## The fundamental theorem of geometric group theory, Part II: proof

1 Oct

A refresher from last week: If a group G acts on a proper geodesic metric space X properly discontinuously and cocompactly by isometries, then G is quasi-isometric to X.  Moreover, G is finitely generated.

Yes, I put “proper geodesic metric space” in italics because I forgot it in the statement of the theorem last week.  [Aside: actually, you only need “length space” there, and proper geodesic will follow by Hopf-Rinow.  But let’s skip that and just go with proper geodesic.]  I also added the second sentence (which isn’t really a moreover, it comes for free during the proof).

At the end of last week I defined proper: closed balls are compact. A space is geodesic if there is a shortest path between any two points which realizes the distance between those points.  For instance, the plane is geodesic: you can just draw a line between any two points.  But if you take away the origin from the plane, it’s not geodesic anymore.  The distance between (1,1) and (-1,-1) is $2\sqrt{2}$, but the line should go through the origin.  There is no geodesic between those two points in this space.

Now we have all our words, so let’s prove the theorem!  I’ll italicize everywhere we use one of the conditions of the theorem.

Since our action is cocompact, we have a compact set K so that translates of it tile X.  Pick a point inside K, call it $x_0$, and a radius R so that K is entirely contained inside a ball of radius R/3 centered at $x_0$.  For notation, this ball will be labelled $B(x_0,R/3)$.

Schematic: K is the red square, special point is the yellow dot, yellow circle is radius R/3, lighter circle is radius R. Cartoon on right illustrates this relationship

We’ll pick a subset of the group G: Let $A =\{ g\in G: g.B(x_0,R)\cap B(x_0,R) \neq \emptyset\}$.  X is proper, so closed balls are compact.  Since the action is properly discontinuous, this means that is finite.  [Reminder: properly discontinuous means that only finitely many group elements translate compact sets to intersect themselves].

Now we’ll show that G is finitely generated, and it’s generated by A.  Choose some group element g in G.  Draw a geodesic in between your special point $x_0$ and its g-translate $g.x_0$.  Now we’ll mark some points on that geodesic: mark a point every R/3 away from $x_0$, all the way to the end of the geodesic.  You’ll have [(length of the segment)/(R/3) rounded up] many points marked.  Let’s call that number n.

There are n blue points, and they’re all R/3 away from each other. Notice the last blue point might be closer to g.x_0, but it’s definitely less than or equal to R/3 away.

Here’s the clever part.  Remember that K tiles X by G-translates (cocompactness), so each of those blue points lives inside a G-translate of K.  Since $x_0$ lives inside K, that means there’s a nearby translate of $x_0$ to each blue point.  And since K fits inside a R/3 ball, each translate is less than or equal to R/3 away from its blue point.

The green points are translates of x_0: I also colored x_0 and g.x_0. The yellow circle indicates the the green point is within R/3 of its blue point.

We can bound how far apart the consecutive green points are from each other: each one is within R/3 of its blue point, which are all R/3 apart from their neighbors.  So the green points are at most R/3+R/3+R/3= R from each other.

Middle portion is exactly R/3 long. So by the triangle inequality, the green points are less than or equal to R from each other.

Remember that the green points represent G-translates of $x_0$.  In the picture above I numbered them $g_0.x_0=x_0,g_1.x_0,g_2.x_0,\ldots g_nx_0=g.x_0$.  We just said that $d(g_1.x_0,g_2.x_0)\leq R$.  Since G acts by isometries, this means that $d(g_2^{-1}g_1.x_0,x_0)\leq R$.  So $g_2^{-1}g_1$ lives inside our set A that we defined above- it moves $x_0$ within of itself.

Here’s a bit of cleverness: we can write $g=g_n=g_0^{-1}g_1\cdot g_1^{-1}g_2 \cdots g_{n-1}^{-1}g_n$, because all of the middle terms would cancel out and we’d be left with $g=g_0\cdot g_n = 1\cdot g = g$.  But each of those two-letter terms lives in A, so we just wrote as a product of elements in A.  That means that A generates G.  We said above that A is finite, so G is finitely generated.

That was the “moreover” part of the theorem.  The main thing is to show that G is quasi-isometric to X.  Let’s try the function $g\mapsto g.x_0$.

