A bunch of teachers have done Log Wars. It seems to came into our community through @k8nowak over at f(t), who got the idea from Denise at Let's Play Math. It has since then been used in a few other's classrooms as well (goes to show that we need @k8nowak to popularize everything for us!)
The idea is straightforward but powerful for practicing purposes.
But I wanted more in the war. So I called in some reinforcements: namely 3 more pages of trig, totaling a deck of 60. I also added an image at the back so it looked nice as well. I had some background template and font from when I did these ninja boards info here and here as well(which other teachers and my students seem to love, but I had some doubts about it, and have not re-implemented it since).
side note: I added in trig because I've been weaving multiple concepts together throughout the course... I will attempt to write about it at some point.
The goods:
Here's the file in pdf
Here's the file again in word
(please let me know if the link doesn't work!)
The reason why I shared pdf as well is because the word document doesn't display the font as well...
So, that's fairly straight forward. Adding trig, nothing special.
War: What is it good for?
I actually created 6 sets of these - each in a different colour, and all of them laminated, then cut with paper cutter.
So... 6 sets... each with 6 pages and 10 cards from each page...
Here's something else:
Then each pieces of paper was cut individually. The longer cuts took longer than the shorter cuts.
What's the first question that comes to your mind? I certainly know what was going through my mind when I was doing all this.
So then...
War: what is it good for?
It's more like: What do you want it to be good for??
Unlike conclusions from Edwin Starr - in my context, it's good for whatever you want it to be good for.
I wrote a bit about harnessing events around us, and I stand by that. There's so much from our day-to-day that we do. And if we pause and think about it, there's a lot of interesting questions to be asked -- problems to be solved!
Nicole also responded below, but @mraspinall continued on with a focus more on the idea of tests
I agreed with quite a lot of what @MathletePearceresponded with:
and said this over twitter as an image, since my comments won't go through:
And then I am unsure if I caught the conversation after that.
I just thought it may be appropriate to share what else I do with "tests." Since, afterall, it isn't the end. In fact, if we make "tests" the end, then more often than not, it will defeat our initial purposes in the classroom.
Let me emphasize first, that "testing" also isn't the only assessment - as I attempted to describe over twitter. It is our way ot understanding student learning, and if it's only from tests, then it's far too limited a source.
Okay, so in no particular order (and I will probably forget some), here are some things that I would do with a test (not all of them with every test):
1. No grades, only feedback
This one is fairly straight forward. No letter grades or "levels" on the returned tests. Just feedback.
2. Returned for discussion
Students get back their written work, it is returned with comments and students get to talk about what they did. Kids like to talk about "what did you do?!" anyway. Since no grades, the discussion is on what they did, and not what a perceived score was.
3. Focusing on the good
In a separate activity, students are asked to identify things that they were able to demonstrate with that specific instance of written assessment. In our Ontario context, it would be the expectations. What expectations were they able to demonstrate? I typically do this with groups. So then, as groups, they were to figure out what expectations their groups were able to demonstrate - as a composite. Of course, I welcome them arguing over what is a good demonstration of expectations.
4. Focusing on improvements
As a separate activity (sometimes related to the third one), I get them to take a look at what challenges the group faced and what their next steps are. I don't want them to just say "learn this better." Instead, I want them to be more specific. How would you show that you understood what you did not understand? What would that look like?
5. Interviews
My students have the opportunity to set up interviews with me in order to demonstrate that they, individually, have achieved a level of understand that they are satisfied with. This is done often outside of class, but sometimes during class as well. I prefer it to not be during class, since I don't believe that if students are working, that I could just sit and twiddle my thumbs (as I mentioned in this post a few years ago). I prefer to be going around and observing (and potentially assessing), and poking, and provoking, and prompting, and understanding.
In any case, students basically come in with the goal of demonstrating their understanding of expectation. They provide both sample questions and their answers, and I pose questions about how they did what they did, as well as some extending questions for them to work on - based on what they did.
These are few of the things that I do. The main thing I wanted to get out there - is that Tests don't have to be the end.
I have been using Pocket for a while now (3 years?) and it has been very handy. For those that are not familiar, it is a read it later application (in fact, I think it was initially called "Read It Later"). It lets you quickly save links that you'd like to revisit. I thought it might be a good idea to start sharing some of the articles that I've kept in my pocket, as well as share some thoughts about it. These will likely be a combination of blog posts, news articles, and journal articles.
Notable quotes "You have to look harder and think deeper to realize that what appears to be progressive instruction sometimes turns out to be more traditional and less impressive than it seemed at first glance."
"If compliance is ultimately valued more than curiosity (even by some teachers who don't recognize themselves in that description), then students may be given directions that are marvelously sophisticated, but the point is still to have them follow directions rather than play with unsanctioned or unconventional possibilities." "the real power [of these passages] emerges when teachers themselves are invited to reflect on their craft and to ask 'Am I doing all I can to nourish students' curiosity, to help them think for themselves and with one another?'"
