Accessibility in Education:

-Dynavox-picture board/speaking tool

-Dragon natural speaking

-Kurzweil 3000 and Firefly

-Reader Pens

-Talking Calculators

-Alpha Smart (writing aid)

-IPads

*ASK for help. LEARN about these technologies*

*As an educator, make sure your lessons are accessible for ALL of your learners….really think about WHAT is it that the students are learning*

*by emphasizing technology, we can appeal to the widest student audience.

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“…mathematics teaching which possesses an ecological sensibility also provokes an awakening for that which is yet unknown. This kind of awareness *re-cognizes* mathematical environments, mathematics, teaching and learning to be coemergent events, events that can only be brought forth as they occur moment-to-moment.”

-Dr.Thom

As the first four months of term come to a close, I find myself more open and accepting of mathematics than I have ever felt in my twenty six years of life. This class has been so interesting and unique: I admit that I expected to be given full lesson examples and ‘real world training’ like so many of my other teacher education courses. This never happened, and I am so very thankful and appreciative I was given a different experience. Jennifer’s job is to push us to **think about how we, as future educators, are going to teach our students; why we will teach certain concepts, and how our perspective affects our students. **Our goal should not be to become a replica of Jennifer and teach her activities (for that is robotic); our goal should be to take the knowledge we all acquired during our 4 months together and connect it to our individual teaching philosophies and then create holistic teaching activities that are meaningful, connected, and leave our students wanting more. This is one of my goals.

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**What about a different character? What about a mermaid?**

No, replied the other person, you have to choose from the giant, wizard or dwarf-there is no mermaid option.

The above is an excerpt from Jennifer’s chapter on “Spaces for unpredictable mathematics.” If we think of the giant, wizard, dwarf and mermaid as metaphors for mathematical learning, we can view the above exchange as an educator asking students to be “boxed” into a category of mathematical learning; one that does not make room for mermaids(who stand for the unpredictable).

Jennifer argues that “…it makes **no sense** that definitive categories or endpoints in children’s mathematical explorations would be desirable or even realistic…”

I wholeheartedly agree with this statement: **WHY** are we creating constraints and trying to push student’s thinking in a directed outcome? To me, this seems akin to rats in a maze, searching for the one correct outcome. That is not learning, that is projecting your desired outcome onto someone else for the sake of maintaining your perspective or ideal. Instead, we need to focus on discovery-based learning, where the teacher is a sort of “guide on the side,” and students take on greater responsibility for their mathematical learning; giving way to quantitative literacy. Through these two means, students understand the role that mathematics plays in the world. This type of learning and teaching will allow for the resurgence of mermaids; who are dwindling in numbers.

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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: http://www.youtube.com/watch?v=yXL4DP_3dJI

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Today’s class focused on math centres.

Jennifer had a few key teaching points that I am including in my journal so that I don’t forget them!

-It is important to start with 3d then move to 1d (3d is more tangible for students to look at)

-We are so used to scaffolding, moving up in small increments, and now literature is telling us to look at the whole and move down.

-We need to teach the following areas:

Comparing and sorting

Conservation

Non-standard and standard units and appropriateness

Estimation- it is developed through experience

Benchmarks: as long as, as heavy as, juice box 250ml, about an hour

Ratios: I’m twice as tall as, half as heavy as

Jennifer also provided us with what educators need to provide in order to create good math problems:

-They must be meaningful and interesting

-Provoke children’s curiosities in creative cognitive and embodied manners (extending their thinking out)

-Be Flexible and Open

-Make use of manipulatives, diagrams, and pictorials

For today’s class, we first learned how to create successful Math Centres, (use once a week with additional help-preview, practice, extensions, can be used for all different grades and mathematical topics, Non-routine problems, adjust the instructions: depending on class competencies, spend entire 20 minutes on each centre.)

We were then able to explore these centres in a hands-on manner (we rotated through 3 centres every 15 minutes.) Being able to work on the math problems during centre time was incredibly eye-opening: I realize how invaluable math centres are to students, as being able to work with a partner and in groups can completely change your own perspective of how something should look (for example, the lidless box activity). Working in the math centres was also a challenge-our group could not figure out Jennifer’s instructions because we kept ripping the most miniscule pieces off of our triangles…which left us with micro pies…After Jennifer came around we all had a great laugh and a good AHA! Moment. As a future educator, I realize the importance of incorporating math centres into mathematics lessons. For myself, I was engaged, enthusiastic, actively problem solving, and was so involved with the activities that I was a bit sad I didn’t get to try out all of them!

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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.

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**Complex understanding** acknowledges that mathematics is more than a collection of parts and how they work together. “Complex thinking involves understanding the role that is played by the whole is necessarily embedded in social and natural environments.” Each time the pieces are put together, something **new** is created. This also means that it cannot be easily taken apart, such as when you bake a cake. You cannot simply go backwards in your procedure and end up with your original ingredients. They are now apart of this new thing, called cake.

Jennifer then postulates that a complex view of curriculum is needed. Okay, so instead of looking at curriculum as separate, distinct parts we need to consider viewing **curriculum as dynamic, flexible, and co-evolving based on interactions between teacher-student-environment. **I understand this concept, and I feel that this is the direction learning is headed toward. I am interested in what this curriculum would look like: Would subjects be taught by guided questioning, problem solving and exploration? Instead of distinct subjects (mathematics, art, science) would we view learning as embedding these elements in every learning experience? At the end of the chapter Jennifer likens mathematics to **residue, **explaining that it “resides seamlessly and all at once with past, present, and future contexts of students’ knowing.” I will admit I feel overwhelmed by this chapter, as you have to completely disregard all previous notions of curriculum and learning. However, the ideas in the chapter ring true for me, and I will work on examining these concepts further.

** **

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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.

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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.

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