I asked a kid what his favorite number was, you won’t believe what happened next :)

A friend from graduate school and her son stopped by yesterday on their way down to NYC.  Her son and my kids have a lot of common interests, including math.  Chit chatting over dinner I asked him what his favorite number was and got quite a surprise:

\pi^{e^i} .

Now that’s a favorite number!

Although his particular favorite number is a little difficult to talk about without getting into things like cos(1) and sin(1), I though that it would be fun to show the kids a little bit about complex numbers.   My end goal was to show the kids the value of i^i rather than \pi^{e^i}, but first we had to have a little discussion about powers:

With some of the simple properties of exponents out of the way, the one last thing we needed to touch on before taking about i^i was square roots. Despite intending to be informal here, I was unfortunately a little too informal and I think that caused a little bit of confusion. It took a few extra examples to get us back on track.

Next up was the fun identity e^{\pi i} + 1 = 0. I didn’t have any interest in deriving this equation, rather I just wanted to use it as a starting point. With just a few manipulations of this equation we can come to a value for i^i. It was super fun to see all three kids react to the surprising value of i^i. At the end I mention the approximate value of \pi^{e^i} as well.

So, a nice little math talk this morning sparked by this surprise favorite number!

Problem Solving

I was lucky to have my early development in math shaped by an incredible high school math teacher – Mr. Waterman.  He lived and breathed math and got out of bed everyday thinking about how to teaching problem solving to high school kids.   I ate it all up back then and the problem solving techniques that I first learned in his classes remain important pieces of my mathematical tool kit even today.   It is actually pretty fun to look back and see how ideas originating in high school math contests can come into play 25+ years later at work.  Although maybe the fact that I still use these ideas every day explains why I think problem solving is such an important part of math education.

For the last few weeks I’ve been seeing a lot of posts on twitter about problem solving.  Yesterday, for example, I ran across this wonderful post from Fawn Nguyen:

Making Problem Solving Part of the Math Curriculum

While others may not, I find it difficult to talk about problem solving in the abstract.  In talking or learning about problem solving a specific example is much more valuable to me than 1,000 words of abstract discussion.  Two of the most amazing specific examples I’ve seen lately come from the Field’s Medalist Tim Gowers and from the head of Art of Problem Solving, Richard Rusczyk.    Gowers decided to “live blog” his attempt at working through problem 1 in this year’s International Mathematics Olympiad.   It is an all too rare treat to see the thought process of one of the top mathematicians in the world in real time:

Tim Gowers walks through an IMO problem

Richard Rusczyk’s example is also a math contest problem – in this case problem #24 from the 2013 AMC 12.  The problem asks a question about the quadratic function  f(z) = z^2 + i z + 1.  I’m sure that as a kid a quadratic with an imaginary component would have really intimidated me, but Rusczyk’s calm and systematic approach to walking through the problem takes all of that intimidation away.  It is such a great example of problem solving – in fact this blog post has been delayed by about 40 minutes as I’ve watched the video twice because I enjoy it so much!

With these two examples in mind, I thought it would be fun to do my own example, though believe me, it isn’t within a million miles of the quality of either of the examples above.  However, what I think is incredibly important about the Gowers and Rusczyk examples is that they show the problem solving process, and I think that the more examples of that process that are out there the better.  So, neither an IMO problem nor a problem from a big US math contest, but here’s the process that I went through thinking about a fun little problem I saw posted on twitter a week ago:

I was actually up in Boston running around with the family when I saw the original tweet, but for some reason the problem stuck with me and I spent the next few days sort of daydreaming about it.

The first thing I did was think through cases that I hoped would be easy – quadratic equations (and also simplified the problem a tiny bit by assuming that the infinitely many perfect squares arose from positive integers n).  Take a polynomial like x^2 + 2x + 5, for example.  We can rewrite this expression as (x + 1)^2 + 4.   Since the only perfect square than is 4 more than another perfect square is 4, this new expression helps us see that x^2 + 2x + 5 will not be a perfect square for infinitely many integers x.

