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!