Thoughts
Kohn discusses how traditional ideologies can be masked by what appears to be progressive thinking in education. This was an excellent read that provides good reflection for anyone who thinks "they are on the right path" (where we are assuming there is a correct path).
This also reminds me of a workshop that I attended a year ago, where the facilitators (the wonderful @robintg and @lkpacarynuk) discussed the importance of having your room reflect your beliefs about mathematics education. In other words, what do you have on the walls? How are your desks set up? What is being celebrated? What does your room say about your practice?
Kohn wasn't writing specifically about mathematics education, but the connection is clear. What I thought about from Kohn's post was more general: the importance of constantly re-examining our actions in order to reflect our beliefs about mathematics education. Of course, our beliefs can also be shaped by this process of reflection and careful thinking. This is not a one way street -- it's more like a busy intersection of ideas and continuums.
Notable quotes "'Mrs. Lathan, you know they’re scared of us and our parents, too. That’s why they don’t be calling home. They just send us to you.'" "...teaching students of color how to navigate a classroom with routines and rules centered in ideals of whiteness, where there is only one “right” way to be a successful student: show in ways recognized by white culture that you respect authority, work to a standard, don’t challenge, don’t make waves, apologize when you do." "Many white teachers are discouraged, believing that they are ill-equipped to meet the needs of students of color simply because they don’t have the same experiences as them. In response, they freeze." "I know that you don’t look or sound like me, but that doesn’t mean that you have no power. My strength in the classroom does not come from my racial identity, and neither does yours. It comes from the way we treat—and what we expect from—kids and families."
Thoughts
Lathan makes a lot of good points with the story she told. I have my own concerns with respect to punishment, detentions, ...etc. But the reality of what she is saying is important. Our students need to have connections with us. The moment that we shift these opportunities for connection to someone else, regardless of the circumstances (race difficulties, behaviour...etc), we lose pieces of the puzzle down the drain. It is often important to involve others - parents, educational assistants, other teachers, other students - in order to build a better support system around them, but we need to be part of that support. The problem doesn't disappear by us handing them off!
I like thinking about assessment. It certainly isn't straight forward, but I find it interesting. In fact, I find it so interesting and important that I have been pursuing graduate studies on this topic. Dylan Wiliam (2013) has suggested that assessment is the "central process in effective instruction...[and] the bridge between teaching and learning" (p.15), and I agree with this sentiment. Usually even the people who strongly disagree with this statement, begin to agree once we establish what we mean by "assessment." It's such a divisive topic. While I understand the historical reasons for why it is divisive, I usually am nonetheless surprised by the differences in how we define assessment in the first place. That's not even talking about how it's practiced or implemented! A few people from Ontario was engaging in related conversations back in October. They planned on meeting at some sort of conference which I was unable to attend because it occurred during OAME leadership conference.
and figured I will spend a few moments to share some thoughts on that post... Please don't take my opinion as any sort of "expert" opinion. It is merely a combination of the literature that I've been swimming in, and my own practices as a teacher. Afterall, I don't believe that there should be "one way" of thinking about, or doing things anyway. These are just going to be my 2 cents as an immediate reaction without supporting every statement with a citation. Okay so I will copy and paste and then write some of my comments in black underneath:
There are a few things I know for sure. That's better than me! I am torn on many things on the front what is true and for sure - as per recent conversation with Brian and Bryan.
Assessment is necessary for growth. Without it how do we know where to go next, how to improve and what we need to work on? I definitely agree with this. But I think that is because our definition of assessment is likely pretty similar - where feedback is a major and integral part of any assessment, and therefore necessary for the development of student learning.
Assessment is subjective. Assessment using the same guidelines can still vary between teachers based on experience and personal opinions. I agree with this, and I think most people would agree with this as well. The problem that people have is usually not agree that it is subjective, but they take issue with the fact that it is subjective. There is good evidence to show that even the best designed - most "standardized" assessments are still influenced by subjectivity. But I think that's a good thing. People often have an unhealthy fascination with "being objective" because they think it carries more fairness.
Assessment influences people in different ways. Some people are motivated to do better based on harsh feedback or assessment, others need positive feedback and gentle nudges to improve their motivation to improve. I am unsure if I 100% agree with this one. I think it depends on the quality of feedback. And by quality I would include: the context in which the student receives the feedback, the task on which the student receives the feedback, the way the feedback is structured and delivered...etc etc etc... I am unsure if we can claim that student A always performs better if they get a harsh feedback.
Assessment needs to take many forms. Students need to be able to demonstrate their learning in multiple ways. They need to showcase creativity, inquiry and good old test taking skills. Hmm... I am unsure about this one as well. I agree that variety is extremely important, but I am unsure about the items which they are showcasing. What would be a "good old test taking skill" and what would it be good for (besides taking the good old tests)?