Above, we wrote as a product of elements of A, so that means that the length of is at most n.  In other words, $d_G(1,g)\leq n$.  Now we’d like to bound it by $d_X(x_0,g.x_0)$.  We found by dividing the geodesic into pieces, so we have $n\leq \frac{d_X(x_0,g.x_0)}{R/3}+1$, where we added a 1 for the rounding.  So we have one side of the quasi-isometry: $d_G(g_1,g_2)\leq \frac{3}{R}d_X(g_1.x_0,g_2.x_0)+1$ (using the action by isometries).

Now we need to bound the other side, which will be like stepping back through our clever argument.  Let M be the maximum distance that an element of translates $x_0$.  In symbols, $M=max_{a\in A} d_X(x_0,a.x_0)$.  Choose some in G, with length n.  That means we can write as a product of elements in A: $g=a_1\cdots a_n$.  Each $a_i$ moves $x_0$ at most M.  If we step between each translate, we have $d(a_i.x_0,a_{i+1}.x_0)=d(a_{i+1}^{-1}a_i.x_0,x_0)\leq M$.  There are steps from $x_0$ to $g.x_0$, and each step contributes at most M to the distance.  So $d_X(x_0,g.x_0)\leq M d_G(1,g)$.

With bounds on both sides, we can just pick the larger number to get our actual quasi-isometry.  We also need the function to be quasi-onto, but it is because the action is cocompact so there are translates of $x_0$ all over the place.

Huzzah!

## The fundamental theorem of geometric group theory (part I), topology

24 Sep

I love the phrase “THE fundamental theorem of…” It’s so over the top and hyperbolic, which is unlike most mathematical writing you’ll run into.  So you know that it’s important if you run into the fundamental theorem of anything.  By now we all have some background on geometric group theory: you’ll want to know what a group action is and what a quasi-isometry is.  (Refresher: a group G acts on a space X if each group element g gives a homomorphism of the space X to itself.  A quasi-isometry between two spaces X and Y is a function f so that distances between points get stretched by a controlled scaling amount + an additive error term).  We say a group G is quasi-isometric to a space X if its Cayley graph is quasi-isometric to X.  Remember, a Cayley graph is a picture you can draw from a group if you know its generators.

Still from Wikipedia: a Cayley graph of the symmetries of a square

There are several more terms we’ll want to know to understand the theorem, but I’ll just do one more before we start.  We say a group G acts on a space X by isometries if it acts on X, and each homomorphism is actually an isometry (it preserves distance).  So for instance, the integers acting on the real line by multiplication isn’t by isometries, because each homomorphism spreads the line out (so the homomorphism of the reals to themselves given by 3 is $x \mapsto 3x$, which stretches out distances).  But if the action is defined by addition, then you’re okay: $x\mapsto x+3$ preserves distances.

Under the red function, f(2)-f(1)=6-3=3, but 2-1=1, so this isn’t an isometry.
Under the green function, f(2)-f(1)=5-4=1, which is equal to 2-1. This is always true, so this is an isometry.

So here’s the fundamental theorem:

If a group G acts properly discontinuously, cocompactly, and by isometries on a proper metric space X, then G is quasi-isometric to X.

You can parse this so far as “If a group G acts by isometries on a space X with condition condition condition, then G is quasi-isometric to X.”  Putting aside the conditions for now, how would we prove such a theorem?  Well, to show something is quasi-isometric, you need to come up with a function so that the quasi-isometry condition holds: for all x,y in X, we need $\frac{1}{K} d_G(f(x),f(y))-C\leq d_X(x,y) \leq K d_G(f(x),f(y))+C$.

So let’s deal with those conditions!  An action is cocompact if there’s some compact subset S of X so that G-translates of S cover all of X.  Remember, each element g in G gives an isometry of X, so it’ll send S to some isometric copy of itself somewhere else in X.  In our example above, the integer 3 will send the compact subset [5,7] to the isometric copy [8,10].  In fact, our example action is cocompact: you can keep using [5,7] as your compact set, and notice that any point on the real line will eventually be covered by a translate of [5,7].  For instance, -434.32 is covered by [-435,-433], which is the image of [5,7] under the isometry given by -440.

This action is also cocompact. Here I have the plane, conveniently cut up with an integer lattice. Can you figure out what the action is? Hint: the red square is a unit square, and the pink squares are suppose to be various translates of it.

G acts on X properly discontinuously if for any two points x,y in X, they each have a neighborhood $U_x, U_y$ so that only finitely many g make $g.U_x\cap U_y\neq\emptyset$.  Let’s look at our example action again.  If I take the points 4365.234 and 564.54 in the real line, I’d like to find neighborhoods around them.  Let’s choose the intervals [4365,4366] and [564,565].  The only integers that make these hit each other are -3801 and -3800.  In particular, 2 is finite, so this indicates proper discontinuity.  If we actually wanted to prove the action is properly discontinuous, we’d want to show this is possible for all numbers, not just these two specific ones I chose.

