Tuesday, July 17, 2018

Fantasy High School

@carloliwitter was tweeting recently about re-imagining High School mathematics and asking for ideas. The topic has caught my fancy even though its a bit off message for the blog.  High School is ripe territory for change while at the same time rife with difficulties for anyone attempting to do so. I'll start with a model that I wish I could have had myself:  proof school.  The high schoolers there do the following:

2 math blocks every afternoon covering a single subject over 6 weeks including:
• Problem Solving + Combinatorics
• Algebra + Number Systems
• Geometry + Topology
• Analysis + Statistics
• Number Theory + Computer Science
The course descriptions sound interesting and they deep dive into number theory, abstract algebra, more advanced Geometry topics etc.

Note: Calculus is not covered.

Ok so that's where my heart's at but I don't think that within a public school setting you could achieve the same thing and the more general challenge is providing a interesting and relevant path for everyone in the system.

So here's some more ideas about changes with the caveat that these are only my impressions and desires and don't reflect a systematic view of the state of the world.

• Reinvigorate Geometry. There's enough material to split into 2 years on its own. De-emphasize all the taxonomy, volumes and surface areas. Instead elevate a proof and computer aided exploration approach. I'd particularly like to see more time to get to cyclic quadrilaterals, powers of a point, Ceva's theorem etc.  That only seems possible if you stretch the course over more than 1 year.   Then its easy to branch out to circle inversions, tessellations etc.
• Explode Algebra II.  The narrative structure isn't there and worse it often ends up being a significant  rehash of Algebra I.   What's core here and what can we add in to make this connect more clearly to future topics? I don't think touring absolute value, exponential functions,  polynomials etc. is working very well.  What's worse is that pre-calc often bleeds into this material.
• Integrate programming more fully as first class component. Maybe the capstone of H.S. should be based around numerical computing.
• Stop doing stats as an add-on over several years. Make this a first class subject and perhaps a joint responsibility of the Science and Math departments.
• Create more "labs" i.e. exploration opportunities.
• Have an applied pathway like Canada. Let's figure out for real what we really need and no more. So Mathematics is not a barrier to those who aren't going into STEM.
• More overall arc to the pathway. Students should have more of an idea of what they are heading towards and what the overall field looks like as well as some of the history of how it developed.
On the flip side what are the systemic issues making this hard to do:

• Choice models. Any model that envisions multiple pathways runs into several immediate concerns. First is that finding teachers to teach all these new subjects well is hard and anytime you have to schedule more classes it makes the logistical process more difficult. Its always going to be easier to schedule math where everyone does the same thing each year so all the teachers and period slots are more easily interchangeable.  Secondly, these models work best at large schools where there is a sufficient number of students to actually support them.  Small schools especially in low population areas may never be able to pull this off. Finally, status issues among classes easily subvert the aim of offering them. If one pathway is perceived to be more useful to getting into college then any such system will devolve into a high/low pathway with all the more ambitious students making the same choices. See: Calculus.
• Integration with College. Any such system has to not impede admission chances and that creates tremendous inertia as well as concern among students and parents.
• Tensions over tracking.   This feeds into my previous point about choice. More radical anti-tracking ideas mean you really cannot provide anything but a single pathway because one of them is likely to be come the de facto "high" pathway if left to develop naturally.
• The realities of the wide difference in relative performance by high school. Kids come into ninth grade at vastly different places within the curriculum and their own personal journey. The system needs to move everyone forward but finding a way to do this is really hard. Most of the solutions so far involve slowing down significantly and basically hanging out in the pre-algebra and then algebra space for much longer than we currently do.  We really need a way to remove math as a gateway to everything post 12th grade. Because systemically we know that isn't working.

[I deliberately didn't really tackle Calculus's place in the curriculum here. That deserves a followup.]

Monday, July 9, 2018

Book Review: Introduction to Number Theory and An Illustrated Theory of Numbers

Over the last 5 months,  I worked through both Introduction To Number Theory and  parts of An Illustrated Theory of Numbers with my son at home so I thought while the experience is still fresh I would write down my impressions.

We started with the Art of Problem solving textbook.  This is considerably shorter than some of their other volumes and its generally expected to take only half a year working through it. (Scale everything by how long it took you to do the Algebra book)  In fact, this was the first AOPS textbook I ever purchased.  A few years back, my neighbor's son was working through a problem set for the online version of this book and needed some help on the problems. I spent a weekend going through the problem set and at the end thought to myself, "Wow these are really interesting problems."  The one below was my favorite.

There are unique integers $a_2, a_3, ... a_7$ such that
$$\frac{5}{7} = \frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!}$$
where $0 <= a_i < i$ for i = 2,3...,7.    Find  $a_2, a_3 ... a_7$

So I ended up purchasing book then and reading and trying out some of the exercises over the winter in the ski lodge while my kids were taking lessons.  Five years later, I've actually had experience with a few of the other books and I now wanted something to do with my son over the back half of the year.

