Red card

As I visited our student workroom one afternoon, a small group of students were killing time by kicking a miniature soccer ball around. As tried to walk by, one student tried to trap an errant pass and gave me a good kick in the shin. Of course, he was mortified.

Naturally, I had the only reasonable response to this stimulus: I quickly spotted a dark red index card and held it silently in the air in his direction.

That Makes It Invertible!

There are several ways of determining whether an $n \times n$ matrix ${\bf A}$ has an inverse:

$\det {\bf A} \ne 0$
The span of the row vectors is $\mathbb{R}^n$
Every matrix equation ${\bf Ax} = {\bf b}$ has a unique solution
The row vectors are linearly independent
When applying Gaussian elimination, ${\bf A}$ reduces to the identity matrix ${\bf I}$
The only solution of ${\bf Ax} = {\bf 0}$ is the trivial solution ${\bf x} = {\bf 0}$
${\bf A}$ has only nonzero eigenvalues
The rank of ${\bf A}$ is equal to $n$

Of course, it’s far more fun to remember these facts in verse (pun intended). From the YouTube description, here’s a Linear Algebra parody of One Direction’s “What Makes You Beautiful”. Performed 3/8/13 in the final lecture of Math 40: Linear Algebra at Harvey Mudd College, by “The Three Directions.”

While I’m on the topic, here’s a brilliant One Direction mashup featuring the cast of Downton Abbey. Two giants of British entertainment have finally joined forces.

Description of Teach North Texas

The following article appeared in the Fall 2012 newsletter of the Forum on Education of the American Physical Society.

Recruiting and Preparing Science and Math Teachers at the University of North Texas

Mary Harris: TNT Co-Director and Professor, Department of Teacher Education and Administration

Jennifer McDonald: TNT Program Advisor

John Quintanilla: TNT Co-Director and Professor, Department of Mathematics

Cindy Woods: TNT Master Teacher

Science and mathematics are fields from which there is a high rate of teacher attrition. Demand for teachers within these high needs fields is growing, with greatest need in schools with diverse populations of low-income students. Compounding the problem, the landmark Rising Above the Gathering Storm (National Research Council, 2007) reports that “middle and high school mathematics and science teachers are more likely than not to teach outside of their own fields of study” (p. 113). The deficiencies found in teaching science and mathematics at the middle and high school levels can be attributed to three primary causes: lack of science and mathematics educator preparation programs that provide strong subject content and pedagogical knowledge for pre-service teachers, lack of support during the first years of employment, and failure of universities to recruit into science and mathematics teacher education programs (National Research Council, 2010).

In response to the state and national imperative for the United States to reemerge as world leader in science, technology, engineering, and mathematics (STEM), the University of North Texas (UNT) implemented the Teach North Texas (TNT) program, a replication of the pioneering UTeach program at the University of Texas at Austin. Combining classroom teaching experiences throughout the pedagogical course sequence, opportunities for professional development and induction, and financial support for its students, TNT has been increasingly successful in raising the quantity and quality of competent and innovative teachers within these high-need fields.

TNT now boasts almost 300 students, including 16 students seeking Physics or Physics and Mathematics secondary teaching certification. Equally impressive, TNT students are an academically talented group, with higher average GPAs and SAT Math scores than college and university averages. We expect to produce approximately 50 graduates annually in the coming years.

So, how are we doing it? Our success is an intricate interweaving of five key components: collaboration, curriculum, staffing, targeted recruiting and retention practices, and community.

Collaboration

Prior to the inception of TNT, our university of over 28,000 undergraduate students produced an annual average of only 8 secondary mathematics and science teachers combined, and a majority of these graduates considered their generalist education courses to be greatly disconnected from teaching in the STEM fields. Our university leaders saw the need for change and pledged cooperation and support for the creation of a teacher education program specifically geared toward mathematics and science.

