Trying something for the first time

Over the summer, I had the chance to see Darius Rucker (and a few other country artists) in concert and one particular song lyric stuck with me. The song lyric goes "When was the last time you did something for the first time?"

Learning a new skill or idea always starts with doing something for the first time. This happens frequently in my classroom when I introduce a new concepts to the class. Students try things for the first time almost on a daily basis in the classroom when they are in school.

However, it can be challenging to try something new. Doing the same thing is comforting. The fear with trying something for the first time is that I might not be good at it or I might fail at it. People often resist change which can be thought of as trying something for the first time. What if its challenging to do? What if it takes a long time to master? What if everyone else is better than I am? What if?????

Yet as a teacher, when was the last time I put myself in their shoes and did something for the first time?

I kept that in the back of my mind for the rest of the summer. As a self-proclaimed life long learner, it was important to keep trying things for the first time.

So this summer, I had the chance to try SUP yoga for the first time.

I also had the chance to run a trail race for the first time (I've actually run 3 so far this summer).

In both activities, there was some anxiety of not knowing what to expect. But with support from the yoga instructor and friends, both events were enjoyable and I would do both again. This support system made the activities successful to me - and success was defined by me. I deemed myself successful in SUP yoga by finally standing on the board at the end of the lesson and not falling in. I deemed success in the trail run with not winning the race but simply completing it and still smiling.

As the summer holidays wrap up and I prepare to return to the classroom, I am making it a goal for the year to keep trying things for the first time. Be it trying a new activity in my classroom or trying a new assessment tool or trying a new sport, I want to make sure that I keep in mind what it is to be a student of life and keep learning! I want to remember that success, however the individual defines it, seems to always be attainable with a good support team.

So when was the last time you tried something for the first time? 

Demonstrating Thinking on Assessments in Mathematics

As we near the end of another school year, I find myself reflecting on my teaching practice over the last 10 months. A piece of "homework" we have been asked to complete has centered around assessments - reflecting on what we have done in our courses this school year. I know that one of my goals for next year is to investigate (and try) alternative ways to assess in mathematics, beyond just using a test. But before we get to that, I want to highlight a success I had on a test this year in my grade 9 (MPM1D) math course.

The unit we had just completed was on linear equations with a focus on graphing lines and determining the equation of a line given different pieces of information (2 points on the line, the graph, a line parallel and a point, etc.). The test addressed the main concepts/skills of the unit and we usually have 1 question that asks students to extend their knowledge in new ways.

Here is the example of the last question on the test.

Note: we had not discussed collinear points in class before the test - hence the definition of the word in the question.

I was amazed at the variety of solutions that students presented and how their thinking was visible but may not have matched the solution I had in mind. There were multiple ways to show their understanding!

Here are some examples of student solutions to this question:
Student #1:



Student #2:



Student #3:


It is clear in all of these solutions that the students have an understanding of what the slope of a line represents and they have not simply memorized the definition/formula. They were all able to apply their knowledge and clearly demonstrate their understanding of points on a line in relation to slopes.

Using VNPS in my classroom, I have been able to observe student thinking throughout the year. This type of question on a test shows that thinking is also possible to be seen on assessments. Furthermore, I sometimes get stuck trying to create "thinking" questions on test attempting to make them the "hard" questions on the test. This particular question helped clarify for me that these "thinking" questions do no have to be particularly harder than other questions. They should, however, be open-ended enough so that students have different entry points to the question that allows them to showcase their understanding of the content. This will be a lens that I will use when creating assessments next year!

Random groupings - allowing voices to be heard

I'm fortunate to teach in classrooms that have white boards around the room. This means students in my classes are often working at Vertical Non-Permanent Surfaces. I've blogged before about the wonders of VNPS but have recently discovered the power of using randomness for assigning groups.

Last week, my department head came to observe one of my classes and he happen to come by as the class was getting started. Students were working in groups and were solving several non-right triangles. It was a review class for their upcoming test. In this class, I assigned students to groups based on ability level - I created mixed ability groups. Because this was a review period, my purpose for assigning groups in this way was to make sure that each group had an expert in the group to help support each learner.

During my debrief of the lesson with my department head (@Domanator19) asked why I chose groups in the way I did and we had a good discussion about the benefits of random groupings. I know they are beneficial and look forward to hearing about them even further at OAME2018 however I was reluctant to use them in this class. I wasn't sure they would be beneficial for every learner. At the end of the debrief, I made a mental note to try using them in an upcoming class.

Today, I used random grouping for the second time in a row with the same class and had a powerful "AHA" moment.

For information purposes, groups were working on problem solving with the midpoint and length formula:
1. Find the length of the line segment from A(-3, 15) to B(5, 12).
2. Find the midpoint of the line segment from C(6, 10) to D(17, -4).
3. G is the midpoint of the line from F(-6, 8) to H(4, 19). Find the length of the line segment from G to H.
4. Point M(-8, 15) is the midpoint of the line segment from L(-15, 2) to N(x,y). Find the coordinates of point N.

