Posts Tagged ‘mathematics’

So, after writing on how I enjoyed “Notes on a Triangle,” I read the chapter on Re-viewing and Seeing Differently and discover that it is about the film and how the students studied the triangles from the film clip! (Funny how things work in life…) I especially like the chapter’s section on how the students noticed that numbers could be triangles: not only were the students constructing triangles from numbers, but they were also forming patterns. Once the students noticed their pattern (the relationship between the increasing length of the hypotenuse and the total number of dots needed to construct the triangle), they were able to use mathematical language and symbols in order to predict the next number in their triangular sequence.  This exploratory lesson allowed for a natural progression to take place in the classroom;building on students’ prior knowledge of triangles affords them a better opportunity to make real world connections (triangles made from squares, soccer drill formations, pyramids, etc.)  I was then reminded of a word from earlier in the year: oikos. Jennifer’s students were creating a sort of family with their number sense: They were able to relate to, and understand that the mathematics they were attending to was connected. The students also seemed to view the classroom as a family, or home: They were confident and comfortable exploring and questioning their environment.

When I become a teacher, I want my students to feel like their classroom is a type of home. I want them to feel comfortable asking questions and challenging themselves. To do this, I need to create an environment that fosters problem solving and open ended questioning. I want to make mathematics more hands on, more interactive, and more engaging; it is through discovery that deep connections can be made.

Sidenote: Notes on a triangle has a partner: dance squared-here is the link:



During class, we were exposed to a variety of geometry activities that connect mathematics to personal experiences and incorporate visualization, kinaesthetic, and creativity (what am I game, building shapes with marshmallows, body drawings). We also watched a short video, titled “Notes on a Triangle.” The video was wonderful! Students could watch first for visual information, and then re-watch to notice patterning, different shapes, definitions of unknown shapes, etc. I immediately thought of the chapter we had to read for today’s class; as Jennifer stresses the importance of connecting mathematics to it, it being ‘more than human world.’ Thom takes the reader back to the snowflake, and the meaningful activities that formed from the snowflakes. While reading, I could see that the students were knowledgeable and understood the concepts of geometric patterns and shapes. They used mathematical language correctly and fluently and also came up with their own idea of how to verbalize their knowledge of snowflakes: through poetry.

I re-read the chapter several times. I did this because reading the passages made me feel good. I felt like I was peeking into what good teaching looks like. I felt that this is what education should look like: guided exploration. This is what successful teaching and learning looks like. Our cohort has gone on multiple school visits, and I have yet to see what Jennifer writes about in her book come to fruition. However, I know that there are educators who want to change what formal education looks like, feels like and acts like. I know that I will be taking these instrumental lessons from the textbook with me as I begin my teaching journey.



In this chapter, the reader is offered glimpses into Jennifer’s mathematics classroom. We see that the students are constructing their own Koch snowflake and identifying interesting patterns that they observe throughout this process. The student examples given in the text are all unique, but also share something in common: each child can write what they are observing using mathematical language and symbols. They start by finding a pattern-some choose sides, others shapes, and others corners. Then, the students describe what is happening to their pattern as more layers are added. Finally, the students revisit their snowflakes later in the year and express snowflake growth with use of symbols and knowledge of number operations.

The idea of guided exploration really appeals to me. The students were all given the same task (observe a pattern and write down what was happening to that pattern), but were allowed to come up with different patterns (corners, triangles, sides) that they could express using mathematical language and symbols.  This allows for creative thinking to emerge, and at the same time, gives the teacher an idea of where each student’s mathematical understanding lies. Furthermore, confidence in mathematical abilities increases due to the exploratory nature of this activity: each student was able to express their pattern in a number of ways, and prove their pattern using language and numbers.

I think it is important to remember that math can be fun and meaningful at the same time! I am sure Jennifer’s students had a great time looking through a magnifying glass to count out their pattern, then making predictions about their pattern’s growth (SCIENCE!) By intertwining mathematics to other subject areas, we are reinforcing this idea that math is everywhere, and we need to embrace it.


Mathematics is not something we have to look up to. It is right in front of us, at our fingertips, caught in the whorl of patterns of skin, in the symmetries of the hands, and in the rhythms of blood and breath. -David Jardine

This quote is a summation of how I view mathematics. Mathematics is a part of our existence. It is everything and everywhere.

