Math Is Everywhere...

Backpacking in 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!

Who Cares? TI-CARES

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!

Tags: Technology, Texas Instruments, Mathematics Education

The Great Calculator Debate Rages On

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

“What Are They Teaching These Kids?”

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:

1. 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.
2. 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.
3. 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

Tags: graphing calculators, math education, math technology

1 + 1 Math is Fun

My sister, Jan Wilton, is the director of the fast growing, progressive Noah's Ark Pre-School in Mesa, AZ. As sisters we are as different as, well, as pre-school to college, in our teaching careers. She is a 4’10” ball of energy whose focus has always been on early childhood education. The teachers at her school are just as enthusiastic and dedicated and have a desire to learn and challenge themselves and their students.

When I was assigned a Texas Instruments Discover workshop in the Phoenix area, I asked Jan if I could work with the children at Noah’s Ark. We had never worked together professionally, so I was even more excited when she suggested a 3-hour workshop for teachers at her school and other pre-schools in the area. She did a great job organizing and promoting the event: 1 + 1 Math is Fun.

So, last week about 20 pre-school teachers, parents, my sister, and I spent three hours on Thursday night sharing, experiencing, and communicating mathematics, technology, and teaching for pre-school aged children. I introduced the appropriate use of the TI-10 calculator and the CBR2 data collection device used with the TI-73 calculator for the teachers to use with the children.

I also emphasized the integration of mathematics with other disciplines. Literature is most appropriate connection for this age group. The teachers were excited to see how easy it is to teach mathematics while reading a story. Some of my suggestions included:

• The Cheerios Counting Book
• Penguins in the Fridge
• Counting on Frank
• Gator Pie
• One Hundred Hungry Ants
• The Greedy Triangle

I was impressed with how quickly they came up with mathematics lessons to accompany the stories.

The next day, I had the opportunity to work with the four and five year olds at the school. It was great fun watching them interact with the technology. They were as intrigued by the overhead projector as they were with the fact the CBR2 mapped their footsteps using sonar as they created “the letter W” on the view screen. They loved “playing the game” with basic addition and subtraction facts using the Problem Solving mode of the TI-10. Most exciting to me was watching the increased confidence and excitement of the teachers as they realized how much mathematics the children were experiencing and how easy it was going to be to continue the teaching with the technology.

The latest report from my sister has been very rewarding. She relates how the teachers refer to my comments regarding the need to integrate mathematics with reading and how excitedly they’ve embraced the school’s goal for next year to build on mathematics and science. The real winners are the children! Thanks to Jan and her teachers, the children of Noah’s Ark Pre-School in Mesa, AZ will become little mathematicians - and that’s the greatest reward for my time!

1 + 1 Math is Fun? Yes, and now they know it too!

Tags: Noah's Ark Pre-School, Texas Instruments Discover, math education

If You Always Do What You've Always Done...

A definition of insanity is doing the same thing, but expecting different results. Some educators seek to stay in their comfort zone by always teaching the same thing, in the same way - unfortunately, the results are the same.

I hear a lot of people say we should prohibit the use of technology in the mathematics classes and go back to more “traditional” teaching. The most common reason cited is the poor computation skills of students. This is like prohibiting the use of electricity and going back to candles because people are not very good at lighting matches. “Shopkeeper” mathematics (computation) is important, but it’s not the most important mathematics for today.

Human nature tends to have us look at the past through rose-colored glasses. Yes, there are some students today who cannot quickly master computation facts, but many students in the past couldn’t either. The reality is that there are a lot of people who were taught in the “traditional” way who hate, avoid, and fear mathematics. Yet, many of these math-avoiders advocate against new teaching methods. It would appear that misery loves company, “If I had to suffer through learning math, then so will you!”

Irina makes some interesting observations in her posting, “New Math, Fun Math”, regarding innovations in mathematics teaching. Most interesting were some of the comments readers made about her post. It never fails to amaze me how hard some people work to stop the use of technology. And, yet, the same people use technology (computers, the Internet, blogging software, etc.) to communicate their comments!

Educators who protest the use of technology must realize the problem isn’t the technology itself, but its usage. I emphasize not just the use of technology, but its appropriate use.

Teachers, look at new ways to incorporate technology into your teaching because, like electricity, it isn’t going to go away. Therefore, we need to embrace it, find ways to help it make us better educators, and make our students better learners.

Remember - If we always do what we’ve always done, then we’ll always get what we’ve always gotten! Let’s start looking for new results by going outside our comfort zone, embracing the appropriate use of technology, and be willing to try something new!

Tags: Mathematics Education, Educational Change, Technology

Techology’s Santa

Techology’s Santa Last weekend I conducted a Texas Instruments PTE (Preservice Teacher Education) workshop at the University of Wisconsin Eau Claire to about 35 enthusiastic middle school and high school mathematics pre-service teachers.

What an uplifting experience to work with young, fresh, teachers-in-training who embrace a new way of teaching! We had a great time exploring how to appropriately use technology to be better mathematics educators. The six hours went by quickly (too quickly) to be able to show them every piece of technology but they did have a chance to explore a couple of graphing calculators (TI-73 and TI-84+) and data collection devices (CBR2 and CBL2) as well as the wireless Navigator system.

