I Hope This Old Train Breaks Down...

My Shiny New School Next Year!

noreply@blogger.com (Mimi) — Mon, 13 May 2013 00:52:00 +0000

This year, job search had been particularly challenging for me, since I was juggling planning for my wedding and the long-distance interviews with schools that typically hire people only after having met them in person. But, in the end, I couldn't be happier to say that I will be moving in July to a great school in Seattle!

Most exciting for me is that I'll be part of a fabulous math department. They are highly collaborative, and they love hanging out with each other on a daily basis. These math teachers are also highly reflective/self-improving, and were especially commended for this during their school's recent accreditation cycle. The school as a whole has a very special culture of sustainability, which is seen through things like the staffers taking all the kids 3 times a week to scrub down the whole school, including cleaning bathrooms and compost bins. Their kids do this (surprisingly) gladly and learn to protect their school environment. They also, on Fridays, serve food in the cafeteria to recycle leftovers from the week, as part of their sustainability theme. They're in the process of building a new green building that is solar-powered, collects rain water, and has an energy counter. When I visited the school, I loved seeing the seamless integration of student art into every corner of the beautiful, historic building. Everyone I had met -- including the class of Grade 11 students that interviewed me -- had asked me tough questions, and I tried to answer them thoughtfully to the best of my ability. It seemed to work out OK, because in the end they had decided to offer me a job on the spot, at the end of the long interview day!! (They said that they don't typically do this, but they had already gone through most of their candidates and were pretty confident that I was the best fit for what they were looking for.) Considering that at that point, I had already fallen in love with the school, I am really glad this was the outcome, because otherwise I would have probably felt totally crushed over a rejection.

I would have been happy to commute for a long time to work at a school like this, but it turns out that this school is actually in a fabulous location, right downtown! I'll be able to walk to work easily, which is an amazing perk for both Geoff and me.

I am very happy with this job-search outcome, and I look forward to a fabulous year! Geoff and I plan to stick around Seattle for a while (probably until our babies grow up to an age where it's appropriate for us to take them abroad), so I'm extra glad that I've found a school that I think I would be happy to stay at, for the duration of that whole time.

PS. By the way, as part of my interview day I had done two demo lessons for them. Despite having run out of time, I really liked the trig lesson that I planned (bit.ly/ferrisTrigWS), and I think that in the future, when I start teaching PreCalc again, it could be fleshed out into a multi-day technology project for the kids. During the demo, since we didn't have time for each kid to build their own ferris wheel animation, I simply pulled this up bit.ly/ferrisTrig to show what is possible, given their understanding of circular modeling and parametric equations. In a multi-day project, we'd start with analyzing / building a ferris wheel together, and then from there on they would create their own story involving circular rotation and minimizing / maximizing distances, and model accordingly.

PPS. An example of why this school is a great fit for me is that they were actually amused and delighted that I had negotiated with them for a better demo lesson topic. For some other schools, that could have been a deal breaker, but for them, they liked that I had a preference and an opinion about the relative boringness of topics, and they also liked that I tried to choose a less-dense topic that allowed me to showcase different ways to engage students instead of requiring me to stand at the board for most of the period. My kind of people!

Mythical Form

noreply@blogger.com (Mimi) — Thu, 25 Apr 2013 21:52:00 +0000

My 7th-graders have been doing some lovely exploration and estimation activities on circles. It took a few days, but I think it was well worth our while, as it helped the abstract formulas make sense to them.

My students today were boggled by the fact that if pi has different digits that go on forever, that means that either the diameter or the circumference is a quantity with also digits that go on forever. That means that we have a "measurable" (ie. finite) quantity that is, in fact, not truly measurable. Trippy, eh? For a moment there, I felt the beauty of abstract math peek its head into our Grade 7 class. The kids now think the circle is a mythical, awe-inspiring form.

Last Week of IB Test Prep 2013

noreply@blogger.com (Mimi) — Tue, 23 Apr 2013 14:53:00 +0000

It's full-on test-prep season, and this year I feel very satisfied with how the test prep went for my Grade 12s (who are off on their study leave this week and have requested for just one last voluntary class session with me on Friday), as well as for my Grade 11s (who are starting their mock IB exams tomorrow).

Some things that I've done throughout the year that I found helpful:

Sequence of repetitive quiz prep/practice, building up to a fairly complex quiz. I did this with my Grade 11s throughout the year, and I found it immensely helpful in repetitively drilling into them ways to think about incorporating graphical analysis into algebraic processes flexibly. I also did this with my Grade 12s regularly throughout this year, in order to go back and fill in some of their procedural gaps from last year. The Grade 12s have said to me that these quizzes have been very helpful, and more importantly, as they began to do mixed review this spring, I didn't feel like they had really any major gaps from last year yet to be filled or reviewed.
Review packets organized by topic for Grade 12s, spiralling back through topics from last year. This year, instead of waiting until the spring to do review for old topics, I started handing out monthly review packets in August and giving detailed written feedback as the packets were handed in to me. I felt that these packets were very useful for me to have a written dialogue with each kid to get them thinking just a little bit further on each studied topic, and the threat of contacting their parents when they laxed on the completion meant that the kids were responding and at least doing some amount of review during the year instead of waiting until April to think about those old concepts.
Weekly lunch time review sessions for Grade 12s starting in January, where they just did full-length old exams. Each week, I would pass out either a new calculator exam or a new non-calculator exam paper (I alternated which type to give them), and we would go over the previous week's exam paper problem-by-problem. The effect of this was that the motivated kids had a chance to try mixed problems on a regular basis, well before we finished learning all the topics in the IB syllabus. So, they got used to looking at full-length papers and feeling that sense of anxiety/uncertainty in their stomach during February, instead of during April. This was immensely helpful in building the confidence of those motivated kids over time.
During the final weeks of concentrated old-exam practice during class, I asked the Grade 12s to identify orally at the start of each class the most common mistakes they tend to make within each topic. (ie. in circle sector problems, not using the correct radian mode; or in solving equations, forgetting that you can solve a complicated equation by simply graphing for intersection) This list helped to provide them with some mental focus even as they sat down for a mixed-problem practice session.
Skimming over/discussing the last semester's mock exam problem-by-problem with my Grade 11s, right before the end of our last class before their new mock exams. Although we had gone over these problems immediately after January, they were more focused now that the stakes were up again. Taking a fresh look at old problems after a few months helped them to focus on thinking about access points into each old problem that they had struggled with, in order to encourage them 1. to go back and revisit the last semester's mock exam and topics during their review 2. to think strategically and flexibly about how to approach each problem type during the test 3. to see how far along they have come in building confidence within those old topics.

I am EXCITED!!!!! I won't find out until July how my Grade 12s will do, but I feel very encouraged by their positive efforts, their confidence, and their general outlook. I also am excited to find out how my Grade 11s will improve from last time. Even a new girl in our class (who has been with us only for about a month, after transferring over from another teacher's class, and who has been seeing me a few times at lunch for help to fill in basic gaps) is shooting for a significant growth from her last test in January. Keeping fingers crossed all around!

Quadratic Function Project Brainstorm

noreply@blogger.com (Mimi) — Fri, 19 Apr 2013 21:47:00 +0000

I'm brainstorming / laying out my end-of-year plans for my 8th-graders. After their end-of-year exam in late May, we will close grades, but we will still have about 3 or so weeks of instruction, which is enough time to do something very rich and not have to coordinate with other classes (since we use the May test to do placement for Grade 9). Last year, I used this extra time to let the 8th-graders define their own math projects, which were plenty of fun, but I wasn't entirely happy with the rigor of their mathematical results. This year, I'm toying with the idea of doing an exploratory quadratic functions unit. (Technically, quadratic FUNCTIONS are a Grade 9 topic for us, but previewing it in Grade 8 is always beneficial.)

I'm thinking of making it largely exploratory, since by then pacing won't be much of an issue and I can let them really take the time to develop their conceptual understanding of quadratic functions, which is the essential access point to a lot of higher-level algebra analysis down the road.... The timing is tight (as it was last year with my other end-of-year projects), but I think it's still doable and has a lot of potential!!!

Let me know what you think. Is it an OK approach for intro to quadratic functions / basic function transformations?? This is based on my rumination about a different way to think about flexible factorization of quadratic functions.

Day 1: Developing the understanding of how to graph y = x² + bx.

Plan - In pairs, kids will be given y = x² + 2x, y = x² + 5x, y = x²- 3x, y = x² - 7x. to graph on the calculator. They will sketch results in their notes, recording all intercepts in the form (x, y), and writing a one-sentence hypothesis about what the graph of y = x² + bx will look like.

Then, they will be reminded that in a function, we can solve for the x-intercept(s) by setting the height of the point, y, equal to zero. They will algebraically show that their hypothesis works for all b values.

Day 2: Developing the understanding of how to graph y = ax² + bx, which is a more general version of the quadratic function.

Plan - In pairs, kids will be given y = x² + 6x, y = 2x² + 6x, y = 3x² + 6x, y = 12x² + 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the graph. They will show how to solve for the x-intercepts using algebra only.

Day 3: Developing the understanding of how the graph is affected by the sign of its leading coefficient.

Plan - In pairs, kids will be given y = -x² + 6x, y = -2x² + 6x, y = -3x² - 6x, y = -12x² - 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the shape of the graph. They will show how to solve for the x-intercepts using algebra.

As part of Day 3, they will do some matching between equations and pictures of graphs and to justify their choices orally.

By the end of Day 3, they should also be able to explain in writing how to graph y = ax² + bx.

Day 4: Developing the understanding of the effect of the constant term c.

Plan - In pairs, the kids will put in a function like y = x², and look at its table in the calculator. They will be asked to generate a second function that would increase all y-values by 1. They will prove their new equation works, by showing the values of both functions side by side in the calculator (y₁ and y₂), and copying down the table values. Then, they will write down the formula for the new function.

They will then look at the graphs of the two functions to determine "what happened" visually to the original graph when the equation got changed that way.

