Even though I’ve taken the past couple of months off from posting, mathematical experiences haven’t! It has been a continued pleasure to watch my students this term gain understanding and ability to see calculus concepts in their everyday life. But a Thanksgiving extra-credit assignment gives their parents and families a chance to see it too.

A favorite extra-credit assignment I give to all students in attendance the last class before Thanksgiving break is to “teach” a new concept they learned in class to a willing audience. This includes preparing a short lesson before leaving on break to ensure they have class notes and their graphing calculator, then finding some time to present the lesson to family or friends. A short paper is due on the Monday after break that details what they taught, to whom it was taught, the reaction of the audience, and what they got from the experience.

I’m not always excited about grading assignments, but this is one that I can’t wait to get to. Students share humorous stories of family traditions and sibling interactions in their reports that lend a personal aspect to teaching.

Their experiences achieve the intended goals of the assignment:

• The parents see evidence their investment is not going to waste! When was the last time you heard your children share evidence they are learning anything new in school? Plus, when was the last time they wanted to sit down and share?

• The students gain a conceptual overview of the semester work and a deeper understanding of the content by teaching it.

I just finished reading this semester’s papers and I wasn’t disappointed! Typical comments included:

“My mom wasn’t understanding everything I said, but she said she could tell I knew what I was talking about.”

“I never knew how hard it was to teach something to someone who didn’t know anything about the topic – I really have a greater understanding for my teachers.”

“I could tell that they weren’t getting it so I used a whiteboard and drew a graph like we did in class. They started to get it then.”

“I found out that in order to teach you really have to know a lot. I didn’t realize how much I had learned this semester until I tried to teach my little sister.”

Even though I suspect some students are astute enough to write what they think I want to hear, the simple fact they knew what I wanted to hear means they got it! One of my students even referenced “Terry – An Unexpected Mathematician” in her paper - now there is an astute student! So I get validation they were listening throughout the semester.

As the semester draws to a close I take opportunities with each class to review the semester’s work and remind the students of how much they gained in content knowledge. That includes the ability to see mathematics everywhere, to identify studied concepts in everyday experiences, and to apply learning techniques to other courses.

What’s my evidence of their long-term achievement? As I tell them, when I meet them on the street ten years from now and I ask them, “What is a derivative?”, I hope they can tell me not only what a derivative is but will share an example of one they’ve observed that day!

I have to brag – the students in my classes this fall at Drake University have taken responsibility of their education in a way that I’ve not seen for quite some time.

It started over the summer when I got e-mails from some of them asking not only for textbook information but also responding to my early messages to them about what they could do to prepare for the first days of class. They were asking how much of the assigned problems they could do before Day 1.

It was truly amazing to have a majority of students ready to hand in homework during the first week. I wasn’t prepared to collect papers that soon so I even had to “make them wait” to hand in the first assignment until Week 2.

Responsibility is one of the lessons I try to instill in my students. Thanks to the diligent and thorough work by Wanda Everage, Vice Provost of Student Affairs and Academic Excellence, and her crew of student orientation counselors, the students in my classes entered their first day knowing that Drake has high expectations for them. At least in this case, they have accepted the challenge!

I’m proud of them. They have started their academic careers off on the right foot and I’m looking forward to being a part of their continued academic growth! Of course, time will tell how the pressure and stress of the first semester fares with them, but I’m optimistic they will continue to rise to the challenge.

Students will rise to meet our expectations - let’s not cheat them by setting the bar too low!

Image from Drury University

Tags: Students, Mathematics, Math Education

Where did the summer go? Classes have started and the leaves are getting a hint of orange. Remember how long the summer months seemed when you were a child? How can it be the same length of time when it goes so much faster - or at least it feels that way?

I have a theory - life is like the counter on a video tape player. Yes, video tapes are a disappearing technology, but I remember the thrill of how they revolutionized the way I enjoyed television.

If you “remember” video tape players, you can also relate how important it was to note the counter number at the start of your recorded show. The numbers at the beginning of the tape seemed to crawl by and it seemed to take forever to advance to past the middle of the tape. However, once past the middle, it took careful watching to not miss the correct number. It was the same tape, yet the numbers raced by as it got near the end.

Of course, mathematics can easily explain this “phenomenon” but it seems to mimic my perception of time as I near the higher-numbered birthdays. Another way to look at this is to observe that the pace of time is increasing at an increasing rate. Wow, it’s a derivative - yet another mathematical connection!

So, my life is like a video tape - the counter is speeding up as I get older. I wonder what technology my students will be referring to when they get to be my age. Any ideas?

Tags: Math Education, Mathematics

This has been a very interesting past five weeks for me: two 1-week workshops, a week of backpacking, and the lost of one of my two dear, loving 11-year old cats. It's good to be back on schedule - at least for now - and to have time to write my mathematical musings again!

I truly believe math is everywhere – even (especially) on a backpacking trip in Wyoming! Steve and I just got back from another adventure in the Bridger National Forest near Pinedale, WY on which mathematical examples were as prevalent as the many gorgeous vistas!

