are the opposite, weak with rote memorization and sequencing, but stronger on concepts, and for these people I think it is particularly important to emphasize the concept and learning through practice rather than through rote. (Others may be weak at both or strong at both.) That's my completely uncaffeinated two cents - I so need some coffee :tongue_smilie:

#29 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 10 September 2012 - 08:46 AM

New Math was a cold war experiment, that didn't work out as well as hoped in the USA.

#30 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 10 September 2012 - 08:55 AM

One thing I don't understand. Professor B and HTT have a HUGE overlap. Professor B claims to be conceptual. HTT does not. I wonder if the the authors just have different ideas of "conceptual". If Professor B is a reaction to "back to basics", and HTT is a reaction to "New Math", maybe they both are middle of the road and the same thing?

#31 lorisuewho

lorisuewho

    Luke 12:48b

  • Members
  • PipPip
  • 1540 posts

Posted 10 September 2012 - 09:04 AM

In Language Arts, it's standard for many of us to teach explicit phonics separate from literature. I'm wondering why it's not common to do something similar in math.


I love this analogy. I guess the question becomes if phonicsis the building blocks of literature/reading, what are the building blocks for "math"? Are the building blocks arithmetic or are they the concepts?

#32 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 10 September 2012 - 09:11 AM

I love this analogy. I guess the question becomes if phonicsis the building blocks of literature/reading, what are the building blocks for "math"? Are the building blocks arithmetic or are they the concepts?


HTT is an attempt to isolate the foundational building blocks of the 3R's. I guess Samuel Blumenfeld believes that the base 10 arithmetic charts are equivalent to the phonics charts that are in the earlier part of his book.

Edited by Hunter, 10 September 2012 - 09:15 AM.


#33 Spy Car

Spy Car

    Beekeeping Professor

  • Members
  • PipPipPipPipPipPip
  • 18047 posts

Posted 10 September 2012 - 09:26 AM

The guy has it all wrong. Teaching math (arithmetic included) without teaching for understanding is an incredibly shallow way to teach. Just memorizing charts of "math facts" is like just memorizing sight-words. Bad practice.

I do not understand why ideas like this get any traction.

Bill

#34 wapiti

wapiti

    Apprentice Bee Keeper

  • Members
  • PipPipPipPipPip
  • 8405 posts

Posted 10 September 2012 - 09:42 AM

I love this analogy. I guess the question becomes if phonicsis the building blocks of literature/reading, what are the building blocks for "math"? Are the building blocks arithmetic or are they the concepts?


IMO phonics are not the building blocks of literature, because phonemes by themselves have no meaning (unlike words, which at least represent something). In contrast, a numeral by itself does have meaning, representing an actual quantity, which itself is a concept. I don't believe the analogy between language and math works. Math and logic? Yes, absolutely. Math and grammar? Possibly, but the basic unit would be a word with an understood meaning. Math and literature? No, I don't think so.

Reading as a "code" is another way to look at it, but still, phonemes have no meaning. Only the sequence has meaning.

Enough of my pre-coffee babbling...

#35 lorisuewho

lorisuewho

    Luke 12:48b

  • Members
  • PipPip
  • 1540 posts

Posted 10 September 2012 - 10:30 AM

IMO phonics are not the building blocks of literature, because phonemes by themselves have no meaning (unlike words, which at least represent something). In contrast, a numeral by itself does have meaning, representing an actual quantity, which itself is a concept. I don't believe the analogy between language and math works. Math and logic? Yes, absolutely. Math and grammar? Possibly, but the basic unit would be a word with an understood meaning. Math and literature? No, I don't think so.

Reading as a "code" is another way to look at it, but still, phonemes have no meaning. Only the sequence has meaning.

Enough of my pre-coffee babbling...


I don't think this is babbling :001_smile:.
I think this is important (in a philosophical way anyway) to think about these things. We can only improve as teachers if we spend some amount of time discussing how we teach and how we learn and not just which curriculum is best.
I need to think on this idea of basic units of understanding more.

#36 lorisuewho

lorisuewho

    Luke 12:48b

  • Members
  • PipPip
  • 1540 posts

Posted 10 September 2012 - 10:31 AM

The guy has it all wrong. Teaching math (arithmetic included) without teaching for understanding is an incredibly shallow way to teach. Just memorizing charts of "math facts" is like just memorizing sight-words. Bad practice.

I do not understand why ideas like this get any traction.

Bill


What if we don't liken it to memorizing sight words but to memorizing phonemes or letter sounds?

