This is a third reminder that this blog has moved! In order to keep receiving free updates on mathematics, teaching resources, and mathematical art, please visit the new website at www.MrHonner.com and subscribe via email or RSS.

I will repeat this message once more and then deactivate this wordpress.com blog.

For more information, you can read the original message below. I hope to see you on the new site!

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Over the past month, I have successfully migrated this blog, and all of its content, to a new self-hosted website: **http://www.MrHonner.com**.

I hope you like the look-and-feel of the new site! I am excited about all the new possibilities self-hosting brings, but I will continue to produce new blog posts on a regular basis.

If you wish to continue receiving updates from MrHonner.com, you will have to subscribe to the new site. (Software limitations prevent me from automatically transferring your subscription.)

Simply navigate to www.MrHonner.com, and enter your email address at the top right, under **Subscribe via Email**. You may also subscribe via RSS Feed by pointing your reader here.

I will repeat this message over the next few weeks, and then I will deactivate this wordpress.com blog.

I greatly appreciate your readership, your participation, and your support! I invite you to join me on my new platform; I hope you will.

Thank you-

Patrick Honner

Filed under: Uncategorized ]]>

This is a second reminder that this blog is moving! In order to keep receiving free updates on mathematics, teaching resources, and mathematical art, please visit the new website at www.MrHonner.com and subscribe via email or RSS.

I will repeat this message two more times over the next 3 weeks and then deactivate this wordpress.com blog.

For more information, you can read the original message below. I hope to see you on the new site!

=============================================================================================

Over the past month, I have successfully migrated this blog, and all of its content, to a new self-hosted website: **http://www.MrHonner.com**.

I hope you like the look-and-feel of the new site! I am excited about all the new possibilities self-hosting brings, but I will continue to produce new blog posts on a regular basis.

If you wish to continue receiving updates from MrHonner.com, you will have to subscribe to the new site. (Software limitations prevent me from automatically transferring your subscription.)

Simply navigate to www.MrHonner.com, and enter your email address at the top right, under **Subscribe via Email**. You may also subscribe via RSS Feed by pointing your reader here.

I will repeat this message over the next few weeks, and then I will deactivate this wordpress.com blog.

I greatly appreciate your readership, your participation, and your support! I invite you to join me on my new platform; I hope you will.

Thank you-

Patrick Honner

Filed under: Uncategorized ]]>

Over the past month, I have successfully migrated this blog, and all of its content, to a new self-hosted website: **http://www.MrHonner.com**.

I hope you like the look-and-feel of the new site! I am excited about all the new possibilities self-hosting brings, but I will continue to produce new blog posts on a regular basis.

If you wish to continue receiving updates from MrHonner.com, you will have to subscribe to the new site. (Software limitations prevent me from automatically transferring your subscription.)

Simply navigate to www.MrHonner.com, and enter your email address at the top right, under **Subscribe via Email**. You may also subscribe via RSS Feed by pointing your reader here.

I will repeat this message over the next few weeks, and then I will deactivate this wordpress.com blog.

I greatly appreciate your readership, your participation, and your support! I invite you to join me on my new platform; I hope you will.

Thank you-

Patrick Honner

Filed under: Uncategorized ]]>

This question relates to mutual funds and investment growth: how much would $1,000 be worth after 35 years?

Filed under: Uncategorized ]]>

To begin exploring the idea, I thought about possible fundamental questions and eventually settled on this: *What are some important content-independent skills that children need to learn?*

I posted the question on Google+, and Don Pata, MrBombastic, Jim Wilder, and Christopher Danielson all offered some great ideas. Here’s the list we compiled through discussion, in no particular order.

**Problem-Solving P****erseverance **— the ability to sustain focus and work through a problem to the end

**Intellectual Discipline **— the willingness to focus and invest energy on learning and development

**Number Sense **— an intuitive understanding of quantity: magnitudes, relationships, and scales

**Reflection **— the ability to objectively self-assess, refine, and adapt

**Communication **— the ability to express information and emotion in a variety of ways, and appropriately interpret and process the expressions of others

**Courage **— the willingness to fail

**Curiosity** — the habit of inquisitiveness and the ability to ask good questions

A good list to start with! Thanks for all the help, and if there are other suggestions, please feel free to leave them in the comments.

You can see the original thread on Google+ here.

Filed under: Uncategorized ]]>

This is an artistic representation of the numbers 1 through 144: each color represents a different prime divisor, and so each stack represents the prime factorization of the given number.

You can read more about this piece here.