The TNT program is firmly rooted in a collaborative vision of excellence. The College of Education (COE) and College of Arts & Sciences (CAS) worked together to implement TNT; and continued support of each college’s dean, our provost, and our president have enabled TNT to grow and sustain a remarkably successful mathematics and science teacher education program.

TNT also collaborates with five local school districts, since all field experiences require the cooperation of district officials and human resources departments. We maintain an extensive network of local Mentor Teachers who open up their classrooms to TNT students for observation and teaching practice. Without their cooperation and feedback, the extensive training we provide students would not be possible.

Curriculum

TNT’s unique curriculum instantly sets our program apart from others. It begins with an invitation for university students who have a declared interest in science or mathematics to explore teaching through a minimal-investment one-credit-hour course. Enrollment in this introductory course does not automatically require completion of the entire TNT program. However, those students who discover a passion for teaching complete a 21-credit hour minor along with their STEM content majors. These students join an educational community solely dedicated to their growth as future middle school and high school science or mathematics teachers. TNT’s dedication does not stop upon graduation; we continue to support our graduates with an induction program throughout the first and second years of their teaching careers.

The TNT minor is a sequence of courses created by COE and CAS that are specifically designed to prepare future science and mathematics teachers for the middle and high school classrooms. Throughout the program, students are trained to use the inquirybased 5E lesson plan model (engage, explore, explain, elaborate, evaluate). Inquiry-based teaching pushes TNT students toward a deeper, more thoughtful understanding of lesson planning and challenges their prior assumptions about how a class “should” be taught. Utilization of the 5E model leads to innovative and creative lessons that ensure student engagement and content understanding.

TNT stresses early and continuous field experiences both to recruit students into the program and to ensure that the teachers we produce are well qualified to teach when they enter the workforce. Few degree programs provide so many opportunities for students to interact with, learn from, and emulate practitioners in the field. TNT requires students to observe and teach at the elementary, middle school, and high school levels. As much as possible, TNT students are placed in classrooms comprised of ethnically and linguistically diverse students and in locations where a majority meet the criteria for free and reduced price lunches. This ensures that TNT students recognize and experience the cultural and emotional developmental needs of diverse populations in and out of highneed schools.

Throughout the TNT minor, students develop and practice inquirybased lessons and use results of student assessment to improve teaching. Initially, for their very first field experience, students think through the questioning strategies necessary to deliver a proven lesson effectively using a kit of materials. As TNT students gain teaching expertise, they are increasingly challenged to utilize knowledge acquired by creating original lesson plans. By the second semester of the minor course sequence, our students are developing and teaching their own lesson plans, and by the fourth semester, they are developing and teaching lessons planned for the high school level.

Furthermore, TNT students use their own teaching as the subject of action research or inquiry. They videotape and study their teaching and study the results of student pre- and post-assessments. This enables our pre-service teachers to create probing questions that link student responses directly to their understanding of the material. This type of self-critical practice characterizes excellence and innovation in mathematics and science teaching.

Staffing

As a joint program of COE and CAS, TNT is led by co-directors from both colleges, both reporting to their respective deans. TNT functions as a small department within CAS but offers a minor comprised of courses from both colleges. Both of the co-directors are released half-time by their departments to lead TNT, and they make requests for funding, development, and research support through both of their colleges.

The co-directors lead a team of seven Master Teachers, who are experienced STEM teachers who possess, at minimum, a master’s degree. Master Teachers are readily available as students prepare their lessons, and, as much as possible, they travel to schools to observe and critique TNT students as they teach. Master Teachers hold the rank of Lecturer, teach multiple TNT courses, and obtain approvals for students’ field experiences with COE and school district officials. The external relationships developed by Master Teachers help the program navigate such obstacles as processing criminal background checks, facilitating appropriate placements for apprentice (student) teaching, and communicating opportunities for employment.