What if the linear term has an odd coefficient, though – say x^2 + 3x + 1.  Writing this expression as  (x + 1)^2 + x solves the problem in a slightly different way than the prior case.  For x > 1, we see that the new expression is greater (x + 1)^2 but less than (x + 2)^2.  Since  it lies in between two consecutive perfect squares, it cannot be a perfect square and so it will not be a perfect square for infinitely many positive integers x.

Since the coefficient of the linear term will either be even or odd, that takes care of the quadratic case. The only way that a quadratic polynomial can satisfy the conditions of the problem is if it can be written as (x + a)^2 for some integer a.  Great, but how does this fact generalize to higher degree polynomials?

The next example I thought about was the 4th degree polynomial x^4 + 3x^3 + 3x^2 + 2x + 1 = (x^2 + x + 1)^2 + x^3. From the quadratic case I thought that transforming the original polymonial into a perfect square plus a polynomial with a degree of 2n – 1 would be the way to go. Unfortunately I couldn’t see what to do, and certainly couldn’t see anything that would help in the case where the remainder was a general cubic polynomial.  What’s stopping a perfect square plus a number described by a cubic polynomial from being a perfect square just by accident? Hmmmm.

As we ran around Boston the problem stayed in the back of my mind. One day I got the idea to look for a different approach to the quadratic case and that turned out to be the key idea for getting my head around the problem. Writing the polynomial x^2 + 3x + 1 as (x + 3/2)^2 - 5/4 also helps you see that this polynomial cannot be a square for infinitely many integers.  Really for the same reason as above – the value is trapped between (x + 1)^2 and (x + 2)^2. This new expression led me to stumble on a similar statement for higher degree polynomials that was a nice little surprise (to me anyway!).

It turns out that completing the square generalizes in a neat way. For the case we are looking at – a monic polynomial of degree 2n with integer coefficients – the generalization is that you can always find a monic polynomial of degree n with rational coefficients whose square matches the first n + 1 terms of the original polynomial. In math symbols, I mean that we can write:

x^{2n} + a_{2n-1} x^{2n-1} + a_{2n-2} x^{2n-2} + . . . + a_0

as

( x^n + b_{n-1} x^{n-1} + . . . + b_0 )^2 + c_{n-1} x^{n-1} + c_{n-1} x^{n-1} + . . . + c_0

where the a’s are integers, and the b’s and c’s are rational.  The the 4th degree case I was looking at above gives an illustrative example:

x^4 + 3x^3 + 3x^2 + 2x + 1 = (x^2 + (3/2) x + 3/8)^2 + 7x/8 + 55/64.

Convincing yourself that this generalization of completing the square is true isn’t all that difficult.  When you square the polynomial with the b_{i}‘s above, you see that 2b_{n-1} has to be equal to a_{2n-1}, so solving for b_{n-1} is exactly the same exercise you do when you complete the square for a quadratic.  Once you have the value for b_{n-1} you see that a_{2n-2} has to equal  (b_{n-1})^2 + 2b_{n-2}, which give you the value of b_{n-2}.  Similarly, once you have the first k b_{i}‘s, solving for the next one just involves solving a linear equation in b_{n - k - 1}.  Basically, you just end up dividing by 2 a lot.

Although the rational coefficients present a small problem, we can use  ideas similar to the ones  we used in the quadratic case to show that for large values of x the above expression cannot be a perfect square.  In the above example of the 4th degree equation, assume that the expression is equal to z^2 when x and z are integers.  Multiply everything by 64 to arrive at the equation:

(8x^2 + 12x + 3)^2 + 56x + 55 = (8z)^2

For large values of x, the expression on the left hand side lies between the consecutive perfect squares (8x^2 + 12x + 3)^2 and (8x^2 + 12x + 4)^2.   For clarity, note that (8x^2 + 12x + 4)^2 = (8x^2 + 12x + 3)^2 + 2*(8x^2 + 12x + 3) + 1.    As long as x is sufficiently large, 2*(8x^2 + 12x + 3) + 1 will be larger than 56x + 55 since the x^2 term will dominate the x term.  Thus for sufficiently large integers x, the value of (8x^2 + 12x + 3)^2 + 56x + 55 (which we’ve assumed to equal (8z)^2) lies between two perfect squares and cannot be a perfect square itself.  So neither (8z)^2 nor z^2 can be a perfect square for sufficiently large values of x which, in turn, means that the original expression cannot be a perfect square for infinitely many positive integers x.