They will keep playing around with this idea, translating upwards and downwards and checking both the table and the graph to observe/verify the effect of c.

By the end of Day 4, they will be given a graph of two functions. One of the functions will have an accompanying formula, and they will be able to see visually what happened to the points on the graph. They will then need to "guess" at the equation of the other, vertically shifted function, and verify it in the calculator.

Day 5: Putting the algebra pieces altogether

Plan - One partner of the pair will have a set of sequenced instructions, to be given to their partner one step at a time. The first step will sound like, "Sketch a graph of y = x² - 9x, labeling x-intercepts with values." Then, after that has been successfully completed, the next instruction will be given: "Now sketch the result of shifting that graph vertically up 4 units, labeling the resulting images of those original points you knew." After that has been done, the partner gives the third instruction: "Now, write the formula for this new graph." Once the partner is finished, they verify their results using the graphing calculator's graphing and table features and write a brief explanation of how they checked their results. Then, they switch, and the new partner has instructions that has to do with a downwards facing function like y = -x² - 6x, and repeat a similar sequence of instructions to generate a new graph, a new / related equation, and to verify all results against the calculator.

Both partners will then work together to complete problems starting with functions of the form
y = ax² + bx and translating those graphs vertically to get new graphs.

By the end of Day 5, they should be able to explain the connection between y = ax² + bx and
y = ax² + bx + c, and explain how to use this connection to graph any standard-form quadratic function quickly in under 1 minute.

Day 6: Practicing/drilling the connection between quadratic function equation and graphs

Plan - In pairs, they will start with a function y = x² - 9x + 1, highlight the first two terms, sketch that function using dashed lines, and then sketch in the "real" final function using solid line. They will repeat this a few times with different functions, until they can fluidly graph any y = ax² + bx + c function. On this day, they'll also learn to visualize the axis of symmetry and to write its equation by inspection of graph.

Day 7: Going backwards from a graph to an equation

Plan - In pairs, they will be given one quadratic graph with two "nice", symmetric integer points being emphasized on the graph, one of the points being on the y-axis. They will be asked to sketch using dashed lines what this function would look like if you shifted those two points down to the x-axis, and be asked to write the function equation of both graphs. They will practice this a few times.

At the end of Day 7, they will be given a quadratic graph whose two "nice", symmetric integer points are both not on the y-axis. This tests them to see if they can figure out that the translated graph would have an equation that looks like y = (x - m)(x - n) + p instead of y = x(x - n) + p

Day 8: Playing around with the idea of adjusting "a".

Plan - In pairs, they will import Dan Meyer's basketball photo into GeoGebra. We will discuss as a class the need to find a modeling equation in order to fully predict whether the ball will make it into the hoop. From there, they will choose two nice integer points, write the equation, and graph. If they notice that the curve goes through those two points but doesn't have the correct steepness desired in order to fit the photo, then they will create a slider value in GeoGebra and toggle the value of "a" until they get a good "fit" around the graph, and record their results.

As a class, we will then go over the idea of solving for "a" using an unused point (x, y) and link it to solving for the y-intercept in linear functions. They will solve for "a" this way to compare analytical results against the technology results.

Day 9: Modeling Individually

Plan - Following a discussion of examples of parabolic applications, each pair will find and import their own photos of "real-life" parabolic shapes from the web. They will then model the function in Geogebra both using technology and using algebraic analysis.

Day 10: Creating posters

Plan - Each pair will create two posters, one with the modeled functions overlaying the photos, and one poster explaining the general process of graphing y = ax² + bx + c and the general process of fitting an equation to a parabolic graph.

Day 11: Practice presentations

Day 12: Math fair for other classes / parents?!

Totally Silly but Works

noreply@blogger.com (Mimi) — Thu, 18 Apr 2013 01:14:00 +0000

I made up a totally silly call-and-response thing this year for practicing exponent rules (after we did the initial exploration, obviously, so that they could understand why the rules work). It's mad cheesy, but the kids totally remember the rules now!! The hardest part is keeping the clapping going, but I'm not sure if it's because of my students being totally off-rhythm in general or what (they're super suburban kids).

So we clap, step from side to side, and I say, "8B, are you ready?" and they chant, "Yeah, oh yeah!"

And then I call on a random kid, "Nora, are you ready?" and she chants, "Yeah, oh yeah!"

and then I call out one of the following: "Power times power", "Power to a power", or "Power over power" while holding up fingers in each hand (up to 5, obviously, in each hand) to represent the original exponents we're working with.

Depending on which one I call out, the kids need to reply with, "You gotta add them up!" "You gotta mul-ti-ply!" or "You gotta can-cel out!" in a sing-song voice, and that kid I named would then have to say the answer (resulting exponent) immediately after. (For example, if I am holding up 3 fingers and 5 fingers, and it's "power times power, you gotta add them up!" then the kid would shout out "8!")

And then we'd resume with me calling on the next kid randomly. It's mad cheesy, but it works! Afterwards, they were all loose and happy when practicing exponent rules. Every practice problem I would put on the board, I'd ask them which rule can be applied first or next, and they'd say it back in that sing-song voice, "you gotta mul-ti-ply!"

Go kids for being good sports!! It helps to make a boring topic a little less tedious! Next year, I'll necessarily add dance moves to help our kinesthetic learners. (I already have them. I came up with them after we did the exercise.)

Yup... I've got little shame left. :)

Thinking About Factorization Flexibly

noreply@blogger.com (Mimi) — Wed, 17 Apr 2013 17:44:00 +0000

I was randomly thinking about this on the way home today and truly fascinated by the teaching possibilities:

Sketching a graph of f(x) = x² + 6x + 7

is the same as sketching f(x) = x(x + 6) + 7

which is the same as sketching g(x) = x(x + 6) and then shifting g up 7 units.

Since the points (0, 0) and (-6, 0) are on the graph of g, the points (0, 7), (-6, 7) must be on the graph of f. This allows us to quickly see the symmetry line at x = -3 without memorizing x=-b/(2a).

Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 1)(x + 5) + 2

So, if h(x) = (x + 1)(x + 5), then we can imagine the points (-1, 0) and (-5, 0) from h being translated up 2 units to get the points (-1, 2), (-5, 2) on f.

Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 2)(x + 4) - 1

So, if j(x) = (x + 2)(x + 4), by thinking about the relationship between j and f, we can deduce that f must have the points (-2, -1), (-4, -1).

Similarly, we can get the partial factorization f(x) = (x + 3)(x + 3) - 2.

If we assume m(x) = (x + 3)(x + 3) and consider the relationship between m and f, we can deduce that (-3, -2) must exist on the graph of f.

So, we can pull together all those points so far to get (0, 7), (-6, 7), (-1, 2), (-5, 2), (-2, -1), (-4, -1), and (-3, -2) as points that must be on f. This way of thinking about graphing quadratics ties together strongly the ideas of factorization and transformation. They're no longer two separate concepts but integrated as one. Since I've never seen this connection in a textbook before, I decided to call it flexible factorization.

One distinct advantage of flexible factorization is that as soon as you are given y = x² + kx + m, you can quickly factor it partially into y = x(x + k) + m, which allows you to quickly determine two points on the graph, (0, m) and (-k, m) and to find the axis of symmetry at x=-k/2. You can sketch the graph roughly in about 30 seconds for any standard quadratic function (this extends to y = ax² + bx + c, as it factors into y = x(ax + b) + c, which means that (0, c) and (-b/a, c) are two points on this graph and the parabola opens in the direction as indicated by the leading coefficient "a".)

Of course, this does not mean that the kids won't have to learn the standard analysis techniques, but I think being able to connect factorization with transformation gives them another tool when modeling and thinking about graphs.

I'm going to keep playing around with this idea, possibly turning it into an end-of-year project in Grade 8. Stay tuned!

Confidence

noreply@blogger.com (Mimi) — Tue, 16 Apr 2013 04:32:00 +0000

I keep circling in my head about this. To help your weak-ability students, the singular gift you can give them is the gift of confidence in math. Yes, it's important to make the lessons relevant. Yes, it's important to have open-ended questions. But, math is inherently fun when you feel confident when approaching a new problem and can see it as a challenge rather than an obstacle, and this is a gift that they can retain even after they leave your class.

Confidence. That's what it all boils down to, for me anyway, when working with weak-ability students. I love this time of the year when I can see the transformations that have occurred from August until now!

Wedding Dress Saga - in Hindsight

noreply@blogger.com (Mimi) — Sun, 14 Apr 2013 06:16:00 +0000

Since I only get married once (I hope), I think it's OK for me to recap the crazy journey that was to find a wedding dress!!! I really dislike German wedding dresses. I think most of them are over the top, unfortunately. Most of the dresses I tried on just overwhelmed me in appearance. I felt like a girl swimming in an ocean of overly heavy fabric and overdone details.

This was one that I liked, if I would have gotten married in a colder climate. My high school best friend loved this one! I liked the sense of movement in the fabric pattern, but I didn't like the fact that I'd have to wear a frame underneath in order to hold up the poofy shape.

Once I started looking at shorter dresses, this was the favorite one that my girls had sent me. It's simple and beautiful (with the gloves) and the cut looks like one of my existing dresses. I thought quite hard about getting a dress like this made from scratch, and even investigated costs associated with this. But, I was a little concerned that since I wasn't planning to wear a veil on the (windy) beach, that the rest of the dress would be overly casual.

Then when I found this dress, I liked it right away. There's something about the simple elegance of the dress that I loved. The cocktail style made me feel like I could walk through a party and play hostess without being bogged down by layers of fabric. But, I didn't love that the left breast was not covered in the same material as the rest of the dress. (It was of a bra material. Why?! German wedding fashion is so weird.) I also didn't cry like the girls in Say Yes to the Dress when they found their favorite dresses.

Unfortunately, when I went back to the store a month later, this dress was already sold. I bought one that was the same style, but 4 or 5 sizes too large. They had to rip it apart, re-make the whole thing to fit me, and also hand-make crinkled fabric in order to cover the left breast to match the rest of the dress.