Filed under: Art, Numbers, Photography ]]>

Below are a few examples of what I consider “bad” questions. “Bad” here might mean poorly worded, poorly conceived, or irrelevant. In addition, there is an example of a question with a problematic rubric.

First, a type of problem that occurs regularly, one that is a pet peeve of mine. From the Algebra 2 / Trig exam:

The concept of “middle term” is artificial and depends entirely on how one chooses to evaluate the given expression. This question does not test an authentic mathematical skill; it tests how well a student executes one particular method of evaluating this particular expression.

Next, an example of a poorly-phrased question, one that confuses mathematical terminology. From the Integrated Algebra exam:

To “solve” a system of equations, one must find the ordered pairs that satisfy the given equations. Apparently this question wants only the *y-values *of those solutions, but the phrasing confuses what it means to “solve a system” and to “solve an equation”.

Students can probably figure out what the question-writer wants to hear in this case, but the lack of precision will only exacerbate confusion about the word “solve”.

Here’s a problem on the Algebra 2 / Trig exam that is simply irrelevant.

This question tests one thing, and one thing only: knowledge of an arcane and largely irrelevant notation, namely, degree-minute-second representation of angles. Would anyone outside the nautical or astronomical worlds consider this even remotely valuable?

Lastly, this question from the Integrated Algebra exam is formulated in a reasonable way, but the official scoring guide poses some unnecessary problems.

This question asks the student to graph an equation and then, using the graph, determine and state the roots of the equation. The correct answer is “2 and -4”, and with appropriate work, is worth three points.

However, if the student gives the answer “(2,0) and (-4,0)”, the student can only earn two out of the three points. So if the student gives the coordinates of the points where the graph crosses the x-axis, rather than names the “roots” of the equation, there is a one-third deduction.

While I believe that the distinction between roots and points is important, losing one-third credit seems seems unnecessarily punitive here. If we want to test student’s knowledge of vocabulary, there are better ways to do it than by sneaking it in at the end of an involved algebra problem.

Moreover, since the question requires that the student use the graph, the student is already being forced to interpret the problem in a geometric context. Penalizing them for thinking of the roots geometrically, then, doesn’t quite seem fair.

Filed under: Uncategorized ]]>

Mathematically erroneous questions consistently appear on these exams. Here are two recent examples, both from the Algebra 2 / Trigonometry exam.

According to the scoring key, the correct answer is (4). This would be the correct answer if the angle were given as -50 *degrees*. Notice, however, that no degree symbol is present. This means the angle is actually -50 *radians*. In degrees, -50 radians is equivalent to roughly -2864.8 degrees, which itself is equivalent to roughly 15 degrees. Thus, the actual correct answer is (3).

The above problem could be considered a typo (although no correction was ever issued), but the most erroneous Regents questions demonstrate a real lack of mathematical understanding on the part of the exam creators. Consider the following question on complex numbers.

None of these answers are correct.

The exam writers believe that (3) is the correct answer. Given a complex number *a + bi*, the conjugate is indeed *a – bi*, provided that *a *and *b* are *real numbers .* But

As emphasis on standardized exam performance continues to grow, a few points here or there can make a big difference in the lives of students, teachers, and schools. The consistent appearance of erroneous mathematics on these exams calls into question their validity as a measurement of “student achievement”.

Filed under: Teaching ]]>

This question is based on the rare tie that occurred in a Olympic qualifying race. In how many different ways could eight racers finish first, second, and third?

Filed under: Uncategorized ]]>

After seeing my TED Talk on Creativity and Mathematics, Maria Droujkova, one of the authors of Moebius Noodles, contacted me and asked if I would put together a piece about how I use weaving to explore mathematical ideas.

My piece is titled “Weaving Your Way Through Mathematics,” and can be found on the Moebius Noodles website.

http://www.moebiusnoodles.com/2012/07/weaving-mathematics/

More resources on mathematics and weaving can be found here.

Filed under: Art, Resources ]]>

So I read “Is Algebra Necessary?“, Andrew Hacker’s OpEd in the New York Times, knowing that I probably wouldn’t agree with much of it, but still prepared to examine my beliefs. Hacker offers up a few of the standard reasons why students shouldn’t be forced to take “Algebra” in high school (“it’s too hard”; “it turns kids off”; “not everyone’s going to be an engineer”), but he doesn’t really bring anything new or substantial to the discussion.

One thing I did find interesting, however, was Hacker’s suggested fix: instead of “Algebra”, we should be teaching courses like “Citizen Statistics”.

