Division. The word instantly brings an image of long division into my mind, but done using an alternative algorithm (the scaffolding method.) As a student, I used this method as a tool to make division a neater, tidier process for myself. My teacher at the time was not so pleased with my problem solving skills: she would not allow me to use my algorithm on an exam, nor was I allowed to teach other students my way of solving division problems. I was deeply confused by her reasoning and frustrated with my inability to fully understand the standard algorithm.

As an educator, we could embrace students’ problem solving skills and exploratory methods used to solve a problem given in class. Does it really matter how a student solves a problem? I feel as long as said student can show how they solved the problem and be able to explain their method, their mathematical understanding is just as good (if not better) than if using a standard algorithm.

This concept is demonstrated in this week`s chapter, where the reader is introduced to Mac, a student who has not been taught formal division. Jennifer`s focus is to set up Mac`s  mathematics side by side, so that he “…opens a space of his own where he relates and integrates the equations in ways that most definitely speak (mathematically) to one another.“

By allowing Mac to explore his knowings of 18, he learns to develop different ways of seeing and expressing his own mathematics:

18-3-3-3-3-3-3=0

6×3=18

18-6-6-6=0

0+6+6+6=18

3×6=18

In doing so, Mac can clearly see that division involves groups and sharing these groups. He can also see that division and multiplication are related to one another: they are opposites.

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This section of the chapter views mathematical understanding from an enactive perspective. I focused in on 2 passages of the reading: the first regards mathematical thinking as a form of communication:

“…our reasons for communicating are not to establish mathematical objects but rather, mathematical objects are necessary for the conceptual development of our own internal thinking and when interacting with others…we do not start with mathematical objects and then communicate, we communicate and through this dialogic process, mathematical objects come into being” (Sfard, 2004 in Thom, pg. 125)

While the second views mathematical understanding as growth:

“…learners are autopoietic beings who determine what phenomena will be experienced as perturbatory and who specify the ways in which they structure their mathematical thinking…mathematical understanding…comes to be through the structural changes within, between, and among learners and their environments.” (Kieren and Pirie, 1992; in Thom, pg. 129)

The first quote makes me uncomfortable. I can reflect on my past experiences and understand that I saw images in my head when thinking about math, and then communicated what these images “were” to me. I have difficulty grasping Sfard’s concept that we communicate then form objects…I think that we see a problem, and envision it in our minds, and then communicate to solve the problem, or understand the problem, or seek further explanation of the problem.

The second quote, although wordy, makes much more sense to me. Kieren and Pirie’s model of mathematical understanding on page 128 of Re-rooting follows with my thinking process as I was trying to understand and sort through Sfard’s theory! In Kieren and Pirie’s model, mathematical growth is structured as follows (primitive knowing as the basis):

  • Primitive knowing
  • Image Making
  • Image Having
  • Property Noticing
  • Formalising
  • Observing
  • Structuring
  • Inventising

This model is dynamic, and allows for understanding to occur on multiple levels at once; as it incorporates “folding back” to allow for deeper understanding. I really appreciate this model, and will use it when planning activities in my future classroom. I also like how this model seems to be in line with how Jennifer has been teaching our class all year: mathematical understanding develops through interactions with, between and among learners and their respective environments.

 

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.

Today in class we watched a video where a psychologist (Varner) observed and asked students to attempt and explain various multiplication questions. This video highlighted a dismal fact about our teachings of mathematics: we teach rote memorization. While watching the video, I noticed that although many of the students could quickly solve 5 and 9 times table questions; questions involving 6, 7, or 8 left most students confused and unable to solve the answer-some were still unable to complete the problem even after Varner gave strategies and hints. I believe that students were able to solve 5 and 9 times table questions because teachers will often show students “tricks” for these number groups. I also believe that students were unable to solve other number groups due to lack of actual strategies and learning comprehension.

In order for conceptual understanding of multiplication to take place, teachers must utilise past experiences (including number sense strategies) and teach both paths of multiplication: grouping and repeated addition.  For this reason, I feel that it is extremely important to teach students a variety of algorithms in order to reinforce student’s number sense; and then encourage problem solving and personal connections in order to make the math more meaningful. As we saw from the video, when students do not have good number sense, they are unable to conceptualize numbers, cannot express their logic, and rely on rote memorization.

 

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?

 

 

In this chapter, Jennifer provides different theories of mathematical understanding. I will focus on mathematical understandings from a Constructivist perspective:

We have two levels of understanding:

Conceptual-knowing what

Procedural-knowing why

Relational understanding relates the procedural and conceptual; and focuses on knowing why they go together. As educators, we want our students to develop relational understanding. How do we do this? The objective of relational knowledge is to combine conceptual and procedural understanding in order to make sense of, and solve a problem. I feel that the use of alternative algorithms can help students make sense of mathematical concepts such as subtraction.

Subtraction can be imagined in two ways: by taking away, and by comparison, or difference.

We can use different algorithms dependent on how we wish to view subtraction:

62-38 = 62-(40-2)
…….. = 62-40+2
…….. = 22+2
…….. = 24

or

62-38=(60-30)+(2-8)=30+(-6)=24

In class, we played the race to 100 game, and it was really fun and challenging at the same time. I have included two templates that use different strategies but still focus on making math meaningful.

 

When I read the title to this chapter, I balked: shape-shifting?! How is the ability of a being to physically transform into another form or being have ANYTHING to do with mathematics? I closed the book. Then, I took a moment to reflect on the title…and postulated that maybe we are not talking about the ability for someone to shift into another form but that we are talking about looking at mathematics as a shape shifter; as it can take many different forms, and have many varied outcomes. Furthermore, when we view mathematics as a shape shifter, we see the need to incorporate more problem solving and open ended questions into our curricula. I read the chapter, and discovered this was Jennifer’s reasoning.

Jennifer begins by explaining that “free ideas” (ideas written in the moment) are very important and highlight the need for individual and group collaboration when problem solving. Jennifer structured a problem as an individual activity, and first had students write out their “free ideas” as individuals. Then, students were grouped together and must collaborate-they had to look at each other’s ideas and understand that someone might have a differing or even conflicting idea than their own idea. Shape shifting comes into play when individuals came together as groups and had to find something that both of the graphs had in common (held true to both). Groups had to shift their free ideas. I really appreciate Jennifer’s theory, and I feel that it is applicable to all subjects. I also feel this to be a good lesson to teach students social responsibility and open communication when we interact with one another (your idea is challenged and you need to shift your idea so it holds true).