I also liked the idea of mixing and matching short and long dresses, so I sent them this picture to ask them to make this detachable train for me from scratch. I wanted a detachable train for functionality and -- so that just in case I hated it in the end, I could still take it off.

When I picked up my finished dress (which costed only 500 Euros even including all the alterations), I was in love. (A few fittings had to occur, remember? It was too small after they resized it, and so they had to resize it again a couple of times.) I was thrilled by how fabulous my seamstress was; you literally couldn't tell that part of the crinkle on the dress was made by hand. I bought a hair accessory to match the modernness of my dress, and a single-pearl necklace to match its simplicity.

Then, because of all the health- and job-search related stress, I lost weight. I don't have a scale at home, but I estimate that I must have lost between 5 and 10 pounds from pure stress. So, I had to bring the dress back, etc etc. get it altered again so that it didn't look like a borrowed dress. In the end, it was such a relief that everything still worked out!! (Reposting final picture from an earlier post... Once I get the professional photos, maybe I'll post an update.) So, whew. If you're a bride-to-be, good luck!!! I hope you can get through the whole process with fewer fittings than me. I estimate that in all, I had to have about 8 or 9 fittings, which is well above the average for a bride-to-be and is a lot for a gal who goes shopping maybe twice or three times a year and who tends to shop like a man (in and out in 45 minutes with a lot of things purchased... Geoff shops way slower than me!).

Final Months in Berlin

noreply@blogger.com (Mimi) — Sat, 13 Apr 2013 21:19:00 +0000

Amazingly, time is flying by and we find ourselves sprinting towards the last months -- ready or not -- before another big life change.

1. Geoff's going back to Seattle after our pre- and post- wedding hanging out time in Berlin. This time, he's bringing with him most of our wall decorations that we wish to keep (paintings, cuckoo clock, vintage posters, marionette, and some other knick knacks from our travels). It's helping the reality of moving sink in...

2. I am in the process of actively looking for jobs in Seattle, with some positive/encouraging progress. In May, I hope to ask for a day off to spend a long weekend interviewing with schools in Seattle. (So far, one interview is for sure. Another one during the same weekend would be fabulous to have, if I can get the extra day off and rebook my flight into Seattle.) These are two really great schools and they both seem quite interested in continuing the conversation of hiring me, so I'm feeling overall pretty hopeful with the job-search prospects.

Coupled with this, I also need to look into the logistics of filming my class just for about 10 minutes. Two different Seattle schools have offered this to me as an alternative to teaching a demo lesson on site. If I can get this working, I could just send the link to other schools that request the same...

3. In other news, since I'm the department chair, I am simultaneously interviewing potential people to hire into my current department. There's something that feels pretty funny about interviewing other people and being interviewed all at the same time, especially when some of those people I'm interviewing are supposed to fill my spot. One person I've interviewed so far is a rock star, and I hope secretly that he'll take the offer so that I can leave my students in good hands.

4. Next week will be my final full week of instruction with my Grade 12s, and also the last week of instruction before the semesterly mock exams in Grade 11. I feel quite excited to see how they will do!!

5. In Grade 9, we're doing my favorite 3-D project again. The kids are making good progress so far -- they've already gotten their designs/dimensions checked off, and half of the groups also had their calculated volumes already checked off. Next week, we'll be working on the surface area calculations, drawing 2-D nets, and starting the construction during class. EXCITED!!!

6. The sun is rolling in, slowly but surely. It's my favorite time of the year in Berlin!!! In May, there will be the annual Karnival der Kulturen, and in June, we're going to try to work out one last trip -- to Kiev (where my friend has offered for us to stay at her condo) or Vienna or Bamberg (with their unique "smoked beer"/rauchbier local breweries) or kayaking through / camping by the beautiful lakes surrounding Berlin. Lots to choose from, but so little time left!! I'm sad just thinking about it.

Survey Project - I'm Back!

noreply@blogger.com (Mimi) — Wed, 10 Apr 2013 15:08:00 +0000

I recently gave a survey project to my 7th-graders, that involved them creating/administering a survey, creating circle graphs with a protractor, drawing conclusions from graphs, and making educated predictions for a larger population.

Here are some clear photos I managed to snap of a few of their posters. I am impressed by how informative they are, considering that the kids didn't turn in any rough drafts. (Note to self: Threatening to turn it into a full-blown writing assignment really helps to bring up the quality of submitted posters.)

See full-sized yellow poster here

See the complete blue poster here.

See the full-sized cream poster here.

PS. Yes, I got married!! Here are a few pics.

Belize looks like this:

Snorkeling with sea turtles!

Our beach ceremony (people sat in a spiral form). It was awesome to have 50 friends and family join us from as close as El Salvador and as far as Sydney and Shanghai!

Newly weds! (My dress worked out fine in the end. It was short, with a detachable train.)

A couple of days after the wedding, we had a sunset cruise with the guests who were still around. It was lovely!!!

Health, Wedding, Keeping in Perspective

noreply@blogger.com (Mimi) — Sun, 10 Mar 2013 14:59:00 +0000

I just wanted to give a quick update. I've been having a lot of health issues lately, which makes it very difficult to look forward to the wedding. In the past few weeks, I've been several times near tears when I tried to talk to people about how I feel about the wedding coming up so soon, and other times I do my best to be optimistic. I'm finally (I think) coming out of the other side of that tunnel, where my health is improving substantially and I'm having a handle on the situation. But, we will wait and see if this holds.

I had an acute eczema outbreak on my face in late January. It started out actually in the first week of January, when I started to have one puffy eye. I didn't know what it was so I went and bought some OTC eye creams to put on it. Within a few days, it got super red, tender, and much more puffy. I got scared and stopped the OTC creams. At this point, after doing some research on the web, I switched over to an all-natural eczema cream for the eye. Probably because of the contact with the OTC creams, however, the allergy opened up and oozed continuously for about two days (during my presentation at AGIS, as all poor timing goes). I kept throwing vaseline and the natural cream on it, and then it almost healed completely, except I got some strong soap in there on accident, and it opened up again and oozed for several days this time. The second time it almost healed, I was using a gentle face wash that Geoff normally uses for his sensitive skin. No matter, the whole thing opened back up, oozed for about 5 days till I got to a dermatologist, who prescribed me a total of 10 days of topical steroids. 3 days were with a strong topical steroid, followed by 7 more days of tapering with a weak steroid. As I applied the creams, obviously the wound quickly healed, but as soon as I stopped and got off the steroids, it came back with a vengeance, opening back up and oozing daily.

In the meantime, I had changed my diet completely, started a new "low histamine" detox plan, and have been religiously drinking herbal teas from my acupuncturist. The detox went well -- I had cut out all processed sugar, all night shade veggies, all citrus fruits, all processed foods, and was only consuming freshly cooked, high-fiber items with little seasoning. When I first got on the detox, my skin went wild. I started getting super itchy, bumpy spots all over my body, and my face was breaking out like mad. I did some emergency research on the web, and found out that this is a common reaction to detox, called "detox crisis." If your change in diet is very drastic, your body tries to expunge all the surface level toxins through your skin because it's unable to get rid of all built-up toxins quickly enough through other means, and so your skin reacts immediately in a negative way. Fortunately, this all resolved itself within a week, and now that I'm two weeks into the detox, I'm feeling much better and even my acute eczema has really calmed down. I find that everytime I "cheat" and eat something that I know I'm not supposed to eat, my body reacts vehemently in negative ways. It's a good evidence that for years, I was feeding my body things that I shouldn't have been, which manifests itself now as a food intolerance.

For example, last night was my Hen Night out with the ladies. They were fantastic -- they brought me curries, caipirinas, Baileys, and all kinds of yumminess to my house before we headed out for the evening. I decided that since this is once in a lifetime, I'd indulge in whatever they brought, and sure enough, a few hours into the evening, my face which had been doing well for two days, started to open up and ooze. sigh. This morning, it's doing better but I had a random, sudden swelling inside my mouth that later subsided after some minutes, which I think is related to the things that my body consumed and considered harmful last night.

At least I have it all narrowed down to food, which is something that can be controlled (although unpleasant). I've done extensive reading on the web, and I've decided that steroid creams are not the way to go. In fact, a lot of people who've used steroid creams on their eyelids, even for a short amount of time, had immediate "rebounding effects" that were like mine, which suggests that steroid creams could be the cause of the subsequent flare-ups and could be causing the condition to worsen. (There is also extensive reports out there for people who use steroid creams for longer, and they become super addicted topically.*) Fortunately, as an alternative I've found many all-natural creams in Germany that do agree with my skin. (Germans are all about the natural products. You can walk into any pharmacy and pick them up, without any prescription.) My favorite products so far are Dr. Hauschka Med, which is a medical-strength line of products developed all naturally with a special "ice plant" herb, that penetrates deep into your skin to offer moisture and protection. I use this on my body, and I find that within days it has made a huge difference in terms of how protected (not itchy) my body feels, and even areas that have not responded to any other ointment/lotion have changed in a short amount of time. My other favorite product so far for my eye is a Urea cream. Urea is the main ingredient in your urine, and urea creams draw moisture to your skin by combining with the hydrogen in the air. They're very gentle and so with my eye area, which is actively irritable, I find it to be the safest product. I alternate between using that and a weak antiseptic salve specific for eye area, depending on whether I am going to sleep or not. (Overnight, I prefer a salve because it's a thicker coating, therefore a better protection against environmental allergens like dust mites.) The herbal teas will also help over time; from my conversations with the acupuncturist and my own research on the web, I can tell that my liver and spleen are damaged from years of imbalanced diet, and they will need time to heal. My acupuncturist thinks it could be a matter of months before my liver/spleen return to good health (in which case I think I could start to eat a more normal diet again), but it's nothing that you can rush, really.