#37 boscopup

boscopup

    Qualified Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11338 posts

Posted 10 September 2012 - 10:42 AM

What if we don't liken it to memorizing sight words but to memorizing phonemes or letter sounds?


I liken letter sounds/phonemes to learning that '3' is * * *.

#38 Spy Car

Spy Car

    Beekeeping Professor

  • Members
  • PipPipPipPipPipPip
  • 18047 posts

Posted 10 September 2012 - 10:43 AM

What if we don't liken it to memorizing sight words but to memorizing phonemes or letter sounds?


That is just a rationalization for not teaching for understanding. One either teach a subject—any subject—for understanding, or one does not.

If you don't teach mathematics for understanding it will eventually bite your children (or adult learners), as it is not a discipline where one can get by on memorization.

Bill

#39 wapiti

wapiti

    Apprentice Bee Keeper

  • Members
  • PipPipPipPipPip
  • 8405 posts

Posted 10 September 2012 - 10:46 AM

What if we don't liken it to memorizing sight words but to memorizing phonemes or letter sounds?


This is what I'm talking about though - letter sounds don't represent any meaning, in contrast to numerals. You can visualize and even manipulate and perform operations with a quantity without even using the numeral (number name), but you can't do that with a phoneme - it has no inherent meaning. It's just sounds that correspond to letters. A phoneme is "made up," whereas a quantity is a real meaning.

This is the reason that persons with certain groups of strengths and weaknesses (e.g., dyslexics) may have a much more difficult time with phonics - that which has no meaning outside of its sequence in relation to other items of the same type is much more difficult to remember because there is no contextual meaning. The "context" of a phoneme is simply its place in the sequence. Sequential weaknesses vs. spatial strengths. In contrast, quantities can have spatial context and can be visualized - they have an essential meaning. Am I talking in circles? :tongue_smilie:

#40 lorisuewho

lorisuewho

    Luke 12:48b

  • Members
  • PipPip
  • 1540 posts

Posted 10 September 2012 - 10:54 AM

I liken letter sounds/phonemes to learning that '3' is * * *.


This makes sense.

That is just a rationalization for not teaching for understanding. One either teach a subject—any subject—for understanding, or one does not.

If you don't teach mathematics for understanding it will eventually bite your children (or adult learners), as it is not a discipline where one can get by on memorization.

Bill


I agree that meaning has to be taught. I don't disagree with this. I don't know what the HTT books says because I haven't read any of it yet.

However, the question that I am asking myself is is there a vein of "mathematics" that should focus on memorization and not just on conceptual understanding. Does conceptual understanding always lead to speed and accuracy in computation? I don't know.

This is the reason that persons with certain groups of strengths and weaknesses (e.g., dyslexics) may have a much more difficult time with phonics - that which has no meaning outside of its sequence in relation to other items of the same type is much more difficult to remember because there is no contextual meaning. The "context" of a phoneme is simply its place in the sequence. Sequential weaknesses vs. spatial strengths. In contrast, quantities can have spatial context and can be visualized - they have an essential meaning. Am I talking in circles? :tongue_smilie:


I understand what you are saying about the phoneme itself holding no inherent meaning beyond the sound and that it needs to be put in context to have actual meaning.

#41 boscopup

boscopup

    Qualified Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11338 posts

Posted 10 September 2012 - 11:36 AM

I think some kids will need to practice facts more in order to learn them.

So far, my two school aged kids have picked up addition/subtraction facts without any drill. They are very different learners. My oldest picks up everything easily. Second child thinks in pictures and has some dyslexic tendencies. C-rods have given him a visual representation of the facts, and he's picking them up very easily. Multiplication may be more difficult. We'll see. My oldest needed to drill those facts a little bit.

Now both of these kids are mathy, and like I said, some kids will need more practice and drill. And I can see some kids doing better memorizing first and then learning ”why”, the same way some kids do better learning sight words first and then phonics. I personally would use that method only for a child that isn't understanding the concepts. I prefer to start with conceptual, falling back to memorization if the kid just isn't ready for the conceptual yet.

#42 wapiti

wapiti

    Apprentice Bee Keeper

  • Members
  • PipPipPipPipPip
  • 8405 posts

Posted 10 September 2012 - 11:40 AM

However, the question that I am asking myself is is there a vein of "mathematics" that should focus on memorization and not just on conceptual understanding. Does conceptual understanding always lead to speed and accuracy in computation? I don't know.


Proficiency with algorithms and quick recall of "math facts" are clearly important, and these are neglected in fuzzy math programs. There needs to be both procedural competence and conceptual understanding.