Essentially the same argument will work for any monic polynomial of degree 2n with integer coefficients, and also essentially the same argument works for showing that the polynomials cannot take on infinitely many perfect square values for negative integer values of x.

So, the generalization of completing the square shows that the only time a monic polynomial in x of degree 2n will take on infinitely many perfect square values for integer values of x is when the polynomial itself is a perfect square.  In that case, all of the values will be perfect squares as desired!

I doubt that my solution is the best or the most elegant, but I had a lot of fun thinking through this problem.  I’m also happy to have stumbled on this generalization for completing the square which I’m surprised to have either never seen previously or (more likely) just forgotten about.

As I said at the beginning of this post, the lessons about problems solving that I learned in the three years I spent in Mr. Waterman’s classes were such an important foundation in my own mathematical development.  Even though I’m not longer in academic math these problem solving strategies play a critical role in my work just about every day.  Away from work I try to communicate those lessons to my kids when we talk about math.

I hope that more mathematicians will follow the lead of Gowers and Rusczyk and give some public examples  of their own problem solving process so that everyone – and especially students – get lots of different looks at the problem solving process.  Seeing that work will help show that mathematical thinking  isn’t  about finding answers instantly and effortlessly, but often involves lots of trial and error, false starts, and most importantly joy at making a little progress.  These are important lessons from math with applications that go far beyond whatever specific problem you happen to be working on at the time.

A fun computer aided number theory project for kids

Not surprisingly, today’s Family Math began with a neat question that James Tanton posted on Twitter:

I chose to not go through the entire problem but thought it would be fun to answer a slightly easier question – what is the smallest positive integer with exactly 100 divisors?

We started by talking through the problem and looking for some ideas on where to start. Turned out that each kid had an interesting idea, and I thought it would be fun to pursue both of them:

We started off down the path suggested by my younger son – start by looking to see if there are any patterns in the number of divisors on the integers. The difficulty is that although there is a pattern, in fact a really cook pattern,  it is a little tough to see. I let them noodle on it for a while and then asked them to talk a little bit about what it means to be a divisor of a number.  That led to my older son noticing a similarity in the prime factors.

Next we moved to looking at the factors of a larger number -> 24.  After understanding the divisors of 24, we built off of the counting project we’ve been doing this summer to see how to find an easier way count the number of factors from the primes. Making this connection was the reason I wanted to do this little project.

My younger son’s idea led us to the rule for counting the number of divisors of of a given integer. Next we move on to my older son’s idea of looking for a pattern when we list out the smallest integers with a given number of factors. To aide in this part of the project we used Mathematica and had a lot of fun looking at lots of patterns in the numbers (also sorry about the camera issue in this video):

Having checked what the smallest integer was that had N factors for N ranging from 1 to 10, we went back to the whiteboard to look more carefully at those numbers. Here my lack of planning ahead came back to bite me a little since the prime factors of all of these numbers had only 2’s and 3’s. To correct for that I added in the smallest number with exactly 12 factors – that integer is 60.

Talking through the pattern we see in this list lets us take a couple of guesses at the smallest integer with exactly 100 factors. Looking at our guesses, the smallest one is 2^4 * 3^4 * 5^1 * 7^1.

Having gone through the difficult part of the problem, we head back to Mathematica to see if our guess is correct. Mathematica gives us a short list of numbers with exactly 100 factors, and our number is indeed the smallest!

Finally, we wrap up by taking a quick look at the second smallest number with 100 factors. When we factor it into primes we see that it looks pretty similar to the number we found – just one prime factor is different.

When you see the original problem for the first time it seems almost impossible to solve. A little bit playing around with patterns leads to the amazing discovery that the problem isn’t as intractable as it seems, though. You also get a look at several really cool patterns relating integers to their prime factors. Definitely a fun project and a fun way to show how computers can be helpful in solving problems, too.

