Blog #4: How Children Learn (Better!) in Algebra

As a middle school mathematics teacher, I often struggle with the fact that over 90% of the eighth grade students enroll in Algebra 1 or Algebra 2. Are all of these children ready to learn this complex mathematical language at the ages of twelve or thirteen?

This past summer, I reconsidered my approach to grading. In my nine years as an algebra teacher, I came to the realization that some students are simply not ready for the rigorous curriculum involving variables, systems, and the quadratic formula, and other complicated concepts. This school year, I decided to place a larger weight on practicing algebraic skills than I ever had in the past. My new policy is that if a student completes every homework assignment in the marking period (following guidelines such as showing all steps of work), he or she will earn at least a B- in the course, regardless of the accuracy or correctness of the assignments. My modification in the grading scale supports the findings outlined in Bransford, Browning, and Cocking’s “How Children Learn” chapter in which the authors not only emphasize the importance of experience, but assert that practicing is also a critical component of skill development.

I have already noticed positive changes in my algebra students’ approach to learning. Students demonstrate increased motivation and determination, as well as higher homework grades than ever before. I have made very few calls home this year due to a lack of effort. A shift has occurred: My students have transformed from entity theorists to incremental theorists. The students have learning goals and understand that if they put effort into attempting the problems, they will be rewarded. Additionally, a surprising effect has surfaced: the averages on the first three assessments are significantly higher than in past years.

Although I modified my grading scale, I have not changed the responsibility placed on the students, nor have I changed the curriculum. Each student is expected to complete homework and perform well on assessments. I believe the focus on the practice aspect of algebra has reduced the so-called math fear or test anxiety within the classroom environment and allowed the student to concentrate on what is really important in math – skill development and independent problem solving. I find that the students ask better questions and are motivated to explore and understand the processes, rather than simply getting an answer.

I encourage the students to “find a better way” to solve problems and I look forward to receiving emails from proud students wondering if their particular method of solving has been used before and/or if it is allowable. These senses that learning is a process and practice is as important as experience have proven successful thus far.

My students are participants in their own learning, no longer spectators, as stated by Bruner in Bransford, Browning, and Cocking. I only hope that the persistence continues throughout the school year while the students set out to improve their mathematical abilities, one practice opportunity at a time.

Blog #3: Learning and Transfer and the PSSA

During a recent meeting with a concerned parent of a struggling eighth grade student, my team of teachers and I brainstormed strategies in which we could assist the student in learning and grasping information, an area in which she has historically experienced difficulty. As we shared our ideas and drafted accommodations, the language arts teacher’s comment surprised me. She stated that it takes twenty-six exposures for a student to commit a term to memory.

Following the meeting, I researched her statement. Although her fact was a bit off, I did find some validity to it. According to Paul Nation (1990), a concept must be introduced five to more than sixteen times before it is learned. This notion supports the Chase and Simon (1973) and Klausmeir (1985) findings cited in Bransford, Brown, and Cocking that it takes a substantial amount of time to learn, regardless of the talent level of the individual, especially if the application or meaning of the information is unapparent.

The Pennsylvania System of School Assessment (PSSA) and its impact on transfer of learning would be an interesting study. Students are tested annually in mathematics in grades 3-8 and 11 to determine their ability to transfer learning in one context (classroom) to another (standardized test) that shared similar characteristics (numbers and operations, measurement, geometry, algebraic concepts, and data analysis and probability). Thorndike and Woodworth (1901), as cited in Bransford, Brown, and Cocking would contend that success of the transfer is dependent upon the learning task and transfer task elements in terms of how well they match. Based on this research, the answer to the question of whether or not the teacher should “teach to the test” is an unequivocal “yes”. The greater the elements match, the greater the success of the transfer. Since transfer of learning typically occurs when a student applies previous knowledge to a new situation, there is a common thread between problem solving and transfer of learning.

Eighth students considered at grade-level in mathematics enroll in Algebra I. Prior to the PSSA, the curriculum did not include probability, geometry, measurement or data analysis. After No Child Left Behind, committees overhauled traditional Algebra I curricula to include a month-long review of tested mathematics concepts. As a result, current Algebra I students lose one month of pure algebraic study, or 10% of the Algebra course. The short-term effects include rushing through the final units and/or not getting to them at all. However, the long-term effects are staggering: These at grade-level students no longer have the opportunity to take Calculus as a senior in high school. Last year, the school district changed the course progression because high school teachers argued that the students were not completing earlier courses to the fullest extent, and therefore, lacked the necessary skills to successfully complete Calculus. High stakes testing interfered with higher level mathematics courses to the extent that an entire course has been removed from the sequence. Priorities, priorities, priorities!