My "aha" moment came when I noticed students that were often quiet in class were now contributing and talking to their group members. When I purposefully make groups of mixed abilities, the strong students often spoke up and directed the conversation. They sometimes took over the problem and presented their solution as the one and only solution. In these situations, quieter students or students that did not have as much confidence in their ability often had a more passive approach to learning. They didn't have time to digest the problem before the solution was on the board. Today, using random groupings (I generated my groups using https://www.randomlists.com/team-generator) these students were randomly placed in a more uniform ability group and had their voices heard. Today, these students had the chance to lead the solution and present their understanding. All students walked away from the lesson with a deeper understanding of midpoint and length.

I guess you can now say I'm converted to using random groupings!

How am I resilient?

"The only real mistake is the one from which we learn nothing." - John Powell

"You miss 100% of the shots you don't take." - Wayne Gretzky

"If at first you don't succeed, try, try again." - Thomas H. Palmer

These are all famous quotes that I often hear and have often said. I've often read about and heard talks on helping students in our current classrooms to be resilient. Merriam-Webster defines resilient as "tending to recover from or adjust easily to misfortune or change".

However, making mistakes, failing, rejection... these concepts are often associated with negativity in our own lives. How many people are anxious just reading these words? How many of us are resilient when faced with these ideas?

I write this post as I ponder on my own resilience. 

Today, the 3rd cohort of the Desmos Fellowship was announced and I was not one of the lucky 40. I am disappointed and wonder what I could have done differently in my application.  So how am I going to be resilient? 

I fret hitting the "Publish" button on this post. How will I be resilient if someone challenges one of my ideas? 

If I want my students to be resilient, should I not be modelling resilience as well? In my teenage years, my father would often be heard saying "Do as I say, not as I do." This was especially true when I was a young driver and my dad was nervously sitting in the passenger seat. If I am not able to be resilient, am I not just telling my students "Do as I say, not as I do."? 

So what does resilience look like for me? Just a few things would be...

I will re-apply next year for the Desmos Fellowship and for other PD opportunities. There is still great learning from the application process and reflecting on my teaching practice. This would be similar to a student not making the team and trying out again next year.

I will teach with my classroom door open and invite colleagues to visit my classroom as often as possible. I don't just want feedback when I think the lesson is perfect - I want continuous feedback along the journey. This would be similar to students showcasing their work and not just focusing on the perfect product. Using VNPS in my classroom allows this to happen frequently.

Memorizing or understanding?

My Grade 10 Academic class wrote a unit test this week assessing student's knowledge of quadratic relations when the equation is given either in vertex form or factored form . 
They were expected to 
- graph the parabola given the equation in either form, 
- determine the equation, in either form, given details about the parabola

Here is an email exchange I had with a student while they were studying:
  
Student Question:

I was looking through the review that you handed out to us, and as I was going through it, I had never seen over half of the questions on it. 

Some of the questions that I was completely confused on were:

Please let me know if questions like these will be on the test, as I have never seen any questions like these in the work that we have done leading up to the test so far.

My Response: 

1.

We have seen questions like this. Rewrite it as . This is a parabola that opens down and has a vertex at (0, 9).

2.

Not a focus on this test. We will see these in Unit 4.

3.

Not a focus on this test. We will see these in Unit 4.

4.  
We have seen questions like this. Rewrite it as . This is a parabola that opens down and has a vertex at (3, 7).

5.

We have seen questions like this. This is the equation in vertex form however the vertex has fraction values . 

Student Reply:

I did not know that we were able to re-write equations, as well as I did not know that x and y values could be fractions that large.

Thank you very much Ms. Gravel!

The reply took me by surprise. Because none of the examples I used in the unit had the vertical shift come first, the student had memorized the "formula" for the equation and not understood the separate parts of the equation. (Disclaimer: Most of my lessons did involve investigations however all the written examples had the equation in the same order).

I shared this exchange with a colleague and they recalled a similar experience with one of their students a few years back. My colleague was teaching the Pythagorean Theorem and a student stated that his example


was not a right triangle.






After many minutes going back and forth, the student finally exclaimed that it wasn't a right triangle because it wasn't in this orientation.



It was then, like for me in this situation, that my colleague realized that he needed examples of right triangles in all orientations to emphasize understanding and not memorization.

Food for thought for the next unit...

Students reflecting on their learning

In order to take the emphasis off of the final mark on the unit test, my colleagues and I decided to try something new in our last grade 9 unit. We were going to have the students reflect on their learning before seeing their final grade on the test.