How best to become proficient with algorithmic procedures and how to gain quick recall of math facts is a different question. IMO, in most situations (outside of certain uncommon learning disabilities/issues), practice using the concept (especially in unique ways, in addition to more straightforward problems) is a better way to commit a procedure/math fact to long-term memory than rote memorization.

Along these lines, take, for example, Rusczyk's perspective (albeit obviously a super-mathy person):

At another contest later that summer, a younger student, Alex, from another school asked me for my formula sheets. In my local and state circles, students’ formula sheets were the source of knowledge, the source of power that fueled the top students and the top schools. They were studied, memorized, revered. But most of all, they were not shared. But when Alex asked for my formula sheets I remembered my experience at MOP and I realized that formula sheets are not really math. Memorizing formulas is no more mathematics than memorizing dates is history or memorizing spelling words is literature. I gave him the formula sheets. (Alex must later have learned also that the formula sheets were fool’s gold - he became a Rhodes scholar.)

The difference between MOP and many of these state and local contests I participated in was the difference between problem solving and what many people call mathematics. For these people, math is a series of tricks to use on a series of specific problems. Trick A is for Problem A, Trick B for Problem B, and so on. In this vein, school can become a routine of ‘learn tricks for a week - use tricks on a test - forget most tricks quickly.’ The tricks get forgotten quickly primarily because there are so many of them, and also because the students don’t see how these ‘tricks’ are just extensions of a few basic principles.



I had painfully learned at MOP that true mathematics is not a process of memorizing formulas and applying them to problems tailor-made for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems.


While the people involved in this quote are not average students, one could apply this idea to average students. (Below-average students are another matter, though I'd prefer to give them the benefit of the doubt whenever possible.) What are our goals in teaching mathematics? To the extent that memorization is taught, is that truly mathematics or is it something else (a means to an end, obviously, but what is the end)?




On another note, and this may be a little controversial, where a student actually understands (not just refreshes their memory) a concept only through performing a procedure over and over again (understanding through repetition, e.g. the philosophy behind Saxon), I question the quality of the initial instruction on the concept. Pattern-seers might find this useful (and I might imagine situations where concept could be inadvertently, implicitly "discovered" in drill), though I still question whether they just follow a pattern that they might forget vs actually understanding the concept, and I'd rather see explicit discussion of concepts at some point, before during or after the drill or memorization.


Way too much thinking out loud. Need another cup of coffee.


Edited by wapiti, 10 September 2012 - 11:42 AM.


#43 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 10 September 2012 - 12:40 PM

I'm getting :confused1:. We are talking about so many things here.

I think different authors are using vocabulary differently, and especially when they are REACTING to a situation that has gotten far off balance. I don't think HTT is advocating not using concepts, but is more reacting to the overwhelming VOLUME of ADVANCED concepts and vocabulary being throws at children who didn't know that 3+1=4. Professor B is reacting to NO concepts. I think they are actually teaching the same thing even though their introductions seem to be saying the opposite.

Separating arithmetic from the other mathematics is not the same as not teaching it conceptually. Mathematics does NOT = concepts. I think recently we have gotten WIDE curricula mixed up with CONCEPTUAL curricula. Professor B is narrow, but conceptual. Other than PB and the vintage curricula, the conceptual curricula tend to be wide, right now.

I think HTT is conceptual even though it doesn't claim to be. It's most of Professor B, but with copywork and flashcards, instead of recitation and fingerwork. I don't think less pencil work means conceptual. HTT is about unburying the phonics and arithmetic from the WHOLE, so that the student can SEE the system and logic.

Phonics are not the building blocks of literature. I agree. I think they are two different strands, just like Euclid said logistics(arithmetic) is different from mathematics. It's funny that whole language arts is attacked here, but whole math is the standard. When Sam Blumenfeld wrote the book, phonics was as neglected as arithmetic, and maybe more so.

Edited by Hunter, 10 September 2012 - 12:54 PM.


#44 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 10 September 2012 - 12:57 PM

I can't wait until people smarter than me READ the book.

#45 Bluegoat

Bluegoat

    Hive Mind Queen Bee

  • Members
  • PipPip
  • 1652 posts

Posted 10 September 2012 - 01:26 PM

I agree that arithmetic should be the foundation and needs to be taught before higher math and conceptually more abstract topics. I utterly disagree, however, that it should be taught with memorization before understanding.
It is perfectly possible to teach arithmetic with conceptual understanding, and in fact, this will lead to better retention, because students who know why they do what they do do not have to rely on memory, but will be able to derive anything they have forgotten.


