Building on Seeing Double

For the past few weeks I have been blessed with the opportunity to work with a fourth grader and really enjoy the learning and relationships that have been fostered. The unfolding story in this entry is to mark the evolution of this journey, knowing there is much more to be written. Enjoy the journey, remark on the learning, and never forget to relish the relationships.

She sat there tapping her clenched fist as waited for her parent to finish the introductions, and then began to share what we had tried to do via Zoom the week prior. My new student had work from her teacher she wanted to share. She had her work from our first interaction she had questions about. Mostly, she had an inquisitive nature with a mix of both question statements and revelations she excitedly buzzed about.

As we moved toward the work we were supposed to be addressing, she shared a statement that centered our work and became the linch pin for our journey. On a worksheet of many subtraction problems, she pointed to a problem and said, “I got that one, because it is really easy…” The problem is shown below:

Doubles

I asked why she thought it was easy and she said it was because they were the same. Since she could see doubles as easy, this gave me a thought, so I tried drawing the same representation with dots. I drew three filled in dots to represent 3 and three empty dots to represent -3. We discussed how these come together to form zero pairs, which she accepted as an easy fact. When I gave her the same problem with one small change as shown below:

Doubles Again

I asked her how she saw this change would affect the result, and how that relates to our picture. Wanting to extend our understanding, we tried making one more small modification, I added one to either the addend or the minuend. After we discussed this for some time, I drew a box around the two problems, and we considered how the previous understanding was being visualized. We also discussed how this helped us make sense of the problems, by seeing what is “new” in the problem and what we have already done in the doubles problems.

It was here where she taught me something new, the connections between the four operations and the Division Algorithm. Since she was unclear how the sum and difference were connected to multiplication and division (her areas of need), she was very excited to see that she’s been multiplying and dividing her whole life. So we talked about the Division Algorithm, is really just a way of viewing how the four operations work together. Noting that repeated addition is multiplication and repeated subtraction is division. In both cases, the remainder, r, is what is the same connective tissue. For example, 4 divided by 3, requires one subtraction of a group of 3, with a remainder of 1, and the product of two groups of 3 and 1 more is the same as the sum of two products of 2 (2×2=4) and adding one more group of 2 plus one more yields 3 groups of 2 plus 1 i.e. 7=4+3=2*2+(2+1)=(3*2)+1. Such a simple statement, but the use of the associative property, combined with groups of and flexibility of this kind of number sense thinking is profound.

Practice = The Games

To provide space for sense making and to make this experience fun, I was stoked to use two of my new tools from the incredible Marilyn Burns: 1) Circles and Stars and 2) Pathways.

https://tinyurl.com/gamesfrommarilynburnsmath

Starting with Circles and Stars, we modified the original game, so that each pair rolled one dice, the first partner rolled for how many stars, the second for how many circles to draw. Each partner took turns, counting the total number of stars and recorded their score. The visual of having the circles as groups of was really powerful, especially when I asked if she could write her total as a sum, as a product, as a quotient, and as a difference. This pushed our thinking quite a bit and we had some great conversations about the various ways to combine numbers. The beauty of our conversations was especially poignant as a tracker of her learning, what was becoming firmer and what was still needing support.

Over time, we moved on to include Pathways, which we use the first example from Marilyn’s entry, and we haven’t gotten to the point of adding the additional complications to the game. The game play is to choose a starting side, from which the opposite marks the finish line. The winner of the game is the first one to connect a touching sequence of squares on the 5×4 grid, where connecting means the two squares share a side or a corner. The players are able to choose any square they wish, and they can claim the square if they can write a product that equals that value. For example, if the square I chose had the number 12, then if I wrote 3×4 or 6×2, I could claim the square. We agreed that we couldn’t claim the square by saying 12=12×1. On our last meeting, the victory here is my student ended up computing both my score and hers. On the pathways to doing so, she was choosing numbers that end if 5, since she could skip count to that number by counting her fingers. We discussed this and we were feeling good about it. Once she ran out of 5’s, she tried the same strategy with 2, and was excited to see anything ending in an even number was divisible by 2. When I chose the square 64, she checked it by counting by 2’s stating 64 takes 32 quantities of 2 to equal the same value. The value of the whole experience provided a deep connections she is seeing in recognizing repeated addition as multiplication.

