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Education: High school physics teacher has perfected the formula for inspiration – latimes.com

This is a really cool story. Amir Abo-Shaeer’s engineering academy was brand new when I was at Dos Pueblos. It wasn’t so much of an academy then, Mr. Abo-Shaeer was able to teach one robotics class. Now, seven years later, Dos Pueblos’ engineering academy has really taken off. Great work Mr. Abo-Shaeer!

Exciting news, I’ll be teaching calculus next quarter!

…kind of.

There is a bit of a catch. I’ll be teaching calculus for business and economics. Commonly referred to as business calc, or (as the TA’s like to call it) busy calc. The catch is that there is no trig (how can you do anything fun in calculus without trig?) and there will be a lot of applications to “business problems”. Regardless, it is still calculus and I am excited!

Keith Devlin is a professor at Stanford but probably better known as the NPR “math guy”. I really like his latest piece on Weekend Edition.

The Way You Learned Math Is So Old School : NPR

Remember going through your multiplication tables in elementary school? Or crunching through long division? Yeah, it sucked and that’s because it’s lame and useless, literally useless. Especially in today’s society where everyone uses a calculator or a computer to carry out arithmetic calculations. Computers do arithmetic for us, Devlin says, but making computers do the things we want them to do requires algebraic thinking. Elementary schools are starting to notice this and it’s changing the way arithmetic is taught. The emphasis in teaching mathematics today is on getting people to be sophisticated, algebraic thinkers. This doesn’t mean that we should stop teaching arithmetic in school; arithmetic is the foundation for strong algebraic thinking, it is not however, the end goal.

The word (almost) every math student hates, “proof”. When teaching a precalculus class I do not introduce the notion of a rigorous mathematical proof, however, I do try to get the students to understand why a particular result is true. I use the phrase “why might this be true” a lot. My goal is to get the class thinking about how we arrive at such a conclusion instead of merely accepting it. This doesn’t always work, in fact it rarely does. Students are too content with just memorizing a formula and knowing how to use it. If it work, it works, who cares why it works. To a certain extent that’s fine, I mean, it is important to know how to apply a result. But my point is this, it’s just as important to ask why something is true and at least try to understand why it is true. Even if you don’t get it completely.

To emphasize my point, I’m trying something new for my class next quarter. In the past when I’ve taught the properties of logarithms, I would simply write a property on the board followed by a brief justification. I had fallen into the mindset of my students. I didn’t care if they knew how to derive the properties of logarithms, I just cared that they could memorize them and know how to use them when solving equations. I know, I’m a hypocrite. For next quarter though, I’ve come up with a worksheet that will combat this.

Section 5.4 Properties of Logarithms

In this worksheet we will derive some important properties of logarithms. These properties will be essential for us when we solve logarithmic and exponential equaitons in section 5.5.

Before we can talk about the properties of logarithms we need to recall a few important facts.

• The definition of a logarithm: $\log_b(x)=y$ is equivalent to $b^y=x$.
• One to one property of logarithms: If $\log_b(x)=\log_b(y)$ then $x=y$.
• One to one property of exponents: If $b^x=b^y$ then $x=y$.

The Inverse Rules

$\log_b(b^x)=\bigstar$

What might $\bigstar$ be? First use the definition of logs to rewrite as an equivalent exponential equation. Then use the one to one property of exponents to find $\bigstar$.

$b^{\log_b(x)}=\bigstar$

What might $\bigstar$ be? First use the definition of logs to rewrite as an equivalent logarithmic equation. Then use the one to one property of logs to find $\bigstar$.

The Logarithm of a Product

$\log_b(MN)=$ ?