Schematic of proper discontinuity: only finitely many g will send the yellow oval to hit the blue blob

Finally, a metric space X is proper if all closed balls are compact.  Balls are intuitively defined: they’re all the points that are at a fixed distance or less from your center.  In the plane, balls are circles, centered around points.  And compact-well, aw shucks I haven’t defined compact and we’ve been using it!  Time for some topology.  We’ll prove this theorem next time around, this post is just definitions and background.  (Sorry for the cliffhanger, but it should be clear what we’re going to do next time: make a function, show it’s a quasi-isometry).

Just like groups are the fundamental object of study in algebra, open sets are the fundamental object of study in topology.  You’re already familiar with one type of open set, the open interval (say, (6,98), which includes all numbers between 6 and 98 but doesn’t include 6 and 98).  I just described another above: balls.  So, open circles in the plane are open sets.  Sets are closed if their complement is open: that is, the rest of the space minus that set is open.  In the real line example, [6,74] is closed because $(-\infty,6)\cup(74,\infty)$ is open (it’s the union of infinitely many open sets, say (74,76) with (75,77) with (76,78) and on and on).

Notice that I haven’t yet defined what an open set is.  That’s because it’s a choice- you can have the same space have different topologies on it if you use different definitions of open sets.  I’ll point you to these wikipedia articles for more examples on that.

A set is compact if every covering of it by open sets has a finite subcover.  That means that any time you write your set S as a union of open sets, you can choose finitely many of those and still be able to hit all the points of S.  From above, the set $(74,\infty)$ is not compact, because you can’t get rid of any of the sets in that infinite covering and still cover the set.  On the real line, a set is compact if it’s closed and bounded (this is the Heine-Borel theorem, a staple of real analysis).

So that’s enough for today.  More next time (like a proof!)  Also, I’m using my husband’s surface to blog this, which means I did all the pictures using my fingers.  It’s like finger painting.  What d’you think?  Better than usual pictures, or worse?

## VEGAN MERINGUES WHAT

17 Sep

I’m late to this vegan meringue game, compared to vegan bakers, but I think I’m still ahead of the game for, yknow, normal people.  (I kid, I love you vegan bakers!  Strangely I have three good friends and an ex who are vegan and love baking).

If you know someone allergic to eggs, or a vegan, or if you make hummus/use chickpeas ever, you should try making vegan meringues!  This time I just went with normal meringue cookies, next time I might try pavlova or a pie.  I’m so impressed by what the vegan baking community has done with “aquafaba”- the liquid leftover from a can of garbanzo beans/chickpeas/chana (as in chana masala the Indian dish, as I learned from wikipedia).  Look at this crazy list of recipes on the facebook group.  And there’s a website dedicated to aquafaba, with the history of it (basically it was a myth, and then someone did it but way too hard molecular gastronomy-style, and then finally someone was like wait you don’t have to do anything fancy).

It is pretty unbelievable how beaten chickpea liquid looks just like beaten egg whites, stiff peaks and all, with nothing added (I would expect agar or something).  So let’s get to it!

It’s a garnanbo! A Zoganba! I meant a garnanza! (Or did I? Everything gets garbled sometimes… )

First, use your can to drain the liquid from the garbanzos.  I used the beans in a crockpot on low with a jar of salsa, a beer, and a pack of chicken thighs all day.  I added some zucchini about half an hour before we ate with tortillas, queso fresco, and spinach (didn’t have lettuce).  Yum.

What if Andre Young is actually a robot series, and this is the 14th iteration of him? He’d be Dr. Dre N.

I realize I’m in the middle of a blog post and I baked these meringues tonight just so I could have a blog post out on Thursday but oh my gosh I just read a great tweet that you should know about.

<blockquote class=”twitter-tweet” lang=”en”><p lang=”en” dir=”ltr”>I can't get over Euclid.</p>&mdash; Ric (@ellopickle) <a href=”https://twitter.com/ellopickle/status/644340277361049600″>September 17, 2015</a></blockquote>

Anyways.  Let’s all calm down.  I laughed so hard at this my husband told me to go to sleep because I’m getting delirious.  That should make a fun post!