So far what I’ve learned or had reinforced through this wonderful experience are: 1) Relationships are key, nothing goes forward without a solid foundation in treating the learner first as a person. 2) By listening to the student, you will find out so much more, including an entry point that builds on a student’s strengths rather than focusing on their weaknesses. 3) There are many deep connections in studying something as seemingly simple as arithmetic, including the underlying observations of the operations and their relationship. 4) The power of visuals to connect learning and being able to see mathematics, not just writing numbers. 5) The power of asking questions, regardless if the student is right or wrong, challenging them on both, and asking for justification, adds a new dimension of struggle and understanding. 6) As always, I end up learning way more from my students than I think I ever actually teach them. And, 7) Working with students in the best part of any work day.

Confessions

“Giving up is the only sure way to fail.” ― Gena Showalter

For most things in life, I completely agree with Gena Showalter’s statement that you may be defeated, you may not have won the day, but only in giving up do you truly fail. The notion of failure tends to have negative connotations associated with it; however, when we view failure as a sign we are trying something, then failure becomes a marker of learning…which leads me back around to Gena Showalter’s statement that the only significant failure is when we fail to keep trying is the moment when we truly lose.

The dramatic entrance and clarification aside, during my first career choice, being a professional student, I encountered many, many barriers where Gena Showalter’s statement rang true. My first encounter occurred in my choice of major, I entered university with the mindset I wanted to find the most challenging subject I could find to major in…which for me was physics.

My story, my road to physics, is through failure and not deciding to quit. My failure occurred during a quiz where I was asked to find the speed of an object falling down a frictionless slide, without any numerical values, and no other information provided (see this diagram below).

While staring at this problem, baffled, I noticed everyone around me vigorously sprawling calculations and notes as if the entire volumes of the library were being scribbled on their paper. I knew then, I had no idea what I was doing in this physics class, and I quit the moment the class got out.

The fire ignited in me was tremendous, and I dedicated the next 10 years of my life to learning this subject. From that humble beginning to coding MCNP walks in a national laboratory for x-ray imaging techniques, physics became the lens from which I viewed (and still view) the world around me. The part that is hidden, the part we often don’t see is the hours of hard work, the hours of struggling, searching for some sort of insight, and hours of conversations with peers, colleagues, and professionals to gleam every possible insight to learning.

Through this journey, I found myself in graduate school in Applied Math (essentially physics) but more mathy. It was during this time that my truth became evident, that I met the first time in my life where I truly failed, I quit. The reason: Abstract Algebra.

Although there were contributing factors to my failure, like I do not understand how to write proofs and I received a lack of support from professors, I failed. I failed to comprehend this math. I failed to have the desire to put in the work like I did for physics.

After encountering another moment like that moment on the quiz in physics mentioned above, I failed to light the fire to try, I quit.

13.

Thirteen years later, I decided to pick up Abstract Algebra again thanks to so many incredible resources now available online. In addition, the burnout had faded while my belief in me had grown. With a tentative step forward, I am back trying to make sense of this landscape. Taking each minor victory in stride and using the change in mindset that this isn’t a barrier, but an incredible opportunity to learn.

The new road is fraught with the same challenges as before, the difference is perspective and a renewed inner fire to light the way. Abstract Algebra didn’t win this war, it just won the first round, proving Gena Showalter’s statement true once again.

Abstract Algebra kicked my ass, that’s my truth. I acknowledge it now and I look forward to being able to return the favor someday soon.

Unexpected Rewards

On a recent rabbit hole of learning, I have been consuming a ton of videos from the YouTube Channel Numberphile, which has ignited a curiosity and love of math that I didn’t know I had. Let me be straight, I have loved math for a long time, but this delightful journey into a variety of topics has proven to shed light on areas I once didn’t understand, connected previously disjoint topics, or shown whole new ideas and concepts that are fun and incredible. In this entry, I wanted to highlight some of that learning and showcase this application.

