# How to make trees in LaTeX

For recent work with Natasha Dobrinen and Rehana Patel, I made five elaborate trees in $\LaTeX$.

I did not find one website which walked me through how to do all this; I pieced it together through a variety of stack exchange posts, as well as trial and error. The TikZ wiki was also helpful, though not all-inclusive. So in this blog post, I will teach you how to make trees with the following components.

• Basic trees
• Custom branching
• Edge colors
• Nodes
• Node labels
• Edge labels (sort of )
• Custom node placement
• Lines and arcs between nodes (and if you’re creative, around nodes)

I’ll caveat this by saying that I make no claims that this is the optimal way to create such trees, but it is has the benefit of not being overly complex. The LaTeX code for these trees can be found on GitHub.

### Basic Trees

We use the package TikZ and the environment tikzpicture.

To create a tree, you first say specify how many levels you want the tree to have, and how far away the nodes on any particular level should be. You then need to define a base node and its children.

\documentclass{article}\usepackage{tikz}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]%Nodes distances per level\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%base node\node [label=$c_0$] {} coordinate (t9)%childrenchild{ coordinate (t0) edge from parent[color=black,thick]child{ coordinate (t00) edge from parent[color=black,thick]}}child{ coordinate (t1) edge from parent[color=black,thick]};\end{tikzpicture}\end{document}

The labels (t9), (t0), (t1), (t00) could have named anything; these are merely labels we assign which will allow us to add nodes/colors/labels later on.

What determines branching then is the placement of the curly braces. Each of these children nodes begins with its own open curly brace, and all of its children must be placed within it, before the end curly brace. You can see that at the end of the line for (t00) there are two end curly braces, one for (t00) — it has no children — and one for (t0), since (t00) was its last (only) child.

It won’t compile without the semicolon after the last curly brace, so make sure you include that. It’s also sensitive to spaces, don’t put any empty lines in between the last curly brace and the semicolon.

### Custom Branching

With tikzpicture, you do not tell it where branches should go. Instead, it branches to accommodate the number of children. If a node only has one child, then the branch goes straight up. If the node has two children, then the branches go left and right. If a node has three children, then it branches left, center, and right, and so on.

Hack: For me, the direction of branching mattered, and there were times when I wanted a node to branch left, even though there was only one child. In that case I created a phantom node to the right, and just colored the edge to that phantom white (see the next section for edge colors).

\documentclass{article}\usepackage{tikz}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]%Nodes distances per level\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%base node\node [label=$c_0$] {} coordinate (t9)%childrenchild{ coordinate (t0) edge from parent[color=black,thick]child{ coordinate (t00) edge from parent[color=black,thick]}%phantom nodechild{coordinate (t01) edge from parent[color=white]}}%endparen for t0child{ coordinate (t1) edge from parent[color=black,thick]};\end{tikzpicture}\end{document}

### Edge Colors

You’ve already seen how we color an edge. We color the edge when we first declare the child node, and you can adjust the color and thickness. The built-in color options are red, green, blue, cyan, magenta, yellow, black, gray, darkgray, lightgray, brown, lime, olive, orange, pink, purple, teal, violet and white. For line thickness, your options are “line width=<dimension>“, or just the built-in “ultra thin” for 0.1pt, “very thin” for 0.2pt, “thin” for 0.4pt (the default width), “semithick” for 0.6pt, “thick” for 0.8pt, “very thick” for 1.2pt, “ultra thick” for 1.6pt (Citation: TikZ wiki). I like to use “thick” because the default edge thickness can be hard to see, especially if you’re going to use colors.

\documentclass{article}\usepackage{tikz}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]%Nodes distances per level\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%base node\node [label=$c_0$] {} coordinate (t9)%childrenchild{ coordinate (t0) edge from parent[color=black,thick]%magenta edgechild{ coordinate (t00) edge from parent[color=magenta,thick]}}child{ coordinate (t1) edge from parent[color=black,thick]};\end{tikzpicture}\end{document}

### Nodes

As mentioned earlier, we start of by defining the default base node. If however you want to work with more nodes — to draw them, color them, or label them — here is how you do it.

After you finish creating all your children, you draw nodes using the relevant labels.