Start beating that bean liquid!  It takes approximately FOREEEEEVEEERRR by which I mean 10-15 minutes.  We worked on our new hobby, PANDA magazine, during this time.  (Brag: we’re in the magazine this issue, because we completed last month’s!  It’s really hard!  Great for DASH fans.  Which should be everyone in the relevant cities.  I’ve DASHed in Davis, San Francisco, Chicago, and will in Austin next year.)

How you bean lately? Oh, same old same old

How bout you, how you bean? Oh, things in my life are a little crazy. I feel like things are getting really mixed up.

Tell me about it. I’ve bean beaten again and again by life.

I thought you’ve bean sort of stiff lately

Bean there, done that is how I feel toward everything nowadays, you know?

Because I can’t read directions I dumped all my sugar and vanilla in at once.

But you shouldn’t do that!  Add about a tablespoon at a time and beat, so you don’t get any sugar clumps and it mixes smoothly.  Then you won’t have any holes in your meringue like I did.  Mine worked out anyway.

If we said “kapull” instead of “kernel” and “lite” instead of “lew tenant”, maybe my friend Thomas (grad student at Temple U) would be meringue

Bake at a very low temperature for a very long time, and they magically work!

1 can chickpeas

1 tsp vanilla

sugar- 1.3 times as much chickpea liquid you get (so a cup or less)

Drain the chickpea liquid.  Start beating it- I did speed 3 (low) for 4 minutes.  Then medium for 4 minutes, then high for 2 minutes.  It’ll form fairly stiff peaks, if not as good as egg whites pretty darn close.

Preheat oven to 250.  Line a baking sheet with parchment paper.

Beat in the sugar 1 TBSP at a time.  Add the vanilla and beat it in too.  Dollop onto your parchment paper sheets.  Bake for 90 minutes, then turn off oven and crack it.  Let cool to room temperature.

10 Sep

I first made these cookies a decade ago, back when I was in high school, and was blown away by how good they are!  They pack a punch with lots and lots of ginger flavor.  I actually messed up baking these (used something that was not powdered ginger) but they still taste pretty good despite the potential garlic-onion in there.  Sounds pretty gross but the molasses and fresh ginger and crystallized ginger sort of hide the strange savory notes.  I also have old baking soda so the cookies didn’t rise as much as they should’ve.  Oops.

Anyways!  They’re even better without any onion flavor in them!  I highly recommend these.  I already had everything I needed to bake a half batch in my pantry; we like keeping crystallized ginger around to chew on if we have a stomach ache.

If you were running a Scottish daycare but didn’t have enough enrollment, maybe you’d put up a sign saying “SPACE FOR MO LADDIES AND MO LASSES”

It generally helps to have slightly softened butter when you start baking cookies, to easily cream the butter and sugar together.  You don’t want it too soft, just to have been out of the fridge for half an hour or so.  In this case I took butter out of the freezer an hour earlier, and it was still a bit too hard to beat.  Thank you extremely strong Kitchen Aid stand mixer– it did a great job beating up the butter.  Cream the brown sugar in, then add the molasses.  Normally you have a pretty fluffy butter-sugar base for cookies, but the liquidy molasses makes it look a little grosser with clumps of butter suspended in the molasses..

If your lawn is dying and you start to see things that look like huge anthills appearing, take another look. You might see some creatures scuttling down into the holes, and you’ll catch a glimpse of some mole asses

Then mix all your dry ingredients, as usual.  Flour, powdered ginger, salt, baking soda.  Toss those in with the wet and mix.  Finally, chop up the crystallized ginger (this is difficult if it’s very dry) and grate some fresh ginger, and mix those in too.

I have fairly large eyes for an Asian person, while my best friend has short eyes. My other friend is named Chris. If he were a redhead I could refer to him as Chris-tall-eyes Ginger

I just learned all about the Kuna, and their traditional garments. Mola says “give us our autonomy!”

After you’re all mixed, toss the dough into the fridge for a while.  I’m not usually a good recipe-follower and I don’t often chill my cookie dough, but it helps these ones hold together (that tricky molasses!).  They’re particularly nice if you roll them in sugar before baking, but I skipped that this time.

What if all the “r”s of the world were suddenly replaced by “l”s? Dilections would be so muddled. We’d all be stuck in  sticky molasses.

