To top off JMM, I went to the AMS panel discussion on mathematics and motherhood hosted by Carrie Diaz Eaton of Bates College. Panelists Karoline Pershell, executive director of the Association for Women in Mathematics, Karen Saxe, the associate executive director of the AMS, and Talithia Williams, an associate dean and professor at Harvey Mudd, discussed everything from how having children affected their careers, family leave policies and informal support networks, role models, and accepting outside help. Here are some highlights from the panel, edited for clarity:

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# Daily Archives: January 18, 2020

## AMS panel on math and motherhood

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## James Tanton’s proofs that 1 = 2

This afternoon at Mathemati-Con, James Tanton proved to us 12 different ways that 1 = 2. Here are just a few of his arguments:

1. Since $i^2 = -1$,

\[ 1 = {i^2 + 3 \over 2} = {(\sqrt{-1})(\sqrt{-1}) + 3 \over 2} = {\sqrt{(-1)(-1)} + 3 \over 2} = 2 \]

2. Take a unit square. The diagonal has length $\sqrt{2}$ by the Pythagorean theorem. But we can also approximate the diagonal using a staircase. The lengths of the horizontal segments of the staircase must add to 1, as must the lengths of the vertical segments. So the staircase, no matter how closely it approximates the diagonal, always has length 2. As we make the approximation better, this means that $\sqrt{2} = 2$ and therefore $2 = 4$ implies $1 = 2$.

3. Write $ 0 = 0 + 0 + 0 + \dots$. Since $0 = 1 – 1$, we have

\[ 0 = (1-1) + (1-1) + (1-1) + \dots \]

But if we rearrange the parentheses, we get

\[ 0 = 1 + (-1+1) + (-1+1) + (-1 +1) + \dots = 1 + 0 + 0+ \dots = 1. \]

Thus $0 = 1$; adding 1 to both sides gives $1 = 2$.

See if you can spot the mistakes in each of the arguments. Tanton ended with the following irrefutable “proof by shopping”: At a store holding a 2-for-1 sale, he asked the sales associate how much it would cost to buy 1 item. She responded that it was the regular price–that is, the same price as buying 2 items. Hence it must follow that $1 = 2$.

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## Advice from Haimo Award Winners

What do award-winning mathematicians Federico Ardila-Mantilla, Mark Tomforde, and Suzanne Weeks have to say about teaching? Each of these mathematicians received the MAA Haimo Award at JMM 2020 and gave 20 minute talks. Here is a summary.

**Federico Ardila-Mantilla** used his time to ask us to reflect by thinking about three “open questions.” First, a story about Federico. A mathematician during the day, DJ at night, it is not surprising that Federico is interested in music. This interest led him to enroll in a *timbales* class at his local community center. His teacher? One of the top timbaleros of the 1970’s. After just two weeks of classes, this timbalero invited Federico to play with him at an event. It was a moving experience for Federico. **What does a similar experience look like in a math environment? When do we see top mathematicians inviting newcomers into their work?**

Undisturbed beech forest behave in a weird but spectacular way. “Apparently, the trees synchronize their performance so that they are all equally successful. And that is not what one would expect. Each beech tree grows in a unique location, and conditions can vary greatly in just a few yards. The soil can be stony or loose. It can retain a great deal of water or almost no water. It can be full of nutrients or extremely barren. Accordingly, each tree experiences different growing conditions; therefore, each tree grows more quickly or more slowly and produces more or less sugar or wood, and thus you would expect every tree to be photosynthesizing at a different rate… [However] The rate of photosynthesis is the same for all the trees. The trees, it seems, are equalizing differences between the strong and the weak.” – The Hidden Life of Trees. **What does this synchrony look like in a mathematics classroom?**

Born in 1952, bell hooks experiences the initial racial integration after segregation was deemed illegal. “Bussed to white schools, we soon learned that obedience, and not a zealous will to learn, was expected of us. That shift taught me the difference between education as the practice of freedom and education that merely strives to reinforce domination.” – bell hooks. **What does a liberating math classroom look like?**

**Mark Tomforde **had four original beliefs that have changed throughout his teaching experiences. They are:

– Old Belief 1: My main objective is to teach math content. The more content the better. Updated belief: **My main objective is to teach my students to think.** In 10 years what will students remember from this course? What do I want them to remember? I want to prepare my students for life, to develop good habits when engaging with facts, to work on improving their skills such as problem solving, attention to details, ability to communicate technical ideas, and to think deeply about a few topics.

– Old Belief 2: The more students I can reach the better. Updated belief: **Quality is more important than quantity.** I am a fixed quantity and the more projects I am involved with, the less of myself I can devote to them. For me, it is better to devote my time and energy to one or two projects and do it well rather than spending little time in a lot of projects.

– Old Belief 3: The best students are our future, teach to them. Updated belief: **Direct teaching to all students paying particular attention to the ones that are struggling.** Are our best students more important? Sometimes we think we are just training the future mathematicians and try and identify the “best students.” But, how many students are we not able to reach? Our goal should not be to find talent but rather to help students develop their talent.

– Old Belief 4: Good students are the ones who can prove theorems, solve problems and do well. Updated belief: **There are many ways for students to be successful and contribute.** When I teach math majors, I regard all of my students as the future of math. This is because math requires a much larger ecosystem than a group of researchers. In addition to researchers, we need teachers, advocates, PRs, writers, and many others are needed. In my classes, I ask students to find a need and contribute to fill that need. Some of the projects students conduct are: leading math circles, tutoring, creating guides for incoming students in the department, starting newsletters, and organizing department picnics.

**Suzanne Weeks **is a co-director of MSRI-UP and PIC Math, and has worked in many industrial projects. Her experiences have taught her it’s important to:

– **Show up** – Take advantage of the opportunities that arise for you.

–** Sit in front** – Be engaged, be involved, be fully in the room.

– **Raise your hand** – Be an active participant, raise your hand, raise your voice, ask for stuff when you need it.

– **Make friends** – You need to have a support group.

– **Be yourself** – Your path is your own. Do what is best for you.

– K**now your worth because you are worthy** – Celebrate your successes.

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