Anyhow, this is really helping me to keep everything in perspective. As we approach the wedding, I just have the hope of being healthy, going on vacation, getting married. I've inadvertently lost a lot of weight over this issue (over the stress and also recently over the detox, I'm sure), so I'm not sure if I'll look as great in my final wedding dress as I had hoped, but as long as I am feeling healthy on the big day, that's the most important part and I know that Geoff'll be happy as well.

Everything else just fades away in importance.

And, as I said, I think I'll get there. I'm already miles ahead of where I was a week ago, and now I have an entire arsenal of natural products, and my detox is going strong (last night notwithstanding). With any luck, we'll get a little bit of sunshine around here that will help my skin to heal. (I hate to say this, but not getting any natural Vitamin D when you live in Berlin is really detrimental to your skin health. I'm not surprised that my doctors and my acupuncturist have all commented on the frequency of acute eczema outbreaks this year, given how terrible the weather has been since AUGUST.)

So, here's to health, to marrying a good man, and to good friends who will take care of you when you are down both physically and emotionally.

*Here is a chronicle of someone's journey in overcoming steroid cream addiction. Go look at the pictures. There are lots of people like him, who suffer repeated, cyclic outbreaks of eczema when they're under steroid treatment, but whose outbreaks become much less frequent (eventually non-existent) over time when they allow their bodies to heal naturally. I'm not anywhere near this, obviously, but I have to be careful because I have taken steroid inhalers in the past for my asthma, and all of those things add up over time... This is the reason that after one visit to the doctor and trying the steroid treatment, even though the next doctor also prescribed me steroid cream, I opted not to use it and to resort to natural products instead.

Independent Theatre in Berlin

noreply@blogger.com (Mimi) — Sat, 02 Mar 2013 07:31:00 +0000

I recently went to a cool little theatre event at the English Theatre in Berlin. (This was only the second time that I have been there. The first time, one of our friends was in a play about the woman who had discovered the double helix structure, Rosalind Franklin.) They're having a festival of theatre in which everyday for two or so weeks, they put on a different set of shows. The festival ends this week, so I decided to organize a little group to go on Thursday night to check out a series of short plays.

The lineup had about 10 or so acts, and I had thought that we would arrive at 8pm and then sit through 10 very short plays. Not so! When you arrive, you get randomly assigned a colored ticket, whose itinerary has already been pre-determined for the most part. Because they have a variety of things going on and the theatre is small, they had to be creative about how they used the space, and plays were going on simultaneously in all parts of the facility. Everyone started off watching a play in the main theatre space, about a woman who had gone crazy. And then after that they sent us off in all directions.

Actually, our friends group got further split up into two smaller groups (they numbered us off, 1, 2, 1, 2...), at which point my friend Elsa and I went together to see a modern version of Joan of Arc. The J of A play we saw was very interesting and it was a two-part play, each part occurring inside a small office space in the back of the theatre. When we sat with "Joan" in her office, we could only hear her side of the phone conversation, and then when we went to the other room afterwards to see the other half of the play, we got to see the other personalities (the two counselors who were putting Joan on trial), in their office. It was really interesting, and we really liked it.

Our other friends, however, weren't so lucky. They were sent off to see a play about child molestation. (Yikes.) So, we decided to leave at 10:30pm, when they said that Elsa and I needed now to switch with Max and Mateja's group.

But, anyhow, the format was really interesting and unique. I really liked it, and thought that was very Berlin!

-----------

Recently I went to another theatre event with these guys. It was also on a Thursday night, I think, and it was called Haus Theatre. As the name suggests, we showed up and it was someone's house! The play took place in a living room, and that (regular-sized) living room had a natural sort of stage layout. All 30 or so audience members sat in rows on one side of the living room, and the actors were on the other side, using an archway connecting to the adjacent room as a way to enter and leave the stage. That play was about friendships, and it was pretty good -- intense but still quite funny at the same time, with a lot of comic relief. The three actors each had a big role, and although they were non-native English speakers (I think they were Spanish, German, and Italian), they were able to interpret those roles brilliantly! The story was about what happens when you feel that your friend has made a mistake (by spending 50,000 Euros to buy a painting that looks just... like a white canvas). Super interesting as the story unfolded and you saw all these opinions they had always had about one another. Afterwards, you could just hang out and mingle with the actors, which was the whole idea. They said that this was the first in a series they planned to do, but they would always look at the space first and then choose a play that fits that space.

Berlin is great!! I love this independent art scene. I'll definitely miss this city.

The Nerd in Me, Looking out for Mr. John Venn

noreply@blogger.com (Mimi) — Fri, 22 Feb 2013 21:21:00 +0000

So my friend posts this picture via Instagram. There's a place in Portland, OR, called Blue Star Donuts, that sells glazed donuts and fried chicken.

True to my math teacher spirit, my immediate, instinctive reaction was: "I dig it, but this is a misuse of the Venn Diagram. YUMMY should really be the superset of glazed donut and fried chicken. Their intersection should be... well, RIDONCULOUS!"

...I've already come a long way from my nerd roots, I promise. I just didn't want Mr. Venn to be rolling over in his grave. Who has two thumbs and is always looking out for the dead mathematicians?

How I Do IB Test Prep

noreply@blogger.com (Mimi) — Thu, 21 Feb 2013 20:16:00 +0000

I don't really like teaching the IB, because I feel as though there are a lot of topics to get through and it's hard to build all the concepts from the ground up (given the time constraints), so you really have to rely on the kids having a pretty strong foundation in the fundamentals prior to arriving in your class.

HOWEVER, I am definitely getting better at it. Kids are pretty happy in my class because they understand the topics and even the convoluted IB problems are becoming more accessible to them over time.

I was thinking about what made the big difference between last year and this year, and one thing is that I am giving consistent practice quizzes at the start of class on a non-trivial skill, through the course of 4 or 5 consecutive class meetings. Following that as a sort of long Do Now, we pick up with "regular" instruction for the rest of the period, on a usually unrelated topic that is concurrent. The quizzes and the instruction run parallel, but they're not usually on the same topic. After 4 or so such practice quizzes, I give a graded quiz on those skills. (Which may or may not be around the time when I give a coherent unit test on the concurrent topic. The kids don't seem to mind that there are always two topics going on at once, because they understand that the quizzes are only to build up specific test-prep skills, and the unit-based learning we do is more coherent and also much more time-consuming.) The key here is that the practice quizzes have to be sufficiently complex, yet similar each time in format and content. The first time they see a practice quiz, they usually cannot do it and they require me to explain the problem to them thoroughly after they try it. But if they're attentive and they keep good notes and they actually try all the practice quizzes earnestly, they can get close to 100% by the real quiz even as I add slightly more complicated parts on the final quiz. This is very important, because 1. it builds their confidence with complex problems, 2. it gives me an opportunity to reinforce the integration of multiple concepts.

For example, recently we did a set of practice quizzes in Grade 11 that looked like this (while we are learning logarithms and log / exponential functions in our "regular" instruction):

"For the function f(x) = 3x^2 - 2kx + k + 1, find all value(s) of k that would...
a.) Cause f to have two x-intercepts.
b.) Cause f to have one x-intercept.
c.) Cause f to have no x-intercept."

As part of this problem, the kids would define the discriminant function in terms of k:
D=(-2k)^2 - 4(3)(k+1), graph it in the calculator, and then use the graph to analyze when the discriminant value is zero (k = -0.791, k = 3.79 when rounded to 3 sig figs). They will then conclude that when k is those two approximated values, f has one x-intercept. They would then test it, by substituting k back into the equation for f... f(x) = 3x^2 - 2(-0.791)x - 0.791 + 1 and f(x) = 3x^2 - 2(3.79)x + 3.79 + 1 both look like they have only 1 x-intercept.

They will then look back at the same discriminant function they had set up, D, and then visually conclude that the discriminant function has a positive value when k < -0.791 or k > 3.79 (the discriminant graph is above the horizontal axis, which means that it has a positive value). And in order to test their hypothesis that k < -0.791 or k > 3.79 would cause f to have two x-intercepts, they will then replace nice integer values k = -1 and k = 4 into the equation for f, to verify that f(x) = 3x^2 - 2(-1)x - 1 + 1 and f(x) = 3x^2 - 2(4)x + 4 + 1 both have two x-intercepts. (And they do.)

Eventually, they repeat this process for the hypothesis that -0.791 < k < 3.79 will cause the discriminant function to be negative (below the horizontal axis), which will cause f to have no x-intercept. They can test it with a nice k value such as 0, 1, 2, or 3.

Well, what is the point of all of this?

1. Notice that I base the problem-solving around graphical analysis, even though we know that we can solve quadratic equations and inequalities without a graph. Research shows that the more you emphasize graphical analysis on the calculator, the better the kids get at visualizing equations and inequalities as graphs. In the long run, when you take the calculator away, they will still be willing to manually simplify their discriminant function, to sketch the graph, and use it to aid their analysis of variable k and its relationship, ultimately, to function f. And that ability to visualize is a powerful tool to have.

2. Specific to this topic, the notion that k and x are related but not the same can be often confusing to students who are not used to doing such sophisticated algebra analysis. A lot of times last year, my (former) 11th-graders used to ask me, "But why can k have two values when they're asking for there to be only one x-intercept for the function f?" Breaking down the process as we did above addresses that specifically, because when the kids substitute the values of k back into the original equation, they can see that the graph they're shooting for works BECAUSE the parameter k has taken on appropriate values. It helps to separate the meaning of k from the meaning of x.

3. For your weaker students, repetition breeds familiarity, which then breeds confidence. They also get to hear me explain the same concept 4 or 5 times, over the course of 4 or 5 classes, which is very helpful for them. They also get to use their notes on all the practice quizzes following the first one, which gradually forces/allows them to be independent while building up to the real quiz. You cannot achieve this type of comfort level by practicing/drilling completely different-looking problems everyday, even if the problems are essentially on the same core topic. Less is more, I think. Once they build an in-depth understanding of a single problem type of sufficient "juice" and complexity, I believe that they can then more easily transfer that understanding to other problems.