A pretty quick bake (10 minutes) and they’ll be done!  These are soft chewy cookies, not crispy.  Again, pardon my old baking soda in the next photos.  The original recipe makes massive cookies, but I like them smaller.  So i did a half batch (original says makes 18) and I got out 27 cookies.  Up to you!  Surprisingly, smaller cookies bake in almost the same amount of time (9 minutes v 10)

6 TB butter

1/2 c brown sugar

1 egg

2 TB molasses

1 C + 2 TB flour

1 tsp ground ginger

1 tsp baking soda

pinch salt

1 TB grated fresh ginger

1/4 c chopped crystallized ginger

1. Cream the butter and sugar until fluffy, then mix in the molasses.
2. Stir together flour, ground ginger, baking soda, and salt.  Add to the wet ingredients and mix.
3. Add the grated fresh ginger and crystallized ginger, and mix thoroughly.
4. Chill the dough in the fridge for awhile.  (I did overnight)
5. Preheat oven to 350.
6. Roll heaping teaspoons of dough into small balls, then roll in sugar.  Place on ungreased cookie sheet, and bake for 9 minutes.

## Race, class, and math thoughts

3 Sep

Yesterday a student walked into a friend’s graduate student office and asked us “Does STEM encourage apathy toward social justice and diversity issues, specifically the lack of black and brown bodies?”  A few things were notable about the ensuing discussion: first, no one was ever interrupted!  There were five of us in the room!  Apparently I have forgotten how to have civil conversations.  I was astonished by the fact that no one ever butted in, and respectfully allowed each person to say their full thoughts, with pauses and everything.  Does this happen in your life?  I think most of my interactions are at the pace of Gilmore Girls or a news channel than, y’know, respect and calm.  Also it was maybe the most diverse in-depth conversation on diversity I’ve been part of: a Chinese man, a Vietnamese woman (that’s me), a white woman, a black man, and a black woman from various socioeconomic backgrounds.

So that’s the context behind this post.  The question she posed was rather overly broad and poorly defined, but we hit on a lot of points that I thought were interesting and unrelated to each other.  Bring in the bullet points and mildly academic-sounding language!  We’re mostly considering the dearth of POC in math graduate school and professorship.

• Local culture/society: higher education isn’t regarded as important in every culture.  Parents who don’t encourage their kids to go to college probably won’t understand a child’s desire to go to graduate school.  The example given here was: “do you play sports?  …no?  Oh.  Okay.”  In this person’s hometown, athletics are more important than scholarship.  How do you change a culture?  Should you try to change a culture?  It’s incredibly important to go into a community and listen/learn first, then try to launch programs/solutions.  Personal notes here: my family is very supportive and basically let me do whatever I want, but they still all balked a bit at my choice to pursue a Ph.D. in math.  They also wanted me to go to UCSD (which would’ve been free + given me money) instead of Yale (which cost money).  My first year of graduate school, I was on the phone with a Yale alum who, after I told him I was a grad student, said “you know you just cut your earnings by a third, right?”  So we’re all from different cultures that have different values.
• Socioeconomic class: graduate school pays very little.  We hover securely above the poverty line (which in 2015 is just under 12k) with about 16-35k stipends (a small survey here), which are generally enough (barely, sometimes) for us to pay for our living expenses.  Plus you often have to pay fees back to the school; mine are about 1.6k per year (just under half a monthly paycheck, twice a year).  If your parents or siblings need help, or if you have children or other dependents, the stipend won’t be enough.  So if you have any of these thoughts in mind, it makes complete sense to not go to graduate school.  For more thoughts on this, check out this blog post about being an academic coming from a poverty background.  I’m astonished at my friends who have a kid on two grad student stipends.  Babies are so expensive!  My baby went to daycare three days a week and it cost almost exactly as much as I made per month.
• Intervention programs: they work.  Reaching out to communities who haven’t heard of/thought of college/graduate school works.  There’s a sad dearth of funding for such programs (often run by private organizations), but they led to some of us being in that room.  Fun fact about me: I’m a Mellon Mays Fellow, which means I had a lot of encouragement as an undergraduate to go into academia.  MMUF is great.  Also so is Upward Bound.
• Pipeline: it’s leaky.  See previous bullet point.
• Math itself is very abstract.  In your day-to-day life, you won’t be confronted by social justice/diversity issues like you might be if you were in, say, anthropology or ethnic studies.  The field doesn’t lend itself to thinking about these sorts of things, so that could explain where that apathy stems from.  I haven’t had a ton of conversations with mathematicians about racial diversity (yesterday’s makes it two or so), but I have had a ton of conversations about gender diversity (and every mathematician knows about the AWM).  That said, I also did EDGE before starting grad school, and I just learned about SACNAS over the summer.  So there are people thinking about these things.

