MATH AQ COURSE REFLECTION

I signed up for this course in order to upgrade my knowledge and skills and increase the number of teachable courses that I have so that I may be eligible for math along with science positions.  With most PD sessions, I figure that it’s been a successful day if I can take away one thing that I can actually use in my classroom.  So, it is with great satisfaction that I have so many things that “have legs” and that I can take away and use not only in a math classroom but many “big ideas” that are applicable to any subject.

I am not one for educational “jargon” and have found it difficult to navigate the new vocabulary that I need in an interview situation and during staff meetings.  Some teachers are really good at using the terms but when asked to explain them, the answers are sometimes vague and not backed up with examples.  Ve clearly distinguished between many terms which I found extremely helpful.  For example, at the start of each class, we used to have to do an “Attention Getter” whereas now it’s a “Minds On” activity.  I now better understand that the “Minds On” should include the interest component and should be related to the topic at hand but should bridge the previous lesson with the current lesson as opposed to just presenting the students with the “wow” factor of the attention getter.  It was also enlightening to hear that the “Minds On” activity does not necessarily have to occur at the start of a class like is expected in my board.  During this class, I have also solidified my understanding of the similarities and differences between assessment as, of and for learning versus diagnostic, formative and summative.  I have been told that “we don’t use the term diagnostic” anymore but the subtleties of these terms have never been clarified to me.  Thus, this course has enabled me to build up my repertoire of current educational language that I feel comfortable and confident using because we have used the terms often in discussion and I feel that I now have a more practical understanding of all of the meanings of these terms.

This course has made me challenge my practice by reflecting on the reasons for doing some of the things that I do automatically.   Education swings like a pendulum but I think its best to take the best of both ends of the swing and amalgamate them into something that is usable.  I think that I have recently been so focused on making individual lesson plans which include all of the parts that administrators want to see (success criteria, learning goals, minds on, action, consolidation plus superimposing assessment strategies and differentiating learning….yes, all the terminology!), that I have not focused enough on what I really want the students to know, do and create by the end.  I also have a very “socratic” way of teaching, but this course has made me wonder how many closed ended questions vs. open ended questions I ask on a daily basis.  I would like to shift towards more open ended questions but I don’t think that I will completely give up those closed ones (I love Kahoot!).  The idea of parallel tasks helped me to think of different ways to differentiate instruction.  Also, I try to make my lessons interesting and have students work in groups, but our discussion of the teenage brain reminded me of the reasons why they enjoy this type of learning.  Despite trying to include all learners, I have never really thought about how to provide different and multiple entry points in a task which is something I’d like to consciously consider when designing tasks.  Also, using technological tools (eg. Desmos) to provide immediate feedback to teens is very intriguing because if I can get them glued to their devices to do math like they are when they play video games, a lot of learning can take place!   This course has given me lots of “food for thought” (and not just the snacks!).

Finally, we “learned-ish” some math….or at least, I remembered some math but learned how to present math ideas not as I remembered learning them but in a way that makes more sense to students and enables them to see connections between different concepts.  On the first day of this course, I stood in the “Happy” with math group and my reason was that I like math because it’s neat and tidy and has an answer.  Well, I have revised that idea after discussing Rich Tasks, especially the messy exploding watermelon activity.  I found these very comfortable because they reminded me a lot of the inquiry method in science and I really liked the open-ended nature of the tasks.  So often in science classrooms, the experiment is more like following a recipe to arrive at a set “answer”.  I want to bring even more rich tasks into my science classrooms as well as try some in math.   I very much enjoyed re-learning math using algebra tiles and investigating using Desmos and in ways that showed me the understanding behind the math instead of teaching me by rote.  Perhaps I learned some of the understanding behind why the square of “the hypotenuse is equal to the squares of the other two sides” but only the “formula” stuck with me over the years.  I’m not sure that I’ve reviewed all of the math that I will need to teach grade 9 & 10 math, but am more confident that I have many more tools to draw on.

During these 3 short weeks, Ve has shown us many great resources that can support student learning.  He has instilled a new way of teaching and learning math using open questions and rich tasks in which students drive their own learning and can achieve success via many different routes.  He has encouraged us to have a growth mindset (yes, I can blog!) but has kept it real by reminding us that it will take some time to implement these ideas (haven’t yet started Twitter).  The last big idea that I have taken away is that Vy really enjoys teaching students, challenging them to think, and works really hard to make learning fun!  Thank-you!

