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% (find-LATEX "2025bad-foundations.tex")
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% (defun C () (interactive) (find-LATEXsh "lualatex 2025bad-foundations.tex" "Success!!!"))
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% (defun o () (interactive) (find-LATEX "2022on-the-missing.tex"))
% (defun u () (interactive) (find-latex-upload-links "2025bad-foundations"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun b () (interactive) (find-LATEX "education.bib"))
% (defun d0 () (interactive) (find-ebuffer "2025bad-foundations.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
% (code-eec-LATEX "2025bad-foundations")
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% (find-sh0 "cp -v ~/LATEX/2025bad-foundations.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2025bad-foundations.pdf /tmp/pen/")
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% file:///home/edrx/LATEX/2025bad-foundations.pdf
% file:///tmp/2025bad-foundations.pdf
% file:///tmp/pen/2025bad-foundations.pdf
% http://anggtwu.net/LATEX/2025bad-foundations.pdf
% (find-LATEX "2019.mk")
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% (find-MM-aula-links "2025bad-foundations" "2" "baf" "baf")
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\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
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%
% (find-LATEX "dednat7-test1.tex")
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%\xyoption{curve} % For the ".curve=" feature in 2D diagrams
%
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% top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
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% «defs» (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
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\def\drafturl{http://anggtwu.net/LATEX/2025-2-C2.pdf}
\def\drafturl{http://anggtwu.net/2025.2-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}
% (find-LATEX "2024-1-C2-carro.tex" "defs-caepro")
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\catcode`\^^J=10
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% «defs-T-and-B» (to ".defs-T-and-B")
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% «defs-caepro» (to ".defs-caepro")
%L dofile "Caepro5.lua" -- (find-angg "LUA/Caepro5.lua" "LaTeX")
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\def\Cahref#1#2{\href{\Caurl{#1}}{#2}}
\def\Ca #1{\Cahref{#1}{#1}}
% «defs-pict2e» (to ".defs-pict2e")
%L dofile "Piecewise2.lua" -- (find-LATEX "Piecewise2.lua")
%L --dofile "Escadas1.lua" -- (find-LATEX "Escadas1.lua")
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\pu
% «defs-V» (to ".defs-V")
%L --- See: (find-angg "LUA/MiniV1.lua" "problem-with-V")
%L V = MiniV
%L v = V.fromab
\pu
% «defs-Verbatim» (to ".defs-Verbatim")
%L dofile "Verbatim3.lua" -- (find-LATEX "Verbatim3.lua")
\pu
% «defs-S» (to ".defs-S")
\input 2025-1-C2-S-defs.tex % (find-LATEX "2025-1-C2-S-defs.tex")
% _____ _ _ _
% |_ _(_) |_| | ___ _ __ __ _ __ _ ___
% | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
% | | | | |_| | __/ | |_) | (_| | (_| | __/
% |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
% |_| |___/
%
% «title» (to ".title")
% (bafp 1 "title")
% (bafa "title")
\title{Bad Foundations \\ and Manipulable Objects}
\author{%
Eduardo Ochs%
%{\large Eduardo Ochs}%
\thanks{eduardoochs@gmail.com}\\
%{\small UFF, Rio das Ostras, RJ, Brasil}\\
}
\maketitle
% «abstract» (to ".abstract")
% Orig: (favp 1 "abstract")
% (fava "abstract")
\begin{abstract}
Imagine a student---let's call him $E$, and make him a ``he''---that
is enrolled in Calculus 2, and who believes that to pass in Maths
courses he only needs to memorize methods and apply them quickly and
without errors. Let's imagine that $E$ is an extreme case of ``bad
foundations'', and that he knows how to solve $x+2=5$ by doing
$x=5-2=3$, but he doesn't know how to substitute the $x$ in $x+2=5$
by 3, and the only way that he knows of ``testing the solution'' is
to apply the same method again and check that he got the same
result.