\documentclass{article}\usepackage{tikz}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%basic tree\node [label=$c_0$] {} coordinate (t9)child{ coordinate (t0) edge from parent[color=black,thick]child{ coordinate (t00) edge from parent[color=black,thick]}}child{ coordinate (t1) edge from parent[color=black,thick]};%drawing nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt] at (t00) {};\node[circle, fill=white,draw,inner sep=0pt, minimum size=4pt] at (t1) {};\end{tikzpicture}\end{document}

Note that each of these nodes needs its own semicolon at the end.

You see that fill=blue makes the blue node, and fill=white makes the empty node. In the case of the empty node, you need the additional parameter “draw.” We also here choose the shape and size of the node. If you don’t specify the shape, it defaults to rectangle. I’m not sure if there are other shapes you can use for the node.

### Node Labels

You already saw above that in the brackets for the node, we can specify a label.

If you want to put a label inside the node, then you would put it inside the (previously empty) curly braces of the \node line.

\documentclass{article}\usepackage{tikz}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%basic tree\node [label=$c_0$] {} coordinate (t9)child{ coordinate (t0) edge from parent[color=black,thick]child{ coordinate (t00) edge from parent[color=black,thick]}}child{ coordinate (t1) edge from parent[color=black,thick]};%coloring and labeling nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt] at (t00) {};%putting a C in the node\node[circle, fill=white,draw,inner sep=0pt, minimum size=4pt] at (t1) {C};\end{tikzpicture}\end{document}

If you don’t want the circle surrounding the C, then just omit “draw”. You should still include fill=white though, because otherwise the edge will overlap with the label.

If instead of having the label on the node you would rather it next to the node, then inside the brackets for that node, write label=degree:thing, e.g. label=180:$v_1$ puts the label $v_1$ to the left of the node.

\documentclass{article}\usepackage{tikz}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%basic tree\node [label=$c_0$] {} coordinate (t9)child{ coordinate (t0) edge from parent[color=black,thick]child{ coordinate (t00) edge from parent[color=black,thick]}}child{ coordinate (t1) edge from parent[color=black,thick]};%coloring and labeling nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt] at (t00) {};\node[circle, fill=white,draw,inner sep=0pt, minimum size=4pt,label=0:$C$] at (t1) {};\end{tikzpicture}\end{document}

### Edge Labels? Node Labels

Whenever I tried to implement edge labels in my trees, it would result in something being displaced.

\documentclass{article}\usepackage{tikz}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%basic tree\node [label=$c_0$] {} coordinate (t9)%trying to label an edge with the word "test"child{ coordinate (t0) edge from parent node[color=black,thick,left,draw=none] {test}child{coordinate (t00) edge from parent[color=black,thick] }}child{ coordinate (t1) edge from parent[color=black,thick]};%coloring and labeling nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt] at (t00) {};\node[circle, fill=white,draw,inner sep=0pt, minimum size=4pt,label=0:$C$] at (t1) {};\end{tikzpicture}\end{document}

I’d believe that there is some way to fix this, or at the very least an alternative way to make trees which handles edge labels better, but in my various poking around stack exchange, I did not find a tweak that worked for my trees.

I instead then decided to add additional node labels, and then just creative with their placement. You can still use \hspace within these labels to move them left or right.

\documentclass{article}\usepackage{tikz}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%basic tree\node [label=$c_0$] {} coordinate (t9)child{ coordinate (t0) edge from parent[color=black,thick]child{coordinate (t00) edge from parent[color=black,thick] }}child{ coordinate (t1) edge from parent[color=black,thick]};%coloring and labeling nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt] at (t00) {};\node[circle, fill=white,draw,inner sep=0pt, minimum size=4pt,label=0:$C$] at (t1) {};%fake edge label\node[label=90:$w_0\hspace{3.7cm}$] at (t9){};\end{tikzpicture}\end{document}

### Custom Node Placement

If you want additional nodes not in the tree, you have to use “positioning” from the TikZ library. Then, you can say where the nodes should be placed in relation to already existing nodes.

\documentclass{article}\usepackage{tikz}\usetikzlibrary{positioning}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%basic tree\node [label=$c_0$] {} coordinate (t9)child{ coordinate (t0) edge from parent[color=black,thick]child{coordinate (t00) edge from parent[color=black,thick] }}child{ coordinate (t1) edge from parent[color=black,thick]};%coloring and labeling nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt] at (t00) {};\node[circle, fill=white,draw,inner sep=0pt, minimum size=4pt,label=0:$C$] at (t1) {};\node[label=90:$w_0\hspace{3.7cm}$] at (t9){};%custom nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt,label=$v_0$,left=4cm of t9] (v0) {};\node[circle, fill=blue,inner sep=0pt, minimum size=5pt,label=$v_1$,below=1cm of v0] (v1) {};\end{tikzpicture}\end{document}

### Curved or Straight Arcs

Let’s say between the additional nodes you make, you want arcs connecting them. Here is the basic way to do it.

