I Taught an AQ…So What???

So I just finished teaching an AQ/ABQ in Intermediate/7 and 8 Math.  The days were long, we had no windows, but wow…what a learning experience.  For those that were involved, I want to extend my gratitude for stretching me and making my think past my current experience.  Here are a few take-aways from the term:

You Can’t Teach an Old Dog New Tricks…Yeah Right!

There was so much experience in the class, people that have been teaching for as long, or longer than I have, and yet everyone walked away with something.  Everyone had some questions, some new conclusions, something that they were excited to try out (including me!)  One thing that resonated with me was that more and more, people were thinking about how to service their students who are yet to come, versus how they have serviced students in the past.  I think a part of this thinking came from me telling a story about a staff meeting where one teacher continued to hold on to an idea that she had.  I knew that the teacher had the best intentions, and in the end it came out that she was scared to let the idea go because she’d be admitting that she had done wrong to so many students in the past.  We talked about how the goal is always to improve and to make things as good as possible for students to come, and eventually she gave herself a pat on the back for accepting the new idea.  Even for me, thinking about things as beneficial for students to come versus detrimental for students in the past has become a conscious effort, and I’ve seen in myself a quicker acceptance to alternative thought.

Open Questions and Parallel Tasks – People LOVE Them!

Come to think of it, people love anything Dr. Marian Small comes up with! I think the most popular take-away this year was open questions and the effect that it has on the un-engaged learner.  (As a side note, I use un-engaged deliberately, because I think dis-engaged implies that the student was never engaged, but un-engaged admits that at one point the student was engaged, and we, as an education system, have done something to un-engage them…not sure who I heard this from, but it’s way too good for me to have come up with.)  So open questions and parallel tasks were really the number one tool according to my class which could be used to re-engage these learners.  I think that people love them because they really give everyone an opportunity to participate.  They allow everyone to “feel the love” in math class.  I remember in my math classes as a kid, I felt the love.  I could answer quickly, I could throw my hand up fast and be confident.  That’s until I got bored, stopped thinking and developed gaps.  Then when I wanted to feel some love, I couldn’t…I wasn’t sure of the answer.  So with open questions, you really give a platform for everyone to participate, and start to build some of their confidence back.

Algebra Tiles – Bring ’em On!

People love algebra tiles (but they have to be taught how to use them!) Oh that day was a roller coaster.  People did not look happy when I brought them out.  Everyone wanted to do my questions using some shortcut they remembered, or some way that they had just taught a class full of kids.  I really had to insist that these little tiles would paint a wonderful picture about algebra, and that we should give them a chance, and one by one, people started seeing the rectangles come out, and started seeing the square get completed, literally.  So…if you’re going to present algebra tiles to teachers, make sure you teach them how to use them.

Technology – We Need It!

I did not expect to be asked so many questions about various tech tools.  I don’t overload the course with tech because I don’t want it to be in the forefront.  I use less in this course than I would with my students, but people wanted more! This is so super exciting to me because it says that teachers are ready to take on this beast, fully knowing that there may be glitches, things may not work out perfectly, and it may be an arena where you aren’t an expert.  But it seems that people are ready none-the-less.  So lets get it! There should be more up-to-date AQ’s about implementing technology in the classroom.  I think that these AQ’s shouldn’t be so much about certain APPs or certain LMSs, because by the end of the course, there’s probably a better app or LMS, but it should be about things like, what makes a good instant response app? What makes a good LMS? What can technology do which pencil and paper can’t???  Hmmm…maybe I’m gonna have a MOOC…stay tuned.

Dan Meyer – We Need More of Him

I think he’s one of my favourites…and I’m sure that I gave a biased account to my class, because they all felt that he was extremely powerful.  I think the big take-aways from him were creating conversation so that math can be used to serve that conversation.  And asking the question first, and then giving information.  I overheard one person say “In textbooks, why don’t they just put it backwards?” Made me really think! A question reads:

The perimeter of a rectangle is 24, the length is 2 more than the width.  What is the area? – Students have so much trouble with this, for many reasons, but I think one main one is organizing the information.  What if the question read:

What’s the area of a rectangle? The perimeter is 24, and the length is 2 more than the width.

I really wonder if that would make a difference??? If you find out, please let me know!

