Tag: Ed Frenkel

Sharing math with the public and especially with kids

My wife and kids are up hiking in New Hampshire this weekend and I’m home with a cat who misses the kids. Yesterday I was watching Ed Frenkel’s old Numberphile interview about why people hate math:

The line about 50 seconds in to the video has always really resonated with me – “How do we make people realize that mathematics is this incredible archipelago of knowledge?” As has the his point later in the video that mathematicians have not generally done a great job sharing math with the public (say from 5:00 to 6:30).

Frenkel’s piece has played a role in many of my blog posts, here are three:

Sam Shah – a high school teacher in New York – wrote a great piece about sharing math that is not typically part of a high school curriculum with kids, and gave some suggestions for projects:

A Partial Response to Sam Shah

Lior Pachter wrote an incredible blog post about sharing unsolved math problems under the Common Core framework. I copied his idea but used math from mathematicans rather than unsolved problems:

Sharing math from Mathematicians with the Common Core

Then when the sphere packing problem was cracked by Maryna Viazovska earlier this year, I wrote about how this was a great opportunity for mathematicians to share a math problem with the public:

A challenge for professional mathematicians

As you can tell, I watch Frenkel’s video quite a bit 🙂 While I was watching the video yesterday I received this message:

Though he isn’t a professional mathematician, this article from Brian Hayes is really close to what I’d love to see from mathematicians.

As are the articles by writers like Erica Klarreich and Natalie Wolchover at Quanta Magazine:

Quanta Magazine’s math articles

and mathematician Evelyn Lamb who has somehow found the time to write more than 150 articles on her “Roots of Unity” blog for Scientific American:

Evelyn Lamb’s blog on Scientific American’s website

There were probably at least 10 to pick from, but here’s an example of how I’ve used one of Lamb’s pieces to talk a little bit about topology with my kids:

Using Evelyn Lamb’s “Infinite earring” article with kids

So, with that all as introduction (!) I was very excited to see Steven Strogatz share an article from Rich Schwartz last night:

I really enjoyed our project with Schwartz’s “Really Big Numbers” and I’m happy to see that he’s writing more about sharing math with kids. Hopefully Schwartz’s article will inspire a few more mathematicians to share some fun math with kids (or with the public in general). I’d love to expand this list of projects beyond 10 🙂

10 Pretty to implement math activities for kids

Does this math course exist?

I’ve spent the last few days thinking about how students can learn about math that is normally outside of the school (both k-12 and college) curriculum.

The topic has been on my mind for a while, actually – pretty much since seeing this Ed Frenkel interview several years ago:

Frenkel’s talk has inspired several of my blog posts.

I wrote this one after seeing a project that Dan Anderson did with his students:

A list Ed Frenkel will love

Then, after seeing Lior Pachter write about how some unsolved problems in math fit nicely into the Common Core:

Lior Pachter’s “Unsolved Problems with the Common Core

I sort of combined Pachter’s idea and my thoughts about Frenkel’s interview into several different posts in the last couple of years:

Sharing math from Mathematicians with the Common Core

10 pretty easy to implement math activities for kids

A partial response to Sam Shah

This week I ran across two new ideas that got me thinking about sharing math, (and not just with kids). The first (I saw thinks to a SheckyR comment on a recent post) is this interview with Keith Devlin:

Keith Devlin’s interview: On learning and what it means to be human

This quote right at the beginning (around 3:40 into the interview) really struck me:

“If the last experience with mathematics is what you learned – certainly up to the middle level of high school – and to a large extent to the end of high school . . . you’ve basically never seen mathematics.”

Then I saw this tweet from TJ Hitchman:

I think the Hitchman and Devlin ideas are connected – if all you are seeing as a student is the math that is part of the normal school math programs (which, at least where I live, seem to be driven by what’s on the state tests) it would be pretty hard for anyone at all to get excited about math.

So, how do we, as Frenkel asks, get students to “realize that mathematics is this incredible archipelago of knowledge?”

A new idea crossed my mind this morning – and it isn’t that well thought out, but . . . .

One of the most influential-after-college classes that I took in college was a year-long physics course called “Junior Lab.” The idea in Junior Lab is that over the course of each semester you’ll do 6 (I think) famous experiments in physics (out of maybe 20 total choices). The website for the course is here:

Junior Lab’s website

After you do the experiments you present the results to your instructor as if you were the one doing the original experiment. As I wrote half-jokingly to my old lab partner, this is the most scary room on campus!

