Focus on Math

Helping children become mathematicians!

Spatial Activity: Tangram Art Bulletin Board November 19, 2012

There is compelling evidence (Reynolds, 1992 and Sfard, 1994) that imagery plays a significant role in mathematical reasoning and that mathematicians use imagery in powerful ways (Hadamard, 1949; Nunokawa, 1994; Sfard, 1994). Mathematics is not just a subject of logic and reasoning, but it is one that is laden with imagery.

Doing activities such involving tangrams (and other similar manipulatives such as pentominoes, dot cards, etc.) gives students a chance to develop their spatial sense in mathematics.

This past Wednesday the students in all of my math classes at Charlie Lake School did tangram art. I provided students with a set of tangrams die-cut from construction paper along with patterns for creating a variety of shapes. Over the years I have collected a variety of tangram patterns in books, but these days many patterns are readily available on the Internet. Put together on the hall bulletin board, the students’ tangram pictures make a delightful display.

When I have enough time, I prefer to have students (especially older ones – maybe not my grade 2’s) cut their own sets of tangrams from 10 cm squares of construction paper, but as my classes run on a fairly tight schedule on Wednesdays, I went with the pre-cut sets. (Download instructions for cutting a set of tangrams from a square here.)

I have found it interesting to observe that some students who struggle with symbolic notation in math “shine” when it comes to visual/spatial activities. I have also observed the reverse to be true: students who easily manipulate numbers cannot always move things in space so easily.

Although students sometimes perceive such as activities as “art” or a day away from doing “real math”, these kinds of activities actually build their ability to use imagery, thus building their math sense.

I encourage you to use some visual spatial activities with your students – of any age and grade level!

Mathematically yours,


BCSTA/BCSSA Conference November 2012 November 17, 2012

Filed under: General Math,Ideas from Carollee's Workshops — Focus on Math @ 8:58 pm
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I had the privilege this past Thursday and Friday attending the joint BCSTA/BCSSA Conference at the Vancouver Convention Center. The plenary speakers, Ben Levin, David Hargreaves, Daniel Wilson and Andreas Schleicher, were excellent and provided much food for thought.

I was honored to be on the program to share a session I called “Adding it All Up: Mathematics for the 21st Century Learner”. One of my main points is that since mathematics is “a science of pattern and order” (as quoted from Everybody Counts), we ought to be seeing the verbs associated with science happening in the mathematics classroom. The BC mathematics curriculum ties mathematical processes with every single learning outcome K-12. Theses seven processes should be visible in our classrooms:


  • communication
  • connections
  • mental math and estimation
  • problem solving
  • reasoning
  • technology
  • visualization

I feel strongly about these processes and contend that understanding “lives” in the processes. Many students do algorithmic calculations daily but do not understand the underlying mathematical concepts. How do they come to deep understanding (one of the four “deeps” we heard about at the conference)? In math, that happens as students muddle through problems applying what they know to figure out what they do not know. It is sometimes messy and complicated, but it is the only way. As Schifter & Fosnot say, “No matter how lucidly and patiently teachers explain to their students, they cannot understand for their students.”

Many thanks to all those who came to my session and filled the room to explore mathematics with me. As I ran out of the handouts, I am making them available for downloading here. Feel free to contact me if you have questions or comments.

Mathematically yours,


Chess: A Power-Packed Game November 8, 2012

My interest in chess was stirred again recently after reading an article in the on-line magazine Education Week. I learned to play the game as a young girl and have always found it to be a fascinating challenge in logical thinking and problem solving.

It turns out that chess is, indeed, it is an excellent way for children to begin to develop thinking skills that will serve them for life. The game provides opportunity for players to anticipate moves, to think ahead, and to begin to solve multi-step problems. The game can help children improve memory, increase their skill in planning and strategizing, and just generally improve cognition.

Salome Thomas-EL, a teacher and principal in the Philadelphia School District for many years, was also a chess coach to many students from schools in low socio-economic districts with great success. In one of the three schools he was hired as a “turn-around principal, 96% of students were living at or below the poverty level. Yet the students excelled, with much of the credit going to the after-school chess program that had a profound impact on how the children think. Not only did he have students go on to win local, state, and national chess championships, but many of these students from impoverished neighborhoods beat the odds and went on to university and graduate school.

El writes this about the impact chess can have on students: “So many young people are raised to question their intelligence. Chess helps shatter that doubt. Chess teaches our young people about rewards and consequences, both short- and long-term. It challenges young people to be responsible for their actions. It cuts across racial and economic lines and allows poor kids to excel at a game thought to be reserved for the affluent. It boosts self-confidence. It is the great equalizer. Students must learn that they are not born smart, but become smart through hard work and the process of growth.”

Some years back I ran a lunchtime chess club at a Duncan Cran Elementary School. I really did not know much about how to teach the game well (El contends that children as young as grades 1 and 2 can begin using a few pieces, or even all the pieces), but I opened my classroom and invited students to come and play each other and myself. Here in the northern Canada we often have “inside days” in the winter when it is deemed too cold for sending students out for recess and/or lunch, and chess club offered a great activity for those inside days. The group that met regularly to play the game was certainly focused on thinking strategies as they faced each other across the chessboards.

I hope you will consider teaching your students (or other children in your life) the game of chess. And to help you do that, you can download some ideas for teaching chess here. There are, of course, many resources available to help you out, many availale with the click of a mouse.

Don’t miss the opportunity to help students learn that they can become smart through hard work – and have some fun while they are at it.

Mathematically yours,