The day the students wrote the test, we photocopied their completed test before grading it. On Friday, students were provided with the photocopy of their own test and a sheet that looked like this:

They then spent the class time working through each question identifying what content of the unit was assessed in that question along with a reflection of how prepared they felt for the question. 

At the end of the period, students were provided with their graded test. 

Some of my take aways from the activity:

-  Students focused more on their own solutions and not just the number of marks they got. Often students who do well on the test only focus on the few questions that contain mistakes and do not take the time to reflect on what they also did well. 

- The conversations between students were focused on learning and not just on marks. If two students got different answers, they worked together to determine where the mistakes arose.

- When the marked tests were returned, there were no surprises or emotional responses to lower than expected marks. And as mentioned above, the students who did well looked through their test with a critical eye as opposed to seeing a good mark and filing it away in their binders. 

I'll admit that this was in reaction to students writing the test and identifying that they felt it was a challenging test. In the future, it would be beneficial to have students go through this reflection activity before the test (even before the review period) so that they can focus their studying on the content of the unit where they feel least prepared. It is definitely something that I would consider doing before the final exam. 

Finally, we did have the benefit of having time to use a class to go through this process. In other units, I hope to use this same reflection activity but may have to assign it for homework the day before I return the test. I could envision students going through this process at home and arriving to class with an educated guess as to how they think they did on the test. 

MDM4U Games - choosing player A or player B

The last few classes have been used to consolidate the concepts we have been working with thus far. Students were asked to ensure they had a definition in their notes for: probability, bar graph, histogram, probability distribution, random variable, discrete random variable, continuous random variable, and probability histogram. I did hand out a worksheet to practice probability and measures of central tendencies. 

Today we played several short dice games (I created these off the top of my head looking for some that have an equal probability of occurring and others that do not). Students were paired up and played each of the following games recording who won each game. 

Game #1:

• Roll the pair of dice. (2 standard dice)
• If sum is Even, Player A gets 1 point
• If sum is Odd, player B gets 1 point
• Play the game 10 times

Game #2:

• Roll the pair of dice. (2 standard dice)
• If the product is Prime, Player A gets 5 points
• If the product is not prime, Player B gets 1 point
• Play the game 10 times

Game #3:

• Roll the pair of dice (1 6-sided dice; 1 12-sided dice)
• If the sum is even, Player A gets 1 point
• If the sum is greater than 7, Player B gets 1 point
• Play the game 10 times

Game #4:

• Roll a set of dice (1 6-sided dice; 1 30-sided dice)
• If one number is even, Player A gets 1 point
• If one number is odd, Player B gets 1 point
• If both numbers are even, both players get 1 point
• If both numbers are odd, both players lose 1 point
• Play the game 10 times

Game #5:

• Roll the pair of dice (1 6-sided dice; 1 12-sided dice)
• If one number is prime, player A gets 2 points
• If the sum is less than 6, player B gets 3 points
• Play the game 10 times

Game #6:

• Roll the set of dice (2 standard dice)
• If the sum is odd, Player A gets 1 point
• If the sum is greater than 10, Player B gets 1 point
• If the sum is less than 5, both players lose 2 points
• Play the game 10 times

Game #7:

• Roll the set of dice (2 standard dice)
• If the product is less than 10, Player A gets 4 points
• If the product is greater than 18, Player B gets 4 points
• If the product is between 10 and 18, each player loses 2 points.
• Play the game 10 times

We gathered the data as a class to see if Player A or Player B had an advantage in each of the games. Based on our results, only game 6 had a clear advantage for player A - all other games were pretty even.

Students were then asked to determine the probability of each player getting points in each game. An interesting outcome was in game #2 where the probability of player A getting points is MUCH less than player B but our class results when playing the games showed that each player won the same number of times. 

This lead nicely to the start of a discussion of fair games and realizing that it is about more than just the probability of an outcome but what the point values are in a game. 

Our next class will be used to define a fair game and look at the mathematics behind determining if a game is fair or not. 

MDM4U - Activity: introducting non-uniform probability distributions

Students were given time to play "Crossing Toronto Harbour". In my de-cluttering of my filing cabinet this summer, I re-discovered a few past editions of the OAME Gazette and found this activity by Kelly Young in the June 2010 Edition.

The main idea: students are paired up and each given 10 "boats" and a 2 dice. Each student rolls the dice and if they have a boat in the dock with that sum, they get to cross their boat to the other side. If that dock is empty, they do nothing and pass the dice to the other player. The game continues until the first player has crossed all their boats to the other side.

My challenge was to find enough items of different colours for students to use as "boats" that were cheap to find but small enough to fit on the game board. I was at the dollar store this weekend and found these packages of decorative pompoms to use.

Students quickly discovered boats in the #1 dock would never cross and that sums of 6, 7 and 8 occurred most frequently.