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You, of course, learn about the experiments, but there are so many lessons beyond that. The class teaches you about the breadth of physics, about experiments not working the way they are supposed to (!!), about presenting and defending results, and about writing papers.

It seems like the Junior lab format would be a great format for showing students math that isn’t typically part of a k-12 or college curriculum. It is a few steps beyond what Dan Anderson did with his “My Favorite” project, but, I think, would give students a totally different perspective on math.

It would be about as far away from a “learn this fact / take this test” type of math class as you can get. The students would have a wonderful opportunity to learn about many different areas of math and math research, and, as I mentioned above, the lessons from this class would reach far beyond the math.

In any case, I was wondering if there is a course like this anywhere. I hope there is because I’d like to think through the idea a little more carefully.

“

A challenge for professional mathematicians

[March 24th, 2016 update – I’m going to link some articles at the end of the blog as I see them. There are two from today. I’m really happy that people are writing about this!]

I saw this article on gravity waves via a Steven Strogatz tweet this morning:

Seeing the article reminded me of the interview that Numberphile did with Ed Frenkel a while back – in particular, the part from roughly 5:00 to 7:00 when Frenkel discuses the need for mathematicians to do better at sharing their ideas with the public:

Frenkel’s point is that even though the ideas in fields such as biology and physics are just as complicated as the ideas in math, these other areas of science are much better at communicating with the public than mathematics is.

I was reminded of Frenkel’s point again this morning when I learned that earlier this month Maryna S. Viazovska solved the 8-dimensional sphere packing problem. Viazovska’s paper on arxiv.org is here:

The sphere packing problem in dimension 8

Maybe I’m a little biased – especially right now because I’ve been spending this week playing around with 4-dimensional shapes with my kids . . .

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but I think that the sphere packing problem (i) is something that can be explained to the public (it certainly seems less complicated than gravity waves) and (ii) is something that the public would find to be interesting. There’s not been much of any coverage of Viazovska’s result, though. Here’s what I found doing a simple Google news search:

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So, it sure seems this new result is something that would be great to share with the general public. There are, of course, many different directions an article could go – just off the top of my head:

(A) Jordan Ellenberg does a great job explaining the sphere packing problem and the connection to things like the Leech lattice and Hamming codes in How not to be Wrong,

(B) John Cook and Keith Devlin both have recent blog post with connections to higher dimensional spheres / cubes:

The empty middle: why no one is average by John Cook

Theorem: You are Exceptional by Keith Devlin

(C) Two years ago, Steven Strogatz shared this wonderful paper on N-dimensional spheres:

(D) The 2-dimensional problem of circle packing is something anyone can understand and is pretty fun to play with – here’s an old project I did with the boys using disc golf discs, for example:

 

Screen Shot 2016-03-22 at 2.51.32 PM

Sphere packing (well . . . circle packing)

Also, a version of the circle packing problem was in Jim Propp’s most recent blog post about mathematical thinking:

Believe it, then don’t: Toward a Pedagogy of Discomfort

So – come on professional mathematicians!! – here’s a great opportunity to promote a neat result and bring some really cool math to the public’s attention. Don’t let the physics crowd have all the fun!

A few articles that I’ve seen:

On Gil Kalai’s blog:

A Breakthrough by Maryna Viazovska lead to the long awaited solutions for the densest packing problem in dimensions 8 and 24

Kalai’s blog post also led to a question on Quora:

Why is the solution in dimension 8 such a breakthrough?

Sharing Math from mathematicians with the Common Core

Last fall Lior Patcher wrote a fantastic blog post about unsolved problems in math and the Common Core math standards:

Lior Pachter’s “Unsolved Problems with the Common Core

This piece made a big impact on how I think about math that is worth sharing with kids.

The general idea of sharing math with kids has been on my mind since I saw Numberphile’s interview with Ed Frenkel:

The line about 50 seconds in to the video, in particular, really resonated with me – “How do we make people realize that mathematics is this incredible archipelago of knowledge?”

In the last several weeks a few other experiences have had me thinking more about sharing math with kids. First, I ran five “Family Math” nights for the kids and parents at my younger son’s elementary school. It was amazing to see how excited and enthusiastic everyone was about topics from pure math. The 2nd and 3rd graders excitement over Larry Guth’s “no rectangles” problem (to be described in more detail below) was incredible.

Second, Cathy O’Neil’s piece How do we make math enrichment less elitist which discusses Peg Tyre’s article in the Atlantic The Math Revolution made me think even more about how to share math with kids – all kids.

