August 8, 2018 at 4:08 pm #1080
Miss LacyKeymaster@Miss Lacy
Should I get unmarked or marked rods? Should I use color names or number names for rods? Why should I care?
I thought it might be nice to create a post that explores the benefits of unmarked rods, which is really using color names verses number names for the rods. Should we be getting kids to count first and understand later? Or understand now and count later?
But really, I think this is more of an argument for algebra first. But then that really leads to the more important question. Are young children capable of algebra?
Gattegno saw this argument earlier on. He tackles it by demonstrating the capabilities of babies to master a language and the need for algebra in order to master the language. (what book did he tackle this @mrs-post?)
The argument for marked rods appears to be based on the student needing something to count. Goutard seems to argue that there is danger in using a manipulative for merely concrete counting. (correct me if I am wrong). It keeps the student from moving into the abstract which is where students need to go in order to become efficient in mathematics. Well, that is part of the arguement.
In this post on Instagram today, I saw again the benefits of algebra first. (a side note, I think a marked rod really doesn’t make the difference. I believe a student can do this with marked rods, which is why I really say the argument is really algebra or counting first).
“What’s 100 squared?” And this is the structure I pull out. Who has enough rods to make a 100 square? Not I said the cat. And neither do I have the time to count out 100 squared. So is math counting rods or is it really understanding structures so you don’t have to really “count” rods. Unmarked blocks let you get beyond the tedious counting and move into thinking abstractly. At age nine, Hannah uses what she already knows about the structure of squares and the nature of numbers to figure out 100 square. I say to her, “If white is 100, do these 10 oranges or orange square equal 100 square?” Hannah says, “yes.” I respond, “Okay, then what’s 100 square?” She then proceeds to figure out that one orange rod is a 1000 and quickly determines that then orange square must equal 10,000. Boom, the benefits of unmarked rods. Which is easier to remember, a bunch of number math facts and algorithms or a few key algebraic structures? Which is easier to do? Tediously count rods or use algebra? The problem rests in one’s faith in a child’s ability to work abstractly. Some believe it and others do not. Some do it to see it and believe. Others do not do it and criticize it and thus won’t ever believe. Taste and see that the Lord is good. #algebra #doitdifferent
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Can a young student do this? I don’t know if my comment shows up but I did add that my 6 (almost 7) year old was able to use the very same principles to figure out 1000 squared.
Algebra presented in Gattegno is so general but its like those jumping boards in gymnastics. It gives students the ability to jump higher and higher. No matter the size of the number the principles still apply. I can square 10, 100, 1000, 10000, etc by looking at orange square. All I have to memorize is the nature of squares and the nature of numbers multiplied by 10, 100, 1000, etc.
Of course, there is another reason for color names but that has to do with language. In relation to fractions, I write about it in this post here: https://wp.me/p7Ob7k-ZJ
@mrs-post You probably have a whole lot more to add to this, and I think it would be a great benefit to hear from you on this.August 8, 2018 at 9:36 pm #1086
Mrs. PostKeymaster@Mrs. Post
In a class I am teaching we’ve been working on the distributive property and how that applies to all kinds of math. We explored the principle and then we applied it to a bunch of areas of math and we generalized it. We didn’t look for a formula we created our own. We noted where the problem spots might be and then each person came up with their ideas and then we shared them in class the following week. The algebra wasn’t just when we were the generalization, the algebra was in exploring how the distributive property works and how we can manipulate it when calculating.
If we just do counting first, each new situation requires that we start from the beginning. Starting the the principle first allows the student to make applications in ways you never expected. Thus the spring board thing you mentioned.
My issue with naming them with number names, in the beginning, is that objects are not numbers. Numbers are abstract. While C-rods are concrete, Gattegno and Goutard were against concrete representations. Thus, they do not use number names for the rods. If you use number names, there is a certain intuition that students develop about numbers which is entirely incorrect. Where it shows up is fractions and particularly in the rod race game. I watch parents have to wrap their minds around what is happening and it is mostly because parents automatically associate the white rod with one. But if you play the rod race game for equivalent fractions the train is much longer than a white rod. Yet they represent the same thing. That doesn’t make intuitive sense, but that’s because we game them number names.August 8, 2018 at 11:56 pm #1087
Miss LacyKeymaster@Miss Lacy
If we just do counting first, each new situation requires that we start from the beginning.
Yes, I would agree. When we did the traditional textbook math, I felt like we were starting over each time the situation just varied a little bit. This was so frustrating to me. We would do 30 problems the previous day and nail it. Then the textbook would add some slight variation, and I would get the deer in the headlights look. I had actually wondered if my kid was playing stupid sometimes.
It wasn’t until we started with Gattegno that I found I didn’t have to start over all the time. I only had to point back to a structure and then they were able to say, “oh yeah, I just have to do this.”
And so I find myself talking less and just pointing and asking a couple questions. It is so profound how seeing a structure and asking the right questions can be all the explanation a student needs.August 12, 2018 at 1:31 pm #1088
Mrs. PostKeymaster@Mrs. Post
I was just editing a video on the Substitution Game using polynomials. One of the things we talked about is solving for an unknown and how would we go about doing it. This is interesting to me, because it is a lot like (almost exactly like) the Guess My Rod or Guess My Train Game. Anyway, we talked about is what the polynomial is saying. If 3x = 9 means that three of some number is the same as nine. What would that number be? We could also say then that x = 1/3 (9). Where do kids learn this skill? They learn it in Chapter 3 of Gattegno in kindergarten/1st grade or if they begin later, at whenever they start Gattegno. If a student is attending regular math courses this is taught in a first-semester algebra course, 9 years after a Gattegno student is introduced to it. Because Gattegno always starts with algebra, the kids get the algebra first and it makes all kinds of stuff so much easier. That has been my experience and not only my experience but also the experience of Dick Tahta who was a close associate of Gattegno’s and who used Gattegno’s method’s of teaching and wrote some of the early material. Steven Hawking said that Dick Tahta best teacher he ever had.
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