One on hand, I have learned the following four unique and interesting ideas I previously didn’t know.

  1. Operations with shapes
  2. Multiplication with halving and doubling
  3. Heroes and Super Heroes
  4. Ptolemy’s Theorem

On the other hand, I’ve learned a lot about different math ideas of which I will highlight some key features here. The first is understanding and accessing graph theory, which is a topic I first heard about during a talk in graduate school. I remember being completely lost in the talk and not understanding any of how the pieces connected or shared together. After watching a few episodes of Numberphile, where graph theory played a central role, I was able to access the understanding well enough to read a book I’d purchased years prior but was unable to access. Reading Richard Traduea’s “Introduction to Graph Theory” covered many of the concepts highlighted in the Numberphile Videos which was fantastic, but the Numberphile videos made the content seem easy and so accessible.

I will write a separate entry for both graph theory and abstract algebra as each warrants its own entry in upcoming discussions.

Returning to the ideas item 1 above, the two items I would like to highlight are the following: 1) Using circles to calculate square roots, and 2) Using parallel lines for multiplication.

Suppose you wish to calculate the square root of 9, you may know that the product of three and itself yields nine, so the square root is three. Here’s the steps:

  1. Draw a number line starting at the origin
  2. Count off, until you have reached the number you are looking to determine the square root of, in our case, we would count out 9 units.
  3. Next, add 1 more to your location, and from 0 to this location (one more than the value you wish to calculate the square root of) will be the diameter of the circle we are constructing. In our case, our diameter is 9+1=10.
  4. Now, we construct the center by marking the spot which is half the radius. For our example, we have the ordered pair (5,0), and a radius of 5.
  5. Draw the circle.
  6. From the location of the point, you wish to determine the square root of, draw a line perpendicular to the axis from the number you wish to know the root of, in our case, we draw a vertical line from 9.
  7. When your perpendicular line intersects the circle you constructed earlier, the length of the line you just drew is the desired quantity. In our case, the length turns out to be 3 as expected.

The next example is for parallel lines and finding products. Construct two lines that intersect at any acute angle. Along each line, equally space the distance between integer values marking a progression of increasing values. Suppose you wanted to calculate that the product of 1 and 3, then draw a line connecting 1 from the “horizontal axis” to the 3 on the angled axis. To calculate, another multiple of 3, say 4 times 3, draw a line parallel to your first line through the point 4 on the “horizontal” axis. The point of intersection between the parallel line through 4 and the “acute angled” line will be the resulting product, 12 in our case.

Now comparing the products of two numbers using the halving and doubling strategy made famous by two different nationalities, which we will compare below.

The “Russian” format takes the idea of double and halving in the following format:

  1. Write the numbers horizontally
  2. Take the first of the two numbers from left to right, and divide it by 2
  3. If you get a remainder from the step above, take only the whole number part
  4. Continue this process on the left until you reach unity
  5. On the right-hand side, double the process until you reach the same level as the one on the left
  6. Look at the left column, throw out any even number rows
  7. Add up the corresponding values on the right column of the remaining rows

For the Egyptian format, it is a similar process with a few key differences. The process is outlined below:

  1. Write the numbers horizontally
  2. Below the number of the left, write the number 1
  3. Continue to double this number until the result is above the original number
  4. Now on the right-hand number, rewrite the same number below it, the 5 in our case
  5. Continue to double the number until you have reached the last row equal to the left-hand side
  6. Now, use the numbers add up to the first number on the left, so we used 16 and 2, the numbers in the same rows as 16 and 2 are added together to create the resulting product, i,e. 16=80 and 2=10, so 80+10=90.

The Egyptian model is nice since it doesn’t require the halving to remove unwanted quantities and requires you to balance the addition of the two sides.

The unexpected rewards of watching Numberphile has opened up lines of inquiry, connections, and understanding I wouldn’t have expected and has me excited about learning a variety of types of math…though it hasn’t convinced me to say maths….yet.