The book itself informally divides into 3 major sections. The beginning sections on integers, divisibility and factorization which culminate in the Euclidean Algorithm for finding the greatest common divisor. There is a middle section  which includes different base number systems and a review of some applications with decimals and fractions. Followed by the last section which deals with modular arithmetic and linear congruences.

Overall the pacing and material is a bit more uneven than some of the other textbooks in the series. I found portions of the earlier and middle chapters repeated topics from the pre-algebra book and some of the problem sets at the end felt a bit repetitive. So we tended to skim some of these chapters.  In particular, if I were revising the book I would replace the decimal/fraction chapters and perhaps even some of the focus on different number bases with a dive into the Chinese Remainder Theorem and a discussion of Diophantine equations and how they relate to the linear congruences.

However, the payoff really was in the last 4 chapters starting with the introduction of modular arithmetic. These are all  well done and have a nice ratio of practice to theory. If one were limited in time, I would focus on this section. I particularly like the work around exponent towers as a motivating problem.

Because there were a few weeks left in the school year when we finished the AoPS book and after seeing a favorable review by Mike Lawler I also purchased Weissman's book.  This is actually a very nice companion to the previous book.  What's most striking is the extremely strong conceptual/visual framework.

We ended up doing the introductory chapter and skipping ahead to modular arithmetic parts. My hope was to make it to the topograph sections but we ran out of time.  The very strong narrative strands really make this book. There are interesting conceptual pieces and data visualizations almost every few pages along with illustrations to go with them.  As  Weissman notes "Most of our proofs are given with visual explanations; geometric and dynamical proofs are preferred"  which makes the treatment fairly different from other texts.

For example, the early chapters have some lovely figurate number illustrations just with the number 100:

and then a section on using Hasse Diagrams to visualize factorization (rather than the more normal trees).

Where this really works well though is in the later chapters.   I love conceiving of the "modular world" and having it bound horizontally via factorization and vertically via powers of prime.

The final chapters which use John Conways topograph construct rather than more familiar approaches to discuss quadratic residues are particularly fun.

Overall, the only weakness I felt in the book was a need for more problems at the end of the chapter. From time to time I also found myself missing interleaved problems ala AOPS as well.  For older more experienced students I might just use the Wiessman book alone but otherwise it made for a very nice complement to the AoPS text.

Wednesday, June 27, 2018

Sometimes representation does matter

This is a small observation based on a post from @samjshah on the topic of the trig double angle formulas:

• $\sin{2\theta} = 2 \sin{\theta} \cdot cos{\theta}$
• $\cos{2\theta} = \cos^2{\theta} - \sin^2{\theta}$

Sam used the construction below to derive the formulas which is a bit different than how I usually think of doing it.

I immediately thought to myself, I can assign x to the base of one triangle and all the relationships must fallout from either the Pythagorean Theorem or similar triangles.  This is certainly true and it looked like this:

So for example using the direct definition of sin and cosine:
$$\sin{2\theta} = \frac{\sqrt{1-x^2}}{1} = \sqrt{1 - x^2}$$

which you can compare to:

$$2 \cdot \sin{\theta} \cdot \cos{\theta} = 2 \cdot \frac{1+x}{\sqrt{2} \cdot \sqrt{1+x}} \cdot \frac{\sqrt{1-x^2}}{\sqrt{2} \cdot \sqrt{1+x}} = \frac{2 \cdot (1 + x) \sqrt{1 -x^2}}{2 \cdot ( 1 + x )} = \sqrt{1 - x^2}$$

That was the expected result but didn't really seem that exciting to me.  But then I looked at  how Sam had done it. Rather than assigning x to the missing piece he directly used the trig value and did this:

This time from the similar triangles BDE and ACF you get the following relationship:

$$\frac{BD}{BE} = \frac{AC}{CF}$$  or
$$\frac{2\sin{\theta}}{\sin{2\theta}} = \frac{1}{\cos{\theta}}$$ which after cross multiplication directly gives
$$\sin{2\theta} = 2 \sin{\theta} \cdot \cos{\theta}$$

So with this representation something much more insightful for me falls out.  Trig while offering a lot of power often has the ability to obscure geometric relationships. But as seen here sometimes with careful usage it can be quite interesting. And of course the importance of the problem setup towards simple vs complex solutions to the same problem is much more universal.

Monday, June 25, 2018

geom walk through 40-40-100

1. Thought process - subdivision creates more interesting angles 2 possibilities 3x/4x or 4x/3x
2. First didn't pan out despite isosceles and angle bisector.
3. Second created more ability to break everything down into combinations of two lengths.
4. Calc as many values as possible to prove the main triangle is isosceles
5. Went down the path of making a parallelogram on the right. Wasn't useful but interesting.
6. Eventually found enough similar triangles to make an isosclese in the center and simplified a bit.

