DEPARTMENT OF MATHEMATICS

REFLECTIONS

The team of writers involved in the researching and compiling of this history of the Laboratory School Mathematics Department included Richard Anderson, Megan Balong, Joan Duea, Dennis Kettner, James Maltas, Earl Ockenga, Lynn Schwandt, and John Tarr. The task was taken seriously by this team. Regular meetings were held using Zoom technology. These sessions were full of light-hearted comments as memories were shared. Time collapsed and it seemed like the team was sitting around a table in the back of a classroom. Lots of reflecting took place.

A critical skill to learn and use in most professions is to be a reflective person. This is especially true in the profession of teaching. The following excerpts from faculty show the nature of implementing teaching strategies and then reflecting on what was learned by the teacher and the learners.

Reflection from PLS Director

Ross Nielsen

Ross Nielsen, Director of Price Laboratory School from 1962 to 1986, made the following statement: “The Lab School should enhance the pre-student teaching experiences for prospective teachers, an area in which we now make one of our greatest contributions. Teacher education students are engaged in classroom observations, sit in with teachers planning their lessons, and participate in all kinds of classroom activities before they actually student teach.  This is an area of training that other schools often cannot provide,” 

Nielsen explained:  “Teaching is an art and a science and apprenticeship should come after a period of study.” 

TELLING versus ASKING

Reflection on Teaching by John Tarr 1961-1998

How was teaching mathematics at MPLS different from that in typical public schools? At one time students memorized facts and were drilled until they could correctly respond immediately. This form of teaching extended beyond basic arithmetic to courses throughout the curriculum. Often students did not understand what they “had learned.” 

In the 1950s, 1960s and beyond, the MPLS program emphasized “discovery learning.” Instead of being told facts and concepts, students discovered them for themselves. This was accomplished by teachers asking questions or posing exercises from which students drew conclusions and made generalizations. The students owned the generalizations.

When compared to students from throughout the state and nation, MPLS students did very well. Results from standardized tests and success in higher education revealed that discovery learning worked. Furthermore, classes were more interesting and discipline was seldom a problem. Students had to pay attention to questions being asked. And they had to think. They had to look for patterns, make generalizations, and draw conclusions.

MPLS Service Project to the Profession: Mathematics Conferences

Merrie L. Schroeder

1960s

The Mathematics Department at the University of Northern Iowa and the mathematics faculty of Malcolm Price Laboratory School (MPLS) partnered for many years to host a mathematics conference at the Laboratory School for mathematics teachers across the state. These conferences were held at MPLS for many years and were so highly attended that the Lab School could not accommodate all the teachers seeking to attend.

I began my career in the very late 1960’s. This was an auspicious time in math education because many significant changes were being made in curriculum and pedagogy. The success of the Russian spacecraft Sputnik lit a huge fire under the national leaders in education, especially in mathematics and science. Put simply, the result was Modern Math...a sweeping change in how mathematics was constructed and understood. It permeated NK-12 and every mathematics subject area. UNI/MPLS faculty did not only embrace these changes in mathematics, but many members served on national boards, councils, and work groups to create and nurture this change. Needless to say, many veteran teachers were anxious about these changes. Many of us novice teachers were eager but naive. It was extremely fortunate for Iowa mathematics teachers to have ground-level help in moving into a completely new way of teaching mathematics. 

As a graduate of the University of Northern Iowa, I was familiar with the MPLS and many of the faculty from my requisite field experiences toward licensure, held in that school. But entering MPLS as a new teacher rather than as a pre-service teacher was a very different experience.

It is amazing how little a pre-service teacher observes when the vision is focused on the task at hand. But as a new teacher with license in hand my eyes were opened to a different place and a different relationship with the school and its people. I was so very proud—and a bit anxious—to walk into my first UNI/MPLS Mathematics Conference as a “real teacher.” My pride was that I had succeeded. My anxiety was that every math professor there knew my quirks, strengths, and weaknesses. Some may have even marveled under their breath that I had made it. But I did. And what I saw and experienced “from the other side of the desk,” as we say, was truly memorable.