Working through some NY State test questions (3/3) 8th Grade

[this is the 3rd in a 3 part series. I’ll use the same introduction for each. The first post is here:

https://mikesmathpage.wordpress.com/2014/08/09/working-through-some-ny-state-test-questions-13-3rd-grade/

and the second post is here:

https://mikesmathpage.wordpress.com/2014/08/09/working-through-some-ny-state-test-questions-23-5th-grade/

Note on this third post – I wrote the post all the way through and then geniusly deleted it rather than publishing it. Sorry if the rewritten post reads like I wrote it too fast. I did.

]

I saw this interesting link posted by a couple of people in the last few days:

How Would You Score On A Third-Grade Common Core Math Test?

To say the least, there’s been a lot written about the Common Core standards and their impact on education in the US. I haven’t followed this debate carefully, nor have I learned much about the standards. With the release of these questions I thought it would be interesting to get a glimpse of testing done in one state. So for today’s Family Math project I asked each kid to work through each of the 15 questions. This post is about the last 5 questions – the 5 from the 8th grade exam. My younger son will be going into 3rd grade this year and my older son will be going into 5th.

Question 1: What is the solution to the following system of equations?  3x + 4y = -2, and 2x – 4y = -8.

For me this isn’t a great multiple choice question, and my younger son shows why.  He is able to answer the question by just plugging the answers back into the equation. No real understanding necessary.

My older son has seen a little bit about solving equations with more than one variable and does recall a pretty good way to solve this problem.


Question 2: Which phrase describes a nonlinear function?

I’ve not covered functions with my younger son, so he does not understand the question. I thought the question was interesting enough to talk through with him, though, just as a very basic (and quick) introduction to linear functions.

My older son recognizes that the first choice – the area of a circle in terms of the radius – is nonlinear. That enables him to recognize that as the correct choice. However, he does not understand the functions described in choices (c) and (d), so we spend a few minutes talking through the functions described in these two choices.


Question 3:  Which number is equivalent to 3^4 / 3^2

Both kids have seen the power notation before, so I do not expect this one to be confusing.  My younger son multiplies out the top and bottom of the fraction and then divides.  I ask him if there is any other way to approach this problem, and he mentions subtracting the powers.

My older son takes an interesting approach and actually writes out the powers as repeated multiplication. He then cancels two threes from the top and bottom leaving 9. When I ask him if there is any other way to approach the problem he explains why the remaining choices cannot be correct.


Question 4: Determine the product. 800.5 x (2 x 10^6)

I haven’t really covered scientific notation with my younger son, so I expected this to be confusing to him. He did have a couple of interesting ideas, but this problem was more of a discussion between the two of us than a solution to the problem.

My older son has seen a little bit of scientific notation and does manage to reason out the solution in a way that was pretty different that what I was expecting.


Question 5: Which expression is equivalent to 4^7 x 4^-5?

This one is so similar to question 3 that I wonder why the Huffington post chose it for their article. I decided to probe a little deeper into their understanding of powers after they answered the question.

My younger son had a little trouble explaining the rule for multiplying powers. It turned out that he was struggling to find a name for the rule, but when I asked him to explain the rule in words he got pretty close. That was nice.

My older son understood that 4^(-5) was the same as 1 / 4^5 and solved the problem from there. I asked him where the other choices came from and he was able to come up with good answers.


Well, 30 problems in a couple of hours was tough, but we got it done. My younger son was able to understand 13 of the 15 questions and my older son, I think, understood all of them. As I mentioned in part 2 of this series, I was a little surprised by the lack of depth in the problems, though the sample size here is pretty small. Really only the question from the 8th grade test about linear functions had any depth to it. The rest could be answered by memorizing rules, or simply checking the answers.

For all the talk and controversy about the Common Core, honestly the questions don’t look that different from the SAT, ACT, and CAT questions I remember from the 80s.

Working through some NY State test questions (2/3) 5th Grade

[this is the 2nd in a 3 part series. I’ll use the same introduction for each. The first post is here:

https://mikesmathpage.wordpress.com/2014/08/09/working-through-some-ny-state-test-questions-13-3rd-grade/ ]

I saw this interesting link posted by a couple of people in the last few days:

How Would You Score On A Third-Grade Common Core Math Test?