Play Time

Following the theme of my last journal entry, we will focus on two simple stories which lead to deep mathematics inside.

Simple Story 1 – A Strange Quirk

I have a strange quirk, once I have started a streak, I tend to be very good about maintaining that streak over time. One such streak occurred during this pandemic, when I noticed the app on my phone counted the number of days I had consecutively read.

Since I only had Steven Strogatz’s book The Joy of X, I would read the book multiple times over, and each time I continued to absorb new information. One concept that hit home was reading about the absurdity of infinite series, and the struggles mathematicians were having with this idea prior to advancements in making sense of infinity and infinite sums in particular.

Most people recognize the terms of the famous harmonic series (or infamous depending on your point of view) as unit fractions, being summed together.

\sum_{n=1}^{\infty} \frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots

As the number of terms tends to infinity the terms are tending to zero, while the sum tends to infinity, i.e. the sum does not converge to some finite number. This feature is one of the reasons the harmonic series is so famous, as this sum does not converge while its terms converge to zero.

The Alternating Harmonic Series

\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots

The Alternating Harmonic Series converges to \ln 2 which one can see a proof of here.

The surprise from Strogatz’s book is that playing around with the order in which we add or subtract terms changes the resulting sum. For example,

1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\cdots = \frac{3}{2} \ln 2

Perhaps the easiest way to see this, is to start with the value we know the Alternating Harmonic Series converges to

1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots= \ln 2

Multiply both sides by one-half

\frac{1}{2} \left( 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots \right) = \frac{1}{2} \ln 2

Distribute the half through

\frac{1}{2} -\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\cdots = \frac{1}{2} \ln 2

Now add the above sum to the original series

\frac{1}{2} -\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\cdots +1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots = \frac{1}{2} \ln 2 + \ln 2

Canceling out terms and combining terms we see we arrive back at our rearrangement

1+\frac{1}{3}-\frac{1}{2} +\frac{1}{5} +\frac{1}{7}-\frac{1}{4} + \cdots = \frac{3}{2}\ln 2

The astute observer will notice that the left hand side is just a rearrangement of the Alternating Harmonic Series, and yet it is now 1.5 times the original amount.

Now we have arrived at the point Strogatz’s was pointing out: The commutative property of addition for convergent infinite sums does not always hold…which is just so weird.

Bernhard Riemann outlined when a convergent infinite sum disobeys the commutative property of addition in a theorem known as Riemann’s Rearrangement Theorem.

Playing with rearrangements to see what numbers we may arrive at is so much fun, which we know we can do as a result of Riemann’s theorem. I am currently using Excel and 1000 terms (truncating the infinite series) and attempting to converge to my favorite number of e.

The best example of this oddity was outlined by Strogatz in the book, by rearranging the series as follows

\left(1-\frac{1}{2}-\frac{1}{4} \right) + \left( \frac{1}{3}-\frac{1}{6} -\frac{1}{8}\right) + \left( \frac{1}{5}- \frac{1}{10}-\frac{1}{12}\right) + \cdots

Simplifying

\left( \frac{1}{2} - \frac{1}{4}\right) + \left( \frac{1}{6}-\frac{1}{8}\right) + \left(\frac{1}{10}- \frac{1}{12}\right) +\cdots

Factoring out a half from the above expression, we see our old friend again

\frac{1}{2} \left( 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots \right) = \frac{1}{2} \ln 2

Mind blown. I know.

The connection of simply re-reading a book to keep a digital count going, lead to a deep inquiry into the nature of commutative properties of infinite sums and uncovering some very strange behaviors. Moreover, it reminded me how this simple operation of addition, when looked at through this lens shared some unexpected behaviors and even added a little more to the fully fleshed-out version.

Simple Story 2 – Neural Networks

Introduction

During our senior year at university, one of my best friends shared they were going to write a neural network that would learn how to play chess, my first interaction with neural networks. As we discussed the project, a very basic idea about what a neural network was and how it “learned” were vague notions and remained clouded until a few weeks ago.