Finally, just yesterday at the gym I was re-listening to Marcus de Sautoy’s wonderful series podcasts: “A Brief History of Mathematics.” One part of the section on Hardy and Ramanugan basically stopped me in my tracks – de Sautoy claims that Ramanujan’s discussion of the sum 1 + 2 + 3 + . . . = -1/12 was one of the ideas that caught Hardy’s attention when Ramanujan first wrote to Hardy. That sum was the subject of an interesting (and quite controversial!) video from Numberphile.

Hearing that story sort of tipped me over the edge and made me want to write about sharing fun math ideas with kids. These ideas don’t require anything too fancy or $1,000 math clubs or anything like that – just an internet connection and some pencil and paper. Finally, with a hat tip to Lior Patcher, I point out how the ideas fit into the Common Core math standards for grades K – 8. All of the information about the Common Core math standards comes from this page: The Common Core Math Standards

For Kindergarten – mathematical coloring sheets:

I first saw the idea of mathematical coloring sheets from the online math magazine Math Munch. They published some coloring sheets made by math artist Dearing Wang. I used Wang’s coloring sheets with my kids here:

If you like great math for kids, check out Math Munch

Then, a Google Plus post from mathematician Richard Green got me thinking more about how you could use advanced mathematical ideas to get kids talking about shapes:

Using a Richard Green Google Plus Post to talk about Geometry with my son

Finally, the amazing book Patterns of the Universe was written by mathematicians Alex Bellos and Edmund Harriss!

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So, I think coloring is a great way to get young kids talking about shapes and patterns. The relevant pieces of the Common Core math standards for kindegarden are:

(i) From the introduction:

(2) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

(ii) From the Standards themselves:

Identify and describe shapes.

CCSS.Math.Content.K.G.A.1

Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

For 1st Graders – A simple version of the Game of Nim:

Last November I had an interesting twitter conversation with Tracy Johnston Zager that came up when she was talking about a simple version of the game of Nim with elementary school kids. (The version of the game is explained in the project linked below.) The kids Zager was working with were asking lots of different questions about the game, and one set of questions was sort of surprising – does it matter what pieces you use to play the game?

I was interested to hear what my kids would think of this question, so I played the game with them and asked them if the game would change if we used different pieces:

A question from Tracy Johnston Zager that caught my eye

I really like the idea of using this simplified version of Nim with kids because there’s lots of nice arithmetic and problem solving involved in the game itself, and I did use it for a 20 minute project during the 1st grade Family Math night last week:

Plans for the K-1 Family Math nights

It was interesting to see that the 1st grade Common Core math standards specifically address the point the kids were asking about – the difference between defining and non-defining attributes. Here are the sections from the 1st grade standards that support using this game with 1st graders:

(i) From the Introduction:

(1) Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop strategies to solve arithmetic problems with these operations. Students understand connections between counting and addition and subtraction (e.g., adding two is the same as counting on two). They use properties of addition to add whole numbers and to create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems within 20. By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction.

(ii) From the Standards themselves:

Reason with shapes and their attributes.

CCSS.Math.Content.1.G.A.1

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

For 2nd Graders – Larry Guth’s “No Rectangles” problem

I probably can’t fully explain how happy I was to see the kids at both the 2nd and 3rd grade Family Math nights have fun playing around with this problem. The problem is pretty easy to state:

Suppose you have an NxN square grid – what is the maximum number of squares you can color in without 4 squares that were colored in forming the corners of a rectangle (with horizontal and vertical sides)?

Here’s how the game went when I played it with my own kids:

Larry Guth’s “No Rectangles” problem

Obviously you aren’t going to talk about 10×10 grids with young kids, but the problem using 3×3 and 4×4 grids captivated the younger kids at Family Math night. It was actually hard to wrap up both the 2nd and 3rd grade evenings because the kids were just running up with lots and lots of different patterns that they thought were maximal solutions.

The 2nd grade Common Core math standards that support playing around with the “no rectangles” game are:

(i) From the Introduction:

(4) Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

(ii) From the Standards themselves:

Reason with shapes and their attributes.

CCSS.Math.Content.2.G.A.2

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

For 3rd Graders – How many times can you fold a piece of paper in half?

This paper folding project that James Tanton did at MIT led to our very first Family Math project:

Toilet Paper used to break paper-folding record at MIT

The project is fun for kids because it is pretty surprising how few times you can fold a piece of paper in half. It is also a great opportunity to talk with kids about both fractions and exponential growth.

The 3rd grade Common Core math standards that support using this paper folding idea with kids are:

(i) From the introduction:

(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

(ii) From the Standards themselves:

Develop understanding of fractions as numbers.