To say the least, there’s been a lot written about the Common Core standards and their impact on education in the US. I haven’t followed this debate carefully, nor have I learned much about the standards. With the release of these questions I thought it would be interesting to get a glimpse of testing done in one state. So for today’s Family Math project I asked each kid to work through each of the 15 questions. This post is about the second 5 questions – the 5 from the 5th grade exam. My younger son will be going into 3rd grade this year and my older son will be going into 5th.

Question 1: Which phrase describes the volume of a 3 dimensional figure?

It takes my younger son a bit of time to understand the question and answers. He eliminates the last two answer choices, though I’m not sure he understood the actual statements. He did seem to understand the difference between area and volume, though, and that helped him talk through the two answers that he thought might be correct.

I liked hearing my older son talk through the answer choices. Right off the bat, this question is making him think him more than the 3rd grade ones did.


Question 2:  What is the value of the following expression?  1,536 / 24 ?

I’ve not talked about long division with my younger son, so I was a little surprised when he started trying to do the long division.  He quickly hit a wall.  However, he didn’t give up and figured out that you could multiply the answer choices by 24 and see if you got 1,536.   That approach helped him find the answer and he was really happy when he found it.

My older son does know long division and was able to get the answer pretty quickly.  I showed him an alternate approach of multiplying and looking at the last digit.


Question 3: Which expression means the same as the following phrase? Subtract 3 from the product of 8 and 5.

I like this question better than the “number sentence” questions from the 3rd grade exam. My younger son spends a lot of time thinking about the four choices, so I was happy that this question got him thinking. My older son forgets the question was asking about subtraction rather than addition, but catches that mistake quickly. That little mistake does show how easy it is to understand the math on these tests but accidentally get the answer wrong.


Question 4: In which number does the 5 represent a value 10 times the value represented by the 5 in 35,187?

We’ve spent a lot of time on place value, so I thought this would be a pretty easy one. My younger son talks through it and gets to the right answer. My older son got a little confused.


Question 5: What is the value of 2/5 + 3/7 ?

I wasn’t sure how my younger son would react to this problem. We talked a little bit about fractions last year, but it has been a while. Both kids took the same approach of getting a common denoninator, though, and worked through the problem without a lot of difficulty. I talked about alternate approaches to answering this question with both of the boys, too.


My understanding is that one of the main selling points of Common Core math is to get kids to think more deeply about math concepts. Though these questions are a bit more advanced than the 3rd grade ones, there still isn’t a lot of depth.  It is obviously harder to emphasize deep thinking on multiple choice tests, but even given that point, these questions are really just facts and computation.

Working through some NY State test questions (1/3) 3rd Grade

I saw this interesting link posted by a couple of people in the last few days:

How Would You Score On A Third-Grade Common Core Math Test?I

To say the least, there’s been a lot written about the Common Core standards and their impact on education in the US. I haven’t followed this debate carefully, nor have I learned much about the standards. With the release of these questions I thought it would be interesting to get a glimpse of testing done in one state. So for today’s Family Math project I asked each kid to work through each of the 15 questions. This post is about the first 5 questions – the 5 from the 3rd grade exam. My younger son will be going into 3rd grade this year and my older son will be going into 5th.

Question 1: Which fraction is equivalent to 2 / 8 ?

Not much of note on this problem. Both kids are familiar with fractions and both solve it in essentially the same way.


Question 2: If each side of a square has a length of 1 unit, which statement about the square is true?

I thought the answer choices for this question were strange – especially since I’m not sure that 3rd graders would understand the difference between mass and weight.  I enjoyed hearing my younger son talk through the difference between area and volume, though.


Question 3: Which number sentence can be used to determine the value of 72 / 9 ?

I was curious to see how this question would go since I’m not sure that either kid has ever heard the phrase “number sentence” before. My younger son was definitely confused in the beginning, but after reading the question again he did manage to make sense of the question. Again, it was nice to see his reasoning.


Question 4: What is 345 rounded to the nearest 100?

Interestingly here, both boys looked to eliminate obviously incorrect answers before finding the right answer. Despite not really doing much of anything in the way of these standardized tests, they seem to have picked up a few test taking strategies.