To derive a formula we make a substitution. Let $x=\log_b(M)$ and $y=\log_b(N).$

a) Use the definition of logs to rewrite $x=\log_b(M)$ as an equivalent exponential equation. Do the same for $y=\log_b(N)$.

b) Multiply together your results from part (a) and use properties of exponents to simplify.

c) Your result from part (b) should be a single exponential equation. Use the defintion of logs to rewrite that equation as an equivalent logarithmic equation.

d) Your result from part (c) should be a logarithmic equation that has an $x$ and a $y$ in it. Plug in $x=\log_b(M)$ and $y=\log_b(N)$ into that equation.

e) What can you say about $\log_b(MN)=$ ?

The Logarithm of a Power

$\log_b(M^p)=$ ?

To derive a formula we again make a substitution. Let $x=\log_b(M)$.

a) Use the definition of logs to rewrite $x=\log_b(M)$ as a equivalent exponential equation.

b) Your result from part (a) should be a single exponential equation. Raise both side of that equaiton to the $p^{th}$ power.

c) Use the definition of logs to rewrite your result from part (b) into an equivalent logarithmic equation.

d) Your result from part (c) should be a logarithmic equation that has an $x$ in it. Plug in $x=\log_b(M)$ into that equation.

e) What can you say about $\log_b(M^p)=$ ?

The Logarithm of a Quotient

$\log_b\left(\frac{M}{N}\right)=$ ?

a) Use the properties of exponents to rewrite $\log_b\left(\frac{M}{N}\right)$ as the logarithm of a product.

b) Use the product rule to split up your result from part (a).

c) Use the power rule to simplify your result from part (b).

d) What can you say about $\log_b\left(\frac{M}{N}\right)=$ ?

I still don’t care if the students know how to derive these properties. Rather, my goal here is to help them understand why these properties hold. In doing that I hope that they will also remember the properties better.

I compiled this post from sources I found online, along with some of my own thoughts. Being the poor grad student that I am, I failed to site my sources; so I can’t take full credit for writing this.

Why study mathematics?

Mathematics is more than just the science of numbers taught by teachers in schools and either enjoyed or feared by many students. It plays a significant role in the lives of individuals and the world of society as a whole. Mathematics is an essential discipline recognized worldwide, and it needs to be augmented in education to equip students with skills necessary for achieving higher education, career aspirations, and for attaining personal fulfillment. Its significance to education is not limited to the following aspects.

Enhances problem solving and analysis skills. Mathematics enhances our logical, functional and aesthetic skills. Problems enable us to apply our skills to both familiar and unfamiliar situations, thereby giving us the ability to use tested theory and also create our own before applying them. By developing problem solving strategies, we learn to understand problems, devise plans, carry out plans, analyze and review the accuracy of our solutions. The methods involved in problem solving develop use of reasoning, careful and reasonable argument, and decision making.

Applied in daily life. Mathematics is not a mere subject that prepares students for higher academic at- tainment or job qualification in the future. It is not all about practicing calculations in algebra, statistics and algorithms that, after all, computers are capable of doing. It is more about how it compels the human brain to formulate problems, theories and methods of solutions. It prepares us to face a variety of simple to multifaceted challenges every human being encounters on a daily basis. Irrespective of your status in life and however basic your skills are, you apply mathematics. Daily activities including the mundane things you do are reliant on how to count, add or multiply. You encounter numbers every day in memorizing phone numbers, buying groceries, cooking food, balancing a budget, paying bills, estimating gasoline consumption, measuring distance and managing your time. In the fields of business and economy, including the diverse industries existing around you, basic to complex math applications are crucial.

Base for all technologies. Anywhere in the world, mathematics is employed as a key instrument in a diversity of fields such as medicine, engineering, natural science, social science, physical science, tech science, business and commerce, etc. Application of mathematical knowledge in every field of study and industry produces new discoveries and advancement of new disciplines. All innovations introduced worldwide, every product of technology that man gets pleasure from is a byproduct of Science and Math. The ease and convenience people enjoy today from the discoveries of computers, automobiles, aircraft, household and personal gadgets would never have happened if it were not for this essential tool used in technology.