Thinking Tools (Manipulatives!)

Today we started by brainstorming a definition of manipulatives.  Our group discussed how manipulatives can help abstract ideas become more concrete but can also be used to take concrete ideas and extend them to become more abstract.  Other groups included lots of examples of manipulatives such as measuring cups, math cubes, algebra tiles, rulers or really any tangible objects.  We also discussed using the term “Thinking Tools” when talking to students about manipulatives.

Even before my experience with Scientists in School, I was a big fan of hands-on activities, but that experience reinforced my understanding of the importance of manipulatives.  I believe that students learn through doing and using manipulatives is a great way to encourage students to become involved and enjoy learning.  We did talk about some negatives which may occur while using manipulatives with students, like flying elastic bands and blocks being thrown.  For those who want to test the limits, there are computer simulations (eg. Gizmo) which allow students to do similar activities using a computer (eg. Geoboards with elastics).  What a concept that would be to have to use an ipad as the consequence of misbehaving with the manipulative!   In all seriousness, I think that a computer simulation of an activity should only be used when the real hands-on activity can not be performed in the classroom due to safety or equipment issues.

Some students don’t want to use manipulatives because there is a stigma in using them or they think that they are “toys for younger kids”.  Often, teachers bring out the manipulatives to help those students who are still struggling with a concept, as if an afterthought.  Instead, the activity should be structured such that all students need to use the manipulative in order to solve the problem.  This will allow all students to become involved and learn by using the manipulatives.  Then, these same manipulatives can be used in the future to build future concepts on past understanding.  For example, using algebra tiles to understand integers with number lines, collecting like terms and then extending that into using the “clothes line” area model of multiplying in grade 9 or quadratics or completing the square or the difference between sin and cos in grade 10.  The best manipulatives are useful across many strands and grades so that students become comfortable using them and can build on previous conceptual understanding. 

I also think that it can be useful for students to develop their own manipulatives.  I had groups in 12U biology make manipulatives in order to explain the concept of DNA replication.  During their presentations to the other groups, it became very obvious if the students had grasped the concepts.  After using the algebra tiles, it became apparent that if the teacher has the students make the algebra tiles, the number of “1’s” can not perfectly divide into an “x” tile but that the “x²” tile needs a length of “x” and a width of “x”.  Thus, some thought is needed if the students make their own math manipulatives.

As I was using the algebra tiles, I became more proficient at being able to visualize the math concepts.  The more I practiced, the faster I was able to represent the math expressions using the tiles.  In our group, we discussed the importance of teachers making themselves comfortable by using the manipulatives before using them in the classroom.  My daughter said that in grade 7, she did not understand algebra tiles at all but in grade 8, the way that the teacher explained the math using the tiles made much more sense and she found them very helpful.   I was surprised at the number of people in our class who have never seen algebra tiles.  I don’t think that I ever used them as a student, but my children have and so they must be gaining popularity with teachers.  I wonder if they are used more in elementary than in secondary?!?  Has anyone used them in secondary math and for which concepts??  Did you meet with success or what would you do differently next time?

Assessment as, for, of learning & Triangulation of Evidence

The AssessPeel document that we read today on Triangulation of Evidence is a great practical resource to support Growing Success.  The primary focus of assessment is to improve student learning and ideally should be for student self assessment.  However, teachers can also use the success criteria to assess the students.  In theory, teachers should observe their students, have conversations with their students and also examine their products in order to give them feedback about their learning.  It’s important to know what we want the students to be able to say, do, and create and then determine the best tool to use for assessment.

I think that giving names to the processes that teachers generally do intuitively makes us reflect more on our practice.  Most teachers observe and have conversations with their students but this formalizes this process and means that teachers have concrete evidence of student learning.  Teachers have historically been very good at giving feedback and evaluating product.  We discussed that collecting hard evidence in many ways to support final evaluations ensures reliable and valid data which can be defended when talking to administrators or parents.