This lead nicely to a discussion of that the probability of any event lies between 0 and 1 and that not all events have the same probability of occurring. Unlike a standard die that has a uniform probability, looking at the sum of two dice has a probability distribution where a sum of 2 and 12 are least likely to occur while a sum of 7 has the highest probability of occurring. Students were introduced a probability distribution chart where all outcomes are listed along with their probabilities. A histogram or bar graph could be used to graphically display a probability distribution depending on the event used.

I also asked students to keep track of what numbers they rolled so that we could continue to distinguish between theoretical probability and experimental probability. Here is a chart of the outcomes rolled in this class:

We will refer to this data to start next class to chat about why it is not an identical match to the theoretical probability distribution.
Homework was assigned from the question bank focusing on probability distributions as well as determining the probability of different sums when rolling 2 dice.

MDM4U - day 2

Today's class had a two main goals:
1. Set up a Google Community for the course.
2. Further investigate the relationship between theoretical and experimental probability.

Task 1:
One of the overall expectations of the course is "demonstrate an understanding of the applications of data management used by the media and the advertising industry and in various occupations." In the past, I addressed this expectation by presenting current articles I have found or asked students to bring in interesting articles and we have had class discussions around these articles throughout the year.

I decided to go digital with this requirement this year. In today's class, students were asked to join a Google Community for this course. This space will be used to collect and store articles students find interesting or have statistical significance. Also, this platform also allows students to post comments on articles their peers have posted to the community. I started by posting an article myself (it was based on what conditions will be required to run a sub 2 hour marathon) and had students post a comment on either a graph or statistic they found interesting in the article. Students were also asked to post, by the Thanksgiving break) an article they find interesting. I envision this being an avenue to generate peer conversations as well as student-teacher conversations in this course. Also, my ultimate goal is to have students post one article a term for a total of 3 in the year.

Task 2:
We played Dice Bingo - with 1 standard dice. Students created a bingo board with only 5 entries similar to this:

Numbers can be repeated.

The teacher rolls the dice. If the number rolled appears on their board, they cross off one occurrence of it (so if they have three 5s and a 5 is rolled, the student only crosses off one of the 5 and leaves the other 5s in play until another 5 is rolled). If the number rolled does not appear, they do nothing. If the number rolled appears on their board but all occurrences of that number have already been crossed off, students add that value to their points total. The game ends when one student has crossed off all the numbers on their board.

At the end of the game, students add up all their points including numbers that have not been crossed off their board and the goal of the game is to get the least number of points.

We played three rounds of the game and then talked strategy.
- What numbers should you use to ensure you get the least number of points?
- What is the max number of points you can get if the game is won in 5 rolls?
- What is the minimum number of points you can get if the game is won in 5 rolls?

We will revisit the game when the teacher rolls 2 standard die and the entries on the board are the sum of the die.

Homework:
Students were asked to post a comment on the article in Google Communities. Students were also asked to determine the max points possible in Dice Bingo if the game was won in 6 rolls; 8 rolls; and 10 rolls. And finally students were given practice questions determining the probability of given events using dice and cards.

MDM 4U - Day 1

The course had a short introductory lesson yesterday where we reviewed the course outline and I explained to the students that this course would be spiralled. 4 main strands (Collecting Data; Organizing Data; Probability; and Statistics) would be incorporated into each lesson and developed throughout the year.

Today was our first day of full lessons. The goal of today's class was to get students used to the format of the class (away from being used to filling out worksheets) and have students begin to problem solve together.

We played a game of "Skunk" with one die. After the winner was declared, we discussed strategies used in the game. Most students identified that if a 1 hadn't occurred in 5 consecutive rolls, then they felt the need to step out of the round because a 1 was more likely to occur next. This lead to the introduction of the idea of theoretical probability. We defined theoretical probability and the idea of patterns that occur in the long run.

Students were then each given a die and asked to roll it 20 times recording the outcome of each roll. We collected this data as a group and noticed that though we should get each number rolled 3.3 times in 20 rolls, not a single student has this occur. However, when we summed out data together (instead of looking at 20 rolls at a time, we now were looking at 220 rolls) we saw that the distribution of rolls was closer to equal for each value.

Finally, students were randomly paired and asked to solve two problems on our VNPSs. The first problem was "The sum of 15 consecutive numbers has an average of 27. Find the average of the first five numbers." I was surprised at how quickly they came up with the answer of 22 but had some challenges explaining their thinking. We discussed how important it is, especially in a statistics based course, to ensure you are using the proper terminology.

The second question was "A band of 10 pirates are going to disband. They have divided up all of their gold, but there remains one giant diamond that cannot be divided. To decide who gets it the captain puts all of the pirates (including himself) in a circle. Then he points at one person to begin. This person steps out of the circle, takes his gold, and leaves. The person on his left stays in the circle, but the next person steps out. This continues with every second pirate leaving until there is only one left - who gets the giant diamond. Who should the captain point at if he wants to make sure he gets to keep the diamond for himself? What if there were 11 pirates?"