Career aspirations. Every branch of Mathematics has distinct applications in different types of careers. The skills enhanced from practicing math such as analyzing patterns, logical thinking, problem solving and the ability to see relationships can help you prepare for your chosen career and enable you to compete for interesting and high-paying jobs against people around the globe. Even if you do not take up math-intensive courses, you have the edge to compete against other job applicants if you have a strong mathematical background, as industries are constantly evolving together with fast-paced technology.

Since mathematics encompasses all aspects of human life, it is unquestionably important in education to help students and all people from all walks of life perform daily tasks efficiently and become productive, well-informed, functional, independent individuals and members of a society where Math is a fundamental component.

Update. November 18. This is a revision of the original post. There is still no conclusion and I don’t know where to put the last paragraph.

This is a very rough draft of my personal statement. I am applying to PhD programs in math education. The one below is for a math ed program at Berkeley. There is no conclusion, I need to add that. Also, it is a bit too long right now. I need to trim some stuff out.

Any and all comments and criticisms are welcomed and encouraged. Thanks in advance to anyone who has any advice, I really appreciate it!

There are two main reasons why I want to pursue a PhD in math education. First, I want to be involved in decision-making concerning equality, reform, and curriculum development for high school math education. Second, I want to better understand issues in undergraduate mathematics education. My experiences as a student and teacher have allowed me to realize these goals and have prepared me for studying math education at the PhD level.

My first experience in math education came as an undergraduate; I worked at UC Irvine’s Center for Educational Partnerships (CFEP) where I assisted in math instruction at Spurgeon Middle School in Santa Ana, California. The goal of CFEP is to increase postsecondary opportunities for California’s educationally disadvantaged students. I participated in CFEP’s Saturday Academy in Mathematics (SAM), an academic enrichment program. The SAM curriculum, written by CFEP and the Irvine Mathematics Project, is interactive instead of lecture based and encourages students to think and work together. Rather than simply teaching math, I want to be a part of designing and implementing new math curricula since I have seen the positive impact it can have on students.

My work as an undergraduate inspired me to pursue a master’s degree in math. As a part of the masters program at Cal Poly, I have the opportunity to teach my own math class. This past year, I have been teaching pre-calculus at Cal Poly. Teaching my own class has allowed me to reach a broader audience of students at a more advanced level. On the other hand, it has also revealed to me certain weaknesses in math education. Many of the students I teach lack the motivation and interest to learn the material. This apathy in many undergraduate math courses concerns me and I want to understand both the cause, and how to better combat it. Moreover, I am interested in the transition from learning high school math to learning college math.

The masters program at Cal Poly has been a good stepping-stone in preparing me for the challenges of a PhD. I know what it takes to succeed in a graduate program. Over the past year, I have learned how to balance teaching, studying, research, and classes. In my first year, I successfully studied for, and passed, a qualifying exam in real analysis. In addition, I have been conducting research with Dr. Vincent Bonini studying the classification of minimal surfaces. As a result of my research, I have authored a brief survey on minimal surfaces, appearing in this year’s edition of the Polymath Newsletter, and I had the privilege of lecturing at the Cal Poly Differential Geometry Seminar. On top of my teaching, exams, and research I have been able to thrive in my classes by maintaining a 3.74 GPA. The masters program has allowed me to hone my studying and researching skills necessary to excel in the PhD program.

Being a senior graduate student this year, I had the opportunity to mentor new TA’s in the Cal Poly Math Department TA mentoring program. I helped new TA’s in developing their syllabi and shared my teaching experiences.

I couldn’t put it any better. This is what I love about math education and why it is so important. You can check out his blog here.

As a part of my measure theory class each student must lecture on a certain topic. Today I lectured on the Jordan decomposition and the Radon-Nikodym Theorem. Lecturing to freshman precalculus students is one thing, lecturing to your peers and professor about measure theory is another. You encourage precalc students to ask questions and be engaged with the lecture. As the teacher you try to make the lecture more of a discussion. This is something you can do because you know and understand the material. However, this can be difficult when you don’t fully understand the subject (like measure theory). The other TA’s and I like to joke about our measure theory lectures, fake it ’til you make it. We might not have a mastery understanding of the material (yet), but we have the fundamentals. From there it’s all about looking and feeling confident when delivering your lecture.