That’s it for now, I’m sure there’ll be a part 2 for the post, but good gosh, I just loved the last 3 weeks so much! Don’t get me wrong, I’m glad its done and I now have the summer to enjoy, but it wasn’t the worst way to spend the first three weeks of summer!

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Make Co-ordinates Come Alive with Google Forms

First of all, I want no credit for thinking of this activity.  It was all Jon Orr’s idea, which you can read here, I just adapted it to use Google Forms and Google Sheets.

So I wanted students to graph the f(x)=log(x) function for the first time, but how boring is plotting x and y co-ordinates? Also, I wanted them to make some big picture connections:

  • An x co-ordinate has a matching y co-ordinate.  
  • We can find that y value by using the f(x) rule.
  • If we all use the same f(x) rule for different x co-ordinates, we will come up with some sort of scatter plot.
  • Hopefully we can draw a curve of best fit for this scatter plot, and predict a pattern.
  • We can describe this curve of best fit with some mathematical properties, so that we can forego the scatter plot in the future, and just draw the graph.

I didn’t tell them these ahead of time or course, but that was my goal.

So I started by giving every student an x-value as they came in, ranging from 0.001 to 12, and I also handed someone x=-1, and someone else x=0.

I then sent students a link to the following form on the class site, and instructed them to complete it:

log form

As they completed the form one by one, the screen in front of the room became littered by random points…

First this:

Graph 1

And then this:

Graph 2

Then this:

Graph 3And finally we had this:

Graph 4I’ll explain how I made that happen live in front of the students in Google Sheets below, but first I want to talk about what came out of the students and their questions.

So first I said that I wasn’t happy because I didn’t know where these numbers were coming from and what they meant, so we talked about labelling the graph.  This is what we came up with:

Graph 5 with titles

Then I asked the students to describe to me in math terms what this graph looks like.  They picked out the x-intercept as (1,0) and one other key point at (10,1) which I was impressed by. Then, they asked me if the y-axis was an asymptote.  I said I don’t know (because that’s usually my answer), so they wanted to check.  They decided to plug in x=0.0000001, and add it to the graph (I just had another student submit a second response to the form that I sent, and the point appeared on the graph).Graph 6 is it an asymptote?

So we decided yes, its an asymptote.  They did a similar process to determine that there’s no horizontal asymptote (even though they said it looks like there may be), but we also decided that this graph increases REALLY slowly.

Ok, so we now had key points and features, and we were all happy.  We then talked about the fact that this looks interestingly similar to the exponential graph, and got into inverses the next day.

So…how did I make the magic happen?

When you create a Google Form, a Google Sheet is created to gather responses.  Each field in the Form gets collected in a separate column in the Sheet.  So, when my students responded to the Form, the populated my Sheet.

Before I sent out the Form, I went to the Response Sheet and selected the two columns where the responses to the Form WILL go.

Sheet empty

I then created a Scatter Plot.

Sheet emtpy w scatter

The columns and the scatter plot were obviously blank to begin with, but as soon as students filled the Sheet with responses, the Scatter Plot filled itself.

Sheet with graph full

When I displayed this to the class, I just zoomed in on the empty scatter plot, and as they entered their responses…bam…magic.

My next goal is to make this work for points of intersection for Linear Systems, where one student will have an x-value which will have a y-value which fits on both lines… I’ll let you know how it goes.

Math Art – Grade 10 Summative

I wanted to share this task because my students were very proud of their work, and they should be recognized…because they’re awesome.

I want to say first that this was inspired by a post that I saw by Mary Bourassa. So thank you Mary!

The task was to first create a rough sketch of a drawing which they think would contain some parabolas, lines, a circle and a triangle, and then to use mathematical equations in DESMOS to create their good copy.  The rule was that they can only use things that we’ve learned in class, unless they could explain to me the mathematics behind something else they’re using.  I’ll show come cool examples of these situations when I go to the examples.