This second question was great because it was unique enough that it leveled the playing field for every student in the class - it was not based on past course material so students who may have a stronger math background did not have an advantage to solving this problem. There was lots of problem solving and collaboration in groups. Each group provided an answer (not all the same answer) and we will verify solutions as a group in our next class.

It was great first class and I hope it has set the stage for a great year!

Spiralling MDM 4U - a great first run through

I did it! I successfully revamped my MDM 4U course following an activity based and spiralled approach to learning.

Was it perfect? No.

Was it messy along the way? Yes - but change is always messy and you have to be flexible to adapt as  the unexpected arises (shortened classes, activities taking longer or shorter than anticipated, etc.)

Will I do it again next year? Absolutely, with some minor tweaks.

HIGHLIGHTS:
My biggest success of the year was the high level of student engagement in every lesson. I found students came into class excited to see what the next activity was. Each student had something to contribute to the discussions and students were more likely to disagree with each other and present their point of view.

I also found I was excited about teaching. After 13 years of teaching, it was a new challenge which was fun.


STRUGGLES:
On the teacher end of planning, one struggle was finding appropriate activities to suit the needs of the lesson. I was grateful for Twitter for ideas. I often took an idea from another grade level and had to tweak it to my content and ability levels. It was also sometimes tricky to know exactly how much time an activity would take.

On the student end, it was a struggle to help them properly document their learning on a daily basis without giving too much information before the activity. As I mentioned in an early post, students are used to filling out worksheets during a lesson which makes it very clear what the unit content is. Taking these worksheets away and not coaching them on how to take their own notes did cause some anxiety at the beginning of the course.


NEXT STEPS: 
I blogged last summer about the amazing Plinko board that my dad helped me build last summer. Unfortunately, I didn't get the chance to use it this year however my ambitious goal is to use it once a month next year (in a non-semestered school, this would be 9 - 10 times in a year).

I also want to provide more opportunities for students to show their knowledge and receive formative feedback on their skills. I picked up an idea at OAME 2017 about "Fast Fours" quizzes and hope to use this idea in my class next year. Once a cycle (every 8 days), students would start the class with a 4 question quiz - one question from 4 main strands of the course (Probability; Statistics; Organizing Data; and Collecting Data).

My plan for next year also involves being more clear on the specific learning goals of each lesson/activity. I now know that my students need more coaching, right from day 1, on how to create their own notes in a math class. By being specific on what the learning goals are for the activity, both as an introduction and a very clear summary, students will have more confidence in their understanding of the content well before summative evaluations.

After presenting at OAME 2017 (here are my slides from that presentation), I've had a few requests for more resources on activities that I've done in my class room. My goal next year is to blog more regularly - hopefully once a month - with details on how the 2nd year is going along with information on what specific tasks I'm using in my classroom.

Finally, I know I need to give more time throughout the year to the final project. Students are asked to pick a topic of interest and analyze data to hopefully prove their hypothesis. I did have to rush part of this at the end of the year and student feedback was that they didn't always understand what they were doing. My plan for next year is to tie it to our classroom activities but rolling it out earlier in the year - having them chose their topic by mid-October.

Happy summer!

A year in review - the value of student feedback

As a teacher, I am frequently giving my students feedback. It could be verbal feedback in class or in extra-help or in conversations. It could be formative feedback when taking up homework or on assessments or on quizzes. It could be summative feedback on evaluations.

As a teacher, I often have conversations with colleagues about the importance and value of good feedback for student learning. How can we learn from mistakes if not for feedback?

But as a teacher, how often do I receive feedback on my teaching practice? And how open am I to receiving feedback on my teaching practice?

I’m very fortunate to work in a school and a department where classroom visits are encouraged. My classroom door is almost always open (I am a loud talker and the door has been known to be closed by other teachers so not to disrupt their class either across or down the hall!). I’m fortunate to work with colleagues who I can throw an idea at and they will critique the idea so by the time it reaches my classroom, the activity is ideal for student learning. These same colleagues are also patient in listening to a lesson that may have flopped and have helped me improve the idea so that it is a success on the subsequent trial.

But how often do we ask students for their feedback? I sometimes fear getting feedback from students as it can often be biased depending on the time of year. But their input is the most valuable in my mind as they are the ones that see my teaching on a regular basis and know what my teaching truly looks like. A colleague could come in for a snapshot of my teaching for a particular lesson or topic but the students are in the room for every lesson. So their feedback is the most important feedback in helping me be the best teacher I can be.