CCSS.Math.Content.3.NF.A.1

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

For 4th Graders – Numberphile’s “Pebbling a Chessboard” game

My original introduction to Numberphile’s math videos came from this presentation of the “Pebbling a Chessboard” game by the mathematician Zvezdelina Stankova:

Our project using this video is here:

Numberphile’s Pebbling the Chessboard game and Mr. Honner’s Square

This is a fun game for kids to explore, and the connection with fractions that Stankova explains in the Numberphile video is probably going to be really surprising for kids to see. It isn’t necessary for the kids to have a complete understanding of why 1 + 1/2 + 1/4 + 1/8 + . . . = 2 (in fact, it would be absurd to expect 4th graders to completely understand this idea) but they can certainly have an intuitive feel for why it is true.

The 4th grade Common Core math standards that support using this project with kids are:

(i) From the Introduction:

(2) Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.

(3) Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving symmetry.

(ii) From the standards themselves:

Build fractions from unit fractions.

CCSS.Math.Content.4.NF.B.3

Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

For 5th Graders – The Collatz Conjecture

This’ll be my one overlap with Patcher’s post – he uses the Collatz Conjecture for his 3rd grade example.

The Collatz Conjecture looks at the following procedure:

Start with any positive integer. If the integer is even divide it by two, and if it is odd multiply it by 3 and then add 1. Now, repeat the process until you end up with the number 1.

The question is – do you always end up at 1? No one knows the answer to this question – the problem is unsolved!

We’ve looked at this problem (and variations of the problem) a few times:

The Collatz Conjecture and John Conway’s Amusical Variation

There aren’t many unsolved problems in math that young kids can understand, so that alone makes this problem fun to share with kids. The extra opportunity kids get to get in a little arithmetic practice also makes this a nice activity.

The 6th grade Common Core math standards that support sharing this problem with kids are:

(i) From the Introduction:

(2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.

(ii) From the Standards themselves:

Analyze patterns and relationships.

CCSS.Math.Content.5.OA.B.3

Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.

For 6th Graders – Numberphile’s “-1/12” video

I know there is are many different opinions of Numberphile’s video about the series 1 + 2 + 3 + . . ., but I loved it:

I’ve also loved talking about this series with my kids – using the idea of “algebraic intimidation” from Jordan Ellenberg’s How not to be Wrong:

Jordan Ellenberg’s “Algebraic Intimidation”

As I mentioned in the beginning of this post, it was Marcus de Sautoy’s description of Hardy and Ramanujan discussing this exact series that gave me the idea to write this post. I think this is a wonderful example to use with kids because it shows them, among other things, that you shouldn’t feel bullied by a bunch of math. That’s why I asked me kids at the end of the project if they believed what we just did.

I also think that you can share this idea with kids, along with some historical context, and they can see some really beautiful ideas in math. I personally tracked down a copy of Hardy’s Divergent Series book to try to understand what was going on after seeing the video. Some other nice background is in this Numberphile video with Ed Frenkel:

Anyway, the 6th grade standards that support talking about this Numberphile video with kids are:

(i) From the Introduction:

(3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships between quantities.

(ii) From the Standards

Reason about and solve one-variable equations and inequalities.

CCSS.Math.Content.6.EE.B.5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

CCSS.Math.Content.6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

CCSS.Math.Content.6.EE.B.7

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

For 7th Graders – the Surreal Numbers

This tweet from Jordan Ellenberg from last August started me down a path that ended last week with an hour long discussion of the Surreal Numbers with the 4th and 5th graders at my younger son’s school:

I wrote about the night (and the prep work) here:

Sharing the Surreal Numbers with Kids

Just as with Larry Guth’s “No Rectangles” problem, I was thrilled to see the kids (and parents) getting so excited about the surreal numbers. I think kids will really enjoy solving the little puzzles that come up – finding the value of the “blue / red” stack, for example. The usual ideas about “infinity” and the “infinitesimal” are tremendously fun, too.

The 7th grade Common Core math standards that support sharing the Surreal Numbers with kids are:

(i) From the Introduction:

(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.

(ii) From the Standards themselves:

Apply and extend previous understandings of operations with fractions.

CCSS.Math.Content.7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

CCSS.Math.Content.7.NS.A.1.a

Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

CCSS.Math.Content.7.NS.A.1.b

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

CCSS.Math.Content.7.NS.A.1.c

Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

CCSS.Math.Content.7.NS.A.1.d

Apply properties of operations as strategies to add and subtract rational numbers.