And because I can’t resist slipping women and math into any post, here’s a great little article about increasing the number of girls in math.  And by “girls” here we actually do mean “female children” (I’ve gotten pretty good at not calling women “girls” but I still say “you guys” a lot).

There were more points but that’s all I remember.  I think I’ll bake something next week, so look forward to that!  Last week for husband’s birthday I said “I baked you a surprise!” and he said “is it pavlova?” and the answer was yes.  Mini-pavlovas!  Just bake for half an hour instead of an hour.

## Unorganized Common core thoughts, math anxiety, also help me?

27 Aug

Disclaimer: I hate high-stakes testing.  Hate it.  This post is not about high-stakes testing; it is about content that happens to be used in high-stakes testing.  Imagine I’m writing about Holden Caulfield, and try to banish all memories of Catcher in the Rye plot pop quizzes from your mind.

My cousin asked me about my thoughts on common core.  Here’s an actual paragraph from the common core website:

These standards define what students should understand and be able to do in their study of mathematics. But asking a student to understand something also means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One way for teachers to do that is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

This sounds awesome!  I often think I understand something well, and then try to write it down only to realize that there are more subtleties than I thought, or that I had a fundamental misunderstanding, and in general that I am wrong.

On March 16th of this year I thought I had an answer to something I’m working on.  Aaaand… today is August 27th and I’m still working on it.  But I didn’t realize my answer wasn’t complete until I started writing, and I wrote “, because” and had nothing to follow it.  This is frustrating and ridiculous, and I don’t expect every kid in America to become a mathematician and work on a problem for six months.  But I do believe that trying to do something, thinking you’re right, and then trying to explain it and realize you’re wrong after six minutes is a good experience and makes you a better critical thinker.  Question your beliefs, analyze your reasoning, explain yourself- all good things.

The above is all theoretical (I was a math and philosophy major in undergrad).  So I looked up some common core examples.

Here’s a comment from this article, as reported by a Washington Post blog:

Try doing multiplication in long drawn out word form like this one: 3, 6, 9 what is the 12th number in this sequence? My son can’t just read that and think the 12th number automatically. He has to write them out. He also cries over this type of math.

The problem itself is great (I’m also not opposed to crying).  Most parents can help their kids with 12 times 3 equals 36 “automatically,” but figuring out why 12 times 3 is 36 is hard.  This question offers justification first, and encourages the student to explore and discover multiplication for himself.  I was quite bad at my times tables as a kid (I still remember nonsense like “Six and eight went out on a date.  When they came back, they were 48!”) because it didn’t make any sense.  But writing out a sequence 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 is just like practicing spelling or lay-ups, and it makes sense.

I started finding more examples of common core type problems, but I think talking about one of my own past experiences is more helpful.

During my first semester of graduate school, I sat down to my first analysis exam.  I’d never been a big analysis fan (though now that I’ve taken it umpteen times I actually enjoy it!), and I felt a bit shaky about the material before I walked in the room.  The professor handed out the exams and said “you have one hour.”  If this isn’t clear to you by now, I’m good at math.  But as I leafed through the problems I started silently FREAKING OUT.  I felt something like vertigo, and also felt very very cold as I thought about trying to finish this exam in the allotted time.  My hands were shaking and I was a little sweaty.  This was my first experience at math anxiety and it was TERRIFYING.  My skin is prickling as I type this, remembering the experience five years later.

I can only imagine what it must be like to feel that with math for years in elementary-high school.  Math anxiety is real, and unfortunately contagious.  There’s been some amount of backlash to common core/shaking up math education, and I’m certain that some of it is rooted in past traumatic memories of math anxiety.  All I can say is, I support students, I believe in them, and that “the most important thing is to stay calm.”

If you google “common core math” you’ll get lots and lots of mild vitriol, but you also get some good stuff.  Here’s a side-by-side comparison of some math problems and explanations of why they switched.  Here’s a cool explanation of the new subtraction.

