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Have you ever seen Math Graffiti?

During our cross-cultural observations in Finland I welcomed the opportunity to visit math classrooms, interview several students and look through math textbooks.  I noted small differences between three particular classes and teaching and learning I have observed in the United States.


Have you ever seen math graffiti? Taken from a street in Rovaniemi, Lapland. Translation: 9 years of primary school.

At Helsinki Normal Lyceum I observed seventh graders studying geometry, computing the volume of cones, rectangular solids and cylinders.  The class began with informal conversation among students.  The teacher then gathered the class attention by asking about difficulties with specific homework problems (from what I observed, the notion that there is no homework in Finland is a myth).  The teacher took note of the confusing points.  Then students completed the correct answers, on the board, to four illustrative problems.  While students wrote on the chalkboard, the teacher circulated throughout the room, checking that other students had completed homework.  She made marks in her grade book. Students were looking at their own and their table partner’s homework, and checking answers on the board. When done with the homework check, the class noted that one of the sample solutions on the board was incorrect and the teacher, through asking the class questions, walked through the correct solution.

Once homework review was complete, one student from each table pair picked up an I-pad.  Students used Socrative software  to compute and then report results for four different problems – results appeared projected from the LCD on the screen.  Once done, the teacher assigned about twelve progressively more difficult area problems from the text.  As an example the students had to calculate the area of a slanted cylinder, or half of a cylinder disguised as a suitcase with a handle.  Students needed to re-think the shape and apply the formula and perform an additional calculation such as doubling or halving. Students worked in pairs.  At the end of the work session the teacher asked the class for attention and students were assigned homework.  Almost all students pulled out their Nokia phones, got up from their desks and took a picture of the assignment which was projected via the LCD.

This class in structure, presentation and topic varied little from a typical class I have observed in the U.S.  However, two differences were noted:

Textbook: The textbook was about a quarter the thickness of your standard U.S. book, and there were only formulas and problems. That is it.  No glossy photos, connections to careers, historical facts, no bright bolded sections, tabs, outlines of notes, no step-by-step tiered examples walking students through prescribed process explanations of how to do the math.

Student Affect: The students were motivated and worked.  They sought each other out.  They laughed, they talked at times while the teacher talked, they asked each other questions.  The room felt very relaxed, and there was no tension, competition, control issues or discomfort between the teachers and/or the students. Two students shared with me that they often had time to work in class, and they would all get through most of the problems.  He said that if they decided to talk to each other and be distracted it was their own responsibility.  He said if they did not do the work then they did not learn.

At Helsingin Suomalainen Yhteiskoulu (HYK) I observed an International Baccalaureate (IB) calculus class that was taught in English.  Students arrived in the classroom at the same time as the teacher (teachers in this school did not have set classrooms, they simply used a rooms for a particular class).  The door shut and locked, the teacher turned on the document camera and asked the students which homework problems they had trouble solving using the product and quotient rule.  Then teacher then sat at one desk, facing the board, as various students came forward to solve problems—both the teacher and other students asked questions about the solutions.  In two of the problems the students had solved the problem in different ways and they discussed how you can simplify/solve various parts of the equation in a different order, but the solutions end up the same.

The teacher then launched into the lesson which was about finding the volume of an oil spill- using the chain rule and determining the change in the radius with respect to time. The textbook was the same text used by the AP (AB) calculus teacher(s) at my home High School  in Jericho, Vermont.  The structure of the class was not that different in terms of review of homework, new material, discussion, notes, and the homework assignment.  One student arrived late with a quiet knock on the door, he quietly said, “Sorry I am late” and sat down.

The students appeared in total control of their learning.  They ran the review of the homework problems, and when a student had a question during the teacher lecture portion of the class, (about the use of inverse, reciprocal and negative reciprocals within the context of the chain rule) the other students explained to her, moving form English to Finish.  It was not just one student but four different individuals chimed in, watching the confused student’s body language, seeing her perplexed look at the explanation by one classmate and adjusting to Finnish when filling in the details to complete the understanding.

Questions: how much learning and teaching is directed by students as motivated independent thinkers?  How much is controlled or directed by the teacher?  How do we create a space and culture in our schools that promotes students taking complete ownership of their learning?

My third classroom visit was a fifth grade math classroom at Innokas Koulumestari—the teacher presented how he used technology in the class, and two students, using a powerpoint and smart board, showed us examples.  Later two other students showed us lego robots they had built as part of their exploration.  IMG_0633

The learning was not divided into distinct content segments, but rather embedded.  The teacher explained that there were also times when students received lessons and completed work from a workbook, but they did not do every assignment in the book.  I took a moment to look through the math textbook sitting on the desk.  First, the physical size of the book was considerably smaller than our Investigations books used at our elementary level.   When I looked closer at the book, I noticed developmentally appropriate progressions in difficulty of the problems, and lots and lots of applications of fractions – number lines, pie charts and music.  There were problems where musical notes illustrated and explained adding and subtracting.  The teacher was a musician and he would use drum beats to teach fractions!

Screen shot 2013-04-04 at 5.18.03 AM One can not make assumptions about math education in Finland based on these three observations; they merely provide a snapshot for reflections on teaching and learning in our own schools. In trying to summarize what was different, it occurred to me that what I was observing was the Common Core Standards for Mathematical Practice. In particular, students were: making sense of problems and persevering in solving them, constructing viable arguments and critique the reasoning of others, modeling with mathematics, looking for and making use of structure, and looking for and expressing regularity in repeated reasoning.  The teacher, however, did not articulate the need or value in these standards of practice, it was simply a part of the learning.

Actionable Items:

  • Create school and classroom environments so that students own learning.
  • Use multiple modes to teach math, bringing in interdisciplinary topics such as robotics, music and other applications.
  • Allow for students to work through answers with each other, to answer each other’s questions, organically.
  • Less is more: utilize math texts and resources that promote independent thinking and solving problems, not prescriptive lock-step methods.
  • Flip the classroom” supports some of the math learning I observed in Finland.
  • Promote the notion of  “80/20” — 80% of the time in the classroom the student is completing the work, 20% of the time the teacher is talking/presenting work. See Building School-based Teacher Learning Communities: Professional Strategies,  By Milbrey W. McLaughlin and Joan E. Talbert.

On a final note, when I was walking on the streets of a small northern city in Lapland, I came across a section covered with graffiti. There was construction with plywood barriers and generators.  On one particular generator there was a math problem, smiling face and peace symbol with the statement concerning nine years of primary school.  What does this say about a culture when such pleasant graffiti, with correct math, mentions school?

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