When we are teaching Calculus to classes that have many students
that are extreme cases of bad foundations we need new strategies and
tools; for example, we can't pretend that ``taking a particular
case'' is an obvious operation anymore---instead we need ways to
make these operations easy to visualize. (Justification)
\end{abstract}
% Para Walter Machado-Pinheiro,
% que não leu e não vai ler
% documento nenhum
\cite{Boaler}
\cite{Hewitt1}
\cite{Hewitt2}
\cite{Hewitt3}
% (find-books "__analysis/__analysis.el" "sfard" "241" "3. Rituals")
% (find-books "__analysis/__analysis.el" "tall-amt" "28" "Most mathematics instruction, from elementary school through college courses,\nteaches what might be called rituals")
% (find-LATEX "2022on-the-missing.tex" "toc")
% (find-es "tex" "tocloft")
\renewcommand{\cfttoctitlefont}{\bfseries}
\setlength{\cftbeforesecskip}{2.5pt}
\tableofcontents
% ;-- degrees
% «degrees» (to ".degrees")
\section{Degrees of bad foundations}
The student E is a case of {\it Extremely Bad Foundations}. The
student E knows, for example, how to solve $2+x=5$ by doing $x=5-2=3$,
but he doesn't know how to substitute the $x$ in $2+x=5$ by 3, and he
believes that the only way to check if 3 is to apply the method --
$x=5-2=3$ -- again, and verify that he got the same answer.
% (find-books "__analysis/__analysis.el" "hewitt-1" "6" "y=2+x")
% lisptreeq(2*3+4*5+6/(7+8));
% lisptreeq(f(a,g(b,c),d));
% ;-- items
% «items» (to ".items")
% (misp 9 "the-conventions")
% (misa "the-conventions")
\section{Items}
\begin{itemize}
\item[(LL)] Learn as least as possible. See:
% (c2m252introp 45 "qca-e-mapa")
% (c2m252introa "qca-e-mapa")
\item[(EC)] Evidence of cheating
% (c2m251introp 42 "nao")
% (c2m251introa "nao")
\item[(OS)] Optimize for speed
% (c2m251introp 20 "semicirculo")
% (c2m251introa "semicirculo")
\item[(OC)] Optimize for clarity
\item[(LR)] Level reduction: see chain rule
% (find-books "__analysis/__analysis.el" "rest-is-algebra" "53" "3.6 Level Reduction")
\item[(JN)] Just notation: the f is ``just notation''
\item[(PM)] Pattern matching: they know how to do it in certain cases
but not in general.
\item[(NF)] New formulas: they don't know how to obtain new formulas
from formulas that they already know.
\item[(JS)] Justification for a step
\item[(SA)] Structured activities
\end{itemize}
\section{The chain rule}
% (c2m251stp 8 "chain-rule-red")
% (c2m251sta "chain-rule-red")
% (c2m251sda "chain-rule-red")
$$\ga{Chain rule (Stewart p.150)}$$
\section{Other}
\subsection{Expandable proofs}
I teach Mathematics ``in a bad campus of a good federal university in
Brazil'' -- see the appendix 1 for what that means -- and the courses
that I teach more often are Calculus 2 and 3 (``C2'' and ``C3'') for
students of Computer Science and Production Engineering. In the last
years the profile of the students arriving in my courses have changed
a lot, and now more than half of my students in C2 are like this:
\begin{itemize}
\item They remember that the chain rule is
$\ddx f(g(x)) = f'(g(x))g'(x)$ and they derive functions like
$\sin 42x$, but they don't have any idea of how to derivate
functions like f(42x) and sin(g(x)),
\item They understand that $\sqrt{a^2 + b^2} = a+b$ is not always
true, but they use steps like $\sqrt{a^2 + b^2} = a+b$ in their
calculations, and when I ask them to tell me what rule that the used
in that step they {\it can't} -- they just say ``oops, sorry, I got
distracted, that won't happen again'', or ``I just followed the
method''.
\item They don't do exercises, don't participate in group discussions,
and don't ask any questions. In the tests a few of them cleary
cheated, and most of the others gave answers that showed that they
understood a few methods from the Calculus book that we used, but
they understand very little high-school algebra -- their answers
were full of ``basic'' errors that they didn't know how to debug.
\item {\sl I don't know how they think}. I've tried several ways to
interact with them, but I've almost always failed miserably. One of
my hypotheses is that they are addicted to ChatGPT and they are
people who lost the notion of ``here and now''; they are mere
spectators in the class, and that "prefer to study at home" using
Youtube and ChatGPT -- but I don't know how true that is.
\end{itemize}
Item (1) means that they don't know how to obtain particular cases,
and don't understand how some rules are consequences of other rules. I
know that they saw some proofs in Calculus 1, but I infer that they
don't know the basic mechanisms behind proofs, and they didn't
understand anything of those proofs.