For those who are interested, these are my notes on the lecture I gave today. The Elements of Integration and Lebesgue Measure by Robert Bartle is the reference for specific theorems, corollaries, and lemmas.

Definition. Let $\lambda$ be a charge on $\mathbb{X}$ and let $(P,N)$ be a Hahn decomposition for $\lambda.$ The positive and the negative variations of $\lambda$ are the finite measures $\lambda^+$, $\lambda^-$ defined for $E$ in $\mathbb{X}$ by $\lambda^+=\lambda(E\cap P)$, $\lambda^-(E)=-\lambda(E\cap N).$ The total variation of $\lambda$ is the measure $\left|\lambda\right|$ defined for $E$ in $\mathbb{X}$ by $\left|\lambda\right|(E)=\lambda^+(E)+\lambda^-(E).$

Note, by lemma 8.3, $\lambda^+$ and $\lambda^-$ are well defined and do not depend on the Hahn decomposition.

Theorem 8.5 Jordan Decomposition Theorem. If $\lambda$ is a charge on $\mathbb{X}$ then $\lambda(E)=\lambda^+(E)-\lambda^-(E)$ for all $E$ in $\mathbb{X}.$ Moreover, if $\lambda=\mu-\nu$ where $\mu$ and $\nu$ are finite measures on $\mathbb{X}$ then $\lambda^+(E)\leq\mu(E)$ and $\lambda^-(E)\leq\nu(E)$ for all $E$ in $\mathbb{X}.$

Proof. We prove the first assertion. Let $E\in X$ and let $(P,N)$ be a Hahn decomposition of $X.$ Then $P\cup N=X$ and $P\cap N=\emptyset.$ We have,

$\displaystyle \lambda(E)=\lambda\left((E\cap P)\cup(E\cap N)\right)=\lambda^+(E)-\lambda^-(E).$

We prove the second assertion. Since $\mu$ and $\nu$ have nonnegative values,

$\displaystyle \lambda^+(E)=\lambda(E\cap P)=\mu(E\cap P)-\nu(E\cap P)\leq\mu(E\cap P)\leq\mu(E).$

Similarly, $\lambda^-(E)\leq\nu(E).$ $\Box$

Theorem 8.6. If $f$ is integrable, $E\in\mathbb{X}$, and $\lambda$ is defined by $\lambda(E)=\int_Ef\,d\mu$ then

$\displaystyle \lambda^+(E)=\int_Ef^+\,d\mu \quad \lambda^-(E)=\int_Ef^-\,d\mu \quad \left|\lambda\right|(E)=\int_E\left|f\right|\,d\mu.$

Proof. Let $P=\lbrace x\in X:f(x)\geq0\rbrace$ and $N=\lbrace x\in X:f(x)<0\rbrace$. Then $X=P\cup N$ and $P\cap N=\emptyset$. Let $E\in\mathbb{X}$. Then

$\displaystyle \lambda(E\cap P)=\int_{E\cap P}f^+\,d\mu\geq0 \qquad \lambda(E\cap N)=-\int_{E\cap N}f^-\,d\mu\leq0.$

Thus, $(P,N)$ is a Hahn decomposition of $X$ with respect to $\lambda$. Now, by the definition of variation we have that

$\displaystyle \lambda^+(E)=\lambda(E\cap P)=\int_Ef^+\,d\mu$

$\displaystyle \lambda^-(E)=-\lambda(E\cap N)=\int_Ef^-\,d\mu$

$\displaystyle |\lambda|(E)=\lambda^+(E)+\lambda^-(E)=\int_E|f|\,d\mu.$

and the theorem is established. $\Box$

Definition. A measure $\lambda$ on $\mathbb{X}$ is said to be absolutely continuous with respect to a measure $\mu$ on $\mathbb{X}$ if $E\in\mathbb{X}$ and $\mu(E)=0$ imply that $\lambda(E)=0$. In this case we write $\lambda\ll\mu$. A charge $\lambda$ is absolutely continuous with respect to a charge $\mu$ provided that the total variation $\left|\lambda\right|$ of $\lambda$ is absolutely continuous with respect to $\left|\mu\right|$.