However, there are many more things I don’t know. A few of my questions include:
What is more important, process or product? Are we talking about learning? or are we talking about assessment? If we refer to learning, then I think the process of learning is more important. The product is more like the icing on top. But also I don't believe that we should separate the two like this. One comes with the other! On the other hand if we are talking about assessment, then I also think the process is more important. If we define the process of assessment with a cycle of elicitation, interpretation, and action, then the entire process is important. I am unsure if there is even a tangible "product" from every assessment cycle. For example, if we encounter a group discussing a mathematical concept, then we interpret what they are saying, and we act by providing some descriptive feedback that moves their learning forward -- what is the product? Is it the feedback? Is it the interaction with the students? Is it the learning that perhaps the students are able to better achieve? I am unsure if this distinction is helpful here!
Why the focus on marks? This one is an annoying topic, but it seems to be a trend through the next few questions. Why the focus? I think it's a combination of history, current societal perceptions, and lack of conceptual understanding of what we are doing.
What exactly does 87% mean? Nothing. Ok, well not completely nothing. Depends on what we are talking about, it might be a bit more than nothing.
How does 75% tell me anything about a student? We have long moved from norm-referenced (where we rank students within the class) to criterion and construct-referenced models (where we make judgements concerning student achievement according to expectations). Ideally, I would like any sort of "grade" to disappear - and in its place, feedback that continuously promotes growth and learning. Even if under a "grading system," I would like it to embody and consider prior knowledge + next steps. But I digress. I think our current model of "75%" would represent a students' achievement according to the curriculum expectation from a specific instance in time, while considering a variety of contexts. It would also be a combination of a variety of different assessments which lead to the enumeration.
Why do we compare students to each other using numerical values? Because we suck. Just kidding. I don't know. Same historical, societal...etc reasons from above, I suppose.
Why do these values matter in real life? The idea that these values can sometimes be high-stakes is quite devastating. I am unsure about this one as well. I don't want it to matter, but it does. Why does it matter? I suppose our political and societal structure is largely influenced by comparisons and motivated (at least partially) by competition.
How do we eliminate “life problems” from assessment? I don't know what you mean here...? I can imagine a variety of things you could be talking about, but each of those would take a bit of time, so I think I will skip it for now and get back to work!
Hoping to continue the conversations :)
Wiliam, D. (2013). Assessment: The bridge between teaching and
learning. Voices from the middle, 21(2),
15-20
I attended a leadership conference put on by Ontario Association of Mathematics Education (OAME) the past three days. I'd like to offer a bit of recap, as well as some reflections.
Below is Day 2 with Jo Boaler (@joboaler). Please see Day 1 here with Amy Lin (@amylin1962), and I hope to soon also summarize and reflect on Day 3
Day 2
Jo Boaler (@joboaler): Promoting Mathematical Mindsets
Jo is a superstar.
If you haven't seen her work, you need to go find her stuff right now. Besides her influential research work, her book What's math got to do with it (North American edition) and The Elephant in the Classroom (United Kingdom edition) has had a lot of impact on educators, parents, and society in general.
She also has this wonderful website http://www.youcubed.org/ which moves to revolutionize math teaching and learning. Also she has facilitated (and still is facilitating) courses called "how to learn math" for educators, parents, and students. I took the course with Jo when it first came out, and it was wonderful. I highly recommend it for everyone.
On day 2, she facilitated two sessions on promoting mathematical mindsets with us. I don't think I would be able to fully describe everything we did on that day. So below is a rough summary of what we did throughout the day.
She started by setting the stage of our current situation.
"There's a big elephant standing in the math classroom - that only some kids will be good at math"
A lot of factors feed into this. Media is one of them. TV tells us that mathematics is a natural ability. I was unable to track down the videos that Jo used, but below is a collection of moments where hollywood characters hate on math compiled by Dan Meyer:
Jo carried on to dispel this myth. It just isn't the case!
Mathematics isn't a natural ability - just as nothing is - and that labeling kids as "smart" is actually devastating for them. Jo provides examples from neuroscience which identifies when learning happens. She draws analogies to footprints in the sand:
She cite the example of cab drivers in London whose brains have structural changes for knowing the roads of the city. We then moved onto discussing ability groupings. Jo shares research which shows that ability grouping not only foster a fixed mindset message to students, this also in turn cause issues in the learning process (and therefore perform worse).
The red line shown in the above graph are where students are streamed by ability. (in Ontario we have pathways: academic vs applied; university vs college). As we can see from the graph, when students are streamed by ability, they show lower proficiency over time.
Jo went on to talk about another huge issue with the perception of mathematics. Often mathematics is being perceived (and, worse, taught as) a right/wrong subject. How exactly can we maintain a growth mindset if math is portrayed this way? Below is a comment from a 6 year old child:
Because the subject is perceived as right/wrong - this implicates what we call "learning" in the classroom!