Alt:
* Per trig is possibility.
* You can translate the left hand subtriangle underneath the right hand one to make an isoscelese parallelogram! That makes its easy since it creates new parallel lines.

Constructions

1. Subdivide $\angle{BCD}$  such that $\angle{BCE} = 3x$   That makes $\triangle{ACE}$ an isosceles and AC = AE = BD.  This implies AD = BE.

Let a = AD and BE  and  b = DE

2. Add a parallel segment DF to CE.  $\triangle{AFD}$ is similar to $\triangle{ACE}$  and also isosceles so  AF = a  and CF = b.

Similar Ratios

3. By similarity of $\triangle{ADF}$ and $\triangle{ACE}$ , $DF = CE \cdot \frac{a}{a+b}$  but triangles CFD and CEF are also similar implying  $\frac{b}{DF} = \frac{CE}{a}$

This simplifies to CE = $\sqrt{(a +b)\cdot b}$

4. Now consider triangles CDE and CBD which are also similar.    So $\frac{BD}{CD} = \frac{CD}{DE}$  or $CD = \sqrt{BD \cdot DE} = \sqrt{(a+b) \cdot b}$ which is the same as CE above.

So CDE is an isoscleses triangle and  and triangle ADC and BCE are congruent.   That means 4x = 180 - 14x  or x = 10 and    $\angle{CAB} = 40$

Friday, June 15, 2018

2018 Year in Review

Unlike the last three years  (2017,  20162015)  I volunteered at the local Middle School for the first time.  So this year's reflection is all about that transition from working with upper elementary students to actual teenagers.   Going into the process I wondered whether they would be easier or harder to lead. I remember thinging "On one hand, they should be more mature and capable of greater focus on the other hand the teenage years can be stressful."   Likewise, because I planned to run one group for all the grades I was very worried about  how to provide accessible content that would be fun and interesting for everyone.

Groups and Questions

However, one of the trends that did follow from the heterogeneous grouping was the older kids tended to work together. From time to time I gently encouraged some mixing but I think also this is understandable and a social fact  I need to keep in mind. I have to maintain enough of a balance of ages in the room so that no one grade dominates and the kids feel like there is a progression where they will have grade peers as they go forward. This sort of parallels how I feel about gender. Its important to maintain enough of a balance that both boys and girls feel comfortable in the room. After that is achieved, if I don't intervene the girls will tend to work more often with each other and vice versa. So I nudge from time to time but I'm still mostly wedded to the idea that in a club where everyone comes voluntarily I shouldn't obsessively force interactions and that regular classes fulfill this function.  If everyone is actively engaged I'm happy.

Along these lines where I'm most  strongly considering a shift is in how I do questions. For the same reasons as groupings I don't normally cold call. Instead I try to actively track who has answered and call on other kids if possible. I also pull kids aside before hand and try to encourage them to go in front of the group. But even using these strategies I wasn't completely satisfied with participation in group discussions this year. There were clearly 4-5 kids who didn't want to talk and I think integrating them more fully in is worth some risks.  So I'm considering just announcing that I will call on everyone for group questions and white boarding as part of our club norms next year and explaining why.  I'm also going to dovetail this with some discussion about public mistakes since that is definitely part of this phenomena.  I still have to think about to this works for introverts (I definitely had some this year)

Curriculum

As I mentioned previously, in the beginning of the year I worried a lot about the huge potential gap in Mathematical backgrounds of the kids I would attract. In theory I could pull kids taking Math 6 all the way up to Algebra II. That encompasses a huge 5 year spread of classes. What happened in practice was almost everyone who joined had completed or was taking Algebra I.  This simplified my planning process quite a bit since I didn't worry as deeply about introducing too much Algebra along the way and I tried to tune for the general 3 year spread in classes.  However, I did have one Math 8 student where this didn't work out well. She was very quiet and reluctant to work with the other kids. Despite trying to work with her each week  I didn't realize until half way through that she felt lost at times and wasn't confident enough to ask questions. For me this was partly a reminder to ask more questions myself. But I think also narrowing the focus down will help here. If I change the club charter to be for kids at the Algebra I level or higher or at least comfortable with asking questions if an occasional Algebra concept is introduced I will be able to plan with better precision.   Long term, If I see more demand for a Pre-Algebra focused club I think it  would sense to find another adviser and run a second group. This split is very difficult to bridge otherwise.