I know this method works, because not only did my 11th-graders do quite well on the mid-year mock exam this year, but generally speaking, when I think back on the topics that we have learned this year, I feel that they don't have collective holes/gaps as a class. We've gone through and filled them all in, using this consistent quiz policy that runs alongside our regular instruction of new topics.

I also use the same practice-practice-practice-practice-then-quiz policy to help my 12th-graders do spiral review from last year. The difference is that I try to integrate even more topics for them. On one set of practice quizzes, for example, they needed to: write a sine function from a graph; describe the step-by-step transformations from y=sin(x) to that graph; graph another quadratic formula within the same grid, labeling all important info; shade the enclosed area between the two curves; write an integral that represents the area between the two curves; evaluate the integral by calculator; then show how they can get the same integrated result by hand. I gave them 5 or so practice quizzes leading up to the quiz on this one, because there were so many different skills involved that they needed to practice. This is how I pull it altogether for them, using a combination of mixed concepts and (still, much needed) repetition.

I feel that we (meaning, my Grade 12s) are in much better shape this year, going into the review period, as a result of this quiz policy that I adopted. We had started the spiraling quizzes on last year's topics, all the way back in August. By now, I have gone through and drilled most of the particular weaknesses that I felt that they needed to see again. This method is systematic, more focused, and much better than just giving them random mixed practice! I feel that, even though we're not yet done learning all the topics (we still have one more to go, which is Vectors), my kids are already pretty OK now with doing mixed IB practice on their own and needing only moderate support from me. If only they can keep up the stamina for doing extra practice on their own during the remaining weeks, then I know that they'll be in great shape by the end of April!

I thought I would share this, you know, because I really think it has made all the difference in my IB classes. In fact, I have been trying to do the same in my MS classes, by spreading out test practice over the course of a week or two, leading up to the test, instead of just giving a single practice test. It feels kind of like a huge waste of time, because the practice quizzes take about 20 minutes each class. But, in the end, you're actually saving time because you're increasing mastery by providing more regular feedback and multiple opportunities to self-assess. This year, I didn't have to spend any extra time reviewing equations in Grade 7 after I taught that unit, because the kids all had the algebra skills down pat. So, yea, do try it if you're not already doing this in your class, and I hope that it helps to address a variety of learning and mastery issues!

Visualizing Concepts

noreply@blogger.com (Mimi) — Wed, 20 Feb 2013 16:31:00 +0000

Here is an MS update. I feel pretty productive lately, as I always do during the second semester. I also feel quite productive with my Grade 11s, and as a result I'm taking on three new kids potentially, at least for a while. Grade 12's are doing OK, but the pressure is sure ramping up for their IB exams, so there's not a whole lot of "cool" instructional things that I can be doing with them...

Grade 8:

Referring to the system {5u + v = 21, 10u + 3v = 48}, one middle-of-the-road kid explained excitedly to her friend, "Oh! I got it! If 5u + v = 21, then 10u + 2v must equal 42. If you put that here into the second equation, then you will still have 1v on the left side. Which means that 42 + v = 48, and v must equal 6!"

Not bad for being day 1 of systems algebra. (Of course, they've been doing the same reasoning using shapes and visual substitution for several days now, so the transition to symbols was seamless. I didn't bother with even any examples on the board this year, and it worked out just fine without one.)

I went up to a boy who was one worksheet ahead, and pointed at the equation {y + 6x = 10,
y + 2x = 2} to ask him what immediate conclusion he could draw based on inspection. He immediately said, "X is 10 minus 2, divided by 4." I had to backtrack to ask him, "How do you know? Because 8 is equal to....?"

He finished my sentence, "4x." Great, now we're talking about the process.

I showed him the traditional notation for showing that work / reasoning via elimination, even though I know that in his head he's doing substitution from one (smaller) equation into the other.

y + 6x = 10
-(y + 2x = 2)
4x = 8

He was not impressed, nor surprised. When you introduce systems via pictures, the symbols become just part of the bigger concept. I explained to him that on the previous worksheet, I had let them just do much of it in their heads, but now we're going to work on the written communication piece, to carefully show our work on paper. I also gave him the hint that sometimes, instead of subtracting the smaller equation, he'll notice that he wants to add the equations instead. I said that he'll know when addition is necessary, because those equations will just "feel different." I left him on just that hint, and sure enough, 20 or so minutes later, when I came back, he had already identified the equations that needed to be added, and he was ruminating over the explicit mathematical reasons why that works. At that point, I felt that he was ready to discuss that if you have additive inverses, then you can just add them to cancel them, so we had a 1-minute discussion about that and I left him again.

I am very pleased with how this unit is going. I think I've gotten it down pretty pat; if I remember correctly, last year I didn't have to change any worksheets at all, and so far I haven't had to alter any worksheet either. This is one unit where I can just sit back and watch the kids' thinking unfold, more or less, and I can be very hands-off in their own building of the concepts of algebraic substitution and elimination. At some point, I looked around the room when I noticed the noise level rise, and saw that it was because literally every pair of kids is engaged in some sort of intense mathematical discussion with their neighbor. Awesome-o!

Grade 9:

In Grade 9, we've been working on learning / understanding / memorizing geometry area formulas via cutting and pasting. We each re-arranged, glued down a parallelogram to form a rectangle, to help them understand why A=bh for a parallelogram. Then, we each re-arranged a trapezoid into a thin parallelogram, to help them understand why A=(b₁+b₂)(h/2) for a trapezoid. This year, I went a step further and highlighted the base 1 and base 2 sides using a green marker, so that the kids can see that they line up within the parallelogram when we do a "move and flip" of the top half of the trapezoid. This simple highlighting technique is superbly visual for my visual learners to see why the new parallelogram base MUST be (b₁+b₂).

Then, I proceeded to give them two practice trapezoid problems. In both cases, I had them draw out the parallelogram that it becomes, labeling the side lengths to emphasize the numerical connection between the trapezoid and its resulting parallelogram. It worked great! My concrete thinkers really latched on after the first example. One of the weakest kids in my class went up to the board and bravely (and correctly) drew out the trapezoid, its new parallelogram dimensions, and the calculated area, without any previous verification that he was on track. I was proud!!

I just love geometry lessons like this, because they involve both tactile and visual learners and help the concept "stick" in their heads. Slowly, I'm getting their geometry concepts up and running in order to do the 3-D project this year.

Addendum March 4, 2013: Here is a visual that Lara has made on Pinterest. Thanks, Lara!

Grade 7:

In Grade 7, we've also been doing Geometry... I'll have to report on that at another time, but I wanted to say that recently (prior to the Geometry unit), I discovered the best technique for teaching setting up of proportions. For ages it used to bother me that kids would not check that two ratios have matching units across the numerators (and across the denominators), and so half the time they would set up incorrect proportions. I figured out a trick this year. The first day we learned to set up proportions, I had them highlight the matching units across the equal sign. For that whole first day, they had to always highlight the units inside the proportions, and check that the matching colors lined up horizontally. After that first day, the highlighters went away and I never referred to them again. In the end, I didn't have a single kid mix up the positions of the numbers inside the proportions on the test. I think that mentally, the colors stuck with them and they're always visualizing that check, even when the highlighters aren't around anymore.

Again, slowing the kids down in the beginning definitely pays off, I think. The highlighters are a trick that I'm trying to use more and more, to incorporate hidden visualization techniques that some of us "math people" tend to internalize in our minds but that teenagers who are used to rushing through things, could need more explicit instruction on. So far, so good.

Substitutes

noreply@blogger.com (Mimi) — Tue, 19 Feb 2013 21:49:00 +0000

I am such a perfectionist; I just hate being absent.

I have to say that I am almost never absent. In the 7 years that I have been teaching, I can count the number of days that I have been absent. If I am absent, it's not because I was so sick and weak that I needed to crawl home. For those cases I would stay at school. If I am absent, it's because:

1. I'm going to a conference or job fair: I missed 1 day this year for a conference, and 1 day 4 years ago because I needed to attend a job fair.
2. I need to catch a flight for a wedding that is out of town: I miss 1 day every 2 or 3 years for this.
3. I'm getting married myself: I am taking 2 days off for my own wedding this year. Hopefully this is a once-in-a-lifetime type of thing.
4. My doctor absolutely isn't available in the entire month except during work hours, and the situation is urgent. I've missed about three or four half-days for this in all the years that I've been teaching.

However, sometimes we need to be absent and we need to have substitutes. And I HATE THAT. I cannot seem to substitute-proof my lessons. I try to give them all the details: all lesson material, with extra copies and neatly labeled answer keys and instructions on what to write on the board and how many answers to check. I try to make the lessons SO easy to run that a non-mathy person can still do it, and I even talk to the kids in advance and prep them for the activity. If there are support teachers in my class, I brief them in advance, send them all the lesson material, and make sure they know what should be happening in case the substitute teacher is unclear. In the end, I come back and the kids are like, "The sub didn't write anything on the board. They didn't do anything." Last time, I made spiffy pencasts and tested them on the computer under another teacher's login, and everything, and the substitute teacher didn't have access on their own login account, and the whole thing was a fail even though in my mind, all they had to do was to hit Play. That was not their fault... however, moral of the story: I cannot substitute-proof my lessons!!

ARGH. So, tomorrow I'm out for a half-day. I planned it so that I'd miss only one single-period PM class, thereby minimizing the damage. I've made all my answer keys and even color-coded them and everything. Keeping my fingers crossed that whoever takes my class will defeat all odds and make me proud.

What do you do to substitute-proof your lessons?

Visits #2, #3, #4

noreply@blogger.com (Mimi) — Sun, 17 Feb 2013 15:41:00 +0000

I've been insanely busy! OMG. This February break will go down in history as the most stressful break ever. At night, we were going out to meet up with old friends in Seattle, but because I was bussing to all these schools, I had to wake up at 6am every morning to get ready (and Geoff also woke up at 6am for moral support). But, I think it was well worth the effort, because I got to see a cross section of different schools and to talk to a bunch of math teachers in the area.