Jo then facilitated an activity where we considered algebra as a problem solving tool. We began by thinking about how we would see the following pattern grow (I apologize that I did not take a picture of these slides because I've actually seen these before during OAME annual conference earlier in the year) I have provided a poor re-creation of the image:
We had an opportunity to come up with our own ways of seeing how the patterns grow, as well as discuss our ways of seeing this. Jo then shared several wonderful animations of how others have seen it grow (apologize no videos here). By beginning with "How do you see it grow," we allow students to all gain confidence in what they are doing -- this is our low floor entry point. As students work onward towards interesting aspects of the problem, and attempt to represent their thinking and justify them to their peers -- this is our high ceiling that students will have the confidence to approach.
Jo then shared an example with three boys working on the problem of predicting number of tiles in pattern 100. The three boys, who have had different successes with mathematics, were completely engrossed with the problem. They challenged and argued with each other about various aspects of the problem. The scenario showed these boys filled with confidence as they tackle the problem with great interest. The audience also offered some thoughts on what they saw from the videos, and lots of wonderful comments came out. e.g. the role of competition in the situation, the expanded need of justification for the boys, the belief that they could do the problem (from themselves, from teacher), the fact that the visual aspect of the problem was important for entry...etc. Jo explains that this would be a low floor high ceiling task, and how important it is to open up tasks instead of having problems that only contained right/wrong answers:
Jo then shared another video example from a group of 7 graders. In the groups, an individual is designated as a "skeptic," and the problem they were tackling was representing 1 divide by 2/3. Jo gave us an opportunity to chat about what we saw in the video. The audience drummed up lots of excellent observations: the existence of the skeptic promoted the need to justify their own ideas and approaches, increased opportunities for discussions across different levels of understanding (at the time...etc.
Jo then dropped a wonderful problem in our laps. This problem got the audience so engaged, that people were working on it well after the day was over! The problem was from the work of Mark Driscoll 2007:
We were all so engaged with the problem, that Jo actually had trouble pulling us back!
My memory is foggy now about when we had a break for lunch. I think it was around this time after the Driscoll problem. I actually had a strange experience at the end of lunch of being called to the front. Needless to say, I was extremely surprised. It seems that Jo and a few others have been wondering who I was! I then had the wonderful opportunity to chat (very briefly) with Jo. While it wasn't the first time I have seen her, it certainly was the first time to actually meet her face to face. I haven't gone to bug her to chat in the past because she is typically very busy being swarmed by lots of people. Because of that, I am very thankful that I had the opportunity to meet her this time.
By the way, we did work out a solution to the Driscoll problem - but I recommend that you all play around with it!
After lunch we began by talking about the mis-association of mathematics and "speed"
Not only do timed tests cause issues like anxiety, and therefore produce these misconceptions of what mathematics is, Jo also provides examples of mathematicians. Mathematicians aren't actually fast at math. They are often slow because they think carefully and deeply about problems. This important quote from Jo Boaler, which I tweeted about, was then shared with us.
I am hoping that it will be shared even more from this point forward. It desperately needs to be heard, echoed, reflected on, and heard again.
Jo then suggests (or was it a story with her own child that she told?) that when a child says "I got everything right" - we respond with "I'm sorry, that means you didn't learn anything." And the important reason is this: mistakes help your brain grow. She gave a great example of how a teacher talks about this with his/her classroom. The teacher gets the students to crumple a piece of paper (to represent the brain), and get them to throw it at the board. As more creases are made, the more connections are made in the brain - and therefore learning occurs.
Jo emphasized the importance of teacher words. She cites a study where students received results with feedback and a comment of "I am giving you this feedback because I believe in you." This resulted in significant achievement gains for the student. This is not to say that we should put this sentence on student papers every time - but to indicate the power of our words. Afterall, with great power comes great responsibility.
She then contrasted two approaches to teaching: traditional vs multi-dimensional. We saw a video example from each of the two approaches. The traditional approach is as you'd expect. Teachers tell, students practice. The multi-dimensional approach video comprised entirely of student talk. Students were presenting their discussions of a problem, and another student even went up and showed how they thought of the problem. Jo then shared the results of the two schools. She found that the students from the multi-dimensional approach scored much higher than the other, despite starting out being lower. Jo describes the conceptual curriculum:
Students spend about 10 times longer on problems in the multi-dimensional approach when compared with the traditional approach. The problems were rich, complex, and allowed for a large number of opportunities for discussions, extensions...etc.
We then carried onto doing another activity that Jo had us do. The instructions were simply to find the perimeter of the following:
We then shared not only our solutions, but more importantly, we shared our approaches. The audience shared several different approaches, and we chatted about the importance of celebrating these approaches. Jo then shared a video of a group of three students working together. The idea was that all of the students must understand enough of the problem in order to explain their thinking process, because the teacher was coming around and randomly (and perhaps strategically sometimes) selecting a student to explain themselves. While the students discussed in spanish, it was a great video to reflect on since it gave us a real taste of the group's interactions, frustrations, triumph, persistence, collaboration. The prompt when the teacher came around was simple. The question was "where is the 10?" But in order to answer this question, the kids had to fully understand the details of their representations, as well as the process of reaching the representations.