I consider myself extremely fortunate to not only have an enjoyable teaching career but also a rewarding “hobby” as a TI instructor of the PTE workshops. The best part is being able to play “Santa” by handing out the free calculators to the student participants of the workshops! In addition to the free calculators the students also receive over $400 worth of technology when they get their first teaching job just for attending a 6-9 hour face-to-face, hands-on workshop! What a deal!!

Can you tell that I’m excited about this program and think that this is of real value?

Tags: Mathematics, Mathematics Education, Texas Instruments, TI-73, TI-84+SE, TI-Navigator, CBR2, CBL2, Technology

If the Calculator Gives All of the Answers, then. . .

Traditional mathematics teaching requires a lot of memorization and little thinking. In the eyes of a middle school student, "Math is just moving numbers around to get the answer the teacher has." and school is, "A place where young people go to watch old people work and are graded on how well they watched."

With the appropriate use of technology, mathematics can be taught interactively utilizing four views: Numerical, Graphical, Symbolic, and Verbal. “USA Today” is a great example of how effective this can be. The NCTM Principles and Standards for School Mathematics is a wonderful resource of how this can be integrated into your lessons.

The problem is it takes more work, initially, to teach this way. It's much easier to write "What is…" questions than to write questions that begin with:

• Explain your thinking…
• What if…
• Interpret the response in context…

I have to admit that as I spend 30 minutes each day setting up and another 30 minutes tearing down the technology for my classes I sometimes envy the professors who walk in a minute or two prior to class; concerned only about an adequate supply of chalk. But, I am willing to spend extra effort and time setting my room because I believe with technology I am a better educator and my students are better learners.

A common complaint about mathematics is that technology/calculators "gives" all the answers. To that I respond, "If the calculator gives all the answers, then what's wrong with the questions?" Technology isn't going away - we have to embrace it and learn how use it to better show students the “whys” and “hows” of mathematics – not just the answers. Only then will we begin to redefine “traditional mathematics teaching.”

Tags: Mathematics, Mathematics Education, Technology, NCTM

3 Steps to Ownership of Knowledge

"I do all of the homework but I can't pass your tests or quizzes." The implication is that somehow it is my fault when, in fact, the problem lies with the student's misunderstanding of the purpose of homework. They don't understand that homework is for their benefit - not mine - and that it takes more than a quick once-through to "do homework" correctly.

Often students have the mistaken idea that putting work down on paper is the same as doing their homework. Correctly they accomplished this by working backwards from the answer, working with another student, or asking an "expert". But, when they stop there, they have not worked to "own the knowledge" of the concepts behind the problems. 

Here are three steps to ensure that time spent on homework accomplishes more than just putting work on paper.
1. Do each problem any way possible.
2. Go back 2-3 hours later and re-work each problem that couldn't be worked without help. This checks to see if understanding was gained.
3. Go back 2-3 days later and try those same problems once again. This third step is to ensure that true learning took place and to give confidence of being able to do the problem after other learning has taken place.

Lack of confidence is a major problem. Revisiting difficult problems a couple of times gives confidence needrd to face similar problems on quizzes and tests.

When I get resistance from students I use an analogy of a music lesson. I took saxophone lessons in grade school from Mr. Blum. He told me from the start that if the day came when I no longer practiced he would drop me as a student. Being dropped by Mr. Blum was the last thing I ever wanted to happen so I practiced my weekly lessons with vigor. I knew that playing through the pieces was not sufficient; I had to practice them each day until I could play them without error. So must students be willing to work each day on difficult homework problems.

Tags: Mathematics, Mathematics Education, Responsibility, Homework

I Hold You Responsible...

At the beginning of the semester I start each class with, "I hold you responsible for all the math you've ever had.? At this point I've never seen more scared students - like deer in the headlights.

Why are my students so scared? One reason is because despite having passed the prerequisites, they know they don't understand the underlying mathematics from their past courses. Why, after having successfully passed other courses, are they not prepared to continue where they left off?

A major reason is that they have been allowed to memorize their way through the other courses and have not been held responsible for understanding the concepts. I tell my students there is little to memorize but a lot to understand.

Another reason is that, historically, too much time is devoted to reviewing past material before introducing the new material. I was appalled when my step-son's Algebra II teacher said the first semester was primarily a review of Algebra I. My step-son needed to be challenged and held responsible in order to value the education he was receiving. Instead, he floated through the year of Algebra II working just hard enough to not fall below a C. What did he learn from that experience? Mostly that he didn't have to work to achieve a grade, just be patient and the teacher would review everything he needed to know for the test.

The hard lessons come very quickly once students enter a college mathematics course that doesn't have the "luxury" of spending half of the semester reviewing the prerequisite content. I see my responsibility as designing lessons which incorporate hands-on, interactive activities to help build my students' conceptual understanding. When students construct understanding for themselves, instead of mimicking lecture processes, they have a basis on which to build additional, deeper, broader understandings of new new concepts.

The Texas Instruments technology helps accomplish this. The TI-84+ family of graphing calculators, the TI-Navigation system, and the TI-SmartView calculator emulator help me bring the numeric, graphical, symbolic, and verbal views of mathematics alive to my students.

It is imperative that we, as educators, reverse the trend of handing everything to our students and that we begin "to hold them responsible for all the math they've ever had." They will thank us down the road.

Tags: Student Expectations, Student Responsibility, Mathematics Teaching, Texas Instruments graphing calculators, TI-84+, TI-Navigator, TI-SmartView, Math Concepts