It could, for example, teach students how the

Consumer Price Indexis computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.

This is indeed a good idea. Ironically, exploring the mathematics of the CPI is largely an algebraic activity.

Discussing *which items should be included* means creating a mathematical model and declaring variables for the unknown quantities we wish to investigate. These fundamental skills are taught and emphasized in high school algebra.

Determining the *weights that these items should be given* essentially amounts to finding the coefficients of some function of those variables and exploring the consequences of those choices. In their basic forms, these skills are also taught and developed in high school algebra.

Thus, it seems to me that Hacker is suggesting we replace Algebra, with, well, Algebra.

There are worthwhile discussions to be had about what we are teaching, why we are teaching, and how we are teaching. But those discussions should be led by people who really understand what’s going on. If Andrew Hacker thinks we should replace Algebra with Algebra, then someone else should be leading the discussion.

*Click here to see more in Teaching.*

Filed under: Teaching ]]>

Here are two examples of 2×2 belt weaves. In both cases, the same kinds of strips are used, but in a different initial configuration.

The 3×3 belt weaves offer more initial configurations, and show more complexity.

There is a rich and interesting structure to explore in these “belt weaves”. For example, these two weaves look similar, but are indeed different.

My students and I had fun exploring the mathematical relationships between the various belt weaves. I will share some of our ideas and results in my series on Weaving in Math Class.

*Click here to see more in ***Teaching**.

Filed under: Art, Teaching ]]>

Students should learn to be on the lookout for things like this. They should develop a quantitative curiosity, exploring the various ways that quantity can be disguised.

And ultimately be able to put together a real quantitative analysis that helps them make good decisions.

This is math that everyone can use!

*Click here to see more in Application.*

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There is nothing more important, especially in the HS classroom, than a teacher who is an expert in his/her respective field. The “tricks of the trade” are second nature for those truly called to this noble profession. A teacher needs passion and patience, but more than anything else she needs to know what she’s talking about. That is what gives the teacher authority.

And here is a slightly edited version of my response, originally posted as a comment on Diane Ravitch’s blog.

*I disagree with the sentiment expressed by your reader.*

*First, it’s impractical to expect all teachers to be masters of their content. If we need 200,000 math teachers in the US, we aren’t going to find 200,000 math experts for those positions, unless we dramatically redefine what we mean by ‘expert’.*

*Second, content expertise is not the source of a teacher’s authority. Being a teacher is more about being a leader than being an authority, and leadership is earned through a combination of respect, effort, enthusiasm, caring, and expertise.*

*Lastly, subject-specific content delivery is one aspect of teaching that can obviously be streamlined by technology. As education evolves, we teachers need to make our case by emphasizing the variety of other tools and expertise we bring to students, not just content.*

The original post can be seen here.

*Click here to see more in Teaching.*

Filed under: Teaching ]]>

From an ordering perspective, it’s nice that all toppings are a uniform $1 in cost. But is it fair?

A burger’s worth of applewood bacon is probably worth a buck; a fried egg is reasonably priced at a dollar.

But how many bread-and-butter pickles would I expect to get for a dollar? Or pickled jalapenos? Way more than I could put on a burger, I suspect!

*Click here to see more in Appreciation.*

Filed under: Appreciation ]]>

This question is based on the the recent historic presidential elections in Egypt. How many more votes did the winner receive?

*Click here to see more in Challenge.*

Filed under: Challenge, NYT, Sports ]]>

These particular story rules sound remarkably similar to techniques of mathematical exploration.

*#7 Come up with your ending before you figure out your middle.*

*#9 When you’re stuck, make a list of what WOULDN’T happen next.*

*#10 Pull apart the stories you like.*

*#11 Putting it on paper lets you start fixing it.*

*#20 Exercise: take the building blocks of a movie you dislike. How would you rearrange them into what you DO like?*

Working backwards, proof by contradiction, taking apart things you understand and trying to put them back together, getting your hands dirty by working out the details–these are all common and useful techniques in exploring and understanding mathematical ideas.

And as a friend pointed out, writing a story is indeed a kind of problem-solving; maybe it’s not so surprising how much that process shares in common with mathematics.

And as a teacher, the following two really resonate, for obvious reasons!

*#2 You gotta keep in mind what’s interesting to you as an audience, not what’s fun to do as a writer. They can be very different.*

*#1 You admire a character for trying more than for their successes.*

*Click here to see more in Teaching.*

Filed under: Teaching ]]>

*Click here to see more in Photography.*

Filed under: Appreciation, Uncategorized ]]>

http://www.youtube.com/watch?v=Rs8masuu5oE

Watching this pattern change and evolve made me think of chaotic dynamical systems, and their representations like the Mandelbrot set.