We packed from the Big Sandy Trailhead to our camp site at Dads Lake. Unlike previous trips, instead of moving to a new camp each day, we stayed at Dads Lake and made day-hikes the next two days.

On one adventure, the trail crossed a rock field and the worn path was no longer visible. We had three ways to solve the problem of finding our way that correspond to problem solving processes used in mathematics: Map, visual clues (blazes, cairns, landmarks), and GPS.

**Map**: Using the wilderness map for that region use identified landmarks such as visible mountain peaks. *When the teacher gives critical information needed to solve a problem and gives the process, then the students have a map.*

**Visual Clues**: Forest rangers or other hikers blaze a mark on trees (picture at right) to indicate the direction where a confusing set of alternate routes may be present. Another way they indicate a trail is to build cairns (picture at left) by stacking smaller rocks often on a large rock or boulder: formations which are obviously not created naturally. *Not all problems in everyday life come with a map showing critical information to be used in an indicated process, but critical information is there if students know what to look for.*

**GPS**: A Global Positioning System device shows the user’s location at a given moment in time and can track the trail taken over time. System maps may show critical information of the area but paths are not indicated. *Just like on our day-hike, sometimes the path to solving a problem is not obvious and the critical information is obscured. This is when we need to see where we are in the moment so we can search for other critical facts of the problem.*

No surprise to anyone who knows me, I was intrigued by and thrilled with the mathematics I encountered on the hikes. I will be meeting my fall semester classes at Drake University in about a month. One of my goals for my students will be to observe where they encounter mathematics (such as calculus) in their everyday life. A simple way for me to help them meet this goal is to share my everyday mathematical experiences. In problem solving it is important for students to know there are multiple ways to approach problem solving just as it was important for us to be able to use more than one way to navigate in backpacking!

Tags: Bridger National Forest, Mathematics Education, Math Education, GPS, Orienteering

In loving memory of Lomy, named for Shalom at a time when I needed peace!

How do you know what brand a company has? Mike Wagner says a company’s brand is a mark of their ownership and is reflected in every aspect of their business.

One company I interact with on a regular basis is the Texas Instruments Education & Productivity Division. Over the years all TI employees with whom I have interacted - from Help Desk personnel to Melendy Lovett, Senior Vice President - have exemplified that TI-CARES.

Last week in Atlanta, I teamed up two other TI National Instructors, Jane Barnard and Jim Haskins. to pilot a new TI course: Foundations of Algebra. Jane and I are on the writing team for this course designed to present mathematics to alternatively certified teachers – those who are teaching mathematics but do not have an undergraduate degree in either mathematics or mathematics education.

So, how does this course reflect TI’s brand?

First, they invested in the development of the workshop. Four mathematics educators spent three days designing, writing, and editing the materials that are being piloted with three workshops around the country.

Second, they brought Jane, Jim, and me to Atlanta for a week. Our time and expenses plus materials to help us present this first pilot workshop is paid by TI and offered free to the teachers.

Third, Melendy and four other TI representatives were there to observe and answer questions from the workshop participants. They were there because TI-CARES what teachers think about the new workshop. How many times have you seen the Senior Vice-President at a workshop?

So – how did TI benefit from this week? Of course they are interested in sales and marketing but more importantly, TI showed they care about increasing student mathematical achievement by helping teachers reach their full potential. Throughout the week instructors showed they care. The teachers saw this and now, they will take that caring brand back to their students where it really counts.

What is TI Education’s brand? It’s in their phone number: 1-800-TI-CARES!

I picked up my car for Terry’s Auto Service – it needed some minor electrical work.

With only a few months of car payments left, I shared with Terry that I was contemplating my option of keeping the car or trading it in. He reminded me that newer cars have even more electronics so I wouldn’t be eliminating all electrical problems – just exchanging the old set for a new one.

Always looking for opportunities to share how important mathematics is in everyday life, I commented how with the electronics they really use a lot of mathematics in their work

Terry’s response made my day, “You know math is not like any other subject. You can’t just memorize it and expect to understand anything.”

BINGO! That’s exactly what I emphasize to my students every day. You can’t memorize your way through math; you have to understand it on the conceptual level.

It’s always a surprise to my students when I refer to them as mathematicians. For most, that moniker conjures up a mental image of an old man who mumbles to himself while filling the chalkboard with unintelligible symbols. But, thanks to Terry, I can add an auto mechanic to my list of mathematicians!

What unexpected mathematician have you met today?

Picture from Images (Note: This NOT Terry)

Tags: auto repair, mathematics, math education, mathematicians

CBAM research and personal observation show that the greatest concern of most new teachers is to be accepted by their students. Successful teachers, however, learn that popularity comes at a cost - a loss of authority that is needed in order to be effectively assessors of student learning.

Each semester it’s always interesting to observe how students respond during the “honeymoon” phase when they are open to instruction and eager to learn - the first couple of weeks before receiving any grades. But what happens after that? The best students stay engaged and take the challenge to broaden their knowledge base and deepen their conceptual understanding. The other students, sadly to say, blame the instructor.