We need to start teaching mathematics in an integrative manner, instead of making it a distinct and separate entity. As such, this chapter has three main themes:

1: Connecting math to the physical world; connecting students with the five senses so that they can see, hear, feel, taste and touch math.

2: Connecting students’ personal experiences and families to the curriculum.

3: Connecting the teacher to the curriculum and then to the students.

These themes are not separate; they are rooted within each other and should be embedded within the mathematical curriculum. I must admit, I am a bit overwhelmed by these themes, as it puts a great responsibility on the teacher: Will I be able to teach math in a way that makes these connections and provides meaning for the students?



Einstein’s theory of relativity states that time and space are not as constant as everyday life would suggest. Following Einstein’s theory, we can state that time seemingly moves faster or slower relative to things like your age or height. This explains why ten minutes to a five year old feels like eternity, while the exact span of time to an adult feels incredibly short. I can think back to when I was younger and I would stand guard at the window, and wait for my dad to come home. Every 30 seconds (but to me it seemed like hours), I would ask my mom, how much longer? And she would reply: “soon, Joanna. I just told you that Dad would be home in 10 minutes.” Clearly, her concept of ten minutes was much different than my own!

How can we apply this concept to the classroom? For students, teachers should focus on creating an engaging and interactive classroom environment that focuses on active, cooperative participation and problem solving. If students are actively involved in their learning, they will be more attentive and time will pass by seemingly quicker than if students were sitting at their desks working on independent tasks (worksheets).

For teachers, we can combine those nasty PLO’s and teach subjects together in order to save time and to show students how what we learn is interconnected and constantly changing; for example, teach fractals by going on a hike and studying fractals in trees (math, dpa, and science all in one lesson!)

Educators also need to help students understand the concept of time, what does time feel like? How does one’s perception of time change dependent on the activity (1 minute of sitting still vs. 1 minute of vigorous activity).  Our concepts of time vary with age and culture. This said, educators need to be mindful of time and how it affects each individual student.

Jennifer suggests there are three spaces for recursion: reflection, creating/relating, and identifying/problem-solving. She uses the game “Can You Guess Our Mystery Number?” to show readers how she creates these spaces.

While reading this chapter, I kept returning to the idea of “folding back.” What did it mean? I needed a deeper understanding, so I did some research and found this:

This document explains the importance of paper folding in mathematics. It is an interesting read, and it also has many samples for the teacher to try out with the students. Now, I know that this is not exactly what Jennifer meant by “folding back,” but by reading about paper folding I was able to visualize what Jennifer meant by folding back from her chapter. And I discovered that it is like physically folding paper! When we fold paper into a shape, we can unfold it to find out how the shape formed-how many folds did we make, where did we make the folds, what are the shape’s characteristics. This thinking is parallel to Jennifer asking her students to fold back on their chosen number to pick out an image that would be true or characterized that number.

Back to the chapter: Jennifer created three spaces for mathematical recursion by allowing the students to first reflect on their understanding of number, then to create a problem for their peers by relating their thinking, and finally, to identify the correct answer by using their understanding of problem solving. Students had to work together and create clues that would lead their peers to the right answer-not too soon though! They had to think flexibly and work backwards in order to create a set of clues that were logical and could work for other numbers (but every clue needed to be true for the mystery number).

2 takeaways from today:

1- I have a newfound interest in Origami and think it is an excellent way for students to create something beautiful using their math skills (and it teaches patience)

2-Backwords thinking and problem solving needs to be integrated more thoroughly into mathematics curriculum. Being able to think flexibly is an important skill.


Mathematical language: It’s a tricky beast, and usually never easily understood (or explained). So why do we teach young students the same terminology that baffles adults? Why are we making mathematics more confusing for kids? And why are we not using simple terminology in order to explain mathematical concepts? We need to start unpacking terminology!


In the above diagram, a Right angle can be interpreted in many ways: to me, it looks like a left angle (L for left, like when we teach children how to know their right and left by looking at the shape their fingers make…)So how is a student supposed to know that this arbitrary angle, is called a right angle and is always set at 90 degrees? Instead of rote memorization, there needs to be more time spent on creating meaning for terms.  Have students get into the corner of a room and pretend they are a right angle (how does this feel, how do you know you are 90 degrees). Have examples of what right angles look like outside of textbook diagrams, or better yet, go for a walk and have students point out how many right angles they can see!

As educators, we need to take the time to teach what matters. Mathematics matters, and needs to be broken down and explained thoroughly so that students are able to build connections and strengthen their understanding of concepts.