Jo then talked a bit about assessment and grading, and she suggested that:
Before sending us on another activity, Jo then shared the roles of the group members as described by Cohen and Lotan. Since group work is often derailed through unequal participation (e.g. few individuals dominate the discussions of the group), something needs to be done! She gave examples of how to mediate these unequal participation (such as group tests, deliberate teacher moves to raise the status of "low status" members...etc). She shares the idea of a participation quiz, and used it on us while we tackle an activity that she had us do. Students are told
1) how to be successful during this quiz
2) What the teachers are looking for during this quiz
Okay and now onto the task itself. It was another patterning example, which is shown below:
Each one of us were required to have a role. There are different ways of separating the roles, here is one of the examples that she shared with us:
There is also an additional role of the "spy" which is a member who goes to check out what other groups are doing and then returns with the information.
During the activity, she came around and noted what we were doing in order to demonstrate teacher actions during this "participation quiz."
Below is an example of what the teacher would note.
What this effectively does, is promote a collaborative community while clearly demonstrating what that looks like. The emphasis wasn't on the right/wrongness of the answers - it was on the process of developing an understanding of the problem.
And with that, our amazing day with Jo came to an end. She was whisked away by people who had dinner reservations for her (likely the conference organizers?). It was unfortunate that I didn't get a chance to chat with her some more, but I'm sure we will in the future.
Thoughts:
So many thoughts to be had here. I am unsure where to begin.
Skeptic role
Maybe I will first reflect on the concept of having a "skeptic role" within each group. Of course, during our discussions, we talked a lot about the positives this brings. I completely agree with the positives, but I think there are also potential challenges worth considering. I frame them as "challenges" because I believe it is worth conquering - and not excuses to not incorporate it (incidentally how I think about a lot of ideas). The situation was that a role of the skeptic would be assign to a student for a prolonged period of time. Here are some thoughts with respect to challenges.
1. Is it possible that those who attempt to justify their ideas would give up in the face of the constant poking from the skeptics? How might this affect group dynamics in a negative way?
2. Don't we want all students within the group to potentially be skeptics? Would the designated role prevent the rest of the group members from thinking critically?
I suppose I am being a skeptic (ha!) - but I won't end there. I believe that thinking deeply about these would involve not only identifying challenges, but also potential ways of approaching the challenges. (of course, we can then think about challenges for these potential ways of approaching challenges... and so on... but I won't go there).
First, the issue of group dynamics. I think our approach won't be an easy one. It would involve many other factors. First and foremost would be to develop a positive classroom culture which would allow for the existence of skeptics without tension. Then perhaps second would be to develop resilience and persistence of students when faced with the task of needing to justify their own thinking. But these two aren't a one-time deal. It's not like once we have a positive classroom culture and persistent students, we stop there. No, I think it will be an ongoing process that maintains these aspects of the dynamic classroom. And so lastly teacher intervention and role modelling is important. We need to show that we embrace the skeptic's comments (and perhaps are skeptic ourselves), and that those questions and doubts are worth exploring. In addition, we would need to emphasize the importance of thorough understanding manifested in the act of justifying our own thinking.
The second issue about getting all students to be skeptic is, I think, a more difficult one. It is more difficult because it deals with the implementation of these "roles." I thought about perhaps not having the skeptic maintaining the "skeptic role" for too long. That it gets switched up periodically -- perhaps even during the same activity. But then this may not offer the same opportunities for the skeptic to be skeptical. Perhaps we would establish these roles in the beginning of the semester, and then take these roles away once each student has developed their own inner skeptic? My thoughts on this is currently quite fragmented, and I will continue to think about this one.
Ok so that was my first thought about the concept of "skeptic roles"
I have also been thinking about group dynamics, and what do we, as facilitators of learning, do.
Status differences
Jo also talked about the status differences within group members. The participation quiz look-fors help a bit. If the group understands that the criteria includes "equal air time" then it may help distribute the conversation a bit. She also suggested randomized grouping which was similar to what Peter Liljedahl suggested (I didn't finish recapping CMEF, now that I look back at it... sad...). A lot of people I know have incorporated visibly random grouping since then. For example, @AlexOverwijk with his posts here and reflecting again here.
Jo's words prompted me to think a bit deeper about the role that "status" plays within the classroom. This is so important to think about. So many factors play into this. For example, each individual's sense of self-efficacy with respect to mathematics or problem solving; social structural surrounding the group; comfort level with each other; prior ideas of what others are like in other contexts...etc.