So what makes a great teacher? I agree with John Richardson’s three practices of a successful teacher: Their teaching style matches the student’s learning style, they give individual attention, and they have a passion for teaching. I would add these:

- Content knowledgeable beyond course requirements

Students can see right through a bluff if a teacher does not know the content. Nothing can substitute for knowledge beyond what you are teaching. However, students also respect honesty which leads to point 2. - Honesty

I love it when students ask questions that I never thought of before - it means that they are actively participating in the class. But, not having thought of the question before means not having already worked through a solution. The most honest answer in that case is, “That’s a good question that requires a good answer. Let’s all work on it and come back with an answer next time.” - A strong sense of self

Students can smell fear. Teachers who lack confidence that they are doing what’s best for the students will back down when confronted. As reported in Delaney Kirk’s post, effective teachers help students be successful and show strength.

So, you want to be a popular teacher? Strive to be an effective teacher - if not the most popular. You’ll be popular with the students who “get it”.

Tags: CBAM, effective teaching, student opinion

A few years ago, I was chatting with a teacher after one of my seminars. She wasn’t too enthused about letting her students use graphing calculators in class because the calculators were giving the answers to the questions on the tests. She’s not alone. A posting by Larry Davidson echoes this frequently voiced criticism.

Since my first posting on this subject, my response remains unchanged: “If the calculator is giving all the answers, what’s wrong with the questions?”

The National Council of Supervisors of Mathematics, the National Council of Teachers of Mathematics, Texas Instruments, and many state mathematics curriculum documents emphasize the appropriate use of calculators and concur that calculators can have positive influences in school mathematics when used with proper restriction and guidance.

Questions asked with a graphing calculator must be generated using higher order thinking skills. All of my assessment items contain my four favorite questions:

Why?

How do you know…?

What if …?

So what?

The last one brings students back to the problem posed and has them explain, analyze, or interpret their answers.

For example, instead of asking, “What is the value of log 4, to the nearest hundredths?” the question becomes:

“Write the exponential statement equivalent to log 4.”

“Explain the difference between log 4 = n and log n = 4.”

“Give the inverse function of log 4 = n.”

Students may use their calculators to test conjectures on similar problems but the calculator is not programmed to “give the answer”.

I challenge all mathematics educators to begin writing new assessment items so the calculator is used as a valuable tool - not an answer machine! After all, technology isn’t going to go away; it’s up to us to learn how to work with it, not against it. Assessment items that are independent of the calculator (that don’t give calculator-savvy students an advantage) assess conceptual understanding rather than rote memorization.

Yes, it’s not “traditional” and it takes more thought, but isn’t that what we’re asking of our students?

Picture from IMAGES

Tags: educational calculators, NCTM, NCSM, calculator debate, state curriculum standards

We’ve all heard a similar story:

*“The power was out so the cash register wasn’t working. My bill was $13.73 so I hand the kid behind the counter a $20 bill. Do you think he could figure out how much changed to give me? No way! What are they teaching kids in school these days?”*

These “experiences” are retold all the time; often starting with, “These calculators are ruining our young peoples’ minds!” The problem isn’t with the use of calculators, but with their inappropriate use!

As a mathematics educator for over 25 years, I’m also alarmed with the seemingly lack of general mathematical understanding of our populace - not just in our young people. Comments such as “I never did understand math! I never use algebra - what a waste of my time in high school.” are commonly heard from mature adults who attended school in the BC (before calculators) days! So, if my generation, who didn’t use technology, feels this way, why do calculators get so much blame for today’s generation’s lack of math proficiency?

To make sure the calculators are used appropriately, I have devised a three question test I use before using any technology for an activity:

**What is the primary mathematical objective of the activity?**

If I lose sight of the mathematical goal for the lesson, the students won’t know what mathematics they were suppose to learn. It becomes a “Golly, gee whiz, wasn’t that fun!” activity with no connections to other tasks have been made, no building on prior learning, and, consequently, no learning of the intended objective.**What can be done without the technology?**

This may seem a bit odd, but I need to ask myself this question to be sure I’m not using technology just for the sake of using it. Since I learned all of my undergraduate mathematics in the BC days, I know 99% of what I teach can be taught without a graphing calculator.**How can the technology enhance the conceptual understanding of the intended objective?**

This is the most important question. If I can’t answer this question with a solid educational reason, then I don’t use technology! Not every lesson or activity needs technology. Sometimes when introducing a mathematical concept such as plotting points on a coordinate plane, a non-technology approach is best. However, after the introduction phase, when students need to use plotting on a coordinate plane as a tool to reach a more complex, abstract intended objective, then the graphing calculator is a must!

Just as we don’t blame the car for a person’s lack of mechanical knowledge of how to change a tire, we shouldn’t place blame on the calculators for student’s lack of mathematical understanding. Calculators can be valuable tools to build conceptual understanding, but only if used appropriately.

So, what are we teaching kids today? If we’re using the calculator appropriately in our study of mathematics – more than they’ve ever learned before!

Image from Flickr