TNT also employs three staff members. With a dedicated Program Advisor, we try to provide a one-stop shopping model of advising our students. Our advisor works with CAS faculty and staff advisors regarding recruiting, enrollment, and degree requirements. The advisor works with COE staff on procedures for formal admission to the teacher education program, admission into apprentice teaching, and teacher certification by the state. The Program Advisor also assists with financial aid issues and scholarship applications and directs Talon Teach, the TNT student organization. Our administrative services officer ensures our unit’s compliance with university protocols with purchasing, payroll, and other similar issues. Finally, our materials manager serves as the program’s quartermaster, tracking and maintaining our large inventory of pedagogical materials that TNT students peruse and borrow as they teach their lessons.

TNT highly depends on existing COE faculty specializing in science and mathematics education as well as CAS faculty who have a high interest in educational issues. Course teams made up of tenured/ tenure-track faculty from COE and/or CAS, Master Teachers, and graduate students meet several times each year and as often as once a week. We organize at least one annual meeting of the entire teaching faculty around program evaluation. At these meetings, data documenting student learning and other evidences of program success or need for improvement are considered, and plans are made for changes to the courses and program and for evaluation of the impact of changes.

Recruitment

In Fall 2012, TNT successfully attracted and enrolled 113 new students. TNT utilizes a combination of guerrilla and conventional marketing strategies with a primary focus on the engagement of students, faculty, and staff to increase enrollment and retention. Recruitment tools include marketing materials, electronic marketing, and word of mouth. The combination of direct recruiting efforts and the maintenance of program visibility have yielded success semester after semester.

The recruitment tactic with the greatest return is presence at all University Orientation sessions. At least one student and the Program Advisor attend all Orientation fairs and are present during one-on-one academic major advising sessions prior to new and transfer student registration. They promote TNT with short program highlights as invited by the hosting faculty advisor. TNT is advertised as a “you can do it all” degree plan. Emphasis is placed on earning a full STEM major along with teacher certification in a “two for one” slogan. Co-presentation by the Program Advisor and a current science student has led to a large increase in science (including physics) enrollment. Follow-up via e-mail reminds students of how to enroll and provides contact information in case of additional questions.

Electronic communication and marketing materials ensure program visibility and engage students, faculty, and staff. Powerful recruiting tools include targeted e-mail solicitation to science and mathematics students, strategically placed fliers with student quotes and photos, and student-produced banners outside of the TNT classrooms. Promotional items such as t-shirts, pens and pencils, notepads, key-chains, and bumper stickers, build program recognition. Faculty, staff, and students are active on the website, Facebook, and Twitter.

Word of mouth has created positive TNT program recognition across campus and within surrounding communities. TNT partners with not only campus administrators but also student advisors from other academic areas, teachers and school administrators in surrounding school districts, and UNT faculty. Consistent recognition of the efforts these partners make for TNT and its students garners continued support from these contributors. Enough cannot be said about the importance of partnerships. Numerous students are referred directly to our program not only by faculty and staff advisors, but also by community program affiliates.

Community

Students quickly learn that TNT is much more than a sequence of courses; it is a community of students, instructors, and staff with a common commitment to STEM teaching and learning. Be it through lounging in the student workroom on a beanbag chair, studying at the worktables with peers for an upcoming exam, receiving lesson-planning assistance from a Master Teacher, or cracking jokes with the Program Advisor after an advising session, our students are engulfed in a supportive environment that is dedicated not only to academic success but also to personal fulfillment. TNT has experienced success in retaining the students recruited by creating an environment in which students want to participate. This is done by using the best resource available: human interaction. TNT faculty and staff, administrators, instructors, colleagues, campus advisors, and most importantly, students all facilitate human interaction. Collaboration and communication, appreciation, and a sense of belonging come with membership in TNT. How we work with the many different people who make up TNT directly impacts the program’s success.

1. National Research Council, (2007). Rising Above the Gathering Storm: Energizing and Employing America for a Brighter Economic Future. Washington, DC: The National Academies Press.
2. National Research Council, (2010). Rising Above the Gathering Storm, Revisited: Rapidly Approaching Category 5. Washington, DC: The National Academies Press.