Learning goals & success criteria are the “student friendly” versions of the overall & specific expectations found in the curriculum.  Learning goals relate to the course expectations and boil down the key concepts & skills that we want students to learn.  They are simplified into student friendly language starting with “We are learning to….”.  Success Criteria are aligned with the learning goal and state what students are able to say, do, and create to demonstrate that they are successful.  The success criteria are written from the perspective of the student, have qualitative indicators and start with “I can…”.   Look for’s are more specific success criteria.  We discussed that in many boards, teachers are expected to post learning goals & success criteria in advance of the lesson but how does a student help to co-create success criteria if they have not yet learned about the topic?  In a math class, it makes sense to provide students with a learning goal based on the mathematical processes at the beginning of the lesson and then co-create success criteria with students at the end of an open lesson or more importantly for rich task process & product.  Assessment should occur when it makes sense and not necessarily on a daily basis.

Today, I learned about Anchor Marks.  I’m not sure that it makes it any easier for me as teacher but marking as levels gives better feedback to the student.  I find it difficult to turn levels into marks and so these Anchor Marks will be helpful.  I’ve also never heard about Task Requirements before, but what a great concept!  These are basic concepts that the student should have already mastered earlier in their learning but which are not part of the current curriculum.  Thus, these tasks need to be done properly before the assignment is accepted by the teacher.  Interesting ideas!  Has anyone used these before?  Any examples to share?

There should be assessment for as, for and of learning.  I really liked the analogy from Damian Cooper that assessment of learning is the game versus the assessment for and as assessment are the practice.  I have also struggled to reconcile the terms diagnostic, formative and summative.  In our discussion, it was explained more clearly that assessment for learning is used to inform the teacher’s approach whereas assessment as learning is for student self-direction.  Both for and as could be diagnostic and/or formative.  Assessment of learning is summative and counted towards the student’s grade.

Our group also had a lengthy discussion about interviews and how it is difficult to feel comfortable using all of these terms when either you have never heard of them or have little real practice using them.  There is always an assessment question as part of any interview these days and so this discussion and day of activities reinforced the current terminology and solidified my understanding of it as well as introduced me to a few new terms.  Being able to say the terms and use them in context will make me more comfortable in an interview setting and also when planning lessons in the future.

How long do you think it will be before the terminology changes again??

Rich Tasks & Contexts

The article by Jennifer Piggott was about “Rich Tasks and Contexts”.  Her question which lead her to write the article was “what are ‘rich tasks’ and why are they important?”.  Piggott partially modeled a rich task in developing her own open-ended question, which is her “problem”.  She started with the information that she knew and then synthesized her own ideas with those of others.  She also connected her knowledge to the NRICH website which, she says, contains many examples of rich tasks and also discussed the Geoboard environment which provides a “rich context”, which is a series of related tasks.   

My understanding of a rich task is that it includes:

PROBLEM DEVELOPMENT:

-rich task environment is key: students drive learning; pose own problems

-accessible to wide range of learners; many entry points

-intriguing, enjoyable problem with real life application

-initial success with different levels of challenges (low threshold; high ceiling)

PROBLEM SOLVING:

-differentiated instruction: different start, middle, end with surprises

-broaden/deepen math content & understanding

-skills: confidence, independence, creative & critical thinking, collaboration & communication

SOLUTION TO PROBLEM:

-elegant, efficient, logical, creative solution & application

-emergence of patterns, generalizations, principles, connections, extensions

-evaluate findings

Teaching using rich tasks reminds me of inquiry based learning in science in which students are given a question based on a big idea (eg. What affects homeostasis in humans?) and then they have to develop their own intriguing problem to solve (eg. What is the effect of the tempo of music on short term memory?). The students then need to develop their own model, test it by collecting data and revisit their hypothesis.  This same task could be used with a math focus in mind and would bring out data collection, relationships between variables, graphic representation and all of the mathematical processes in the curriculum.  In science, we are always connecting ideas to real life applications.  Dr. Marion Small believes that teachers should present non-contextual math problems along with those based in real life.  She also believes in creativity and divergence in mathematical problem solving and to not just link math to the arts (or science, in my case!).  I found the idea of solving a problem just for the sheer challenge and enjoyment of the process quite freeing.   

Dan Meyer stated that many teachers don’t like the ambiguity of open-ended rich tasks.  Generally, our vast experience as students of mathematics has taught us that the product is what is valued and thus it is more comfortable to retreat to that same method of teaching math (ie. formula, plug in #’s, spit out answer).  Dr. Jo Boaler stated that our past and current model of education mostly prepares students to do calculations but with the advent of computers, jobs needing only these skills no longer exist.  Thus, we need people who are problem solvers.  In order to change our mindset, Jennifer Piggott suggests that doing rich tasks means that the teacher has to “let go” because this type of teaching is student driven instead of teacher directed.  Dr. Jo Boaler cited a study which found that after students struggle with a problem, they are more ready to learn the mathematical method because their brains are primed.  In class, we discussed that if each teacher increases their emphasis on process over product, that eventually the educational system will change and students will become better problem solvers which is what our society needs.