Besides the great school that I saw during visit #1 (which, by the way, now has an official MS opening posted on their website), I saw another great school during visit #2, in a location that is not within walking distance of downtown but is fairly central, within easy bussing or carpool distance. I took the bus there and it took me about 30 minutes. It's a small and pretty new school, maybe just about 300 kids from K to 12. They seem to have really wonderful administrators, and they even asked me to go back to do a demo lesson the next day. (This felt funny to me, of course, because I typically think that most of the hard work is in preparing for the lesson, not in delivering it. And it had been many years since I had been asked to do a demo lesson, so I was actually a bit nervous!) In the end, the demo went fine, I think, but it's too early in their interview process for me to know if they're really interested. (They actually hadn't posted the job yet, so I was the first candidate they were speaking to about the job, and that was only because I had emailed them out of the blue, requesting a visit.)

Visit #3 was to a large public school, and this one was pretty far, well north of the U District, if you are familiar with Seattle. The principal was very down to earth and kind, and she showed me around their math hallway (there was a lot of integrated technology -- smartboards, doc projectors, etc.) and told me that they have a few National Board Certified math teachers. I was pretty excited about the possibility of teaching at a good public school, but unfortunately they're not hiring at the moment. She said that if I am willing to, I could apply to the district and get an offer from the district (city) level, if I am willing to be dispatched anywhere in the district. She also alluded to the fact that they don't do cool, integrated teaching of mathematics like they do in all the other subjects, because math is high stakes and kids have to pass end-of-course state exams in order to graduate. That's a shame to hear, and it reminded me that I have to be very careful about public schools, in order to not end up somewhere that forces me to teach to the test.

Visit #4 was much more relaxed, because I visited a friend from PCMI on her home turf. She teaches at a coed Catholic school, in the boonies. It took me close to 2 hours to bus out there, and then on the way back, because I had missed the 1:10 bus, I had to wait 2 extra hours for the next bus to come. At least I got to check out the ponies at the street corner while I waited. (No joke.) But, her school was lovely! The campus was huge and beautiful, the technology was wired through the roof, and the kids seemed well-balanced with interesting personalities and "strong academics", as she said. It was a nice school to check out, because it helped me realize that no matter where I work, it's always going to be a tradeoff -- schools in the boonies will be equipped with great facilities, but the tradeoff is that I'll have to commute that far out from the city. Schools closer to the center will be less space-equipped, so when I evaluate them, I need to use a different set of criteria...

But, anyhow, this trip has been very productive! It has helped Geoff and I make the decision that we'll definitely go through with this cross-pond move, even if I may have to be flexible and to take whatever may come (including doing part-time gigs till I find a full-time job). So now, it's off to wedding planning and job searching all at once. Yikes.

Visit #1

noreply@blogger.com (Mimi) — Tue, 12 Feb 2013 23:13:00 +0000

I went to visit my first Seattle school today. It went well, I think. I loved the school. The kids were very nice; the teachers went out of their way to make me feel welcome; and I saw a lot of great math going on, from Grade 6 math to Multivariable Calculus. The location of the school is also fantastic -- within walking distance of my old hipster neighborhood. If this school is interested in hiring me, I would be thrilled!

But, so that I am not keeping all my eggs in one basket, I am visiting at least two more schools this week in hopes of just networking, seeing what is out there.

Job search is always tricky. I know that if they decide to hire me, I will adapt to whatever situation it is, but the question is always how to get that foot in the door, when you don't know what other candidates have to offer?

Anyway, I can only be myself and then hope for the best! Good thing there is that whole wedding planning thing to distract me.

They Like Logs!

noreply@blogger.com (Mimi) — Thu, 07 Feb 2013 23:25:00 +0000

I don't know if it's a coincidence, or if there are other forces at play. But today, during class, I noticed that all of my 11th-graders are solving all kinds of log and exponential equations fluidly without accepting any help from me. They were even completely comfortable finding inverse equations given a function like f(x) = a*b^(cx - d) all by themselves.

Amazeballs. This is the first time that I think my students as a whole really understand logs!

I still believe that the credit goes to this. Sometimes, it just pays to slooow theeem dooowwwn.

Good Things and Bad Things

noreply@blogger.com (Mimi) — Tue, 05 Feb 2013 20:52:00 +0000

Good things:

I've been thinking about little changes that have big impacts. For example, recently my colleague asked me for some articles on teaching with technology. When I was reading up on various research done about teaching via graphing calculators, I learned that how the teacher teaches with the calculator actually has a great impact on student learning and flexible problem-solving. If a teacher always emphasizes the connection between algebra and graphical analysis using the calculator, then even when you take away the graphing calculator, more of the students are able to think flexibly of multiple modes of solving problems. So, I have been pushing my Grade 8 students to be more and more reliant on the calculator as a daily tool, rather than just irregularly incorporating it.
This change has allowed me to take on an even more passive role in my Grade 8 class (which is good, because that means they have to be even more independent). Now when I go over answers to worksheets, we only go over a subset of the answers, during which I call on a student, they provide their answer, and then I turn to the class and say, "Does everyone agree?" If they agree, we go on, and I never have to say true or false. If they disagree, then I pick a person to say step-by-step how they did the problem, and after each step I ask the class, "Do you agree with everything on the board?" Eventually, the class helps them to find their mistake, or we all agree on their answer and other kids try to figure out their own mistakes. After reviewing about half of the worksheet answers, I give the class another 10 or so minutes to verify the rest using a graphing calculator. My 8th-graders have become really good at graphing a function on the TI, adjusting window range, and then using the numerical-entry feature of Trace to quickly verify (x, y) pairs on the graph. They also know that they need to graphically check 2 points on a line in order to verify its equation, and they know how to verify their predictions along the line such as checking the value of k in (1000, k), or checking the value of n in (n, 849). On the test, I built in extra time for them to just check everything on the graphing calculator, and in the end, the kids said that the test really wasn't so bad. (Even though it had at least one quite tricky PSAT problem and other parallel, perpendicular, collinear testing problems that are fairly complex for Grade 8.)
My 7th-graders are getting very communicative about math. Today, we played a modified Bingo Game to review for our test on Thursday. I had them write in integer values of -5 to 9 in a 4-by-4 grid, with 1 "freebie" space anywhere. Then I started writing questions on the board, one at a time. Nothing special, except we weren't going over the answers like we normally would. Once they determined the solution to a problem, they can cross that solution off of their grid, but they had to put the problem's letter (A, B, C, .... etc) next to the crossed out number, so that if they got Bingo, we could verify that they actually had all the correct answers associated with the correct problems. Sometimes I noticed while walking around that the kids were getting stuck on a problem, so I would ask, "Who can give a hint for how to start this problem?" and kids would eagerly raise their hands to offer hints. Along the way, they offered many hints like, "Cross multiply!" "Reduce before you divide!" "Find the common denominator!" "Check by putting the values into the equation!" and they also helped each other set up the percent increase/decrease problems as proportions, multiplying decimals, and finding "weird percents" like 0.1% of 3000 or 400% of 0.5. These 7th-graders are not just getting really good at algebra, but they're getting all the descriptive terminology down, too! Sometimes, they noticed that they had marked the same number as being called twice during the same game, and they had to go back to figure out which problem was solved incorrectly, and that was another way of having them self-monitor instead of me monitoring them. Eventually, when someone called out, "Bingo!" they would give me the problems and the solutions associated with those problems, and instead of me saying whether each answer was correct, I would ask the class. If the class agreed, we'd let the kid go on to the next number. Else, we stopped to go over the problem on the board. Again, I keep thinking about how I can hand over more and more of the "correctness" control to the kids, and today was a good day in Grade 7 for that.
I recently started my weekly lunch review session with my 12th-graders. I told them right off the bat that these sessions are totally voluntary, but the kids who come tend to do a lot better on the IB exam. It's not the one-hour studying during lunch that makes the difference. In fact, when they come, they just sit and do independent mixed practice using old exams without my help really. I am helping to model what it should look like to study at home, and my physical presence builds their courage to try unfamiliar problems, I think, knowing that I can be there to help if they do get terribly stuck. The first session went very well last week. I plan to alternate between non-calculator paper and calculator paper each week, in order to build up their ability to switch gears and to think in a different mode during a different setting. So, this week we'll be doing a calculator paper. Whatever they don't finish, they'll just take home as additional homework, since I expect that they're now putting in at least a couple of hours each week to do mixed practice on their own. I have seen them grow a lot during the last year and a half, and I know that they will do well if they put their minds to it.

In the end, I received some very positive feedback from those of my 9th-graders who had put in a lot of work into their videos project. They said that even though in the beginning, they weren't totally comfortable with the topics that they had chosen and the problems that they needed to explain, by the time that I had made them re-do and re-do it, they thought the concept was very easy in the end. The question that remains is only how I can manage this in the future for all kids, even those who put in minimal effort, and how to extend this level of articulation to all topics, and not just the one that they chose at the semester mark.

Bad things:

I am sick and still allergic, and I feel like I am walking around in a fog. I really hope that I get well by Saturday, since I'll be seeing Geoff for the first time in over a month! (He has been working away from Germany, and finally I'll be visiting him during my February break.)
I also lost weight recently, probably due to stress and all that jazz. It's definitely not intentional, but now my wedding dress is too big and I will probably have to take it back to the store again. I am feeling quite anxious about this, because now the clock is ticking and I don't want to risk another alteration. blah.

Berlin Dining Scene

noreply@blogger.com (Mimi) — Sun, 03 Feb 2013 15:41:00 +0000

Geoff and I love food. Geoff loves experimenting with new restaurants, and I always have my favorites no matter where we live, that I frequent on a regular basis. Between the two of us, we have tried a good amount of places here.