My colleagues and I were facilitating a lesson highlighting computer science and we used neural networks as a vehicle for the lesson. The lesson sparked a curiosity in me, as in that old conversation with my college friend, I was left unclear and trying to decipher what the pieces are the comprise a neural network and how it works.

After diving into neural networks, I am pleasantly surprised that the concepts are quite simple, but have a ton of depth with many, many fantastic features. Neural networks are one of those rare gems that can be both complex and complicated, so let me define those two terms to illustrate the point.

Complex vs. Complicated

When I refer to an object that is complex, I mean it is something that is often amorphous, ambiguous and requires concerted thought to make sense of it. My favorite example of a familiar complex task is raising a child, a process that is amorphous, ambiguous, and requires lots of mental work to be successful.

In contrast, a complicated task is concrete and fairly clear understand in the sense of when and how the task is completed, i.e. it’s made of discrete pieces. Complicated tasks tend to be skill-based and specific, and can be stacked on top of each other to create more complication. My favorite example of a familiar complicated task comes from my high school social studies class when we had to memorize all 50 states and their corresponding capitals. The task of memorizing a pair of state and capital was complicated, and stacking that task 50 times made the complicated task hard, but this task is not complex.

With that distinction clear, a neural network is complex in the sense that it is adaptable and capable of learning. Moreover, the application of creating a neural network to solve a problem is a complex task, figuring out what to measure, what variables are important for the learning, and what are not so many ambiguous features, this part is very complex. The understanding and application of backpropagation is also a complex feature, with complicated components.

The complexity aside, a neural network becomes complicated when we consider the interactions within the network itself, which will play with the structure and function momentarily. Considering a very basic model with no hidden layers, the activations from the input, the weights assigned to the input, the synapses, the neuron, and the outputs are singularly a series of single finite steps, with a straight forward path, i.e. they are complicated….that’s not to say neural networks become very, very complicated very, very quickly, which we’ll see as we discuss the structure and function next.

Neural Network: Structure and Function

The following two sources were very helpful in getting my own head wrapped around Neural Networks.

  1. Simple Neural Network in Python From Scratch (from YouTuber Polycode)
  2. Neural Networks Playlist (from YouTuber 3Blue1Brown Channel)

The first resource is just enough to get a solid basic understanding, while the playlist in the second does a fantastic job of expanding on the original idea and highlighting both the complexity and complicated nature a neural network can take on.

The main idea is to walk away with is a neural network that has an input series, that connects like synapses to a neuron. The neuron connects to the inputs to make a decision and then is mapped by a function to output. Each input has an associated weight to it and each neuron has a bias connected to it, this helps to train the network through a function that tries to balance the weights and measures for the system, through a process called backpropagation. The image below is my own attempt to connect all these pieces.

Neural Network Schematic

So….you might be wondering how is this related to play time?

Aside from the utterly fascinating world of beginning to understand neural networks, the idea of play came to mind in the “training” of the network. That is, a network only learns from iterating through large numbers of trials to gain the insight we need for the network to have. To undergo training has a sense of play to me, and learning should always incorporate as many elements of play as possible.

There is much more I want to say about Neural Networks, but I will digress for another entry.

Conclusion

While one person’s version of play may look very different, the idea of learning being a fun undertaking is something we tend to forget, especially in a school setting. I cannot express the joy these two examples of learning have brought me, and I am curious what bit of play have you had to enjoy in your learning?

What does play look like for you?

The Simple Deep End

“The most complicated skill is to be simple.” Anonymous

The Simple Start

One of the joys of exploring mathematics is coming home to elementary ideas like addition and discovering a whole new world is waiting to be explored. Experiences like this has been a frequent occurrence as I have tried to read every blog entry from the incredible Marilyn Burns’ blog.

The post that made me stop reading and decide to play for quite some time is here, in which Marilyn follows up on a post about the number eleven. The genesis of the post was a comment from one of her readers, in which she noticed the sum of consecutive numbers five and six equal eleven. In addition, the difference of the squares is also equal to eleven, i.e.,

Difference of Squares

The question and Marilyn’s inquire if the difference of the squares of two consecutive numbers is always equal to the sum of the consecutive numbers. A process that is easy to check via an algebraic approach. Let n be any natural number. Then m, is the next natural number satisfying the relationship m=n+1. The sum of two consecutive numbers always yields an odd number which is easy to check since, m+n=(n+1)+n=2n+1.