For 8th Graders – Terry Tao’s “Cosmic Distance Ladder” lecture

Terry Tao’s “Cosmic Distance Ladder” public lecture at MoMath is an absolute gem:

In the first 45 minutes you learn how some basic ideas from geometry helped Greek scientists  find good approximations to (i) the radius of the earth, (ii) the size of the moon, (iii) the distance to the moon, (iv) the size of the Sun, and (v) the distance to the Sun. All without any technology at all – just math ideas 🙂

This lecture is a fascinating history of science and an incredible opportunity for kids to see a lecture from one of the most respected mathematicians alive today. On top of that, you get some fantastic examples of how elementary geometry helped us understand “the real world.”

The 8th grade standards that support sharing Tao’s lecture with kids are:

(i) From the introduction:

(3) Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.

(ii) From the Standards themselves:

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

CCSS.Math.Content.8.G.C.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

 

In addition to the projects above, we’ve looked at ideas from mathematicians like Katie Steckles , Barry Mazur , Jacob Lurie , Laura Taalman , and too many more to name, I guess!!

One thing that makes me really happy is that so many mathematicians are sharing great math ideas on line. I’m excited to do a bit more thinking about how to share more and more of those ideas with kids.

 

Amazing math from mathematicians to share with kids

About two years ago I saw this Numberphile interview with Ed Frenkel:

One of the ideas that Frenkel mentions in the interview is that professional mathematicians haven’t done a good job sharing math with the general public. Although I’m not really the kind of professional mathematician Frenkel was talking about, I took his words to heart and have been on the lookout for math to share – especially with kids.

It turns out that there are some fantastic ideas that are out there for kids to see. Some surprising fun I had sharing Larry Guth’s “no rectangles” problem with kids earlier this week (see below) made me want to share some of the ideas I’ve found in the last couple of years, so here are a few examples:

(1) One of the most incredible lectures that you’ll ever see is Terry Tao’s “Cosmic Distance Ladder” lecture at the Museum of Mathematics in New York City:

I used Tao’s video for three projects with my kids – but there are probably 20 math projects for kids you could get out of it.

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Terry Tao’s MoMath lecture part 3 – the speed of light and paralax

(2) The Museum of Math’s public lectures are a great source beyond Tao’s lecture.

Here’s a project based on Bryna Kra’s lecture:

Angry Birds and Snap Cubes – Using Bryna Kra’s MoMath public lecture to talk math with kids

Eric Demaine’s lecture was part of our Fold and Cut theorem project:

Fold and Cut part 3

and I can’t say enough good thinks about Laura Taalman’s work – she’s inspired dozens of our projects.  Just search for her name on the blog:

(3) and Speaking of Fold and Cut . . .

Katie Steckles and Numberphile put together an incredible video about the Fold and Cut theorem. I used the video this week for project with 2nd and 3rd graders at my younger son’s school earlier this week.  Steckles’s presentation is so incredible – this is the kind of math that really inspires kids:

We used it for three projects (including the Eric Demaine one above):

Our One Cut Project

The Fold and Cut Theorem is Awesome!

In prepping for the grades 2 and 3 projects I also totally coincidentally ran across a “fold and punch” exercise that is a great activity to try with kids before trying out fold and cut:

(4) Another great success with the 2nd and 3rd graders was Larry Guth’s “no rectangles” problem. I had a great time playing around with this problem with my kids, but nothing prepared me for how enthusiastic the kids in the two programs were about this problem.

Larry Guth’s “No Rectangles” problem

After the 3rd grade night, Patrick Honner sent me this picture that I used to wrap things up with the 2nd graders.

(5) The Surreal Numbers

I’d seen John Conway’s surreal numbers previously via an amazing Jim Propp blog post:

The Life of Games.

and I wanted to revisit them after finally reading Donald Knuth’s book:

Revisiting the Surreal Numbers

Infinity + 1 and other Surreal Numbers

Playing with the surreal numbers via checker stacks is an incredibly engaging way for kids to learn about mathematical thinking.

(6) Speaking of John Conway –

In the 2014 edition of the Best Writing in Mathematics Conway had an article about variations on the Collatz conjecture. It was a fascinating article that even gave us the idea to translate some of the math into music.

The Collatz Conjecture and John Conway’s “Amusical” variation

I’ve also talked with the boys about the standard version of the Collatz conjecture:

It is a great way to introduce kids to an unsolved problem in math while also sneaking in a little bit of arithmetic practice!