Mathematical teddy bear

Small print: have you HUGGED a mathematician lately? (haven’t they earned it?)

Source: http://www.cafepress.com/kleinfour.22599164

Mathematical T-shirt

Source: http://maa-store.hostedbywebstore.com/MAA-ALL-GOOD-T-SHIRT/dp/B008VGK9NS?class=quickView&field_availability=-1&field_browse=2673528011&field_product_site_launch_date_utc=-1y&id=MAA+ALL+GOOD+T-SHIRT&ie=UTF8&refinementHistory=brandtextbin%2Csubjectbin%2Ccolor_map%2Cprice%2Csize_name&searchNodeID=2673528011&searchPage=1&searchRank=salesrank&searchSize=12

Fourier transform

Source: http://www.smbc-comics.com/index.php?db=comics&id=2874#comic

More on divisibility

Based on my students’ reactions, I gave my best math joke in years as I went over the proofs for checking that an integer was a multiple of 3 or a multiple of 9. I started by proving a lemma that 9 is always a factor of $10^k - 1$ . I asked my students how I’d write out $10^k - 1$ , and they correctly answered $99{\dots}9$ , a numeral with $k$ consecutive $9$ s. So I said, “Who let the dogs out? Me. See: $k$ nines.”

Some of my students laughed so hard that they cried.

There are actually at least three ways of proving this lemma. I love lemmas like these, as they offer a way of, in the words of my former professor Arnold Ross, to think deeply about simple things.

(1) By subtracting, $10^k - 1 = 99{\dots}9 = 9 \times 11{\dots}1$ , which is clearly a multiple of 9.

(2) We can use the rule

$a^k - b^k = (a-b) \left(a^{k-1} + a^{k-2} b + \dots + a b^{k-2} + b^{k-1} \right)$

The conclusion follows by letting $a = 10$ and $b =1$ .

From my experience, my senior math majors all learned the rule for factoring the difference of two squares, but very few learned the rule for factoring the difference of two cubes, while almost none of them learned the general factorization rule above. As always, it’s not my students’ fault that they weren’t taught these things when they were younger.

I also supplement this proof with a challenge to connect Proof #2 with Proof #1… why does $11{\dots}1 = \left(a^{k-1} + a^{k-2} b + \dots + a b^{k-2} + b^{k-1} \right)$ ?

(3) We can use mathematical induction.

If $k = 0$ , then $10^k - 1 = 0$ , which is a multiple of 9.

We now assume that $10^k - 1$ is a multiple of 9.

To show that $10^{k+1}-1$ is a multiple of 9, we observe that

$10^{k+1}-1 = \left(10^{k+1} - 10^k \right) + \left(10^k - 1\right) = 10^k (10-1) + \left(10^k - 1\right)$ ,

and both terms on the right-hand side are multiples of 9. (I also challenge my students to connect the right-hand side with the original expression $99{\dots}9$ .)

$\hbox{QED}$

Divisibility tricks

Based on personal experience, about half of our senior math majors never saw the basic divisibility rules (like adding the digits to check if a number is a multiple of 3 or 9) when they were children. I guess it’s also possible that some of them just forgot the rules, but I find that hard to believe since they’re so simple and math majors are likely to remember these kinds of tricks from grade school. Some of my math majors actually got visibly upset when I taught these rules in my Math 4050 class; they had been part of gifted and talented programs as children and would have really enjoyed learning these tricks when they were younger.

Of course, it’s not my students’ fault that they weren’t taught these tricks, and a major purpose of Math 4050 is addressing deficiencies in my students’ backgrounds so that they will be better prepared to become secondary math teachers in the future.

My guess that the divisibility rules aren’t widely taught any more because of the rise of calculators. When pre-algebra students are taught to factor large integers, it’s no longer necessary for them to pre-check if 3 is a factor to avoid unnecessary long division since the calculator makes it easy to do the division directly. Still, I think that grade-school students are missing out if they never learn these simple mathematical tricks… if for no other reason than to use these tricks to make factoring less dull and more engaging.