It has been a custom of mine to ask for student feedback at the end of each course. I’ve done this every year of my 13 year teaching career. In the beginning it was done on a piece of paper where I would ask students to provide an answer to these 3 key questions:

1. What was one thing Ms Gravel did well this year?
2. What was one thing Ms Gravel could improve next year?
3. Any other suggestions for Ms Gravel?

I’ve kept most of this feedback and have found it helpful. However this was often rushed and done on the last day of classes.

Over the years, this practice of collecting student feedback has evolved to an electronic format and formalized to a department wide Google Form. The questions have become more elaborate and focused on key aspects of the course and not so open-ended as my paper version years ago. Ideally, I would collect student feedback a few times a year so that I could use the feedback and improve the course as the year progresses but in the very least, obtaining the feedback once a year is a success for me.

I’ve now collected this feedback from my students for this school year. As much as I appreciate the “Everything was great” comments, I actually enjoy reading the constructive criticism more. In a year where I have revamped a course, this feedback was most valuable. The feedback did give me insight that I need to be more clear in expectations and has given me more direction for next year. It has also given me a bit of insight on student learning in my class as well.

So as I wrap up this school year and start to look at next year, this student feedback has given food for thought just in time for planning for next year!  

Spiralling MDM 4U - Term 1 review

It's now been a full term of spiralling through MDM 4U and I thought this would be a great time to reflect on what has gone well, what were some struggles, and where I'm heading with the course. 

HIGHLIGHTS:

The students' involvement in classroom discussions and willingness to try even if they are unsure if they are correct have been highlights of the course thus far. The students have been engaged in the activities we have done in class and were able to clearly show their knowledge of games of chance in their "Unfair Game" project. 

It has also been interesting to easily connect terms and concepts that would be separate in the textbook in different sections. In particular, it was great to see the connection between permutations and combinations and how their formulas are so interconnected.

We ended the term with a review sessions where students were asked to work in pairs and write down all the terms and concepts (with formulas) that we had covered since the beginning of the term. Here is an example of one list:

After the list was made, students were asked to self-assess their confidence with each topic. They were asked to put a check mark beside concepts they knew, a - beside concepts that they needed a bit more clarification with, and a 'x' beside concepts that they were still struggling with. It was interesting that most of the concepts that had 'x' beside them were concepts that we had just recently introduced in class. 

STRUGGLES: 

One of the most common comments from students is that they are struggling to keep track of what we are learning in each class. In previous mathematics courses, students would have been given worksheets to fill out and that dictated what they were learning and what they needed to know for assessments. One of my dilemmas with spiralling has been how to keep track of our learning without filling in a worksheet. It's not that I am completely opposed to worksheets (I use them in my other courses) but I feel they defeat the purpose of discussion based learning. With a worksheet, students only want to know what to fill in on the worksheet and not explore the concepts being taught. With my approach to MDM 4U, I want the learning to be authentic and the students to be in the learning driver's seat. 

I do write key terms on the board and we often work through examples as a class but I believe the concern is around what students should write in their notes and quite simply how to make a proper note in math that is not a worksheet. I believe I may have a solution to this dilemma that I will be implementing in the new year upon our return from the holidays. My plan is to use a sheet like this:

Students would have access to these sheets at the beginning of class and would then fill it in as they need as the concepts arise. A summary would then be emphasized at the end of the lesson or the beginning of the next lesson so that it is clear what concepts are being covered. 

WHAT'S NEXT:

The remainder of the year will continue to be activity-based and discussion based learning in this course. We are moving towards more statistics based concepts and not as much of a focus on probability. My goal is to create at least one 3-Act math task to use in a lesson to either introduce a new concept or reinforce a concept.  

Students have also selected their topics for their year-end project. It will be exciting to see them collect their data (either primary or secondary data) and apply all their skills from the entire course to analyze and answer their key questions. 

Creating Art in Math

We decided to let our students demonstrate some of their creative abilities for our inverse assessment in MCR 3U. The key skills we covered were
- what is an inverse?
- relationship between domain and range of a function and its inverse
- relationship between any point on a function and the corresponding point on the inverse
- relationship between the transformations applied to a function and the corresponding transformation on the inverse

Instead of assessing these skills in a typical tests, the following assignment was used.

Though the instructions are quite straight-forward, it does take some thinking and understanding of functions and transformations. Having just completing an entire unit on transformations, this assignment was a chance for students to apply these skills to a unique situation.

Here is my attempt at an exemplar:

The biggest challenge was making sure that the functions they created were close enough together to create a closed figure as their final product. Most students got around this by creating more of each type of function to create the image they envisioned. I could not complain as a teacher as students were further practicing their skills and providing even more evidence of their understanding of functions and transformations by creating several equations of each type.

The art segment of this project was marked using a rubric.




Following this, students came to class and were asked to determine the equation of the inverse of one of their functions (quadratic, radical or rational). This created a unique assessment for each student as they had each created different functions in their art project. 