What I finally saw was a whole NK-12 school on tip-top display for Conference attendees. The school was sectioned by levels of learners, and so the very youngest children’s classrooms were not actually used for the conference since none of the attendees could fit into the chairs. Nevertheless, their areas all impressively displayed the math learning that was happening in those areas

The classrooms that had chairs big enough to seat adults were commandeered by the mathematics faculty as areas for breakout sessions. Needless to say, these areas were also lively with displays of student learning and goals for learning. Some areas were reserved for lounging and refreshments for attendees, and other areas were reserved for speakers to regroup. I can only imagine how much energy and camaraderie went into preparing the classrooms and building for the onslaught of NK-12 mathematics teachers to come and learn and share. This happened every fall during the weekend of the neighboring city’s farm fair. So MPLS teachers knew the drill and seemed happy to be a showcase for the weekend.

Most importantly, as a young teacher I began to know my professors as colleagues. This was a major shift for me because it signaled to me that I was soon to become a contributor to the body of work. I was no longer a sponge. I needed, in due time, to consider myself as one who had something to share. This was a bit daunting but also very exciting. 

It was daunting because the work of these UNI/MPLS men and women of mathematics education was recognized in national journals and in materials published by major companies. Their names were known across the state and nation, for they were experts in curriculum and in model teaching. 

It was exciting to realize that my classroom experiences would be considered as colleague-to-colleague. My work could be reviewed, improved upon, and become part of the research for improving mathematics teaching and learning. Eventually, I might even be invited to be a presenter at a future conference. What a goal!

The relationship with these faculty members was exhilarating. They not only sent teachers home with tried-and-true teaching materials, but they were eager to share with us and to give us techniques for success. As well known as these faculty members were, they treated all participants as good friends and peers. They listened to our issues and concerns about this “modern math thing,” and they helped us think of strategies and activities that would make the transition smooth and allow us (and our students) to gain confidence. This is a critical attribute because so many times teachers go home from conferences feeling frustrated because their session leaders are “up on the university pedestal, not knowing what we go through.” As a result, the materials or strategies they hear at conferences go unused for lack of faith or trust in the session leader. This was definitely not the case at the MPLS Mathematics Conferences.

Over time, MPLS moved the conference to the new Schindler Education building. It was nice, but it wasn’t the same. At MPLS, we were in classrooms of NK-12 students who were learning from the materials we were able to take home. The atmosphere at MPLS was so authentic that one could feel the spirits of the students in the rooms in which we were also learning. For teachers to see the work of the students the same age as their own students displayed on the walls was telling and encouraging. And to learn from professors who were teaching us from the same environment as the ones in which they taught their own students just felt real. That can’t be replicated in a different setting, be it university classrooms or conference centers.

Looking back on my beginning conference experiences at MPLS, I would have to admit that I had a very special launch to my professional life, one that is no longer available to the vast majority of beginning teachers. 

We study history, so we are told, so that we will not replicate the bad and that we can replicate the good. MPLS has a history that could and should be replicated. I count my blessings.

Teaching In The 1970S AND 1980S

A Faculty Reflection by Earl Ockenga

The PLS Math Department had a reputation for developing problem-solving materials that were shared with teachers at conferences and workshops and in publications. Finding ways of using the materials to generate student enthusiasm for problem solving was the fun part for me.

One of my favorite activities using problem-solving cards was to select 25 problems having numerical answers and display them on a chalk tray. I would have students, working in partners, fill in a Bingo board (a 5-by-5 grid) by randomly writing the provided answers in the 25 empty spaces. The game rules were these: partners would take turns selecting any problem card on the tray, solve it, and mark (X) on the answer on their grids. The first partnership to get a Bingo (five Xs in a line vertically, horizontally, or diagonally) won.

What I liked about the activity is I didn’t have to do much—just enforce the rules of only one problem card at a time and no running to the chalk tray. What made the activity special was once there was a Bingo, the other partnerships wanted to keep solving problems to see who would be the first to get a double Bingo.

Another variation was Problem Solving Tic-Tac-Toe using an overhead projector (a device that is now a relic). It was the same idea as Bingo: students filled in a tic-tac-toe board with nine provided answers, the teacher projected a transparency (another relic) of a problem card, students solved the problem and put an X on the answer in the tic-tac-toe. Three Xs in a line won. And like the Bingo activity, there usually would be the request to keep playing to see who would get the first double tic-tac-toe or a black-out with every space with an X. Yes, everyone won!