To check the difference of these two consecutive numbers, m and n, consider

As shown in Marilyn’s entry, the difference of the two consecutive numbers is the odd number corresponding to the sum of the two consecutive numbers. This was an excellent exploration on Marilyn’s blog, and I thoroughly enjoyed the journey, especially her visual approach creating a colorful visual model.

The next step in this journey, then led me to the idea of considering a parameter, where the whole number value is arbitrary and what information might we uncover in this format. Again, let n be from the set of natural numbers, and let m be defined in the same manner, with a parameter value d (with d coming from the set of natural numbers as well) such that m=n+d. The sum is

where d=1 in our previous experience. Now let’s consider the difference of squares

Notice that the difference of these squares yields the exact case as Marilyn discussed in her blog, when d=1. Comparing the two results the sum of the two numbers is twice the original number plus the value of the parameter (i.e. how many integers away the second number is from the first), and in the difference of their squares it the same as the sum times the value of the product. So the ratio of the difference of squares and the sum of the two is equal to the value of the parameter, d, i.e.

Interestingly the ratio of the difference of their squares and the sum of the numbers is a constant once the parameter value is chosen, and it will also be a whole number since our field we pull n and d are whole numbers…that’s an unexpected and very cool conclusion.

For d=2, the purple line indicates the sum of the two numbers, the green line indicates the difference of squares, and the blue line indicates the ratio, which is 2 since d=2 in this case. When d=1, the green line and the dotted purple line are colinear.

The Deep End

This led me to a classic investigation I lead my Integrated Math 3 students through, which starts off with the question: What numbers can be written as a sum of two consecutive numbers?

Students usually have trouble getting started, understanding that two natural numbers, m and n, are said to be consecutive if they satisfy the relationship m=n+1. When students realize that adding one more to the previous number creates the condition for consecutive numbers, then addition of the two to yield the sum as 2n+1 always results in an odd number is met with grins and celebrations.

The next step in this is to pose the problem that leads to a difficult discovery and many opportunities for struggle. The question is: Given the input of a starting natural number and the number of consecutive integers, what is the resulting sum without having to perform the addition?

The question may take an example, or two, to catch on:

  1. Suppose we start at the number 10 and we had two consecutive numbers, the sum would be 10+11=21
  2. Suppose we start at 172 and we had four consecutive numbers then the sum would be 172+173+174+175=694.
  3. Suppose we start at 19293 and we had seven consecutive numbers then the sum would be 19293+19294+19295+19296+19297+19298+19299=135072.

So the question is asking for a function with the two inputs of starting location and number of consecutive terms, and outputs the sum as a result. This investigation was a ton of fun and was inspired by the book

Fostering Algebraic Thinking

I’ll leave the exploration as a fun journey to explore, but I love how the connection plays out in both exploring and the generalizing that arises in understanding the structure of the number system.

I love how simple ideas of addition and consecutive numbers can lead to such deep explorations of mathematics.

Here are some possible ideas for extending these activities:

  1. Expand the domain from natural numbers to all integer values and find what processes hold true and what other things generalizes.
  2. Explore the visual representation of this process, see what visualizations uncover.
  3. Ask a different question in these regards, but I’m not sure what that new question might be.

What curiosities do you have?

Let’s continue this conversation, like on Marilyn’s wonderful blog, a comment could start an entire new exploration.

Hello World

“The secret of getting ahead is getting started,” said Mark Twain.

I’ve been waiting to start a blog with a single focus for a long time, as my interests vary with each passing season, especially during these crazy COVID times. So, this blog is going to focus on the learning that is currently occupying my mind.

Currently, my focus is split between learning more in chemistry and exploring the world of the calculus of variations. The goal of these two pieces is leading to my desire to revisit the mathematics in thermal physics.