(7) Occasional contest math problems

I happened to run across another MoMath lecture yesterday – this one by Po-Shen Loh. He was talking about “Massive Numbers.” I thought maybe he’d be talking about the book “Really Big Numbers” by Richard Evan Schwartz:

A few projects for kids from Richard Evan Schwartz’s “Really Big Numbers”

or maybe Graham’s Number:

An attempt to explain Graham’s number to kids

The last 4 digits of Graham’s number

but instead he talked about a neat problem from the 2010 International Mathematics Olympiad:

His presentation is fascinating and I even talked through the first version of the problem with my younger son:

Another math contest-like problem I really enjoyed talking about with the kids was this one:

Show that any positive integer n has a (positive) multiple which has only the digits 1 and 0 when represented in base 10.

A challenging arithmetic / number theory problem

(8) Building off of popular books by mathematicians as well as public lectures

I was surprised at how much great math writing and speaking there has been for the general public in the last couple of years.

Jordan Ellenberg’s “How not to be Wrong” inspired several projects – probably my favorite was using his idea of “algebraic intimidation” to talk about the famous 1 + 2 + 3 + . . . = -1/12 video by Numberphile. :

Jordan Ellenberg’s Algebraic Intimidation

Jacob Lurie’s Breakthrough Prize public lecture inspired two projects about a year apart from each other:

 

Using Jacob Lurie’s Breakthrough Prize Lecture to Inspire Kids

Using Jacob Lurie’s Breakthrough Prize talk with kids

And, Ed Frenkel, who got me thinking about sharing advanced math with kids in the first place has inspired a few projects, too:

Fine Ed Frenkel – you convinced me

Ed Frenkel, the square root of 2, and i

and one of my all time favorites:

A list Ed Frenkel will love

(9) Finally, it would be impossible to write a post like this one without mentioning the work that Evelyn Lamb is doing writing math articles for the general public. I’ve lost count of how many projects she’s inspired, but it is probably well over 20. I’m especially grateful for her talk about topology which have generated really fun conversations with the boys. For example:

Using Evelyn Lamb’s Infinite Earring with kids

Evelyn Lamb’s fun torus tweet

and

Henry Segerman’s Flat Torus

which arose after Lamb pointed out this video:

 

So, I’m really happy that mathematicians are sharing so many amazing ideas. I think this is the sort of math promotion that Frenkel had in mind. Hopefully it continues for many years to come 🙂

No, for goodness sake, don’t stop teaching math

Sorry that I had only 20 minutes to write this because I had to run out the door at 3:00, but I felt it was necessary to give some response to today’s Bloomberg article . . . .

Saw this Bloomberg View article via a Keith Devlin re-tweet this afternoon:

Frenkel’s Slate piece is a good response to today’s article, but this old blog involving another of Frenkel’s ideas is actually what came to my mind:

A list Ed Frenkel will love

In that post you’ll see a simple question that Frenkel asks about math – “So how do we make people realize that mathematics is this incredible archipelago of knowledge?”

Dan Anderson’s project with his students is, I think, one great answer to that question. Dan’s students gave short presentations on math-related topics that they choose because they wanted to learn more about them. The list of topics is absolutely wonderful. I’ve always imagined Frenkel sitting through those presentations with a big smile on his face.

He’d smile because math **IS** this incredible archipelago of knowledge and kids **DO** want to learn more about it. It is hard for me to understand how someone could even suggest that we stop teaching math.

Just a few weeks ago I had the incredible opportunity to sit and watch a 6th grade girl make some amazing shapes out of Zometool sets just playing around as I sat in our kitchen eating dessert with her parents – I see a remarkable young geometer at work here:

A fun Zometool Story

Here’s one of her shapes, for example:

IMG_1728

Just last week my kids and I were talking about square roots – my older son made this wonderful observation about the graph of y = \sqrt{x}

You just never know what kids are going to notice or wonder or think about, or what is going to suddenly capture their imagination. How could you even think that we should stop teaching math?

And, finally, give me just one minute more of your time – if you can watch my kids learning about the Chaos game and think we should stop teaching math, well, I guess there’s nothing I can say or show you that would ever convince you otherwise:

A great piece on Grothendieck by Ed Frenkel and a nice problem for students interested in math

[note: home sick with some stomach bug for the last two days – sorry for what is surely a bit of a rambling post]

Ed Frenkel published a nice piece in the New York Times today on the life and work of Alexander Grothendieck.