Question 5: What number goes in the black to make this number sentence true? 12 x 2 = ( ? x 2 ) + (2 * 2)?

Again, I wondered how the kids would react to the phrase “number sentence,” but it turned out that they basically ignored it. Both of them simply solved it by calculating. When I asked my older son for an alternate explanation, he showed how you could solve the equation by factoring a two out of the right hand side.


Not much to say about these questions. I didn’t feel that these questions were probing any deep concepts or taking any sort of unusual approach to elementary school math. They did get my younger son thinking, though, so I thought the exercise of going through these questions with him wasn’t a waste of time.

 

If you want to get people talking about math – talk about how to divide fractions!

Today my twitter feed has been filled with talk about this article by Professor Marina Ratner of U.C. Berkley:

http://online.wsj.com/articles/marina-ratner-making-math-education-even-worse-1407283282

Dividing fractions is a subject with a bit of history for me. On the funny side, as a kid I missed the week of school when this particular subject was taught and I never seemed to be able to catch up from that missed week. My pals on my high school math team loved giving me a hard time about always having to go back and figure out how dividing fractions worked. Even now they’ll needle me about it when it comes up in one of the videos I do with the boys.

When I was going through this subject with my older son I’m sure my approach was pretty much all over the place. The primary reason is that I’d never had to explain dividing fractions in detail to anyone – much less a kid. That alone assures a lot of stumbling around. Another reason is that although we were following Art of Problem Solving’s Prealgebra book when the subject came up formally, much of the teaching I do with my kids doesn’t follow a textbook and many important subjects come up almost out of the blue as we discuss various math problems. I certainly wasn’t aware of the different (and sometimes strongly held) beliefs about teaching fraction division when I was talking about it with my son.

Teaching the same subject to my younger son was a little different. Hopefully I’d learned a little bit from going through this subject once already (ha!), but also I’d begun to follow lots of teachers and math ed folks online so I’d seen some approaches to teaching fractions that were different that what I’d done on my own. I still used the approach in Art of Problem Solving’s Prealgebra book as the starting point, but I supplemented it with a couple of other ideas. Here are those three approaches from back in January. Having watched all three of this videos again just now, I’m perfectly happy with what we did and I believe that all three approaches have merit:

(1) Art of Problem Solving – define division as the reciprocal of multiplication and understanding fraction division just boils down to understanding what the reciprocal of a fraction is. Note also that he notices that you need multiplication to be commutative for this approach to work (!):

(2) Talking about patterns. Here we look at this sequences of divisions: 8 / 8, 8 / 4, 8 / 2, and 8 / 1 to help form a guess about what the value of 8 / (1/2) might be. We also use dimes and nickels to illustrate the division:

(3) Drawing rectangles / using snap cubes to talk about division. This is the approach that appears to have motivated today’s WSJ article -“Who would draw a picture to divide 2/3 by 3/4?”

The best response that I saw to the WSJ piece was this simple tweet from David Radcliffe:

Finally, if you don’t find fraction division to be an interesting topic to think about, perhaps you’ll be more interested in this delightful problem that Radcliffe posted earlier in the week:

Using divisibility rules to build number sense

Yesterday we played around with Jo Boaler’s Number Talks idea and that conversation got me thinking about other ways to build number sense. We’ve done a couple of different projects over the last few years that were at least partially motivated by building number sense. Probably the most fun was a talking about counting in bases other than 10 – even building binary adding “machines” out of duplo blocks! We’ve spent so much time counting in other bases over the years, though, so I wanted to try out a different idea. The idea I settled on was divisibility rules.

We’ve have talked a little bit about divisibility rules in the past. Chapter 3 of Art of Problem Solving’s Prealgebra book, for example, has a brief section on these rules. Like many topics in math I’m sure that not all of the details sunk in the first time through, so going back through some of the divisibility rules would be both worthwhile and fun. As an added bonus, my younger son really likes the divisibility rule for 7 for some reason.

We started with divisibility rules for 5 and 10. The goal was just to get them talking and thinking about numbers.