Here’s an article called “The Ten Dumbest Common Core Problems” that I’ve run into a lot [I dislike it because it doesn’t give credit to where any of the pictures/examples came from], and my thoughts on 1-10.  Again, not an elementary-high school educator, just a math person:

1. 7 + 7 = 10 + 4 = 14.  Maybe adding a word like “number bonds” is strange and new, but memorizing that 7+7 = 14 can be hard (I’m a terrible memorizer, but a good reasoner).  Breaking 7 into 3 and 4, and then pairing up to get a 10?  That’s how I add large numbers quickly, and I had to figure that out myself.  Helping kids learn the trick is great.
2. Awesome, visualizing subtraction.  See above explanation on the new subtraction.
3. This is a worksheet with a typo.  So… unclear why it’s included.  If it didn’t have typo, it’d be a good sheet.
4. This worksheet’s picture doesn’t make sense.  But the math does.  (Figure out what’s unknown if you have parts of a whole)
5. More visualization of addition and subtraction.  I literally did this with 12 year olds when I was 15, using quarters on a table.
6. I like this one too.
7. I always hated “carrying the one” so I’m all for the new addition.
8. I don’t understand this.  This doesn’t mean I hate it, it means I want to know what it says.
9. This isn’t new; I did this worksheet as a child.
10. Another typo

Final anecdote in this jumble of a blog post: how I learned to divide fractions.  They tell you to just multiply by the reciprocal.  Like I said, I’m bad at memorization, so I’d often do random nonsense that seemed sensible (divide the tops, multiply the bottoms).  No matter what I tried on my own, I felt like I just couldn’t get this dividing fractions thing. One day, my dad drove me to Baker’s Square to buy a pie.  He asked how school was going and I told him that I would never get how to divide fractions and I’d always get it wrong and math is dumb and I was just a sad little 8 year old sitting in that passenger seat.  He said, okay, well, if we split this pie among our family, how much pie does each of us get?  And I said “one fifth.”  What about if just you and I take the pie?  “One half.”  What if I give you the pie?  “One.”  What if half a person gets a pie, how much pie does one person get? “Tw-OHHHHHHHHHH.”  Crystal clear, this concept that had been frustrating me for weeks.  Why do we use the reciprocal?  Because if half a person eats a pie, that whole person eats TWO PIES.

That’s my happy story of understanding the madness behind the method.  I’ve seen dozens of students who know to multiply by the reciprocal, but who don’t understand why, which makes word problems quite difficult.  Conclusion: so far, I’m all for common core math.  The important thing is to stay calm.  And ask questions.  And maybe not help your kids with homework?

Oh one more thing: I’m interested in giving back to the community/doing some volunteer work and putting my skills to use.  I told my advisor yesterday that I don’t have any skills, and he responded that I have math skills and I should go find something like the free clinic for math.  Do you have any recommendations for me to help people with my math skills?

## Cinnabon mots

20 Aug

I made these fantastic and very large cinnamon rolls last week.  They are fluffy and light and soft and airy and buttery and pretty delicious and remarkably un-vegan.  Personally I prefer my pumpkin vegan cinnamon rolls, which are a bit heavier, smaller, and denser with a stronger cinnamon/ginger/nutmeg flavor and filling, and they don’t really require frosting.  These white cinnamon rolls definitely require frosting- the cream cheese/butter combo makes them super delicious and ties together what’s basically a soft white roll + brown sugar and cinnamon.  It’s just down to your preferences- are you a sugar cookie or a spicy ginger cookie kind of person?    This was from a recipe called “Clone of a Cinnabon” and they are quite similar to that mall staple.

If you don’t have three hours to spare/you want cinnamon rolls RIGHT NOW, check out these super fast banana ones that I’ve made over and over again.

I feel a lot of pressure to come up with butter and butter puns… maybe I should start doing repeats and milk them for all they’re worth

I microwaved my milk for thirty seconds and put it in the mixer bowl, then added the yeast- you want it to be warm but not too hot.  Tickle the yeast, don’t kill them.  Also, turn on your oven to 350 while you prepare the dough, so you can have a warm place for the dough to rise.

Another way the kitchen aid stand mixer has changed my life- no more ten minutes of kneading required!  Just use the dough hook and the mixer does it for you!  I’ve never been a good kneader so this is great.  I did use a whisk to mix the liquid stuff first though (the yeast with the milk, then the egg and melted butter).

LOOK AT THIS ONE HANDED ACTION SHOT! Do you know how hard it is to crack and open an egg with one hand? The eggstremes I go to for you.

Now that my measuring cup was empty (because the milk was in the mixing bowl), I could use it to melt six tablespoons of butter- another 30 second zap.

Ghee whiz it’s easy to melt butter

Add that in to your mixing bowl, then measure out your dry ingredients: flour, sugar, salt.  Toss them in, and use that incredible dough hook to knead the dough.  Or knead it yourself on a floured surface for ten minutes, until it makes a nice soft ball.

It’s so great that after you bake something, you get to eat a baked good! If you try sometimes, you just might find you get what you knead.