I am reminded of the situations that I commonly find myself in. I look younger than my age - though regardless I am still young in my 30's. Typically in a group of teachers, my experience is usually called to question. Of course, polite conversations dictate that these sentiments are not brought to the surface, but I can often sense disparities in how my words are treated (and maybe the fact that these sentiments aren't brought to the forefront is what makes it even more uncomfortable). This may have an impact on how often I share my ideas -- not because I don't believe in sharing, but I feel like my ability to share the ideas have been ineffective.my own roles in groups, how to "raise status" - what if teacher intervention actually lowers status... etc.
This wasn't the case at this conference. I felt very comfortable with the groups that I interacted with, and therefore was able to gain confidence in what I was saying, in sharing ideas and in facilitating discussions during activities.
Despite this...
(I was even called up to meet with Jo... needless to say I was fairly embarrassed)
This prompted me to think a bit more deeply about how to create an atmosphere where each individual student in the groups feel comfortable in sharing their ideas - without being shackled by other dynamics.
Despite thinking a lot about this, I think I came to the conclusion that it isn't impossible to remove all factors. All we can do is help facilitate and create an environment where those factors don't matter as much. And the following may help:
5. improve accountability in all members in the group - through the idea of "equal air time"
6. develop a growth mindset within individuals in the classroom (de-emphasize right/wrong dichotomy)
I am sure there are more that would help... but the above is a decent list that I have generated just now. I welcome any additions!
In any case, the idea of status is so powerful, that it is important for us to recognize and tackle that daily.
Human interactions may naturally engender status differences (due to various reasons) - and recognizing this occurrence may help us lessen its negative impacts.
Assessment
It probably doesn't come as a surprise that approaches to assessment would be one of the issues that I reflected a lot about. My own research interest is basically around exploring (and perhaps empowering) different ways of obtaining an understanding of student learning besides paper-and-pencil assessments. I also had a few conversations about this with others at my table. Our conversations about change seems to always have this sense of helplessness caused by the current systems in place. I have often been under the impression that we actually have quite a lot of freedom with respect to assessment, since our curriculum document supports and indicates the importance of the mathematical processes (shown below):
I have always felt that the existence of these process expectations helps empower us as teachers to broaden our tools for better understanding student learning. This is not to mention that the details of expectations often lend themselves to methods beside paper and pencil anyway (how else would you effectively understand students' ability to investigate, explore for example from the expectations)
But the principals and superintendents, that I had conversations with, have often faced teachers who had felt pressured by these "tests" as the only way to legitimize assessments. That is certainly devastating.
How do we, as a society, move away from this?
In any case, I think I've rambled on enough. I hope that the above recap and reflection has been useful to somebody :)
Don't hesitate to ask if you need any aspects clarified!
I attempted to post this as a comment at @blaw0013 's post from 2012 but it was causing me all sorts of issues since what I wrote was too long... and so I decided to post it instead and invite conversation
~~~
If all interactions are inherently coercive, and in my mind
similar to gravity, I would like to add my own mass in here and invite you to
think with me on a few things. Before I
go on, I wanted to clarify that I do not have a set position on this yet, and
am hesitant in stating that I have decided on a position. In fact, I am unsure that I would agree with
the idea of taking a position - unless it is my own dynamic position that I
have subjectively decided on.
First, let me begin by providing some context on how I got
here. Bryan Meyer and I have been
conversing over DM (twitter was a poor medium for in depth discussions) for a
while about a variety of different topics.
More recently I asked about how he has identified his position with
respect to learning and knowing. To
this, he responded that he identifies with forms of radical constructivism, and
stated that the "social" was accounted for through the idea of
intersubjectivity. A few more
back-and-forths later, he asked if I thought of mathematics as invented or
discovered (yes, we were still conversing through twitter at this point...
silly us).
Looking back, I think my answer would have differed if he had
asked whether I thought the "learning and knowing" of mathematics was
invented or discovered. Regardless, this
is what I said:
"Here are my current thoughts on that: I think it's
both. I think mathematics is
individually invented, but collectively discovered. With respect to mathematics as a subject, I
like Shapiro's view of structuralism, where mathematics exist as structural relationships.
To make sense of ideas in mathematics, one must "invent" concepts
from being situated with the rest of existing knowledge, or resolving cognitive
dissonance. At the same time,
mathematics as a social practice is discovered collectively. If mathematics is a landscape of buildings,
then individuals are inventing different parts of the buildings on their own,
but others are inventing the exact same things.
The superimposed image of mathematics is the structural relationship
that is discovered."
We left this a bit (because we were still on twitter and it
was difficult to flesh everything out) and we went on to having him describe
his personal epistemology. To this he
stated that no knowledge is external to the knower, and therefore it is invented.
I responded with:
"...back to my thoughts on how mathematics is
"individually invented, but collectively discovered." I am basically attempting to make a case for
a duality between invention and discovery.