Are you game?

Spiralling in MDM 4U continues and we've reached the end of the second spiral. With a great focus on probability and counting techniques, students have had the opportunity to experience and demonstrate in these two spirals knowledge of permutations, combinations, factorials, binomial and geometric distributions, probability and odds.

As a summative assessment, students were asked to create an unfair game. They could use any materials available in the classroom or provide supplies from home as needed. In addition to creating the game, students were also asked to analyze the game - determine the probability of winning, show that their game was unfair, along with a few other questions.

Most students chose to revolve their games around dice and cards. These were manipulatives that we had used repeatedly in class and they were comfortable determining probability with. 

Some examples included:

Game #1: 
A player pays 3 tickets for the dealer to roll a 20-sided die on a flat table.

- If the die lands on an even number (red) excluding 20, the player loses the 3 tickets they bet with
- If the die lands on an odd number (black) excluding 1, the player loses 1 ticket they bet with and 2 tickets are given back to them
- If the die lands on either a 1 or 20 (green), the player wins 10 tickets, but paid 3 to play so overall gains 7 tickets

Game #2:

The name of the game is Cross the River, this is because the pathway takes the students across a beautiful river. It costs the student 2 tickets each time they play this game. It guarantees that the student wins every time! He or she will start on the start sign and will start off by rolling the two dice once. If he/she rolls an odd sum then he/she will move up two spaces, but if he/she rolls an even sum then he/she will move up one space. They will get four rolls in total. This game guarantees that they win because if they roll four even sums, which is the minimum, they will at least get to the fourth space, and this will give them one ticket back. Technically they are winning, but in reality they are losing one ticket in the sense that they pay 2 tickets to play, but only get 1 ticket back.

Game #3: 

The name of my game is “The Role Of Fortune”. I choose this name because I was hoped that the familiarity of the name, derived from “The Wheel Of Fortune” would grab the students attention and draw them towards my game. My game is a very simple game which will captivate and interest younger students because of its ease to play and enjoyment from playing. My game costs 3 tickets and involves rolling 2 die. The rules are simple, your objective is to roll a sum of 2, 3, 4, 7, 10, 11, or 12. Any other role will result in the loss of your 3 tickets. A sum of 2 or 12 results in the grand prize of 3 tickets. A roll of 3 or 11 results in a prize of 2 tickets and finally a roll of 4 or 10 results in the prize of 1 ticket.

We then had the opportunity to host the grade 8 math class for a Games Day. Students used the opportunity to collect experimental data for their games. Following the Games Day, students were asked to analyze their experimental data and compare it with their theoretical calculations.

When students began planning their game, they focused primarily on making it unfair - as stated in the expectations of the project. After the Games Day, students were then contemplating how to make their games more "interesting" and more fair. As part of their final reflection about the project, students commented on improving their game by making it straight forward (not too many steps) but also fun and not too obvious.

Finally, students were asked to create a sign for their game so that players knew the rules of the game, the cost of the game, and the probability of winning the game. They struggled with how to state the probability of winning their game without giving away that it was unfair. It was another example of a reassuring theme in the course that we must be analytical with data to ensure we have the whole picture and do not simply believe all that we read. 

A sweet 3-Act Math Task

The other day we had the opportunity to host our Grade 8 students in the Senior School to showcase what Grade 9 math is all about. To give them a taste of linear systems, John Doma (@Domanator19) and I created a sweet 3-Act Math Task.

Act 1:
Act 2:
Act 3:

Though I have used several 3-Act Math tasks in my MPM1D class, this seems to have been one that created the most opportunities for different solutions. Though unintentional, the data that we provided in Act 2 allowed for multiple solutions. This lead to the majority of students being engaged in the activity and willing to contribute their ideas (maybe it was also because we gave them a small box of Smarties to use in their problem solving!).

Here are some of the ideas that students brought forward:

a) 2 cm = 215 Smarties. Therefore 1 cm = 107.5 and so 15 cm = 15(107.5) = 1612.5 Smarties
b) 5 cm = 485 Smarties. Therefore 15 cm = 3(485) = 1455 Smarties
c) 11 cm = 1147 Smarties. Therefore 15 cm = 11 cm + 2 cm + 2 cm = 1577 Smarties

My take away from this activity was making sure to truly think about what data to provided. Going forward, I will definitely be reconsidering the data I do provide in Act 2 taking into consideration multiple pathways to creating an answer.

Developing a classroom of inquiry

My MDM 4U course has been an adventure this year. My goal, as mentioned in previous posts, has been to spiral the course or at the very least deliver the content through activities. It's now been two months and I thought I would share some highlights thus far.

1. Students are more engaged and willing to participate.

• They absolutely love playing games - even if its as simple as "pick a number from 1 - 6 and once your number is rolled, sit down". They are starting to ask questions that help lead the lesson as opposed to being told what they need to know.