Next came the divisibility rules for 2, 4, and 8. Here we build a little bit on the place value ideas we talked about with the divisibility rule for 5. That idea gives us a slightly more precise way of talking about the divisibility rule for 2. Then we extend that rule to 4 and 8:

Next come the divisibility rules for 3 and 9. Developing these rules uses the a slightly different rule that the the one we used in the two previous videos. The difference here is that 10, 100, 1000, and etc are not divisible by 3 or 9.

I’m not intending to be rigorous here, so the important step of introducing remainders is informal. Looking at the remainders helps us understand how to build up the divisibility rule in these two cases:

Now comes one of my favorites – the divisibility rule for 11! I remember seeing this for the first time in 10th grade and being absolutely amazed. To help understand this rule I have to introduce the concept of negative remainders, which probably seems like a pretty strange concept when you see it for the first time, but will become an important concept much later when we study number theory:

Finally, as I mentioned at the beginning, the divisibility rule for 7 is one of my younger son’s favorite bits of math. Going through the construction of this graph is definitely worth a Family Math day all to itself. For now we have to be satisfied with using the picture from this Tanya Khovanova blog post:

Tanya Khovanova discusses the divisibility rule for 7

We wrap up the talk by walking through a few examples using her picture:

This seemed like a pretty fun activity. The number talks seem to help build understanding of arithmetic and place values. Talking about divisibility rules involves a lot of arithmetic, but the arithmetic is not the main focus. Instead this idea spends a bit more time on the ideas of place value and remainder. Feels like a pretty nice follow up or extension of the number talks idea.

Trying out Jo Boaler’s Number Talks idea

Yesterday (August 1, 2014) I saw this video on Jo Boaler’s YouCubed site:

I hadn’t come up with any ideas for our Family Math talks for this weekend, but after seeing this video I thought it would be fun to try out this idea with the boys this morning.

5×18 first

Younger son (will be in 3rd grade):

Older son (will be in 5th grade):

It was interesting to me that both of the kids approached the problem essentially the same way the first and second times. The subtle difference, which I think is somewhat interesting, is that 5*2*9 = 10*9 = 90, is a slightly easier computation than 5*9*2 = 45*2 = 90. At the time it didn’t occur to me to ask either kid why they chose to multiply in the order that they did.

Next comes the more difficult problem of 12 * 16. Rewatching Boaler’s video as I was writing this up, I see that I didn’t remember the second problem correctly – she used 12*15. Hopefully the difference between these two problems is not a big deal.

Older son first this time:

Younger son:

Again interesting to see how similar their approaches were. The first approach involved factoring, though the multiplication after that was different. I was surprised to see the way my younger son multiplied out 3*64. The second approach from both of them involved the distributive property, and the next two videos show some fun geometric ways to understand multiplication.

First, using our snap cubes we take a closer look at the picture I drew for my older son that connects the distributive property to squares and rectangles. We also talk about how thinking about multiplication this way helps understand multiplying polynomials. Naturally, I miss an easy opportunity to ask what happens in my algebraic example when x = 10. Oh well . . .

Finally I use the same geometric idea to show one way to understand why an negative number times a negative number is a positive number. Instead of viewing 12 * 16 as (10 + 2) * (10 + 6) we look at it as (20 – 8)*(20 – 4) in this video.

So, after going through this, I’m not sure what I’ll be doing differently going forward. I certainly agree completely that developing a good sense of numbers and arithmetic is extremely important and also see that this exercise is a good way to do that. I tend to focus on developing that number sense when we are working through problems, though, and personally prefer to work on it that way. That said, as reasonably quick and easy way to help kids develop number sense, the approach outlined in Boaler’s video seems pretty good.

Echoing and maybe amplifying a point in Keith Devlin’s latest article

Today (August 1, 2014) Keith Devlin published a nice article touching on both mathematical ideas and math education:

Most Math Problems Do Not Have a Unique Right Answer

If you have any interest at all in math or math education this article is well worth the 5 minutes it’ll take to read. I want to extend one of the points in the piece a little, though. Maybe it was my own poor reading of what he wrote, but I felt that one of the ideas near the middle of the article did not come through as clearly as I would have liked:

“Knowing how to solve an equation is no longer a valuable human ability; what matters now is formulating the equation to solve that problem in the first place, and then taking the result of the machine solution to the equation and making use of it.”