I wonder if there is mathematics that models systems like this and describe how they behave?

*Click here to see more in Resources.*

Filed under: Appreciation, Resources ]]>

By overlaying all the small letter *a*‘s, say, from all the different fonts, one can take a *visual average* and create a new letter *a*. Repeat for the whole alphabet, numerals, and punctuation marks, and *voila!*, you’ve got **Averia**.

The idea of taking a visual average may be a bit mysterious, but the author describes a few different approaches in how to combine the images. Essentially all of the instances of a particular symbol are placed on top of each other, and the the darkest parts of the new image are where the instances intersect the most. The result is then smoothed over to create a readable letter.

And the font looks pretty nice, if not too exciting. Just what you might expect from the *average font*.

*Click here to see more in Application.*

Filed under: Application, Appreciation, Representation ]]>

*Click here to see more in Appreciation.*

Filed under: Appreciation ]]>

This question deals with the high number of adults in prison in Louisiana. Of Louisiana’s 4.5 million residents, how many are in jail?

*Click here to see more in Challenge.*

Filed under: Challenge, NYT, Sports ]]>

Click.** **The light came on.

Click. “This one’s *long delay. ” *

Click.** **“This one’s *boost .” *Click.

I started to get confused. Then it hit me: there are

possible combinations of effects!

So I asked to go back to the bass.

*Click here to see more in Appreciation.*

Filed under: Appreciation ]]>

It’s a familiar story to anyone who has ever contemplated *teaching to the test*. As rankings/ratings/grades become more and more important, colleges/schools/students (and teachers) tend to focus more and more on those metrics, perhaps at the expense of what’s really important (whatever that might be).

A perfect rating system, presumably, would compel the rated parties to meet and expand the standard of excellence. But in practice, it seems difficult to come to a consensus about what comprises excellence, and even harder, then, to construct an appropriate rating system.

So how should we measure a college or university?

*Click here to see more in Teaching.*

Filed under: Challenge, Teaching ]]>

*Click here to see more in Photography.*

Filed under: Photography ]]>

*Click here to see more in Appreciation.*

Filed under: Appreciation, Art, Photography ]]>

At the very least, I think this is an open-and-shut case of fraudulent advertising!

*Click here to see more in Appreciation.*

Filed under: Appreciation ]]>

Bressoud starts with the encouraging news in the STEM fields: Science, Technology, Engineering, and Mathematics. In the past fifteen years, colleges have seen a 33% increase in students in these majors. However, those numbers may be dominated by particularly large increases in Biology and Psychology.

As a mathematician, Bressoud is interested in **math-intensive** majors, and so he looks more closely at mathematics, engineering, and physical sciences**. **As total college enrollments and STEM majors have increased, these math-intensive majors attract a consistent percentage of students. In fact, Bressoud notes that this percentage has been stable for the past 30 years, as math-intensive degrees have shown no growth as a percentage of overall college degrees.

This is curious, given the increasingly quantitative nature of modern society, industry, and academia. Are greater percentages of students in other countries pursuing such degrees? Or do we only need 0.5% of our college students studying math-intensive fields?

*Click here to see more in Teaching.*

Filed under: Teaching ]]>

You see, if a rider could evade the conductor at a consistent rate, it might make mathematical (if not ethical) sense to gamble on paying the higher fare every so often. For example, let’s say you can successfully sneak a free ride once every three attempts. If the ticket in the station costs $5, then the price of the on-board ticket should be at least $7.50 to discourage you from attempting this cheat.

I never found out the price difference in Portugal. But I do know how it works on the Long Island Rail Road.

Returning from vacation, we were rushing from the airport to the train station. We didn’t have time to purchase tickets from the machine beforehand as the train was literally pulling into the station as we arrived. After a long day’s travel, we were happy just to make it on board our final mode of transportation.

And it turned out to be nearly a 100% increase. Instead of the usual $6.25, the on-board charge was $12. I guess that means they think fare-evaders can get away with it a little less than half the time?

We were happy to get home in a timely manner. And I was happy to have one more open mathematical question resolved!

*Click here to see more in Application.*

Filed under: Application ]]>

But there is still work to do.

*Click here to see more in Appreciation.*

Filed under: Appreciation, Teaching, Technology ]]>