Mathematics as something to be individually constructed in the process
of sense making -- which I think is where you are coming from, and I agree with
this. On the other hand, I believe that
there is an existence of mathematical properties prior to my construction of
concepts. e.g. the concept of addition.
My thinking is that in the process of our individually making sense of addition
(for ourselves), this is a kind of collective discovery of some sort of
structural property of mathematical objects - in the sense of e.g. Shapiro's
concept of structuralism. But I am not set in stone with respect to my personal
epistemologies, learning theories, and theories about the nature of
mathematics...I think your question is - then how is it possibly inventing if
it already exists? I am currently
reading about the phenomenology of practice which I think may have had an
influence on my current thinking.
Imagine the concept of a bridge.
Imagine an individual who has no knowledge of this concept. As this individual becomes familiar with the
concept of a bridge - in whatever way they choose to define bridge - they are
constructing their own idea of what a bridge is - in relation to the rest of
what they understand. And this act is invention to me -the construction of a
concept with relation to an existing body of knowledge that the individual has
constructed so far... I am unsure if you have seen the movie "the gods
must be crazy," but, the idea is hat a coke bottle falls from a plane to a
village, and then the villagers began to use the bottle for things outside of
what the bottle is typically used for in our culture. The development of what it is and what it is
used for, would be invention. However, at the same time, there is pre-existing
properties of the bottle that allows it to tend toward certain inventions. The
process of understanding and identifying these properties - would be
discovery."
After a few more exchanges, we recently moved onto e-mail,
and his post (and your response here) was brought up, along with a few images
of quotes from von Glaserfeld's (1996) book on Radical Constructivism (studies
in mathematical education).
Ok, and now that we're in the present, I hope the above has
invited you enough to our conversation!
Feel free to comment on my existing analogies and half-baked ideas.
First, let me address a wondering from this post &
comments.
"I would argue that as soon as you ask the question,
you have created an unjust space between people--there is no way that one could
be deemed not correct, nor more correct. Each is assumed to be "correct,"
i.e. viable, for that other autonomous being."
I wonder about the degree of justification for
"viability." Even if we stand
on the platform of individualistic ways of knowing, is it not still possible to
identify degrees of "correctness" to the rest of his/her own networks
of constructions? As a simple (and
perhaps ill-formed) example, if I believe in 1+1=2, then if I state that 1+2=2
I would have a less degree of "correctness" to someone who believes
1+2=2, and then state 1+2+1+2=4? My
example is flawed in many ways, but I hope I am somewhat illustrating my
wondering.
Now I'd like to return to the attempt of clarifying my own
tentative position (and in fact without this act of doing so, I would have no
position to speak of).
On mathematics:
As I attempted to illustrate through the copy-and-pasted DM
conversation, I currently envision mathematics as invented and collectively
discovered. I think my current state of
opinion has to do with a combination of different experiences. Thinking back, in an effort to clarify, I
think the aspect that I believe is "invented" is the learning of
mathematics. The aspect that I believe
is "collective discovered" is the nature of mathematics as structural
relationships.
What I am saying here is actually related to my wondering
earlier. Let me attempt another
example. Let's say I am an individual
who has constructed the concept of 1 as a lone object with several properties. This makes sense to me as it relates to my
other constructions through my interactions with the world. As I construct a new concept of 2, I relate
to my previous constructions for the concepts of 1, as well as all of its
innate properties of one-ness. By doing so,
improving my own "correctness" for the concept of 1. The structural relationship that I have
"invented" would then be that 1+1=2.
At the same time, I arrived at this conclusion not only
based on my own constructions of other concepts, but also relying on the
inherent properties of one-ness and two-ness.
I am unsure if that example was any good... I suppose this relates to my tentative
thinking that while abstract mathematical "objects" do not exist, the
structural relationships amongst objects - do.
Thoughts? Comments?
This is all getting very foggy for me now haha.
*note* This whole discussion actually is reminiscent of when
I was studying the various philosophies of mathematics (which I am unsure I
have decided on a strongly set position there either).
I attended a leadership conference put on by Ontario Association of Mathematics Education (OAME) the past three days. I'd like to offer a bit of recap, as well as some reflections.
Below is Day 1, and I hope to also summarize and reflect on Day 2 & Day 3
It's quite an involved read, but I hope you will all be able to get lots out of it :)
Day 1
Amy Lin (@amylin1962): Tales of Passion
Amy shared some tales of passion. She had a wide variety of examples and resources that she shared, and she told the stories structured through Peter H Reynold's (@peterhreynolds) inspirational books. I will structure my recap in a similar way. I've found some videos that correspond to the books from Peter Reynolds that I would highly recommend (unless you've read the books)
The Dot
Amy didn't show the following video, but a quick search on Google came out with this reading of the book that I thought would be wonderful to share:
What do we do about students who come in and says that they just can't do it?
Amy then shares her thoughts on this, and talked about visualization. She cited an IKEA example to illustrate the power of visuals and images as a way of developing understanding.