The Lives of Alexander Grothendieck, a Mathematical Visionary

In addition to Frenkel’s perspective on Grothendieck, what caught my attention was an almost off-hand observation about complex numbers that is really fascinating. I know it would have been quite a head scratcher for me in high school so I thought it would be fun to write about. Here’s the comment about the equation x^2 + y^2 = 1:

“One can show that the solutions of [x^2 + y^2 = 1] in complex numbers are points of an entirely different space; namely, a plane with one point removed.”

Students familiar with the equation x^2 + y^2 = 1 probably have only thought about this equation when both of the variables x and y are real numbers (when the solution is the familiar unit circle). The extension to complex numbers is a nice mathematical surprise.

So how can you think about Frenkel’s example? An excellent starting point is Richard Rusczyk’s sample solution for problem #25 of the 2013 AMC 12. The video below is a great way for students to see the power of geometric reasoning with complex numbers:

An approach similar to what Rusczyk outlines above is also a good way to start thinking about Frenkel’s equation. Try a few examples first – if x = 10i, for example, what values of y will satisfy the equation x^2 + y^2 = 1 (remember that both x and y are complex numbers)?

Now, if you have a generic value of x, what values of y will solve the equation? You’ll find that there are 2 values of y for most values of x, though importantly, not all.

Next is a real geometric leap – if every point x in the complex plane paired with exactly two points in Frenkel’s equation, seems as though the solution to the equation would be equivalent to two copies of the complex plane (possibly glued together in some strange way). Though it is challenging for sure, it is fun to think about what’s different from the situation I just described – in what way is the situation Frenkel describes similar to a plane with a point missing?

Away from this fun example of geometry with complex numbers, it was nice to see Grothendieck’s work described to the public. Another recent article about mathematicians written for the public was Michael Harris’ piece in Slate about the Breakthrough Prizes in math:

Michael Harris on the Breakthrough Prizes in Math

One of Harris’ points caught me off guard:

“Tao—the only math laureate with any social media presence (29,000-plus followers on Google Plus)—was a guest on The Colbert Report a few days after the ceremony. He is articulate, attractive, and the only one of the five who has done work that can be made accessible to Colbert’s audience in a six-minute segment.”

I was surprised to hear that Harris thought that the work of Jacob Lurie, Richard Taylor, Maxim Kontsevich, and Simon Donaldson really could not be made accessible to the public. Surprised enough, actually, to ask Jordan Ellenberg on twitter if he agreed with the statement:

Though his answer was not really a shock, it still disappoints me a little that work of these researchers is so inaccessible to the general public. Hopefully Frenkel, or other mathematics writers, can find a way to bring the beauty of their work to the public. I’d love to know more about Lurie’s work, or any of their work, frankly.

More public lectures like the one Terry Tao gave at the Museum of Math would be great, too. I’ve already done three projects with my kids already based on that lecture. It is amazing for them to be able to learn from Terry Tao!

Terry Tao’s MoMath lecture Part 1 – The Moon

Terry Tao’s MoMath lecture Part 2 – Clocks and Mars

Terry Tao’s MoMath lecture Part 3 – the Speed of Light and Paralax

It would be wonderful if there were more opportunities like Tao’s public lecture to introduce kids to research mathematicians and more article’s like Frenkel’s, too. Despite being home sick, Frenkel’s article made my day today.

A list Ed Frenkel will love

Came home from work tonight to see this incredible post by Dan Anderson:

link in case the link in the tweet doesn’t work:

http://blog.recursiveprocess.com/2014/07/29/my-favorite-for-the-students/

Since reviewing Dan Meyer (and colleague’s) new piece Central Park  yesterday, I’ve had my mind in a bit of a muddle thinking about the difference between the kind of math that I think kids will like to see and the kind of math that other folks think that kids would like to see.

Dan’s post went in a much better and much more productive direction – what kinds of math do kids find exciting when you ask them?  Well timed, Dan, well timed, and what an absolutely awesome list.  It is so amazing to see the mix of math theory and math that connects with the lives of the kids: topics ranging from Brower’s Fixed Point theorem, Symmetry, and Graham’s number to golf handicaps, diving scoring, and formatting yearbooks (which I bet looks a lot like the “Central Park” task). I’ve got no idea at all what “Mole Train Woot Woot” is, but I want to know. Badly.

Earlier this year Ed Frenkel did a nice interview with Numberphile:

He has several interesting things to say about math education:

“So how do we make people realize that mathematics is this incredible archipelago of knowledge?”

I think that Dan’s project is an tremendously exciting and inspiring way to approach Frenkel’s question.