Example. Let $f\in M^+$ and $\lambda(E)=\int f\chi_E\,d\mu$. Recall, $\lambda$ is a measure by corollary 4.9. If $\mu(E)=0$ for some $E\in\mathbb{X}$ then $f\chi_E=0$ $\mu$almost everywhere. Thus,

$\displaystyle \lambda(E)=\int f\chi_E\,d\mu=\int 0\,d\mu=0$

which shows that $\lambda$ is absolutely continuous with respect to $\mu$.

Example. Let $\lambda$ be Lebesgue measure and $\mu$ the counting measure on $\mathbb{R}$. Then $\mu(E)=0$ if and only if $E=\emptyset$. Hence, $\mu(E)=0$ implies that $\lambda(E)=\lambda(\emptyset)=0$, which shows that $\lambda\ll\mu$. However, if $E=\lbrace x\rbrace$, the singleton set, then $\lambda(E)=0$ but $\mu(E)=1$. Thus, $\mu$ is not absolutley continuous with respect to $\lambda$.

Absolute continuity can also be characterized as follows.

Lemma 8.8. Let $\lambda$ and $\mu$ be finite measures on $\mathbb{X}$. Then $\lambda\ll\mu$ if and only if for every $\epsilon>0$ there exists a $\delta(\epsilon)>0$ such that $E\in\mathbb{X}$ and $\mu(E)<\delta(\epsilon)$ imply that $\lambda(E)<\epsilon$.

Proof. If $\lambda(E)<\epsilon$ for any $\epsilon>0$ then this is necessary and sufficient for $\lambda(E)=0$. Conversely, suppose that there exists $\epsilon>0$ and sets $E_n\in\mathbb{X}$ with $\mu(E_n)<\frac{1}{2^n}$ and $\lambda(E_n)\geq\epsilon$. Let $F_n=\cup_{k=n}^{\infty}E_k$ so that $\mu(F_n)<\frac{1}{2^{n+1}}$ and $\lambda(F_n)\geq\epsilon$. Since $(F_n)$ is a decreasing sequence of measurable sets, we have that

$\displaystyle \mu\left(\bigcap_{n=1}^{\infty}F_n\right)=\lim_{n\to\infty}\mu(F_n)=0 \qquad \lambda\left(\bigcap_{n=1}^{\infty}F_n\right)=\lim_{n\to\infty}\lambda(F_n)\geq\epsilon.$

Hence, $\mu(E)=0$ does not imply that $\lambda(E)=0$. $\Box$

Recall, corollary 4.9 states that if $f\in M^+$ then $\lambda(E)=\int_Ef\,d\mu$ is a measure. Conversely, when can we express a measure $\lambda$ as an integral with respect to $\mu$ of some function $f\in M^+$?

Corollary 4.11 showed that a necessary condition for this to hold is that $\lambda\ll\mu$. It turns out that this condition is also sufficient when $\lambda$ and $\mu$ are $\sigma$-finite. We state this result as a theorem.

Theorem 8.9 Radon-Nikodym Theorem. Let $\lambda$ amd $\mu$ be $\sigma$-finite measures on $\mathbb{X}$ with $\lambda\ll\mu$. Then there exists $f\in M^+$ such that $\lambda(E)=\int_Ef\,d\mu$ for $E\in\mathbb{X}$. Moreover, the function $f$ is uniquley determined $\mu$-almost everywhere.