Once the art was complete, students had to submit two things along with their artwork:

The Equations page:

They had to select 15 equations from their sketch and identify where I can find them.  the 15 equations had to include a variety of:

  • Parabolas in Vertex and Factored form
  • Lines (horizontal, vertical, and slanted)
  • Circles (one was sufficient, and it was allowed to be centred and the origin)

The Calculations page:

Students had to submit some calculations to me:

  • Show algebraically the lengths of the sides of one triangle in your artwork
  • Show algebraically all of the angles in that triangle
  • Determine algebraically the point of intersection of two lines in your artwork
  • Determine algebraically the point of intersection of a line and a parabola in your artwork
  • Show me how you convert one of your parabola equations to standard form

The Interview:

Finally, students were given the chance to explain their thinking orally.  Individually, they were asked questions like:

  • (I’d point to a parabola in their artwork) Tell me an estimate of the equation of this parabola in vertex form, and explain your choices.
  • What would the equation look like if I wanted to make this parabola look like (then I’d draw for them a new parabola similar to theirs, but slightly different)
  • Describe using transformations the parabola here…(point to a parabola)
  • Why did you choose to use the Cos law here (pointing to their calculations)

I’d usually ask two questions about their artwork, and one question about their calculations.

Their summatives were amazing, including all aspects.  This is the actual assignment that we gave them.

Here are some interesting examples:

I love how this student used screenshots to identify where the functions are on the Equations page!

Winter Winter descriptions

This student went ahead and learned how to create ellipses on her own.  She said that she just assumed that if you multiplied the bracket by a number, it would stretch or compress the circle because that’s what it does to a parabola.  She also felt that it was important to colour in the artwork, so she did it by hand!

power puff rough
The Rough Copy
powerpuff
On a grid
powerpuff colour
Coloured by hand!

This student’s artwork had a title.  “Nonno” – Grandfather in Italian.  When I saw the rough draft, I told her to not be married to the details because she may not be able to fit it all in, but I obviously underestimated her.

nonno rough Nonno

A few students exported their image into PAINT and then used the fill function to colour in the final product.  I was amazed at the amount of effort that went into this.

minnie not filled  Minnie colour

Some were just fun to see…

monkey rough  monkey

clown rough  clown  castle

And others were very interesting artistically.  When I saw the rough copy, I had no idea that the final product would be this awesome…

eye rough
Eye Rough Draft
eye with axis
Eye With Grid
eye no axis
Eye Final

So thanks to my students for doing these for me, and if you’ve got any ideas or modifications, please let me know!

10 Good Things – A Reflection

So my man Jon Orr tagged me as one of the next people to try this #10goodthings reflection in his post.   Love the idea, for a few reasons.  First, it makes me feel good to say out loud the good things that I’ve done… I think we all need a pat on the back once in a while, and thanks to Mr Orr, I’m about to give myself one.  Also, I love reading others, as they give me ideas…so steal away!

1. I gave myself a new course, MHF, and its going really well.  The students love the open-ended questions and challenging tasks.

2. I invited George Couros to our school, and we started the hashtag #CastleProud which has taken off with staff and students showing their pride in what the school does.

3. I’ve started blogging about my ideas… it really makes me flush out my thinking. In my blog, I’ve organized some 3 act tasks which I think are, and will continue to be hits…Tim Horton’s XXL coffee, and The Ferris Wheel…reimagined.

4. In one day, I met the 4 most influential people in my educational thought: Dan Meyer, Sugata Mitra, Jo Boaler, and Marian Small.

5. I taught an AQ course for Intermediate math…really sparked my love for math pedagogy again…I would rate this as my best experience for the year (professionally).

6. I’ve tried a lot of tech tools…and I’ll keep doing that.  I think my fave for the year is the entire GAFE (Google Apps for Education) platform…so many things that it can do.

7. My students and I have great rapport, still.  They entertain me, and I’m pretty sure I do the same for them.

8. I’ve shared my interest in creating a culture of Growth Mindset with many others around the school and the board…many staff said it was the best PD they’d been to.  I see that as really making a difference.

9.  I’ve remained zen in the face of stress.  Others in my school have asked “Do you guys meditate up the math office all the time?” with respect to how calm the department is, and I like to think that it has something to do with my leadership.

10. I got out to almost every football practice this season! Whoot! 4-3 record in our second year in existence! Go Dragons!

So…go ahead and pat yourself on the back and tell us…what are your 10 Good Things?

Quadratics Summary – Make a Website

So I had my students create a website to summarize their understanding of quadratics.  I was amazed at what they produced.  They used their own choice of platform, but most went with SMORE, Wix, Weebly or Google Sites. This is the rubric that I used:

Website

My favourite part of this process was having students comment on each others work on an LMS.  They gave each other (for the most part) pointed feedback with great ideas on how to fix things.  Then I allowed them to resubmit after taking the suggestions into account.