In the 1970s, Joan Duea and I were asked to provide math activities for the IDEAS section of the Arithmetic Teacher magazine. Seeing the enthusiasm our students had playing Problem Solving Tic-Tac-Toe, we submitted ready-for-teachers-to-use versions for one of the monthly issues, knowing it would be a good day in the classroom for the teacher who used this lesson.

Math problems my students wrote were problems I enjoyed sharing with teachers at workshops and conferences. A cartoonish drawing of a semi-truck zipping down a highway and a request for them to write a math problem for classmates to solve involving the truck would result in a dandy collection of multiple-step story problems about what’s in the truck, who’s driving, where it has been, where it is going, speed, distance, and even roadkill. It was a quick way to generate a set of problem cards for other classmates to solve (or the need to ask the student author for missing information). Kids were complimentary to student authors, which was neat. I wish I still had the student-written problems.

The Problem-Solving Animals poster in the Arithmetic Teacher (an activity we submitted for the IDEAS section) created a lot of student interest in story-problem writing. The Instructions were to pick three animals, three colors, three things to do, and use the picks to write a story problem. We asked teachers to send their students’ story problems to the Zoo Keepers and some did. Joan and I received hundreds of story problems and zoo art that we posted at the 1977 National Council of Teachers of Mathematics (NCTM) annual conference. The problems appearing on the IDEAS poster, shown here, were written by Joan Duea’s students at that time.

Problems requiring additional information in order to solve were also fun to solve. A favorite was: What’s the speed of hand squeezes? Is it faster than the speed limit posted in front of the school? Gathering the necessary information required students to line up to see how long it would take to pass a squeeze from hand to hand, which they insisted on doing more than once just to pass the harder-than-expected squeezes back the way they came. It was an activity I would do with teachers at workshops and the adult hand-squeeze speed would seldom reach student speed, which I recall approached 15 mph.

What it felt like to be Michael Jordan was another active problem-solving experience for students. Given a life-size photo of Jordan’s hand (which barely fits on an 8.5 by 11-inch sheet of paper) and a pop can, students were challenged to construct a scaled-down paper pop can that they would feel in their hands the same as a real pop can would have felt in Michael’s hand. Continuing the activity to determine which ball in the assortment of balls from the PE department best fit their Michael-size hand wasn’t the best idea—way too much dribbling and not enough math.

It has been 22 years since I’ve been in a classroom and things have changed. The problem-solving materials used then are outdated now and my memories of how effective the activities were, have likely been enhanced with the passage of time. But they are good memories.

Calculating at PLS

John Tarr 

1950s-2012

In the early days of the twentieth century nearly all calculations were done without the aid of machines. Students were drilled in adding, subtracting, multiplying, and dividing to the exclusion of most other mathematics. More advanced students learned a step-by-step process for calculating square roots of numbers. 

In the 1950s a model of a slide rule was mounted above the chalkboard in one of the Lab School mathematics classrooms and students learned how to use slide rules for multiplication and division. This activity was part of a unit in logarithms.

During the 1960s the MPLS department was given 25 Olivetti office calculators. Students will remember that the machines were quite noisy and sometimes had to be unplugged to be shut down. MPLS also used Monroe calculators and the faculty wrote workbooks to be used with them.

During the Iowa Problem Solving Project in the 1970s, classroom sets of battery-operated handheld calculators were purchased through grant money. The Project mailed sets of 25 calculators along with problem-solving booklets and problem cards to schools participating in the project. Keeping fresh batteries in the calculators was the biggest problem.

The development of instruction in computers was a whole new story. MPLS teacher Lynn Schwandt was a pioneer in computer instruction. Input into computers was done through punched cards. Schwandt developed sets of pre-punched cards (UNI kits) that the students selected and assembled to accomplish their objectives. Schwandt took the cards to the UNI computer where the students’ programs were run overnight; they got their results the next day. With this early instruction, some students continued into careers in the field.

Now most people have a mobile phone that has more computing power than a room full of computers in mid-twentieth century. We have come a long way. 