During our conversation, I showed the group the NRICH website.  Jennifer Piggott’s article was actually found on the website of which she is a proponent.  I didn’t find it easy to “search” for a particular topic (I was looking for a pendulum activity for our “rich task”) but found it easier to just “browse” through the website.  There are lots of great tasks presented on the website, but I wondered if they were all met the criteria of being “rich”.  Since I used one of these activities in my 3 part lesson plan, I brought it in and we tried it as a group and then discussed if it fit the criteria of a “rich task”.   The group liked the “Simplifying Doughnut” Go Fish game, and although it was intriguing and fun, it did not allow for enough diversity to make it a rich task.  I asked the group how we could make this activity into a richer task.  One idea was that given the dominoes, the students could devise their own set of “rules” to play the game.  This would allow for more student direction and diversity and they would own the task. 

I also asked our group if they thought that rich tasks support adolescent learners during this time of brain development.  We talked about how rich tasks would allow adolescents more creativity, choice, risk taking, independence (from the teacher) and sociability which they find rewarding.  As long as the task provided enough entry points and the students met with success, the challenge of the process of problem solving should reinforce the levels of dopamine to make it a pleasurable experience.

Rich tasks are done in group settings.  I asked how we should structure the groupings of students.  We concluded that mixed ability groupings would be the best choice theoretically because a rich task should provide lots of entry points and challenge all students, but practically, there may be times in which homogeneous grouping would allow for a more equal sharing.  I also posed the question to the group regarding how we assess students individually regarding their process when they are working in a group setting.  We are all hoping to gain more insight into the practicalities of assessment of rich tasks during tomorrow’s class!

I think that it is easier for me to let the students drive their own learning in science which I have taught for many years because I am already comfortable with the content.  I myself am capable of seeing the connections and big ideas in the science curriculum and thus am able to make the connections when consolidating their learning.  It is definitely a scarier thing to think of presenting a rich task in a math class which I have never taught nor have a good grasp of breadth of the curriculum.  As is backed up by the research, I believe that teaching using rich tasks supports more learning, for both the students and myself….I will just need to “let go”!!

Zootopia, Differentiated Instruction and Growth Mindset

One thing that I took away from our group’s discussion about differentiated instruction is that everyone is looking for tangible methods of teaching open ended questions and growth mindset in our classrooms in the fall. 

Last night, we had “family movie night” and all watched Zootopia.  For those of you who have not yet seen it, Zootopia is a Disney computer animated film in which all animals live in harmony, at least on the surface.  The main character, Judy Hopps, is a rabbit who wants to be a police officer.  The first part of the movie is all about stereotypes and fixed vs. growth mindset.  None of the characters believe that a “dumb bunny” could ever become a police officer and all show fixed mindsets.  The exception is Judy herself, who has a growth mindset and believes that she can become Zootopia’s first rabbit police officer.  Judy unfortunately has some stereotypes of her own as seen when she meets Nick Wilde, the fox, who she assumes will be “as sly as a fox”.  Unfortunately for Judy (spoiler alert!), Nick turns out to actually be a wily, sarcastic con artist.   During Nick’s childhood, he was bullied and learned the hard way that others expect him to be sly, and so has grown up meeting their expectations.  Both Nick and Judy have a fixed mindset about foxes but as they become unlikely partners in solving a crime, both develop more of a growth mindset about Nick’s potential (I will leave it for you to guess Nick’s future career!).

  

This is definitely a movie to show, discuss and reflect on during class, especially in alt ed or elementary where you are able to link to a variety of subjects!  I would show the first part once the characters are introduced and then pause it to discuss growth vs fixed mindsets of the various individuals (Judy, Nick, Judy’s parents, the other police officers) and stereotypes. I would then return to this topic of discussion after the movie.   As well as teaching students about growth mindset and stereotypes, there are links to other topics as well.  There is the predator-prey discussion as the animals are all living in harmony as the movie’s title suggests but as the predators begin to revert to their “salvage ways”, we wonder who might be behind this (horrible or natural!?) transition to a feral state.  The predators are the minority group which can become a social justice conversation but also an ecological discussion as to why there are fewer predators than prey.  As they are solving this crime, Judy and Nick visit the various districts in Zootopia which are essentially different biomes (eg. Tundra, rainforest).  This can serve as a starting off point to ecology or geography.  There is also an original song by Shakira with the message “Try Everything” in which she sings about the importance of making mistakes!