I think that Devlin has condensed a lot of information into the phrase “formulating the equation” so I’d like to un-condense it a bit. I worry that it is easy to think (as I did at first) Devlin’s comment means that you just write down some equations, head off to the computer, get your results, and charge full speed ahead.

The mathematical thinking that Devlin focuses on lies in understanding the results, not in simply writing down the equations. Rather than discussing this idea in the abstract, though, I want to give some fairly concrete examples from my own life and development in understanding this process.

The first example comes from my undergraduate thesis in college. In the spring of 1992 researchers (who would go on to win the Nobel prize in physics for their work) released the first detailed map of the cosmic microwave background radiation. I sat in the auditorium for this presentation and couldn’t help getting caught up in the excitement. After the talk I asked my adviser, Professor Ed Bertshinger, if I could try to study some piece of these results for my undergraduate thesis. He liked the idea and mentioned that he’d been wondering how the microwave background radiation would be distorted if the entire universe was rotating. Seemed like a fun problem – especially since my interest in physics was more math-y than hands on physics-y, so I dove in.

After 6 months or so I presented my findings to Professor Bertshinger. I was quite happy to have both an equation and some computer modelling. Though we had spent plenty of time talking over the course of the year, this was the first time that he saw the final results. Upon seeing the equation he drew a couple of diagrams on his chalkboard and eventually arrived at a picture nearly identical to the one drawn by my little computer model. It was stunning for me to see – six months of work for me and he drew the damn picture in about a minute just by waving his hands! Given enough time I could write down the equations, but he understood them. Seeing first hand the difference between those two acts was a powerful lesson for me.

The next examples are two fun billion dollar prize promotions that I’ve have come across my desk at work. The first was Pepsi’s “Play for a Billion” in 2003 and the second in Quicken’s billion dollar bracket game from this past spring.

In the Pepsi game 1,000 contestants took a guess at a six digit number from 000000 to 999999. So, 1,000 guesses at a number selected at random from one million possibilities. The chance of someone guessing correctly was 1,000 / 1,000,000, or more simply 1 in 1,000. As long as you’ve got the right security to prevent cheating, the math behind this game is not particularly hard (or even particularly interesting). This is a case where understanding the formula and solution does not take much time.

The Quicken promotion involved contestants trying to predict the outcome of 63 basketball games. Since you don’t know the precise chance of picking the outcome of any one game, the math you need to use to understand this promotion is pretty different than the math you needed for the Pepsi promotion. A purely formulaic approach is going to have a variety of problems – not the least of which is small changes in your assumptions lead to large changes in the estimate of the odds of someone winning this promotion. Worse, it is not at all obvious what the correct assumptions should be to begin with! Nonetheless, given the amount of money on the line, you need to be confident that you have analyzed the problem correctly and that’s where the mathematical thinking comes into play. Many of the articles about this promotion assumed that each game was a 50/50 chance, and thus sadly missed an opportunity to write a really neat and math-related article. Working through the various mathematical ideas behind this promotion (even the “birthday paradox” came up!) was one of the most interesting problems that I’ve ever worked on.

I guess the last example that’s not too hard to understand comes from problems in the financial markets. Roger Lowenstein’s incredible book “When Genius Failed” talks through the problems which arose when just one hedge fund got into trouble in 1997. More recently, we are all probably way too familiar with the problems that arose in the financial markets in 2008. Not all, but many of these problems came from groups of people using mathematical models and formulas that they did not fully understand.

The financial crises, and all of the terrible consequences that resulted, probably explains better than anything else why I wanted to clarify Devlin’s point. When Devlin writes about mathematical thinking, he isn’t talking about just writing down equations. Anyone – from undergrads writing about physics, journalists writing about basketball promotions, to derivative traders playing with other people’s money can write down equations and play with computer models. If we don’t understand the equations, solutions, or limitations, we get into trouble. That trouble can range from writing a dull senior thesis to causing the collapse of the world’s financial system. When we learn a little bit about mathematical thinking, hopefully we’ll do neither!