Nation, I.S.P. (1990). Teaching and learning vocabulary. New York: Newbury House.

Blog #2: Coach Kopp - An Expert Teacher

I have been blessed with many teachers who cared for my well-being and deserve recognition for their part in helping me achieve my goals. However, there was one teacher who set himself apart by displaying principles noted by Bransford, Brown, and Cocking.

Michael Kopp is a veteran basketball coach, ready to begin his 29th year of coaching the Central Catholic High School girls’ team. He has amassed over 600 wins and is recognizable state-wide for his accomplishments. Not only is he an extraordinary coach, he is a humble man who shies away from the spotlight and teaches greater lessons than basketball to his players.

Flash back to 1995, my senior year: My teammates and I were decent players, but none of us were superstars, 1000 point scorers, or even Division 2 prospects. It was deemed to be a “rebuilding” year at CCHS.

Coach Kopp undoubtedly knew the “talent” he was inheriting, but he quietly began his quest for another winning season. Coach, a mathematics teacher by day, noticed patterns – who worked with whom best and in what situation, which defense to play when certain players were on the court, and whom to play during key moments of the games. Coach Kopp’s choices reflected his deep understanding of the game of basketball. His knowledge was conditionalized and transformative, dependant upon the opposing team’s strategy.

At the end of a game, Coach Kopp coolly and confidently retrieved knowledge in answering reporters, only of course after he expressed his pride in his team.

Coach Kopp led us to the AAAA State Basketball Championship Game at the Hershey Park Arena, a feat which no one predicted of at the onset of the season. This “rebuilding” year team knocked down powerhouse teams with Division I prospects and went all the way to the top. How? We were led by an expert.

Coach Kopp organized his thoughts around the “big ideas” that Bransford, Brown, and Cocking noted – how to work together as a great team to beat five (more) talented individual players. A novice coach may require the memorization of plays, but an expert like Coach Kopp taught his players how to read defenses and trusted us to alter offensive plays, especially when the other team had our offensive scheme committed to memory.

Coach Kopp’s playbook was not “a mile wide and an inch deep”, a criticism of curricula by the Third International Mathematics and Science Survey within Bransford, Brown, and Cocking. He kept it simple – five offensive plays and four out-of-bounds plays – and did not change his approach during my four years at CCHS. His vast knowledge confirmed that good execution of a few plays was more critical than having numerous plays in the book.

Coach Kopp understood the game of basketball more than any other coach with whom I have worked. He is an expert in coaching and I only hope he stays in the game long enough so that my daughter is able to experience his greatness firsthand.

Blog#1: Grassroots Video & McLuhan

The Horizon Report (2008) was both exciting and intimidating at the same time. I was instinctively proud (which admittedly felt a bit strange!) when I realized that I incorporate grassroots video in my classroom, without previously knowing the technical name “grassroots video”. I use Teacher Tube, a downloadable educational video site that offers video lessons, groups, and teacher blogs. It is a good resource to use if a student has been absent for several days and needs to hear and see a particular mathematics lesson in progress, instead of copying and reading notes in order to catch up with the class again. The videos are also useful for a substitute teacher to play for the class on a large screen. In general, I am not a proponent of showing videos when I am absent. However, it is very difficult to find substitutes for math classes, and showing a short math lesson video and following up with a check for understanding activity may increase the likelihood of the substitute returning to a math classroom and increase student knowledge using a different teacher resource.
In my opinion, Marshall McLuhan would argue that grassroots video, in this capacity, extends the mathematics teacher. Instead of instructing one-hundred and fifty students in one classroom each school day, these online videos allow the teacher to reach thousands or even millions of learners across the world. Learners experience the lesson in real-time and follow along as if they were members of the actual classroom. However, McLuhan states that each extension mitigates another component. I believe he would contend that grassroots video has the ability to make teachers and physical classrooms obsolete – a frightening prospect.
As a result of grassroots video, learners retrieve a sense of control over their pace of education, traditionally dictated by the classroom teacher and/or district curricula. A learner could potentially complete Algebra I in one week, instead of an entire school year! If overextended though, McLuhan may hypothesize that grassroots video would negatively impact communication and collaboration skills of the learners. Since the videos are one-way (not interactive), learners lose the ability to ask questions to remediate or enrich the lesson. In addition, learners would not have the opportunity to be grouped homogenously or heterogeneously to supplement learning and to gain the important skills and benefits associated with group work.
The Horizon Report (2008) validates the importance of staying current with technology trends, both personally and professionally. The report left me wondering what is possibly out there for future technologists to design. We must proceed with caution, however. As a Japanese proverb states, "Better than a thousand days of diligent study is one day with a great teacher."