“Students failing algebra rarely recover”

Jill Tucker

Published 10:22 pm, Friday, November 30, 2012

If student’s don’t get algebra the first time, repeating it doesn’t work. The article specifically says teaching it again the same way doesn’t work.

The researchers challenged school officials to rethink when and how students are assigned to certain math classes and come up with new ways to teach the topics, especially to those who didn’t get them the first time around.

Take students out of the standard algebra curriculum and put into alternative logical foundations. They don’t get what algebra is about in the current approach. Current algebra teaching is procedural and it doesn’t form memories they can retain and use.

Current teaching in math is mostly memorization.

There are large gaps in the concepts. These gaps were filled in 19th century New Math of Boole, Grassmann, Dedekind, Peano.

However, this good new math was dropped in favor of an incomplete version that is mostly useless. 19th century new math was invented for arithmetic’s logic. this part is ignored in teaching new math and common core.

Common core takes Euler 1765 algebra and sprinkles a few words from set theory that have nothing to do and so are just ignored. Set, relation and function are not used in the way that Dedekind does. Is addition a function? Yes. What function is it? Can you write it in function notation?

x+y

f(x,0) = x

f(x,y’) = f(x,y)’

From this we can prove

f(0,y) = y

f(x’,y) = f(x,y)’

Are you completely lost? Think about it. Addition as a function is something you never heard of and never saw written before. However, this was developed in the 19th century to fill the logical gap in what addition is. yet modern teachers and students have never heard of it and are stupefied when they see addition written as a function.

Do you have any idea how to prove the second set of equations from the first set? This is fundamental to what addition is and why it works.

These two equations define addition

f(x,0) = x

f(x,y’) = f(x,y)’

Where ‘ means the next node in a chain from a starting node, 0.

This is counting along the chain. The f function here is the counting on function for the chain. this is addition. Addition is the counting on function for a chain of nodes on a line, i.e. satisfies the 5 linear chain axioms, i.e. the Peano Axioms.

Addition and multiplication of whole numbers are functions for counting along a chain of nodes. They depend on the chain of nodes being linear from a starting point. If we had a different geometry of the chain of nodes, we would get different functions for addition and multiplication by imposing these equations.

Let’s call f(x,y) as S(x,y) now. The S(x,y) function lets you move along the chain of nodes faster or in jumps like a Queen in chess.

The function S(x,y) has properties that make it what we expect of addition. Take S(x,y’) if you increment one of the inputs, the output increments by one. In fact,

S(x,y+z) = S(x,y)+z

If we increment an input by z, the output increments by z.

S(x,S(y,z)) = S(S(x,y),z)

Which functions solve this? Addition does. The addition function, S(x,y) satisfies

S(x,0) = x for all x

S(x,1) = x’ = S(x) for all x

Addition is a way to move along a chain of nodes by more than one node at a time. Addition is equivalent to going one node at a time.

Addition is repeated successor.

Learning to think in these terms is what algebra means and how it ties into arithmetic, a subject you may think you understand, but don’t. Those gaps were filled by Dedekind and others in the 19th century. My materials at this webpage and ebooks fill those gaps and are easier to understand than Dedekind or axiomatic set theory books.

This essay may seem cryptic to most. My materials in my ebooks help make it less cryptic. This is really what is going on with addition of whole numbers. This is what Grassmann, Dedekind and Peano showed when they filled in the logical gaps in Euler’s 1765 algebra textbook.

Yet current teaching in algebra is the same as Euler’s 1765 textbook. All the logical gaps found in the 19th century and filled in are ignored by modern teaching. They just sprinkle in the word set a few times and think this is the same as teaching the careful concepts in the Dedekind 1888 book. It isn’t.

Euler’s 1765 Algebra is a procedural text, i.e. a cookbook. Current math texts for K-12 are the same style of procedural cookbook. They don’t teach concepts and they don’t explain. They pretend to, but don’t. We know that from the Dedekind book and from axiomatic set theory books like Patrick Suppes.

We know some students can’t learn the recipes for algebra problems in current classes. Those classes try to go up a steep learning curve of procedural problems going to polynomial functions and rational functions including factoring.

This is not as important as understanding counting by one and how counting by one leads to the addition function for whole numbers. This is the logic of recursion and induction. This is mathematical thinking. This is what the struggling students need. This gives them insights that give them hooks to remember.

The best students can memorize procedural activities and repeat them on exams. They still don’t understand the underlying logic, but they can pass the procedural exams given. The books and materials are little more than rote memorization. But good students can slog through this amazing barrier of rote memorization material.