2. Students regularly contribute and offer answers, even if they are unsure if they are correct.

• Most activities require students to contribute their findings or results of an activity. I found that by having each begin the course by contributing in this way, I have created a space where students are more willing to contribute.
• Its okay to say "I don't know". It's even okay to offer an answer and when asked why you chose that number to say "I don't know" but I thought...

3. Students offer solutions without definitely knowing the correct formula.

• I have students who have prior knowledge and are stuck on having a formula. It's been great getting them away from the need for a formula and focusing more on explaining why something is happening. Students in my class are getting better at communicating their understanding.
• Students are making connections between activities and again are not so focused on just memorizing formulas.

So until the next post, here is to creating more activities to discover statistics.

Favourable outcomes

We have spent a lot of time rolling dice in my MDM 4U classes so far this year. One day we played 3 different short games all involving being in pairs and rolling a pair of dice.

Here are the details of the three games.

Game #1:Roll 2 dice and consider the sum of the dice.
Player A wins 1 point if the sum is EVEN. Player B wins 1 point if the sum is ODD.

Game #2: Roll 2 dice and consider the product of the dice.
Player A wins 1 point if the product is EVEN. Player B wins 1 point if the product is ODD.

Game #3: Roll 2 dice and consider the sum of the dice.
Player A wins 2 point if the sum is EVEN. Player B wins 1 point if the sum is greater than or equal to 7.

This lead to a discussion of which player would you choose to be in the game and why.

For Game #1, I had a justification for 3 different outcomes:
1. Choose Player B because the mode sum is 7 so there would be more odd outcomes.
2. Choose either one because Even and Odd are equally likely (but not sure why they are equally likely)
3. Choose Player A because Even outcome more frequent
Even # + Even # = Even #
Even # + Odd # = Odd #
Odd # + Odd # = Even #

It was great that students were willing to share their thought process and we had supporters and opponents of each theory. We then developed a chart to show all possible outcomes of the sums and saw that both Even and Odd outcomes are equally likely. Probability was introduced.

In Game #2, based on their results of playing the game, each student would choose to be player A. Similar to game #1, by examining a chart of the possible products, it was clear that Even outcomes are more likely.

We started to discuss the idea of fair games and mutually exclusive events.

Homework for the class was to determine how to chance Game #2 so that it was equally likely for Player A or Player B to win the game. Students came up with some alternate version of one of the following:

- Player A gets 1 point for an EVEN product and Player B gets 3 points for an ODD product (changing the point value of each player)
- Player A gets 1 point for a product of 1 and Player B gets 1 point for a product of 36. All other products result in no points for either player. (only selecting some of the products)

Further discussions in upcoming classes will focus on fair games.

Spiralling MDM 4U - Day 1

I've made it through my first day adventuring down the path of pseudo-spiraling with MDM 4U. And it was actually quite successful and fun! My first disclaimer is that what I am doing may not be true spiraling but I can definitely say that it is activity-based learning. The students have been warned that each class may look disorganized but there is a master plan! My biggest leap of faith was leading a lesson without having a nice handout for students to fill in as we went through the activity.

Today's lesson began with each student being given a dice and asked to roll it 30 times. They were asked to keep a tally of each value rolled in a chart. Students were then asked to chat about their outcomes where they discussed why a few of one number and more of another arose and the discussion led to each value should be represented equally.

We then tallied everyone's responses in one large table and discussed these findings. There was some excitement when the first two columns (# of 1s and # of 2s) were the same and then further intrigue when there were almost double the number of 3s. Again, the discussion led to why each value was not represented equally.

A comment was made about the size of our sample (287 total rolls) and if this was large enough. This is where we distinguished between a population and a sample and started to lead to the idea that the closer the sample represents the population, the better representation it is of the population.

The data was then graphed in a histogram (first individually by each student and then together with the large group). One question that arose was "why can't we draw a scatterplot?" I was really excited that this came from the group and I wasn't telling them right from the start that a histogram was the way to go because of ...

Again, we discussed what should the histogram look like in an ideal situation and why use a histogram over other graphs. The terms discrete random variable and continuous random variable were introduced. Outcomes and events were discussed.

Students were then given a second die that was different than 6-sided and asked how their results would differ if the same activity were repeated with the new die. The idea that the larger the number of sides on the die, the less uniform the histogram would be for 30 rolls. Furthermore, the larger the number of sides in the die, the more rolls would be required to see a uniform distribution.

We conlcuded the lesson by reading a current article and discussing the need to critically assess the numbers we read. In the future, I will give the article for homework before discussing it to ensure all students have enough time to fully read the article.

Key Terms Introduced in the Lesson:
Population; Sample; Discrete Random Variable; Continuous Random Variable; Event; Outcome