Investigations in Mathematics

A Reflection on Teaching

Dennis Kettner

1975-2009

When I arrived at the Malcolm Price Laboratory School in 1975, I was given the privilege of teaching several secondary mathematics courses. The most unique and unconventional course was entitled Investigations in Mathematics. It was a semester-long elective course, which was offered to juniors and seniors. Students had the option of taking it for one or two semesters.

Investigations in Mathematics was composed of a variety of different, non-sequential mini-units, often involving different modes of learning. For instance, the ratio/percent unit was composed of a series of worksheets with narratives, problem-solving examples, and sets of ratio/percent problem-solving opportunities. The problem-solving items gave students the repetition to master the concepts being taught in the unit. Students read and studied the narrative materials and problem-solving examples, and then completed the sets of problems using paper-and-pencil methods. Answer keys for the sets of problems were provided to the students so that they could receive immediate feedback as to the accuracy of their answers. To conclude the ratio/percent unit, students took a written comprehensive test to assess their mastery of the content of the unit.

Another example was the Probability unit. This unit consisted of an instructional booklet commercially written and published by a life insurance company. The booklet included instructional narratives and problem-solving activities, while making connections with probability theory and life insurance concepts. Not only did students learn basic probability concepts, but they also became acquainted with the basic concepts of life insurance and the way in which premiums were determined. Again, students took a comprehensive written test at the end of the unit to assess their understanding of the content covered in the unit.

A third example was the Triangle unit, where students listened to a series of prepared cassette tapes that talked about properties of a varying number of triangles. Students performed a variety of activities using paper-and-pencil methods as they listened to these tapes. Students were often asked to solve problems using the properties of triangles. To conclude the Triangle unit, students took a written comprehensive test to assess their mastery of the content of the unit.

Each mini-unit was assigned a prescribed number of weeks of credit prior to the student starting the unit. For instance, each student knew that the ratio/percent unit was worth 2 weeks of credit when completed, whether it took 1 week or 3 weeks to complete. To gain credit for the Investigations in Mathematics semester course, students had to complete 18 weeks of credit by the end of the semester. Those students who completed their 18 weeks of credit prior to the end of the semester could complete additional mini-units for extra credit. Those students, who did not complete 18 weeks of credit by the semester’s end, were given an incomplete grade until such time that the required work was completed. The late work was accepted with a lower grade. 

Students were encouraged to work in learning groups on each unit. Groups of 2 or 3 were most productive. Working in cooperative learning groups encouraged students to communicate mathematically with the other members of the learning group, while developing critical thinking skills and sharing those ideas with others. At the conclusion of each unit, students were encouraged to choose different class members as their learning group for the next unit. In this manner, students learned to work cooperatively with a variety of different learners.

Upon completion of 18 weeks of credit during the semester, each student experienced a different and unique mathematics curriculum, since they had the option of choosing different mini-units. Because the students had a significant voice in choosing their units of study, they felt ownership in those mathematics concepts they wished to study. Ownership was a major factor in motivating students to learn and excel in this educational setting.

The Investigations in Mathematics course was designed to meet the learning needs of the college-bound student who was not necessarily going to major in mathematics or a mathematics-related field. Students who had successfully completed algebra and geometry often chose Investigations in Mathematics as their next mathematics course. Non-college=bound students were also encouraged to take the course as it was deemed important that all students be exposed to mathematical ideas throughout their secondary experience. In general, the course was well received by students as evidenced by the fact that usually 20 to 30 students chose to take the class each semester.

Investigations in Mathematics was a student-centered class. The teacher of this class was primarily a facilitator. As students worked in small learning groups on their mini-units, the teacher went from learning group to learning group answering questions, clarifying misunderstandings, or solidifying each student’s mathematics foundation. Much of the teacher’s time outside of class was spent doing administrative tasks such as correcting quizzes and tests, maintaining student records, doing creative work such as searching for materials for new units of study or writing new units of study. 

As in any educational endeavor, the needs of students as learners changed as time went by. Throughout its existence, this course was in constant development. Various forms of technology, including calculators and computers, were integrated into the course both as content and modes of learning. A research component gave students the opportunity to explore topics that were mathematically oriented and inspirational to them. For example, one of the most informative research papers that was written explored the relationship between mathematics and the musical scales. The student who explored this idea was a gifted musician whose interest in mathematics was tenuous, but studying their connections was greatly motivating.