\documentclass{article}\usepackage{tikz}\usetikzlibrary{positioning}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%basic tree\node [label=$c_0$] {} coordinate (t9)child{ coordinate (t0) edge from parent[color=black,thick]child{coordinate (t00) edge from parent[color=black,thick] }}child{ coordinate (t1) edge from parent[color=black,thick]};%coloring and labeling nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt] at (t00) {};\node[circle, fill=white,draw,inner sep=0pt, minimum size=4pt,label=0:$C$] at (t1) {};\node[label=90:$w_0\hspace{3.7cm}$] at (t9){};%custom nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt,label=180:$v_0$,left=4cm of t9] (v0) {};\node[circle, fill=blue,inner sep=0pt, minimum size=5pt,label=180:$v_1$,below=1cm of v0] (v1) {};%arcs\draw[thick] (v0) -- (v1);\end{tikzpicture}\end{document}

If you want the arcs to be curved, add “out” and “in” degrees.

\documentclass{article}\usepackage{tikz}\usetikzlibrary{positioning}\begin{document}\begin{tikzpicture}[grow'=up,scale=.6]\tikzstyle{level 1}=[sibling distance=4in]\tikzstyle{level 2}=[sibling distance=2in]\tikzstyle{level 3}=[sibling distance=1in]\tikzstyle{level 4}=[sibling distance=0.5in]%basic tree\node [label=$c_0$] {} coordinate (t9)child{ coordinate (t0) edge from parent[color=black,thick]child{coordinate (t00) edge from parent[color=black,thick] }}child{ coordinate (t1) edge from parent[color=black,thick]};%coloring and labeling nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt] at (t00) {};\node[circle, fill=white,draw,inner sep=0pt, minimum size=4pt,label=0:$C$] at (t1) {};\node[label=90:$w_0\hspace{3.7cm}$] at (t9){};%custom nodes\node[circle, fill=blue,inner sep=0pt, minimum size=5pt,label=180:$v_0$,left=4cm of t9] (v0) {};\node[circle, fill=blue,inner sep=0pt, minimum size=5pt,label=180:$v_1$,below=1cm of v0] (v1) {};%arcs\draw[thick] (v0) to [out=0,in=0] (v1);\end{tikzpicture}\end{document}

You can play around with those degrees.

Note: These arcs won’t work for nodes which are on the same horizontal line; one node must be at least a little above or below the other one.

Hack: For one of my trees, I needed arcs to go around nodes, rather than connect them. I didn’t find a straightforward way to do that. What I did instead was create teeny tiny nodes slightly offset from the actual nodes, and then I made curved arcs to connect these little (basically invisible) nodes.

### That’s all, folks

That’s it! You now have the basic tools needed to make the trees I made. Again, you are welcome to refer to my code on GitHub to see more details.

# The Effect of Testosterone on Cognitive Reflection

My husband is a Caltech alumnus, and I recall him bringing to my attention a 2017 Caltech magazine article which described a study on the effect of testosterone (this was relevant to Caltech because one of the authors is a Caltech professor). The study participants, all male, were administered a gel which was either a placebo or which would elevate their testosterone levels. The participants were then given a series of puzzles to solve, and the performance was compared between the two groups. The researchers controlled for a variety of factors, including age, mood, and math skills (based on a basic math skills test they administered).

When my husband first mentioned this study, my first reaction was dread – Please don’t tell me that the group with elevated testosterone did better. Because then people would feel more justified in saying, “Men are just, on average, better at math than women.” Besides being problematic on its own, this thinking can have negative sequels, such as underestimating the abilities and accomplishments of women in mathematics. Moreover, some may then say: Should we be dissatisfied with the current gap in performance of male and female students in mathematics? Maybe female students are, on average, performing relative to their male counterparts as well as can be expected?

But the study’s authors did not seem to have in mind comparing mathematical thinking between men and women. First of all, instead of comparing men to women, they compared men to men who had been administered additional testosterone. Due to the existence of the testosterone replacement therapy industry, the effects of additional testosterone is a relevant question today. Secondly, they were not testing mathematical thinking specifically – they were testing “cognitive reflection,” i.e. the “capacity to override incorrect intuitive judgments with deliberate correct responses.”