“If you want your kids to understand and appreciate the beauty and power of mathematics, we have to connect it to our daily lives”

You definitely see this connection in some of the projects, but it is also nice that many projects involve interesting ideas from pure math. Maybe the students are not able to fully understand all of these ideas, but these ideas from theory are still somehow capturing their imagination, and that is so great to see.

“In the case of mathematics [students] are not even aware of the masterpieces of great mathematicians the way that they are are of the existence of the masterpieces of the great artists.”

Well, Ed Frenkel, on this point I think that Dan’s post will really make you happy.

Ed Frenkel, the square root of 2 and i

A few weeks ago, some thoughts on twitter from Michael Person inspired this talk with my kids:

https://mikesmathpage.wordpress.com/2014/04/19/imaginary-numbers/

Last weekend I picked up the audio book version of Ed Frenkel’s “Love and Math” and Frenkel’s discussion of \sqrt{2} and i made me want to revisit this conversation about properties of numbers.

We started with \sqrt{2}.   Their reaction to hearing that we were talking about \sqrt{2} was to talk about why it was irrational, and since they nearly remembered the proof from last time, this proof made for an instructive start to the conversation today.  It is always nice to review some of the ideas behind these simple proofs with them and watch their ability to make mathematical arguments develop.

Next we moved on to talking about i.  They remembered a few basic properties about i, though my older son still thinks that it is something that math people just made up.  I’m not terribly bothered by that for now, but the ideas in Frenkel’s book are giving me some new perspective on how to present some of these more advanced concepts to the boys.  Hopefully this new perspective is going to lead to a much better approach to teaching them math.  In any case, here’s what we said about i:

The next two videos are the main point of the talk today – in what ways are \sqrt{2} and i similar?  This question is a specific example of the broad question of symmetries in math that Frenkel discusses in his book.  I felt like the book walked up a couple of stairs and then hopped into an elevator to the top floor, though.  The ideas were inspiring, but I was left (i) wanting more and (ii) wanting to fill in a few more details.  One focus of these math conversations with my kids over the next few years will be spent on (ii).  I’ll work on (i) by finishing the audio book on a drive to and from Boston this weekend!

For today, though, let’s just stick with some similarities between \sqrt{2} and i that Frenkel highlights:

So, without digging too deep into the details, it looks like the set of numbers that we get by adding \sqrt{2} to the rational numbers has some nice, simple properties.  If we add or multiply, we seem to never leave the system.  Pretty neat.  i seems to have the same property.  Frenkel make the point that is we aren’t too bothered by \sqrt{2}, we shouldn’t be that bothered by i.  This is a nice point, obviously, and a fun idea to share with kids.  I really loved that my older son made the connection between i and x from algebra.  Only one step away from polynomial rings . . .  ha ha!

So, definitely on the theoretical side, but a definitely a fun morning.  Looking forward to plucking a few more ideas  out of “Love and Math”  to share with the boys.

Fine, Ed Frenkel, you convinced me . . . .

(Morning note – I wrote this way to fast trying to get out the door for work.  Sorry for the likely typos and formatting erros, will edit later)

Yesterday Ed Frenkel of UC Berkeley published this op ed in the LA Times:

http://touch.latimes.com/#section/-1/article/p2p-79479491/

He also followed up with this twitter link to a Richard Feynman video:

In the video Feynman talks about some conversations with his cousin about solving linear equations using algebra.  By unlucky coincidence, I’m right in the middle of that topic with my younger son right now so the video really did hit home.

Though I haven’t read Frenkel’s book “Love and Math” yet, it does seem that he and I are pretty much on the same page when it comes to math education.  Even to the point where I’ve been using Rubik’s cubes to teach all sorts of fun little math to the kids – ha.

https://mikesmathpage.wordpress.com/2014/01/09/a-little-fun-and-a-little-math-wtih-rubiks-cubes/

I took Frenkel’s weekend posts to heart today and instead of further work on solving linear equations, I decided to cover a fun math application with my younger son -> maps.

First we took a close look at our wall map of the Earth and talked a little bit about the distortions in the map near the poles:

Next, using a lacrosse ball and a strip of paper, we walked through how you might try to make a map of a sphere on your own and the problems that you’d encounter:

Finally, we tried to recreate an amazing video that Henry Segerman made about stenographic projection:

The link where we found Segerman’s sphere (including his video which is much better than ours) is here:

http://www.thingiverse.com/thing:202774

Definitely a fun morning of math, so thanks to Ed Frenkel for the inspiration.  I hope he succeeds beyond his wildest imagination in convincing the world that there’s lots of fun math that is accessible to young kids.