One of the difficulties in offering a non-sequential course such as Investigations in Mathematics with its flexible and ever-changing content was communicating to colleges and universities the degree of difficulty of the course’s content and whether these institutions of higher learning would accept it as a legitimate credit on the student’s transcript. Seemingly, an annual clarification from our counselor’s office to institutions of higher learning was necessary, even after many modifications of the course description had been implemented.

No “Gotchas”!

Merrie Schroeder

1992

Malcolm Price Laboratory School (MPLS) was a launching pad for preservice teachers. This meant that every teacher-preparation student from our university was required to fulfill their first of three field experiences in theLaboratory School. Often, subsequent field experiences could be completed there as well, including student teaching.

If asked to capture the spirit of mathematics teaching at MPLS, it could best be summed up in one phrase: “No ‘Gotchas’!”

This phrase was espoused and demonstrated in all math classrooms at all times; it was the spirit of the entire department. Teacher preparation students would be carefully mentored on the meaning behind “gotcha.” Simply put, it is when a teacher tries to catch a student being underprepared, not paying attention, or just not understanding the concept. The purpose of “gotcha” is to embarrass or humiliate a student into “doing better.” This type of teaching can be fairly fun for a teacher who likes to exhibit superior standing with the students. It often leads to the familiar term “math anxiety,” an emotion that can cripple a student for future math experiences. It is unfortunate that some mathematics teacher candidates choose teaching math because they hope to emulate a teacher who played “Gotcha” with them. Those models of teaching can last a lifetime if a better way isn’t inculcated early and constantly. 

“A better way” was the spirit of the MPLS mathematics department. Students who are “caught” are trapped. But at MPLS, students were taught in a manner that released them to higher possibilities—to fly as high as they could. “No ‘Gotchas’.”

One might ask exactly how this spirit was manifested. In part, the answer is through the constant writing of interesting, rigorous, and relevant curriculum. Another part was through peer evaluation of teaching methods that allowed faculty to modify strategies as well as curricula. In both curriculum and pedagogy, student success was the center of every decision. 

(It is interesting to note that there were no materials written by MPLS mathematics faculty on the topic of classroom management. Apparently, there was no need, despite the fact that this topic is of highest interest to beginning teachers, and, thus, teachers in preparation.) 

Beyond developing curriculum and modeling solid teaching methods, describing the actual spirit of teaching mathematics at MPLS is not possible since it was not visible, thus, not measurable or countable. It was about heart. 

How does one transplant a heart? Perhaps by starting with authentic student-centered curriculum materials and with reflections on pedagogy with model teachers.

Helpers Along the Way

by John Tarr

2021

I was a mathematics teacher at Malcolm Price Laboratory School (MPLS) in Cedar Falls, Iowa for a very long time. It was a wonderful life for me. Teaching is a service profession; we help students learn. But who serves the servers and who helps the helpers? For me there were many. Here I mention several who were instrumental in my growth in my career.

My introduction to MPLS came in the late 1950s when I observed mathematics classes at the school as part of my teacher-training courses at Iowa State Teachers College. This was followed by my student teaching in the MPLS Mathematics Department. Ross Nielsen was chair of the department and one of my supervisors. Throughout my tenure at MPLS, Nielsen was instrumental in my development. Nielsen’s influence came primarily after he became head of the Department of Teaching and Director of the Laboratory School. He worked tirelessly for the school and served as helper to countless teachers behind the scenes.

My primary student-teaching supervisor was Don Wiederanders. After Nielsen left the MPLS Mathematics Department to become Director of the school, Wiederanders became chair of the department. Through his leadership, the department continued to flourish. He brought out the best in us — creativity, sensitivity, productivity. Like Ross Nielsen, Don Wiederanders often worked behind the scenes to make possibilities happen.

My other student-teaching supervisor was Joe Hohlfeld. We later became colleagues and friends. I learned from Hohlfeld to put students first. He was able to see good and potential in every student. Hohlfeld eventually got a doctor’s degree from Indiana University but he never lost his humble attitude.