To that end, the puzzles given to the study participants were ones which have tempting, though incorrect, initial potential answers, which would need to be overridden in order to arrive at the correct answers. The questions asked were:

1. A bat and a ball cost $1.10 in total. The bat costs$1.00 more than the ball. How much does the ball cost?
2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?
3. In a lake, there is a patch of the lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

Answers to these questions can be found in both the original article and in the Caltech article.

In other words, the study tested the effect of additional testosterone on a type of impulsivity, and how it affects critical thinking.

Interestingly, the authors found that overall the testosterone group had 20% fewer correct answers than the placebo group (more detailed results are described in the study itself).

Though it was not the explicit goal of the researchers to compare men to women, it is certainly tempting to attempt to extrapolate these results; men do, after all, have on average more testosterone than women. And it is in line with many studies and informal observations – that men are more confident that women, and will often answer questions more quickly, but that does not translate to more correct answers. This is perhaps reassuring to women – just because the men around you seem to feel more confident than you do, that does not mean that you know less or are less qualified.

If this topic interests you, there is a fascinating “This American Life” episode consisting of stories on the effect of testosterone.

# Women in math, part 1

Although it is widely accepted that there are disproportionately few women in mathematics, the numbers for math majors do not seem to be as stark as I first thought. David Bressoud’s MAA article from 2009 uses data which is a few years old now, but gives a thoughtful and comprehensive review. In it, he cites the US Department of Education’s National Center for Education Statistics which says that in 2007, women were 44% of math majors, which is higher than I would have guessed. Still, those numbers do decline quite sharply as the education level and prestige increase. According to the AMS’ 2014 preliminary report of new doctoral recipients, only 30% of the recipients are women, and according to the AMS’ 2013 departmental profile report, only roughly 12% of tenured faculty members in mathematics departments which grant doctorates are women.

There are of course many factors that influence the number of women in mathematics, and rather than try (and fail) to identify them all here, I will examine one issue at a time.

I would like to discuss an empirical difference in the way men and women students behave in the classroom, which was first brought to my attention in a bio page I read of the mathematician Katrin Wehrheim. Dr. Wehrheim is now at Berkeley, but she used to be at MIT and their Women in Mathematics page still contains her bio.

In particular, I found the following passage insightful:

“…In Katrin’s experience, women bring a different culture of thinking into mathematics that helps deal with these issues.

It’s not that all men and women have distinct ways of doing math, but she notices that many women tend to focus on what they do not understand, while their male colleagues often rush to push together pieces they do understand and just take certain things for granted along the way. She sees it all in the time in the classroom. Women may be left behind while concentrating on a source of confusion unless they have the confidence to ask the question. If they do, the resulting discussion often clarifies a deeper issue for the entire class. So, Katrin believes that a higher representation of women, and people of diverse backgrounds in general, could produce an educational climate in which communication and clarity are valued higher.”

Two points here struck me. First of all, Dr. Wehrheim made an explicit  argument for the benefits of diversity in the classroom. Rather than just saying “diversity is important” or even “different perspectives in the classroom deepens the conversation” (which is true), Dr. Wehrheim is precise. In general, the women in the classroom seem to focus more on what they do not understand than on what they do understand. These nuances can sometimes be overlooked by the men, so when the women do feel comfortable to ask questions, it leads to the whole classroom having a more sophisticated discussion.

The second point that struck me was that I have seen the dynamic Dr. Wehrheim describes played out in the classroom. This may be expressed to positive effect – the women students are aware of and comfortable to voice a potential concern – or to negative effect, where the women students constantly struggle with self-doubt. Unfortunately, I see the latter quite often. Even with upper-level classes, I see women students disengage when confronted with intentionally challenging problems. They may say, “Oh, I didn’t understand xyz in class. I’ll need to review my notes/the textbook before I can do this problem.” This is of course problematic. After clearing up one or two quick items, they typically would be ready to tackle the questions, which is the whole point.

Dr. Wehrheim’s comments have stuck with me over time. Do I do this? (Without question.) What is the psychological effect of always seeing what you do not understand rather than what you do? Could that have been why more of my female undergraduate peers chose not to continue with graduate-level math?