To relate this to differentiating instruction using the ideas presented by Marion Small, I tried to develop some open ended questions based on the movie:

Tell me about the mindset of a character throughout the movie

Discuss how realistic this movie is

Which character do you relate to and why

Describe some similarities and differences between Judy and Nick

…and math links….

In the movie, Judy has 275 siblings, how many siblings would she have now (someone actually did the math based on the science of bunny reproduction – see link….yikes!)?

https://www.reddit.com/r/zootopia/comments/4djt2y/judy_doesnt_have_275_siblings/

Create a sentence using the words: greater, population, rabbits, foxes

If you can think of other open ended questions or parallel tasks for this movie or other movies with similar themes, please add your ideas!!!!

 Despite analyzing the movie based on the topics that we have been discussing all week in class, I really did enjoy all the Disney “good being triumphant over evil” messages in Zootopia and would look forward to watching it again in class with my students!

Mind Sets & Equitable Education

In my second year of university, during a microbiology exam, I almost changed my major.  I contemplating this change while I was writing a 3 hour exam consisting of all multiple choice questions based only on memorized facts.  This was not what I expected of my university education.  I did stick with it, and by fourth year, I was shocked when (finally) a prof wanted me to actually think.  However, it seemed as though the university had taught those of us who made it through the first couple of years to survive by memorization then they expected us to be able to think without any preparation. 

I have always put a lot of effort into my studies.  When I started teaching, I believed that everyone can learn, they just have to work hard, like myself.  I had a growth mindset.  After teaching for a number of years, I realized that not everyone was like me.  Some didn’t want to work hard and others had learning challenges which needed much support that they were not receiving.  I still believe that everyone has the ability to learn, however, some individuals will use a different method of reaching the same goal.

I try to teach problem solving using whichever subject that I am teaching.  I believe that the process is more important than the product and I encourage students to have a growth mindset.  It is, however, more difficult to quantify process (perhaps we will explore this further during the assessment day?!?) and the students and parents (and many teachers who need to collaborate & be consistent in their assessments) value the mark on the final product more than the process.  A balance between process and product is important. 

I always tell my students and own children that I value their Learning Skills above the actual mark on their report cards.  Generally speaking, if they do well on learning skills, then they will achieve high marks, but more importantly, if given structured, safe environment, teens with a growth mindset will be able to think critically and solve problems in a novel way.  These teens will become creative, critical thinkers who will be able to solve the big problems (eg. social, environmental) of the future.

The Biology of Risk Taking

Teenagers.  Invincible, reckless, impulsive, selfish.  Many times adults, the media, and society stereotype teens negatively.  When given the wrong set of circumstances, teens can definitely find themselves in bad situations, and perhaps in jail or dead.  It is possible for teens to engage of risky behaviours such as drunk driving, unprotected sex or elicit drugs.  The emotional limbic system develops later in teens than the rational prefrontal cortex.  The limbic system includes the nucleus accumbens which is involved in sexual arousal and is the target of recreational drugs as well as the amygdala which is involved in selecting appropriate responses to stimuli as well as the evaluation of faces.  On the other hand, the prefrontal cortex controls motivation, impulse, thinking, problem solving, and the ability to time travel.  The dopamine produced by the VTA in the limbic system is part of the reward system and teens crave the pleasurable experiences that it produces.

Teens are generally self conscious and some have low self esteem.  They enjoy taking risks but only in safe environments.  An educator or adult who has a fully developed pre-frontal cortex can provide that structured, safe environment.  It’s important that students feel as if their ideas are valued and they are included but not necessarily single out, in a group poll, for example.  In these environments, teens full of passion can use their impulse to take risks to become involved in collaborative inquiry based learning to develop solutions to open ended problems.  These teens are the future and we, as educators, have a responsibility to encourage them to take risks, make mistakes, and learn to solve the big problems in the world.