The struggling students can’t memorize an algebra book of procedures. So why not try teaching them the concepts of arithmetic using algebra and set theory? Then they might catch on to the meaning of what they are trying to learn.

Procedural math for factoring polynomials is not as important as conceptual math of how addition of whole numbers is defined as a function using recursion. That is indicated above in the equations for f which is renamed S. My materials here and in ebooks do this at a slow pace with many problems. They are the slowest paced set of material to teach this subject. They also have the most examples and worked out problems for Peano Arithmetic.

I tutor secondary-school and college students in Math and was certified in Florida and Massachusetts. I agree with all of your points; in my case, you are preaching to the choir. I would add “Mathematical Reasoning and Plausible Inference, vol. 1” by Georg Polya and Felix Klein’s 3-volume series, “Elementary Mathematics from an Advanced Standpoint,” to your list..

I encounter certain challenges repeatedly; in sum, though, all of them result from the approach that teaching is identified as training. Training is a manufacturing process: raw children in, skilled children out. First, the classroom environment is not conducive to thinking deeply about anything, especially mathematics; the instructional model is, “I do; we do; you do”. Second, most teachers don’t understand anything you’re talking about, i. e., they don’t know any math beyond what they’re teaching and never developed any insight into the nature of mathematics (set theory is understood as the arithmetic of finite sets plus some cute facts about infinity plus, in a few cases, some properties of continuity). They’ve forgotten it because they don’t use it. I’ve encountered no high school teachers who even recognize the names Dedekind, Grassman, or Felix Klein. Some teachers recognize Polya (from a math ed. class), but are unfamiliar with his work. Third, the curriculum is driven by the perceived need to cover a list of so-called core topics. (If there is such a list that makes sense, the current list isn’t it). Thus, there is insufficient time for discovery, mistakes are punished, risk-taking is thereby discouraged. Generally, the wrong outcomes are encouraged and the right ones are ignored. Fourth, I sometimes encounter challenges associated with the child’s development. Some of my students are 12 years old. Some of them struggle with pattern recognition and anchor themselves to the operational approach emotionally. Actually, all of my students do this and I have to work very hard to encourage them to haul anchor and start to explore. Some just aren’t quite ready to make logical (not pattern) connections, yet, but, I keep pulling them forward as long as I can. Finally, their parents learned math the old school way and expect their children to show progress in the same way. This problem is huge.

The dominant objective in tutoring, from the client’s (parents) perspective, is to improve their child’s academic outcomes, i. e., grades and standardized test scores. I have, generally, smart (some are VERY smart) and, at the least, very caring parents whose children want to “succeed” in everything they attempt. A story: Just this week, a parent (who’s a project manager for IBM and whose prior background was writing code, C, C+, C++, etc. for a company IBM acquired) said to me, “I wish he’d use the textbook. It’s really very good. I help him from time to time and I use it to refresh my knowledge of Algebra (I).” She meant that this textbook is a good cookbook and the recipes are usually easy for her to follow. Granted, this book is an “honors” textbook and it has some nice “enrichment” topics. Her assessment, nevertheless, is spot on. It’s a very good source of recipes for solving math first-level math problems. The authors have the focus of this book backward. Discovery and enrichment are “extra”; operations are essential. They fail to realize that operations must and will be learned if discovery is employed, emphasized and encouraged, but, discovery will be discouraged if operations are employed, emphasized and encouraged. Students (like us, their parents) will seek the path of least resistance. Our job is to make the path to deep understanding that path and to create obstacles to operational learning (following recipes). This transition is difficult and emotionally painful, yet, profitable.

Keep up the work. I know, you feel like you’re screaming into a wilderness, but, some hear you. It’s hard to move a large object that’s at rest; it takes a lot of force, at first, then, the mass becomes part of the momentum…:). I had no intention of writing all this, btw, but it just kept coming.

Thanks, your words help.

Thank you for writing this. I’m determined not to have my kids limited to and frustrated by the cookbook, procedural approach to mathematics, but I’m without any sort of roadmap. Currently taking Keith Devlin’s MOOC “Intro to Mathematical Thinking” and searching for deep understanding of arithmetic and algebra (not just procedural application) so as to be the kind of teacher for and with my kids, that I couldn’t otherwise access the services of.

You are welcome. I am glad someone appreciates it. The Devlin course is a good start. Also I have a set of books on Kindle.

I have also started a series of worksheets on Kindle on rules to draw number lines as a way to understand the Peano Axioms in a more approachable way.

http://www.teacherspayteachers.com/Product/Five-Rules-to-Draw-Number-Lines-Introduction-722596

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