The concept of women struggling more with self-doubt than men is not novel. However, Dr. Wehrheim’s description of what this may look like in the math classroom made me more conscious of it as an educator. What to do with this information is still not clear to me, and that will have to be a topic for a future blog post.

I have been active as an undergraduate and now as a graduate student in various student activities. There is certainly much that can be said about undergraduate mentorship; here I want to share resources to which I have exposure and which I found to be particularly helpful for mathematics majors.

• Undergraduate club – Clubs build community among the undergraduates, which of course is very important. In particular, they build a bridge between the senior students and the more junior students. Talks given there to the undergraduates introduce them to interesting concepts and subject areas.
• Undergraduate advisor – This can be a faculty member who takes this role seriously or it can be a staff member. (As an aside – there was a staff member with this responsibility in the [rather stressful] physics department of my undergraduate institution and she became like a mother to the students, who not infrequently burst into tears in her office.) The undergraduate advisor helps students make course selection decisions, gives them career advice, and perhaps even has connections to industry which can then be shared with the students.
• Faculty advisor – The advisor oversees the student’s mathematical growth. They connect the student to the larger mathematical community, including recommending graduate programs, if relevant, and writing recommendation letters.

I would like to discuss the Directed Reading Program for a moment, since that is likely the least familiar to readers and the most specific to mathematics. It is still fairly uncommon – I am only aware of 11 universities and colleges which run it. The Directed Reading Program (DRP) pairs a motivated undergraduate stent with a graduate student to learn together weekly on a topic of the undergraduate’s choosing. After an initial meeting, the graduate student helps the undergraduate find an appropriate text, and then together they will discuss the material. At the end of the semester, the undergraduates will each give a 15-minute presentation on their topic.

The DRP is particularly helpful for math students, because math textbooks often do not provide much motivation, background, or context. Rather, they can fall into a definition – example – theorem cycle. A graduate student can help fill in these gaps. For many of the Rutgers undergraduates I’ve seen participate, this is their first exposure to sophisticated mathematics. Moreover, the DRP establishes a connection between undergraduate students and graduate students, and this mentorship can then continue.

The graduate students at Rutgers who have participated in the DRP have all done so as volunteers, and so the program requires little to no funds. I would strongly encourage math department with graduate programs to consider starting their own DRP.

What mentorship programs have you benefited from?

# Why learn math?

Why learn math?

This is a common question. It is often voiced, in one way or another, by frustrated students or adults with unpleasant memories from their math classes.

A standard answer to this question is that math is useful – which it is. It is true that if you want to do anything technical – engineering, computer science, physics, chemistry, etc – you will need math, and likely quite a bit of it. It is also true that even if math is nowhere in your job description, it can benefit you personally and professionally. For example, a better understanding of algebra can help you to decide which loan to accept. A better understanding of statistics can help you to sort through the plethora of data we all receive on a daily basis. And a better understanding of probability would certainly help you as a game show contestant.

However, if your goal is to avoid math, you can. You do not *have* to know math, or at least not much. You can get by with basic numeracy skills, mostly elementary algebra.

I would like to put forth a less-quoted reason for learning math: math informs the world around you. Galileo is attributed with saying, “Mathematics is the language with which G-d wrote the universe.” It is certainly true that if you understand this language, then you will be able to more easily navigate and manipulate the world around you. You will be able to make more informed decisions, and you will be able to see what is at a problem’s core and then to creatively problem solve. But this inherent utility is not the sole reason for studying math.

I liken learning mathematics to learning history. One learns history to better understand our complex personal and societal relationships. History is the context for all our interactions. History helps us to understand a politician’s speech by providing background on the speaker, the location, the words said, and the words unsaid. Moreover, the knowledge of a city’s history changes it from a collection of roads and buildings to a living place, with its own foibles and quirks, challenges and successes.

Similarly, to know mathematics is to find a pattern and see sense in it. To notice in the world around you symmetries and structure. To understand that dysfunction is often an oversight of mathematical principles, and harmony is an embrace of them. To know mathematics is to find order in a seemingly chaotic world. Understanding mathematics opens you one of the driving forces of life.

So how can you bring more mathematics into your consciousness? Reading blogs like this is a start!

There are several energizing and and educational TED talks on mathematics. Here is a list of some of them and I would also like to point you towards Roger Antonsen’s talk Math is the Hidden Secret to Understand the World and Dan Meyer’s Math Class Needs a Makeover.

Why do you learn math?