In Berlin, my favorite/highly recommended restaurants are:

Goodtime, which is a fairly expensive Thai restaurant with great food and great ambiance. My favorite location is the one in Zehlendorf. But, it runs quite pricey there. For an entree with a pot of tea, that can cost you over 20 Euros per person. I've also been to the Goodtime located in Mitte, and that was nice as well.
Since Goodtime is quite expensive, I have found a cheaper option that is equally tasty and located right in my neighborhood. Papaya is also a mini-chain, but the location on Kleistpark is by far my favorite of the two locations I've been to. The food there is so flavorful, exquisite, and spicy that, this summer when I was traveling in actual Thailand, I was craving the Thai food from Papaya. Papaya is not cheap compared to a lot of Berlin places, but definitely a cheaper option than Goodtime. An entree with tea will run you around 15 Euros. My favorite dishes from here are Ped Pad Ki Mau (fried duck with Ki Mau soy sauce, basil, chili, fresh peppers, etc) and Penang curry.
Addendum April 21, 2013: There is another nice Thai restaurant called Sida that is great for a group (if you make reservation early) and has good food across the menu. Their foods are flavorful and very affordable!
Yogi Haus is by far the best Indian restaurant in Berlin. We've been to several others, including some famous ones from Tripadvisor. Yogi Haus is huge, but it gets very crowded on Friday and Saturday nights, completely packed sometimes including all indoor and outdoor sitting areas. The mango curry there is to die for, and the price is great at this place.
There is an Ethiopian restaurant that is located half a block away from my house. It's called Abyssinia, and it's right around where this restaurant used to be.They have excellent service, and their food is delicious. I regularly order their Doro Wot, which is a type of red curry with chicken and a hard-boiled egg. If you're a fan of Ethiopian food, I highly recommend this place. They're so relaxed and so great about letting you hang out there, too.
Addendum April 21, 2013: There is another famous Ethiopian restaurant in our neighborhood called Bejte Ethiopia that has great reviews, but honestly it's not nearly as good as Abyssiniea.
Besides that, a couple of blocks away from me has the most delicious Chinese restaurant I've yet been able to find in Berlin: Chi Chi Kan. They have some dimsum type of things, but those are just OK. The best things that they have, in my opinion, are their lamb chops with bok choi, and also their Exotic Chicken appetizer. It's a bit on the pricey side; if I order a starter, a pot of tea, and a main course, it's a lot of food but it can run close to 20 Euros.
Mustafa's on Mehringdamm is definitely not overrated. There are kebap places everywhere in Berlin, but their unique combination of crispy toasted bread, juicy meat, stirfried vegetables and potatoes, spiced fresh "salad" (lettuce and tomatoes and onions and such toppings), and flavorful sauces makes this place a magical kebap place even though the lines are so, so, SO slow-moving. Even if there are only 15 people in line, expect to wait for about 40 minutes. But the food is so tasty that it's definitely worth the wait.
The best German Sunday brunch places I have been to are Cafe Morgenland and Deponie Nummer 3. The former is always impossible to get a reservation, and the latter is always quite free to go at the last minute.
Addendum April 21, 2013: We have tried a few other great American-style brunch places. The California Breakfast Slam and its sister brunch place, the Chicago Breakfast Slam have absolutely banging! breakfast! The Mexican-styled breakfast with beans and eggs and tacos, guac, sour cream are to die for, and their French toasts are complex and mouth-watering as well.
The best Berlin cappuccinos I have tried are from Double Eye on Akazienstrasse and Maxway Cafe near Winterfeldtplatz. The latter, unfortunately, is in the middle of training new baristas, so sometimes your coffee can be very disappointing. If you care more about the ambiance than the quality of the coffee, then Cafe Bilderbuch is your best bet. The back part of the cafe just feels amazing, like you are sitting in someone's livingroom.
I am not a big fan of German cuisine, but Marjellchen is very delicious. They somehow turn the traditional German fare into juicy, flavorful affairs, and the atmosphere is very relaxed. Of course, you should anticipate to pay some extra money, because this is a restaurant that is popular on TripAdvisor. But, it is well worth a visit.
It's funny to come to Berlin to eat burgers, but if you live here, The Bird is a staple. This place is crazy; you have to make reservations even on a Tuesday night at 10pm, if you want a seat!! Once I went with my friend at 10pm on a Tuesday night with no reservations, and we had to stand at the bar to wait until someone kindly gave us their seat. Not only do they use real steak as meat, but their fries are also intoxicatingly good. The burgers are huge, so go with an appetite! Maybe afterwards you can walk through Mauerpark (the Bird is next door to the park) to burn off some calories.
My Japanese friend Mamiko recommended to me two Japanese restaurants, both of which I really like. Sasaya serves traditional Japanese food, and is quite expensive. The one time that I have been there, I ordered an eel rice, and it was the best eel rice I've ever had. Full stop. You need to reserve a spot though, like a week in advance, because the restaurant is quite small. Cocolo is a delicious ramen noodle bar, also small, near Hackeschermarkt. They have a small menu, but everything on their menu is mouth-wateringly good. In the winter time, expect to wait outside to be seated, because the restaurant really is that small and that popular. Next door to Cocolo is another hip sushi restaurant called Kuchi. Mamiko and I intend on checking it out tonight, so I'll keep you posted on our assessment.
Addendum April 21, 2013: Mamiko and I actually didn't go to Kuchi, but went instead to a place called Hashi, which means chopsticks in Japanese. It's a Japanese snacks place -- and Mamiko absolutely loved it! She said the foods are super authentic and make her feel like she's back in Japan.
Of course, if you're already shopping on Ku'Damm anyway, the 6th floor of KaDeWe has lots of bustling gourmet food stalls. You should check this out, because it's a great touristy experience.
Nocti Vagus is an eating-in-the-dark gourmet cabaret restaurant. When we went, we had a lovely time, and both the food and the service there were excellent. It was a set menu (you get to choose from vegetarian, meat, or "surprise menu") of about 50 Euros per person before drinks, so definitely prepare to spend a fair bit of money if you plan to go. There is always a performance during dinner, and for us that was a lovely little surprise to hear the musicians perform in the dark.
Of the high-end restaurants that we have tried here, Don Camillo is probably my favorite in terms of its combination of food and service. They don't have printed menus. Instead, they bring you the ingredients and just describe to you how it's going to be made. We sat out in the garden on an autumn evening, and it was very comfortable pace for a very expensive meal (a couple of hundred Euros per person). Definitely something for a very special occasion. Geoff also likes Remake, which is on Big Hamburger Street (Grosse Hamburgerstr), funnily enough. We went to Remake for celebrating our engagement, and there their specialty is in finding new twists in old ingredients. Semi-recently we also went to Tim Raue, where the food was great but the service was horrible. (The chef basically came out to shout at us for the waitress's mistake in ordering me food that I had already said in the beginning that I was allergic to.)

I hope that if you visit Berlin sometime, you'll find this list to be helpful! :) Mmmm food...

Trying Acupuncture

noreply@blogger.com (Mimi) — Fri, 01 Feb 2013 21:02:00 +0000

This week, I went to my first acupuncture appointment. It was fabulous. Actually, I was so surprised by the experience that I came home and did some extensive googling on both acupuncture and my specific acupuncturist. I was very satisfied by both results.

I have had asthma since I was young. As I grew older, the number of things I am allergic to seem to increase every year. Dogs, cats, dust mites, some types of trees, cockroaches, shellfish, alcohol... The list goes on and on, and those are only the things that I've tested positive to, officially. I haven't had an updated allergy test now for a while. My asthma got really bad when I was living in New York (with all the cockroaches, I guess); once, I landed in the ER after being sick with a bad cough for a month, when my asthma simply stopped me in my tracks and I lost the ability to breathe in any air. At some point, my skin started to itch really bad all the time, in some variation of eczema (skin allergy). Recently, a whole area of my face swelled up and just cracked open, which is another form of eczema allergies. It's mostly healed now (with very diligent care on my part), but I told myself after this embarrassing ordeal that I'm tired of living like this, and that I would try and fix my allergies at the root.

That's where my opinions differ from that of most Westerners. I believe that allergies can be fixed. You don't have to live with allergies, if you find the right traditional doctor who can suppress your body's unnatural reactions to environmental stimuli. Steroids and antihistamine are just there to cover up the symptoms of your poor health.

I find that Western medicine can be very limiting in issues that deal with internal, non-surgical medicine. In the past, when I had recurrent infections, I went to Western docs and they just kept feeding me antibiotics. I would go on an antibiotic regimen for 2 weeks, feel better, and then my meds would run out and that infection would come back. I'd go back, and this time they'd give me a stronger antibiotic, which would drive the infection away for maybe an additional week before it comes back. The cycle kept repeating itself, which was driving me mad. It was out of sheer desperation that I turned to traditional medicine. The only thing that eventually cured me was when I went to a doctor in China (during a well-timed visit to my parents), and the doc decided to feed me herbs that would make my body an unwelcoming environment for bacteria. After I started taking those herbs, which was about 3 years ago, I've never had any problems since. (I had to take them for about 2 months, let's say. But, it was TOTALLY WORTH IT.)

So, this time, I did some research and I found out that there was a great acupuncturist in Berlin, who came highly recommended by everyone for treating a variety of illnesses, from sports pains to allergies to digestion problems to menstruation issues. I went to her today, and it was a great experience. She was super professional and attentive, and asked me many questions -- unlike the recent dermatologist I went to who had only wanted to hand out some steroid cream and push me out the door. We did the needle thing, which was actually totally weird, cool, and relaxing. She also gave me some herbs, that I'll have to pick up from a traditional pharmacy soon. (I am quite amazed that I could even do that here! I cannot be happier!!!) I know it'll take several months for me to see whether this thing really works, but for curing what has been more or less a lifelong allergy/asthma ailment, I am very willing to be patient and experimental.