A mathematical magic trick

In case anyone’s wondering, here’s a magic trick that I did my class for future secondary math teachers while dressed as Carnac the Magnificent. I asked my students to pull out a piece of paper, a pen or pencil, and (if they wished) a calculator. Here were the instructions I gave them:

Write down just about any number you want. Just make sure that the same digit repeated (not something like 88,888). You may want to choose something that can be typed into a calculator.
Scramble the digits of your number, and write down the new number. Just be sure that any repeated digits appear the same number of times. (For example, if your first number was 1,232, your second number could be 2,231 or 1,322.)
Subtract the smaller of the two numbers from the bigger, and write down the difference. Use a calculator if you wish.
Pick any nonzero digit in the difference, and scratch it out.
Add up the remaining digits (that weren’t scratched out).

I asked my students one at a time what they got after Step 5, and I responded, as the magician, with the number that they had scratched out. One student said 34, and I answered 2. Another said 24, and I answered 3. After doing this a couple more times, one student simply stated, “My mind is blown.”

This is actually a simple trick to perform, and the mathematics behind the trick is fairly straightforward to understand. Based on personal experience, this is a great trick to show children as young as 2nd or 3rd grade who have figured out multiple-digit subtraction and single-digit multiplication.

I offer the following thought bubble if you’d like to think about it before looking ahead to find the secret to this magic trick.

What the magician does: the magician finds the next multiple of 9 greater than the volunteer’s number, and answers with the difference. For example, if the volunteer answers 25, the magician figures out that the next multiple of 9 after 25 is 27. So 27-25 = 2 was the digit that was scratched out.

This trick works because of two important mathematical facts.

(1) The difference $D$ between the original number and the scrambled number is always a multiple of 9. For example, suppose the volunteer chooses 3417, and suppose the scrambled number is 7431. Then the difference is

$7431 - 3417 = (7000 + 400 + 30 + 1) - (3000 + 400 + 10 + 7)$

$= (7000 - 7) + (400 - 400) + (30 - 3000) + (1 - 10)$

$= 7 \times (1000-1) + 4 \times (100-100) + 3 \times (10-1000) + 1 \times (1-10)$

$= 7 \times (999) + 1 \times (0) + 4 \times (-990) + 3 \times (-9)$

Each of the numbers in parentheses is a multiple of 9, and so the difference $D$ must also be a multiple of 9.

A more algebraic proof of (1) is set apart in the block quote below; feel free to skip it if the above numerical example is convincing enough.

More formally, suppose that the original number is $a_n a_{n-1} \dots a_1a_0$ in base-10 notation, and suppose the scrambled number is $a_{\sigma(n)} a_{\sigma(n-1)} \dots a_{\sigma(1)} a_{\sigma(0)}$ , where $\sigma$ is a permutation of the numbers $\{0, 1, \dots, n\}$ . Without loss of generality, suppose that the original number is larger. Then the difference $D$ is equal to

$D = a_n a_{n-1} \dots a_1a_0 - a_{\sigma(n)} a_{\sigma(n-1)} \dots a_{\sigma(1)} a_{\sigma(0)}$

$D = \displaystyle \sum_{i=0}^n a_i 10^i - \sum_{i=0}^n a_{\sigma(i)} 10^i$

$D = \displaystyle \sum_{i=0}^n a_{\sigma(i)} 10^{\sigma(i)} - \sum_{i=0}^n a_{\sigma(i)} 10^i$

$D = \displaystyle \sum_{i=0}^n a_{\sigma(i)} \left(10^{\sigma(i)} - 10^i \right)$

The transition from the second to the third line work because the terms of the first sum are merely rearranged by the permutation $\sigma$ .

To show that $D$ is a multiple of 9, it suffices to show that each term $10^{\sigma(i)} - 10^i$ is a multiple of 9.