Item (2) means that they only have a very shallow notion of what is a
valid step in a calculation; they believe that they need to memorize
lots of rules and apply them without errors, and that's it. See
\cite[p.22]{McGowen} ("rules without reason").
% (find-books "__analysis/__analysis.el" "rest-is-algebra" "3" "Sepideh Stewart and Stacy Reeder")
% (find-books "__analysis/__analysis.el" "rest-is-algebra" "5" "lack of sense regarding denotation")
% (find-books "__analysis/__analysis.el" "rest-is-algebra" "19" "Mercedes McGowen")
% (find-books "__analysis/__analysis.el" "rest-is-algebra" "22" "rules without reason")
In the beginning I didn't know what to do with these students. Now I
believe that an approach based on ``manipulable objects'' might work.
I will call the students above ``students with bad foundations''. Some
cases are worse than the others; see \cite[p.5]{StewartReeder}, that
uses the expression ``lack of sense regarding denotation''. I will
refer to the more severe cases as ``students with {\sl very} bad
foundations''.
\subsection{Manipulable objects}
\subsection{Set comprehensions}
My favorite way of presenting variables is by starting by the set
comprehensions in the appendix A, in which all variables are bound and
vary over small finite sets, and all the resulting sets are easy to
draw; this lets us postpone most of the problems in p.6 of
\cite{EllermeijerHeck} (``The meaning of variable is variable in
mathematics''). The sets
$$
\begin{array}{rcl}
B & := & \{(1,3), (1,4), (2,4)\} \\
C & := & \{(1,3), (1,4), (2,4), (2,4)\} \\
\end{array}
$$
``are'' different drawing instructions that yield the same drawing;
this hints at an idea of equality to which we will return later.
The points of a set like
${(x,y) | x∈{1,...,5}, y∈{x,6-x}}$
correspond to the outputs of this program
% https://en.wikipedia.org/wiki/Cooperative_board_game
%V for x=1,5 do
%V for y=x,6-x do
%V print(x,y)
%V end
%V end
%L defvbt "for x for y"
\pu
$$\vbt{for x for y}$$
and they can be calculated by hand by using a tree. Discussions become
easier if we fix a standard way of drawing these trees as tables, and
the two tables below both represent how we calculate the points of the
set comprehension above:
$$[Tbl1] \; [Tbl2]$$
Most students find the the second table easier to follow than the
first one, but the columns in the first table were chosen in an
obvious way, and it is not so easy to explain how we chose the columns
of the second table. This is a nice example of how explanations can be
expanded and contracted.
Set comprehensions are also a good tool for introducing how to make
and test hypotheses. Suppose that a student has found this as the
result of the exercise 2J:
$$[Compr] = [fig]$$
We can rewrite this as:
\begin{tabular}{lll}
Let S2J = ... \\
Let S2J' = ... \\
Hypothesis: S2J = S2J'. \\
Let's test the point bla. \\
$bla ∈ S2J?$ (yes) \\
$bla ∈ S2J'?$ (no) \\
So $S2J != S2J'$. \\
\end{tabular}
The appendix C shows a way to structure that as a game.
% ;-- expressions as trees
% «expressions-as-trees» (to ".expressions-as-trees")
\subsection{Expressions as trees}
% ;-- proofs-with-justifications
% «proofs-with-justifications» (to ".proofs-with-justifications")
\subsection{Proofs with justifications}
For example, a student with very bad foundations can write a
``sqrt(3)=20'' on his test and then show me this when reviewing his
grading,
% (%i) sqrt(3)=20;
% (%o) sqrt(3)=20;
to try to convince me that sqrt(3)=20 is not an error, {\sl because
Maxima said that that is true}.
% ;-- appendix-comprehensions
% «appendix-comprehensions» (to ".appendix-comprehensions")
% (mpgp 8 "comprehension")
% (mpga "comprehension")
\section*{Appendix A: Set Comprehensions}
\cite{RestIsAlgebra}
\cite{McGowen}
%\newpage
% «references» (to ".references")
\printbibliography
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
% «make-with-bib» (to ".make-with-bib")
% (misa "make-arxiv")
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
cd ~/LATEX/
which biber
biber --version
make -f 2019.mk STEM=2025bad-foundations veryclean
lualatex 2025bad-foundations.tex
biber 2025bad-foundations
lualatex 2025bad-foundations.tex
# (find-pdf-page "~/LATEX/2025bad-foundations.pdf")
% (find-pdfpages2-links "~/LATEX/" "2025bad-foundations")
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