If kneading yourself, turn off your oven beforehand so it’s the right not-killing temperature for the dough to rest.  Otherwise, turn it off, then put the dough back in a bowl, cover it with a kitchen towel, and stick it in the off-oven for an hour.  Go do something else, like attend to the little monster who’s been crying at your feet this whole time:

Worst sous chef ever

Anyways, when you come back to it the ball should be twice as big as before and so fluffy.  Plop it onto a clean floured counter, leave the kitchen towel on it, and make your filling: this is just a mixture of brown sugar and cinnamon, but I like adding white sugar to cut the molasses-ness of it.   This time I used the Cinnabon method and spread butter, then topped with sugar/cinnamon, but I actually think it’s better to mix the butter with the sugar/cinnamon ahead of time and use that as a filling, like we did with the pumpkin rolls.

Roll out your dough into a nice thin rectangle and fill it.  I used to use wine bottles or tomato sauce jars for this, but we got this awesome fancy rolling pin as a wedding pressent and I love it.  It’s too large for a drawer and sits on our counter looking mildly rustic/pretentious and I love it.  It’s like you’ve been eating jelly beans your whole life of various quality, and then someone gives you some Jelly Bellys.  That’s how I feel about this rolling pin.  (I love Jelly Belly jelly beans a LOT.  If you’re ever near Fairfield, CA I highly recommend the free Jelly Belly factory tour.)

Swing your partner round and round… Now dough

See…

Dough!

Notice that I don’t put the cinnamon sugar all the way to the edge- this helps keep the filling from leaking out.

Now roll that log up (long side rolls up) and pinch the ends and the seam.

Don’t be fooled- this would make a terrible rolling pin, even though it looks about the right size and shape.

Cut it in half, then cut each half in thirds, and cut each third in half.  If that sentence was hard to read, just cut it into 12 pieces in a reasonable way.  Put them into a buttered 9×13 (I used a pyrex), toss it back into that warm oven for half an hour.

Notice how far apart they all are- feeling a little bit uncomfortable, a little bit like middle schoolers at their first dance.

They grow up so fast. This is more like the club when they’re leaving college.

Take out the big cinnamon rolls, then heat the oven to 350.  Bake for 20 minutes, until very lightly browned on the top.  Meanwhile, switch to the regular paddle (not the dough hook) for your mixer and beat some butter with some cream cheese until homogenous, then add a bunch of confectioners sugar, a dash of vanilla, and a bit of salt.

The proeblem with studying for speelling tests at the last minutee is you might end up addding extra letteers to words. Creamming is not the answer.

Let the rolls cool a bit before your frost them, so the frosting doesn’t totally melt everywhere.  Like 5-10 minutes.  Then frost and eat these beauties.  We actually split them usually because a single one is so huge.

Fake cinnabons, adapted from “Clone of a Cinnabon” on allrecipes

1 c milk

1 packet of yeast

2 eggs

6 TB melted butter

4 1/2 c flour

1 tsp salt

1/2 c sugar

Filling:

3/4 c brown sugar

1/4 c white sugar

3 TB cinnamon (+ginger, nutmeg if you like that)

6 TB softened butter

Frosting:

1 package cream cheese (8 oz)

1/2 c butter (1 stick)

3 c powdered sugar

1 tsp vanilla

1/4 tsp salt

1. Heat up the milk until warm, then mix in the yeast.  Turn on oven (to whatever temperature).  Mix the egg and melted butter with the yeast-milk, then add the flour, sugar, and salt.  Mix until it turns dough-like, then knead until soft (10 minutes by hand, 3-4 minutes by stand mixer).  Turn off oven, and cover the kneaded dough and put into oven for an hour.
2. Filling: mix all the filling ingredients together: you’ll actually want to cut the butter into the sugar so you’ll end up with a crumbly delicious mess.
3. Take out your dough from the oven once it’s doubled in size, and plop it onto a floured surface.  Let rest for ten minutes, then roll it out into a big rectangle (12″ by 8″ or so), and spread the filling on it.  Roll up from the long side, tightly, and pinch the seams and end.  Cut into 12 pieces, and place into a buttered 13″x9″ pan.  Cover and put back into that warm oven for half an hour
4. Take out the rolls, heat oven to 350, and bake the rolls for 20 minutes until very lightly golden on top.
5. Meanwhile, beat the cream cheese and butter together, then add the sugar, vanilla, and salt and beat until smooth.  Frost the rolls about 5 minutes after they’re out of the oven.