It probably sounds a bit funny to you, but I hope that you are reading this and finding that there is hope to cure whatever ailment you are having. Don't let the limitations of Western medicine stop you from exploring other possibilities. Very rational, scientifically-minded people that I know are big fans of traditional treatment. The difference is in how open you are to unfamiliar experiences, that's all.

I cannot wait for my next acupuncture appointment. :)

Grade 9 Videos Review Project

noreply@blogger.com (Mimi) — Wed, 30 Jan 2013 19:58:00 +0000

I am learning from a not-great experience with having kids create videos to recap the skills that we had learned during the first semester. I'm jotting it down here so that you can read about how it went and help me to make it better next time.

I still think the idea was good. We had done so much great multi-stepped, contextual algebra practice during the first semester (with my "low" Grade 9 class), that I didn't want them to just leave it all behind as we move on to new Geometry topics in the second semester. I didn't want to have to come back in June to re-teach them everything they knew, but I knew that retention would always be a problem for these kids.

So, I came up with the idea that we'd divvy up all the topics from the first semester, and each group would be responsible to make some explanatory videos on one topic. They'd upload the videos to the web, and I would provide links for all the kids to access these instructional videos. Then, during Spring Break, I'd assign as vacation homework for the kids to watch each other's videos and to do just two or so practice problems related to each video. This way, they're somewhat refreshed on the old concepts over time, and it also takes the pressure off of me as the "all-knowing info source" when it comes to review time.

Sounds good in theory, except I totally underestimated what a huuuuge task this would be for a group of kids who cannot really self-monitor their progress very effectively. They really did try; that I am impressed by. I had helped them prepare for the filming last week prior to leaving for the AGIS Conference, in hopes that they could just use my day of absence to film their videos on the iPads and to upload them. Little did I know that it was not going to be so easy. When I got back this week and looked at the videos, I was pretty disappointed. Most of their videos either had inaccurate mathematics, or the problems they picked were too easy (or, sometimes, too ambitious). I blame that on myself; if you want it to be done right, you simply have to closely supervise the kids in order to give them just-in-time feedback as they are filming and pulling those last pieces together. So, today in class I gave them another 80 minutes to re-do and re-do their mathematics and their videos. This time, I checked in with every group to make sure that their math was production-ready by the time they started filming. Even then, they still had to repeat the filming a bunch of times just to avoid all the careless mistakes. It was just so tough for them to master the simultaneous communication and solving of a multi-stepped problem. In the end, it was really good practice for them to zoom in on their own mistakes and to keep re-doing to correct them, even though the final video quality was not great. (On the iPads, the audio and the video are both of weak quality.) Fortunately, since the kids had mostly selected (with my help) the topics that they had individually struggled with on the semester test, this was a great remediation strategy for them to have to create these explanatory videos, regardless of video quality.

In the future, I really need to think carefully about what technology to use, how to set up the room so that different groups can be filming at the same time, how to help them rehearse prior to filming, etc. There is too big of a range between groups who put in a lot of effort into this project to make a good video, and those who just kind of slapped something together and called it a day. I also need to think about the timing, because unfortunately, this project takes too much time in class, as the kids need support all the way through (including the filming parts) and part of me just wants to move on to new topics already, knowing how much ground we still have to cover.

It's quite a shame that this project didn't work out to have superb products, because I think the process was definitely worthwhile and many of the kids learned a fair amount while doing this. It was definitely challenging for them, and I think some of their frustration came from how challenging this task was. I have hope though, that with some restructuring, I can find a way to make this work much better the next time. One of the restructuring ideas I had, for example, was that instead of everyone doing review videos all at once after a whole semester, after each unit I'd pick a small group of students (who had performed weakly on the exam) to do the videos for just that recent topic. This will ensure that timing is less of an issue, because they'd just work on it outside of class with my help, say at lunch time, and it'd also ensure more immediate remediation. I would also be able to ensure the videos are of better quality, since I am only focusing on managing one group at a time. Have you ever done something like this? Can you share any tips with me to make this a more successful experience in the future?

Addendum: Here are a few samples of produced videos. http://bit.ly/linesReview1, http://bit.ly/linesReview2, http://bit.ly/linesReview3, http://bit.ly/anglesReview1, http://bit.ly/anglesReview2. The kids in my "low" group had to create videos for lines, quadratics, midpoint/distance word problems, drawing geometric diagrams, and analysis of angles. Blip.tv only lets me upload 3 videos each day, so I'll post more links later!

Intensive Feedback for Every Kid

noreply@blogger.com (Mimi) — Mon, 28 Jan 2013 13:34:00 +0000

I have said this before and I will say it again: I find that mini whiteboards are wonderful in giving immediate feedback to students and receiving immediate feedback from them. This year, after I started using mini-whiteboards on a semi-regular basis in Grade 7, I have seen my students growing leaps and bounds in their accuracy. They absolutely love those lessons because they love to be recognized for being correct. (I usually say after a complicated problem, "Pat yourself on the back if you got that one completely correct." They love patting themselves on the back. heehee. And come on, who doesn't?) By now they're used to the idea that when I say something during the lesson, it is going to help them during the mini-whiteboard practice, so their ears actually perk up to listen. That is night-and-day compared to their attention span on other days.

In one 80-minute lesson last week, we reviewed: multiplying a 2-digit number by a 1-digit number in our heads; finding the common denominator of two fractions like (3x + 5)/24 = (6 - 2x)/7 as an application of this arithmetic skill; cross-multiplication and why it works (this was following DAYS of fractional equations practice, so I felt that at this point they were ready to bypass the denominator part and ready to see why the numerators would change as such); solving various proportional equations using cross-multiplication; solving percent word problems using proportions. In fact, they were so great with this exercise that they were able to figure out that something like (3x - 5)/6 = (2x + 7)/4 would have no solution, which is a topic from a while ago that I just threw in to the mix.

In one lesson, basically all the kids practiced and understood all of these skills. Of course we'll go back to individual/paired practice this week as we build up towards a formal assessment, but having their intensely focused attention for 80 consecutive minutes and receiving/giving constant individual feedback from/to every kid is simply priceless. It does wonders for their progress towards mastery.

In fact, the word has gotten out that I use these mini whiteboards regularly and that I love them... Other-subject teachers on my floor have started to borrow them from me to use in their classes. Great!

...Wow. We're so ready as a class to move on from basic algebra to explore basic geometry. I can actually feel the anticipation! :)

Gestures, Language, and Student Understanding

noreply@blogger.com (Mimi) — Sun, 27 Jan 2013 10:50:00 +0000

I went to a fabulous AGIS session this morning on sign language in the elementary classroom. It was led by Armin Martin and Johnnie Wilson from the Munich International School. They showed videos of kids who use their hands to touch different parts of their heads (front, back, left, right, top, bottom), in order to figure out how many "faces" a cube has. The kids are able to link the mathematical word "face" to layman's definition of "face", in order to bridge the gap between the concepts.

A very interesting point that was made during this presentation was that signs can be used as an intermediary between normal language and academic language, or between home language and school-instruction language. Intentional incorporation of appropriate signs can be a strategy that works not just with our ESL / EAL population but even with our normal kids, and it ties nicely into math because when you gesture in space, you are quickly illustrating and bringing in extra dimensions that are hard to do/experience on paper. The presenters presented research that said that even when you later take away the gestures, the kids still retain the primitive, physical understanding that they had achieved through gesturing. So fascinating, because this discussion/session got me thinking about a lot of different things that previously I had thought to be disconnected:

1. Recently my 7th-graders have been working on percent word problems such as "64 is 40% of what number?" Sure, some kids can easily navigate the proportional reasoning --> 10% of that number must be 16, so 100% of that number must be 160. But, for many kids they need a different strategy, and so we have been practicing setting this up as a proportion. Even then, for kids to read a problem like this and then to consistently set it up correctly, is not trivial. They need to be able to:

a.) Parse the verbal description.
b.) Correctly associate the value given (in this case, 64) with either the fractional percent (in this case, 40%) or the whole (100%).
c.) Set up proportion accordingly.
d.) Solve algebraically.

Of this, part B is the most difficult for 7th-graders. I found this year that when I went around to conference with kids about this process and to help them get started on the assignment, I can just point at the value within the problem (64), and then gesture to them using distance between my hands to ask, "Is this the part or the whole?" This has helped them tremendously, because they can associate the rather multi-stepped numerical operations to a simple visualization, and then they only need to focus on part A (re-reading the question to themselves) in order to make that determination and to carry out the rest of the steps by themselves. This simple gesturing was able to shrink my conference time with each kid to under 1 minute; they immediately would say, "Oh, I get it now," and then proceed with the other questions which were not always phrased in the same way or giving the same information. ("42 is what percent of 70?" "What is 20% of 95?" etc.)

2. When I taught 8th-grade back in New York, I taught one particular 8th-grader who was very economical with his words. I would always put explanation questions on the test, and he was so concise with his explanations that he could always write down the correct answer in about half of the word that I would need to use. This is an incredible skill, because in order to do this, the kid needed to:

a.) Know/master the concept and relevant vocabulary.
b.) Prioritize information in his head.
c.) Formulate his understanding in as few words as possible using the prioritized list.

3. I have read somewhere that babies can already understand physical rules in the world. If you play an optical illusion on them that is against their normal experience, they will keep staring at the place where the ball disappeared after dropping. At this stage, their understanding is far beyond their ability to verbalize it. So, I think it is definitely true that we understand a lot more than our words are able to describe, especially at a young age or as a language-learner.

So, in short, I think gestures are a fabulous way to help kids understand concepts when their language is not yet developed enough to explain or describe a concept (old or new) fully. But, this intentional incorporation of gestures should lead in mathematics to a formalization of those concepts, and attachment of specific language. Because in doing so, we are teaching the value of specificity and prioritization, which are very important skills for the older students to have/develop in the long run.

Hope this little reflection was helpful to you in jogging your brain about how to bridge abstract concepts quickly to visual/kinesthetic understanding for kids. Please share with me if you have had similar success in other examples of utilizing intentional gestures to build intuitive understanding!