I don't think this is just a matter of helping when kids forget either. I have a very hard time forming memories of things without a conceptual framework, it takes me forever and does not stick. I think I am probably a visual spacial learner, and my memory of things is architectural - anything that doesn't have an architecture gets lost.

Any students like me who are taught facts without context or algorithms without concepts are unlikely to remember them for long.

#46 Bluegoat

Bluegoat

    Hive Mind Queen Bee

  • Members
  • PipPip
  • 1652 posts

Posted 10 September 2012 - 01:40 PM

This is what I'm talking about though - letter sounds don't represent any meaning, in contrast to numerals. You can visualize and even manipulate and perform operations with a quantity without even using the numeral (number name), but you can't do that with a phoneme - it has no inherent meaning. It's just sounds that correspond to letters. A phoneme is "made up," whereas a quantity is a real meaning.



I think you are confusing things by saying "letter sounds". Letters and sounds are different. The letter "s" represents a sound, a concrete thing, used in one way or another. The sound can be used alone, say to imitate a snake, (or maybe the snake uses it to express his nature?) or we can use it with other words to make an expression, like "similitude".

The number on paper or word for the number is a symbol. A "3" represents something concrete, a quantity of some kind. It can be used alone to represent a particular quantity (say three birds) or as part of a larger expression (say 3 +3 meaning six birds.)

Or to be more concise, a representing a sound is a meaning.

#47 regentrude

regentrude

    Qualified Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 14575 posts

Posted 10 September 2012 - 02:05 PM

Separating arithmetic from the other mathematics is not the same as not teaching it conceptually. Mathematics does NOT = concepts.


:confused:
Mathematics is ALL about concepts.
Without concepts, no arithmetic is possible.
What is "the other mathematics"?

#48 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 10 September 2012 - 02:53 PM

:confused:
Mathematics is ALL about concepts.
Without concepts, no arithmetic is possible.
What is "the other mathematics"?


What I mean is that mathematics is not a synonym for concepts.

Euclid was the first one--that I have heard of--to separate logistics(arithmetic) from "mathematics". I wish I knew the Greek word for "other than arithmetic" which he supposedly "exercised a strict taboo about". HTT says "arithmetic and mathematics lead towards two different divergent paths of interest and activity".

Supposedly Euclid started all this, not me :lol:

I'm in way over my head. I'm interested, but WAY over my head.

#49 Hunter

Hunter

    Amateur Bee Keeper

  • Members
  • PipPipPipPipPipPip
  • 11462 posts

Posted 10 September 2012 - 03:04 PM

In teaching the child arithmetic, it is important to convey to him the genius of the system itself. Next to the alphabet, it is the greatest mental tool ever devised by man. Therefore we should approach the subject with the excitement and interest it deserves. Any teacher who makes arithmetic dull does so because he or she does not understand its beautiful simplicity, its logic, and its facility which permit us to do so much with so little. The ten-symbol place-value system is perhaps the diamond of human intellect. It, and the alphabet are the child's greatest intellectual inheritance. Both sets of symbols represent the distilled genius of the human race. It is therefore obvious that such gifts must be presented in such a way as to make the child appreciate what these symbols can do for him in furthering his own potential and happiness.

Blumenfeld, Sam (2011-12-16). How To Tutor (Kindle Locations 1845-1851). Paradigm Company. Kindle Edition.

... students scarcely become aware of the decimal place-value system as a complete arithmetic system quite separate and distinct from the rest of the subject matter in elementary mathematics. The result is that students learn arithmetic very poorly and very haphazardly.

Blumenfeld, Sam (2011-12-16). How To Tutor (Kindle Locations 1854-1857). Paradigm Company. Kindle Edition.

Edited by Hunter, 10 September 2012 - 03:06 PM.


#50 NilaGroveSchool

NilaGroveSchool

    Hive Mind Larvae

  • Members
  • 28 posts

Posted 10 September 2012 - 03:14 PM

In my opinion How to Tutor does approach math / arithmetic conceptually:

pg. 188 bottom paragraph "What is arithmetic? Arithmetic is simply the art of counting. All arithmetic functions (addition, subtraction, multiplication, and division) are merely different ways of counting. In addition we count forward. In subtraction we count backward. In multiplication we count in multiples, which is merely a faster way of counting forward when dealing with great quantities. In division the same principle is applied in the reverse direction." ...the then proceeds to discuss how we historically advanced math with the Hindu place-value system, etc.



Also tagged with one or more of these keywords: how to tutor, htt

What's with the ads?