When I joined the MPLS faculty in 1961, following a year teaching at Franklin Junior High School in Cedar Rapids, my colleagues welcomed me and continued to help with my development as a teacher. George Immerzeel had been at MPLS prior to my early experiences at the school. He returned to the school after writing/working for Scott Foresman Publishing in Chicago. Immerzeel was the most creative person I ever knew. He was a leader in the teaching of mathematics nationwide. It was a privilege to work with him.

Lynn Schwandt joined the MPLS mathematics faculty soon after I did. In the mid-1960s Schwandt was awarded a summer fellowship at Stanford University and shared his enthusiasm with his colleagues. I applied and was accepted for the program the following summer. The Shell Oil Company funded the program for two mathematics/science teachers from each state west of the Mississippi River. My family and I enjoyed the summer of 1965 at Stanford, and I gratefully attended classes taught by outstanding professors.

In my early years at MPLS, I was primarily a teacher. While colleagues were involved in research and writing, I taught more so they could teach less. Later I became active in writing and service to the university and had a lighter teaching load. Dennis Kettner joined the MPLS Mathematics Department and spent nearly as many years there as I did. Kettner is exactly 15 years younger than I am, so much of his tenure occurred in my later years. I valued our relationship as colleague and friend.

Earl Ockenga was a creative, highly-productive member of the MPLS Mathematics Department later in my career. Ockenga and I were colleagues on several projects, often with Immerzeel and Joan Duea. Ockenga probably successfully published more materials than anyone in the department. His work effectively improved the teaching of mathematics throughout the country.

Another of the helpers along my journey was the guidance counselor Dick Strub. If I had a problem or critical decision to make, I knew I would get a sympathetic ear from Strub. If one of my advisees or math students had a situation beyond my experience, Strub helped us make the proper decision. Like most teachers, Strub put students first.

I worked with some other MPLS teachers who shared their knowledge and talent with me. Lynn Nielsen and I wrote tests together for a new history textbook. He had a strong background in history; I had experience in test writing. It was a good collaborative effort.

Lou Finsand, a member of the MPLS Science Department, wrote grant proposals that funded several workshops and classes for the university. I worked with Finsand teaching teachers about the metric system for weights and measures. I appreciated Finsand’s organizational abilities. He made my task easier, and I probably learned more than my students.

Throughout my time at MPLS I taught at every level from first grade through graduate school, but predominantly at the high school. My college teaching was off-campus through the UNI Extension Service. My elementary-school teaching occurred during summer programs. I especially remember my first-grade experience. Fortunately for me, Judith Finkelstein was teaching another first-grade class in the adjoining classroom. Together our two classes held a mathematics-oriented Olympic Games with every student winning a ribbon.

There are many other teachers at MPLS who helped me along the way, too many to list. There are others in the UNI Mathematics Department, the College of Education, and other colleges who helped throughout my career. Some administrators were especially helpful to me. Ross Nielsen (mentioned earlier) was most important, but there were others. Jim Albrecht was principal of the high school during most of my time at MPLS. What I remember most about Albrecht was that he respected the teachers and allowed them to make important decisions, giving his advice when asked.

When I joined the MPLS faculty, J.W. Maucker was president of the college. As a young teacher just beginning my career, I was in awe of the president. I was amazed and relieved when Maucker was so personable to me and supportive of the work of the MPLS Mathematics Department throughout his presidency.

James Martin was provost of UNI during the John Kamerick presidency. Martin asked me to chair the self-study for a North Central Association (NCA) evaluation of the university. I was given a reduced teaching load and worked with departments throughout the university. The self-study was well received by the NCA evaluation team. Years later when the next evaluation was conducted, Martin again named me to head the self-study. 

Gordon Rhum was dean of the Graduate College at UNI when I was the College of Education representative on the Graduate Research Committee. The committee reviewed applications from faculty members for modest funding of research projects. Members of the committee and Rhum developed a camaraderie that made our work enjoyable. When a graduate dean position became available at another university, Rhum encouraged me to apply. I was honored to be asked, but said I thought I had found my niche as a teacher of mathematics. 

I have limited my list of helpers to some of my colleagues at MPLS and UNI. Others could have been mentioned. So too could many friends. Most important of all are my family who helped me throughout my life and forgave my mistakes along the way.