If $\sigma(i) > i$ , then $10^{\sigma(i)} - 10^i = 10^i \left( 10^{\sigma(i) - i} - 1 \right)$ , and the term in parentheses is guaranteed to be a multiple of 9.

If $\sigma(i) < i$ , then $10^{\sigma(i)} - 10^i = 10^{\sigma(i)} \left( 1-10^{i-\sigma(i)} \right) = -10^{\sigma(i)} \left( 10^{i-\sigma(i)} - 1 \right)$ , and the term in parentheses is guaranteed to be a (negative) multiple of 9.

If $\sigma(i) = i$ , then $10^{\sigma(i)} - 10^i = 0$ , a multiple of 9.

$\hbox{QED}$

Because the difference $D$ is a multiple of 9, we use the important fact (2) that a number is a multiple of 9 exactly when the sum of its digits is a multiple of 9. Therefore, when the volunteer offers the sum of all but one of the digits of $D$ , the missing digit is found by determining the nonzero number that has to be added to get the next multiple of 9. (Notice that the trick specifies that the volunteer scratch out a nonzero digit. Otherwise, there would be an ambiguity if the volunteer answered with a multiple of 9; the missing digit could be either 0 or 9.)

As I mentioned earlier, I showed this trick (and the proof of why it works) to a class of senior math majors who are about to become secondary math teachers. I think it’s a terrific and engaging way of deepening their content knowledge (in this case, base-10 arithmetic and the rule of checking that a number is a multiple of 9.)

As thanks for reading this far, here’s a photo of me dressed as Carnac as I performed the magic trick. Sadly, most of the senior math majors of 2013 were in diapers when Johnny Carson signed off the Tonight Show in 1992, so they didn’t immediately get the cultural reference.

Bad puns

I thought I saw an eye-doctor on an Alaskan island, but it turned out to be an optical Aleutian.

No matter how much you push the envelope, it’ll still be stationery.

A dog gave birth to puppies near the road and was cited for littering.

A grenade thrown into a kitchen in France would result in Linoleum Blownapart.

Two silk worms had a race. They ended up in a tie.

Time flies like an arrow. Fruit flies like a banana.

Two hats were hanging on a hat rack in the hallway. One hat said to the other: ‘You stay here; I’ll go on a head.’

I wondered why the baseball kept getting bigger. Then it hit me.

A sign on the lawn at a drug rehab center said: ‘Keep off the Grass.’

The midget fortune-teller who escaped from prison was a small medium at large.

The soldier who survived mustard gas and pepper spray is now a seasoned veteran.

A backward poet writes inverse.

In a democracy it’s your vote that counts. In feudalism it’s your count that votes.

If you jumped off the bridge in Paris, you’d be in Seine.

A vulture carrying two dead raccoons boards an airplane. The stewardess looks at him and says, ‘I’m sorry, sir, only one carrion allowed per passenger.’

Two Eskimos sitting in a kayak were chilly, so they lit a fire in the craft. Unsurprisingly it sank, proving once again that you can’t have your kayak and heat it too.

Two hydrogen atoms meet. One says, ‘I’ve lost my electron.’ The other says, ‘Are you sure?’ The first replies, ‘Yes, I’m positive.’

Did you hear about the Buddhist who refused Novocain during a root-canal? His goal: transcend dental medication.

There was the person who sent ten puns to friends, with the hope that at least one of the puns would make them laugh. No pun in ten did.

And now for some math puns:

What’s purple and commutes? An Abelian grape.

What is lavender and commutes? An Abelian semigrape.

What’s purple, commutes, and is worshipped by a limited number of people? A finitely-venerated Abelian grape.

What do you get when you cross a mountain goat with a mountain climber? You can’t — a mountain climber is a scalar.

How does a linear algebraist get an elephant in a refrigerator? He splits the elephant into components, stuffs the components in the refrigerator, and declares the refrigerator closed under addition.