"why can't we use pictures to talk about mathematics?"
Beyond this, she also gave examples using videos and verbal communications.
She shared her experience of how she's used her dog Kipper in several investigations within the classroom. She then also shared @fawnpnguyen 's wonderful website called visual patterns.
p.s. Fawn is awesome, it's not just her website. You need to follow her blog on a regular basis if you don't already.
After sharing a few more success stories from students of various grade levels, she went on to her next section
Ish
Again, the following video wasn't shown, but I thought it may be helpful to those who haven't seen the book:
Using Ish was a launching pad, Amy suggested that we could do things that are math-ish, and challenged the way that the audience thought about "math."
She cited Fawn's example of "why wait for calculus," and followed with her own class examples where she has been attempting to spiral the curriculum through activities (I am unsure if she was inspired by @AlexOverwijk or not, but Al works a lot with what he calls "activity-based teaching." See some wonderful examples here).
Sky Color
The next section that Amy drew from was Sky Color by Peter Reynolds. This was a story about a girl who did not have a color blue and discovered that she could still draw the sky. This was her lead into talking about "how am I going to make the sky without blue paint?" which translated into "How am I going to teach without worksheets and textbooks?"
She shared some examples of how she's been attempting to do this,
I'm here
Lastly, she left us with the story of I'm Here by Peter Reynolds, which is about an autistic child's story.
The above video does not read out the story unlike the previous one. But it's effective as well - perhaps a great example that we can use images and videos to convey ideas.
And this is the powerful image from Peter Reynolds that Amy left us:
Thoughts
This was a great start to the leadership conference. While we did not get a lot of opportunities to chat with each other during the presentation, Amy had some great words and examples to share. I loved the books that were shared, and have not seen them before. I think for a lot of secondary school teachers, the most difficult task would be developing the Sky Color. There are a lot of excellent stories, resources, and examples being shared around the #MTBoS, and more definitely needs to happen. In my personal experience, I have found it significantly easier to uncover curriculum expectations for the junior grades (9, 10) through students investigating problems and teasing out the conceptual understandings. The senior grades is a slightly different beast, and the implementation is arguably more difficult. I think this stems from a variety of challenges. I will list two big ones:
1. External pressures
Students, teachers, parents, administrators - for these grades levels - all are facing some external pressures. The pressure of acceptance to colleges and universities, the pressure of marks, the pressure of the unknown future...etc. These weights heavily influence the teacher's pedagogical decisions, as well as the dynamics in the classroom. The film (which we screened the next day) Race to Nowhere gave a powerful example of these pressures. Trailer below:
2. Context & Content
With respect to curriculum content in mathematics, it seems to a lot of teachers that we climb this ladder of abstraction. As we reach grade 11 and 12, there are (perhaps) more abstract representations and abstract connections within the curriculum. It isn't impossible to "concretize" these abstractions, (e.g. Dan Meyer's thoughts from two years ago) but it definitely becomes a bit more difficult. For example, we may still be able to lead in with an interesting #act1 hook that perplexes students and allows them to develop questions, but it takes longer in order to tease out the abstractions that is expected of students. Perhaps it also requires more prompts, build-ups, and follow ups. My own example of farming the painted cube makeover scenario:
.
I have been meaning to blog about that activity but haven't had time. This is a good opportunity to very briefly chat about that. It wasn't hard for students to develop questions and for us to explore them. I am not going to spend much time recounting the specifics of what we did during class (maybe in a different future post), but basically it follows #3act format with respect to developing perplexity, interest, then generating student process & solution. If you are unfamiliar, Dan Meyer gives a good breakdown here in terms of teacher moves.
There's lots of discussions and learning to be had from questions like "what does the next one look like" or "how many of each" or "how can we compare the number cubes of different colours." These are certainly all worth exploring, and we definitely did this in my class. There are, however, a lot of rich extensions that we'd want to get into - if we want to reach the curriculum expectations of grade 12. For example: intersections (when do we have the same number of the cubes), rational expressions (what does that look like), function behaviours (what is each colour of cubes doing), inequalities (do I have enough cubes to build shape 100?)...etc. These extensions are not unreachable. They are there. They can be explored. They are certainly interesting problems worth exploring. However, the challenge is often that some of those problems aren't natural. In other words, students don't think of those questions right away when looking at them. Rational expressions is a good example. Why would you normally think about the side lengths dividing each other and think about what that means? Or if you consider number of different colours - why would you normally consider what a division mean?
It may be that the image itself doesn't lend itself to easily consider those types of questions, and that a better image is needed - but the difficulty (or at least, perceived difficulty) lies in reaching a specific abstraction that may not be interesting, or not easily accessible.
Which made me think about whether it's worth exploring those questions or topics at all if they are not made interesting.
In any case. despite challenges, I believe these are worth tackling.
Coming up: Day 2 - Jo Boaler's (@joboaler) wonderful all-day session!