Based on specific feedback from the kids I was pleased several of the topics I took risks experimenting with like  the King Chicken problem from (here) or the day on Polynomial Deltas from ( here)  were well received.   That's encouraging because I have a full year of meetings to fill and some kids have been with me now 3 years in a row.  I'm thinking about how to mine more Math Circle topics from various books and also on the lookout for some more art projects to intersperse. Tessellation based images are one of the first to come to mind.  I think the math history day (here) was also successful and I will think more about ways to bring history in from time to time.

Contests

I transitioned away from several local contests  I did not really like this year and replaced them with 3 new ones:

• MathCounts
• Purple Comet
In general these all were improvements.  For the most part I think I get more bang for the buck doing a contest during a club meeting. That maximizes participation which by itself is worth a lot. But in addition, it allows the potential for reviewing problems together afterwards as a group and de-emphasizes the award/ranking ceremonies.  If the kids win they of course love the trophies but for the most part I think the potential for discouragement remains high. In addition, we can still leverage all the benefits of competition in houses.  Calling something a contest still creates drive and excitement and often brings out the best work in students.

Despite being traditional in format Math Counts was a great success in particular. The kids mentioned it repeatedly when I asked for feedback.  The question banks are generally superior to the local ones so I'm mostly happy with it. However,  I did have some kids fixate on a rivalry with another school. So framing the competition and how we treat it remains an issue to work on next year.

Finally, I had one request for a training schedule for the various competitions. In the past I've assumed that the kids really wouldn't have the time or drive to do large amounts of additional practice at home so I haven't done more than send out links to old tests.  Next year I'm going to try an experiment and make a light suggested practice schedule for AMC8 on a spreadsheet and let the kids signoff on what they tried.  We'll see from there if there is more interest that I realized.

This Year's Topic Map

http://mymathclub.blogspot.com/p/2017-2018-topic-map.html

Friday, June 8, 2018

More connections

I've been thinking alot about polynomial deltas recently. See: http://mymathclub.blogspot.com/2018/05/polynomial-deltas.html.  It turns out, that there are a variety of problems where its fun to use them. Basically anywhere you think you have a polynomial function and you can curve fit is a good candidate.

For example:  Find a formula for $\sqrt{n\cdot (n+1) \cdot (n + 2) \cdot (n+3) + 1}$

You could do the algebra and factor cleverly or you could calculate the easy values around 0,1,2 ... and calculate the deltas to do a quick fit.

But I thought of another scenario this morning where I think they come in particularly nicely and answer a long standing philosophical question of mine. There's a class of formulas that are usually proven inductively where one's often left asking: "How did someone find the original pattern to test?" As a student I would just play around, but now I see these more as curve fitting exercises.

A good example of this is the sum of squares $\sum_{i=1}^{n}n^2 = \frac{n (n+1)(2n +1)}{6}$

The inductive proof is not hard, and there are some beautiful visual versions (link to proof ) but it was always hard for me to think how this was actually discovered.   Enter the deltas ....

When looking for a formula we just need to generate enough values and see if the deltas resolve. If they do its a nth degree polynomial and we can work out the coefficients.

n   sum-of-squares  deltas

0    0
1
1    1            3
4           2
2    5            5
9           2
3   14           7
16
4   30

This shows its a 3rd degree polynomial of the form  $Ax^3 + Bx^2 + Cx + D$

• from f(0) = 0 we see D = 0
• from the deltas we see  $A = \frac{2}{6!} = \frac{1}{3}$
• We can then substitute in f(1) and f(2) to get a simple system $B + C = \frac{2}{3}$ and $4B + 2B = \frac{7}{3}$
• After solving we find: $f(x) = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{6}$  which combines to exactly our original  $\frac{n (n+1)(2n +1)}{6}$

Note: you could also treat this like a linear system if you can tell what degree the function is likely to be but that's actually more work anyway in many cases.

Wednesday, June 6, 2018

6/5 Woven Math

I've been wanting to do this math/art project since I first saw Allison's artwork on twitter. I finally had enough time to practice and find the supplies.  Once you gain experience its possible to make all of the Platonic solids but 1 or perhaps 2 basic shapes are probably a good target for the first time.

Notes:
• 2 large poster boards sufficed for 15 kids and I think could have easily made twice as many structures.
• We spent about 45 minutes and everyone made 1 or 2 projects.
• Keep repeating over/under at each step and make sure to look for the correct woven pattern before taping.

Preparing strips ahead of time would be great but it was too time consuming so I had the kids bring rulers and scissors and built in time for prep work. We ended up dividing into drafter, cutter and weaver teams. As the strip cutters built up enough supply, I switched them over to weaving.

We used 1/2 inch strips that were 5 inches long.  You could probably go an inch or so longer. The kids liberally secured each joint with tape.

The basic weaving joint.

Add 3 more to make a double cross.   And then 2 more to make the first triangle.

2 more strips to reach the second triangle and 1 more to make the half ball. Finally join  the tricky last 4 strips.