Here are a few examples, you’ll be able to notice the difference between the students that really took the time to understand and appreciate the visual representations, vs the ones who just learned the rote and regurgitated the information.

Sample 1 – this student kept asking me about decomposition because he wanted a shortcut to trial and error for factoring trinomials… I wouldn’t tell him, so he learned it himself… quite well too

Sample 2 – this website is so well organized and the student did a great job including visuals that are easy to follow.  Actually did a great job teaching everything Quadratic… Take that purplemath.com!

Sample 3 – a simple site, but again went into great detail to explain all of the understanding that she’s gone through.  

So that was it… I mean, I can’t say that everyone did amazing work, but I have to say that I was impressed by most of the stuff submitted.  Just amazing what they picked up in class, and also how they went out of their way to fill in some of the gaps that they felt they had.  If you’ve done something similar, please share with me your rubric as I’m not too happy with this one… 🙂

3 Part Lesson – Tim Horton’s XXL Coffee???

I haven’t tried this with my class, but I’m looking forward to it.  If you do try it, please let me know how it went 🙂

Here is a link to a folder which holds all of the images used in this lesson.

And here is a link to the Google Slides presentation with pictures already included .

Act 1 – What questions come to mind?

Tims Cups

Show the students this image and ask them what questions they have… I am anticipating answers along the lines of: How much is in each one? Does it go up by the same amount? Is the  XL worth the price compared to the small?

Key talking point – I think its really important to dig into the “Does it go up by the same amount” question if it comes up.  Do you mean height? Does that matter? Do you mean volume? Are these Cylinders? How do we get around the fact that they’re not??? Also, what is measurable…can we measure the volume? What about the radius? What measurements would be more accurate?

By the end of this all, I hope to get to solving the following problem:

“I love my coffee, and an XL just doesn’t cut it by December.  I’d like to have you design a XXL coffee for me.”

Act 2 – The information

Ask the students what information they would like in order to solve the problem.  You can provide as much or as little information as you’d like, and you could have them calculate or estimate the rest.  This is all of the info I collected.  The images are available in the link provided at the start.

Width of the TOP

Tims Top Width XL Tims Top Width S Tims Top Width M Tims Top Width L

Width of the BOTTOM

Tims Bot Width XL Tims Bot Width S Tims Bot Width M Tims Bot Width L

Heights

Tims Heights

Prices

Tims Prices

Volumes

I think I wouldn’t give the volume measurements.  I’d like them to find the volumes, and maybe I’d release this to check their answers.

Act 3 – So what does this look like?

Here, I’d like to see presentations of each group’s product, with mathematical reasoning as to which one is the best.  As a group, I’d like to come up with the criteria for what the best one will look like, and then we can decide on the winner.

Some of the mathematical discussion I anticipate will be around variables, so for example if two groups look at cost in these ways:

Size vs. Cost
Size vs. Cost
Volume vs Cost

So which one is better? The top graph compares the size to the cost (S=1, M=2 etc), and the bottom one compares the actual volume to cost.  Which one will we use? Do we want to be mathematically fair, or make more money?

So that’s the plan.  Let me know how it works, or what you think of it. Cheers.

Patient Problem Solving – Scaffold Problem Solving, Not Problems!

We want students to do challenging things in mathematics, but we also want them to be successful.  Its a constant battle between scaffolding too much and not scaffolding enough. I’m always worried about not being clear in what I want, and asking students to do very confusing tasks, but I also don’t want to take them through things step by step, because then they’re not thinking about the math themselves.

What does not help?

Telling students what to do.  Giving students exact directions on what to do next.  If we have to do this on a daily basis, then we haven’t taught students how to actually think.  If we have to do this, then we have to scaffold each problem for our students, because each problem is different.  This obviously doesn’t help long term, because eventually we hope that students can do things without us. This involves instructions like:

  • create an equation for this sentence
  • measure the slope of the line

What does help?

Scaffolding thinking.  Using things like the Mathematical Processes to help our students think about big picture mathematics.  Having students do things like reflect on the reasonableness of their answer is useful in every problem.  Having students understand that we can represent mathematical models in various ways also helps in a variety of problems. Another way to use scaffolding is to scaffold in time during the lesson to have students discuss their own next steps.  Perhaps its a few minutes to talk about what should be done next.  Perhaps its some time soft students to look up vocabulary words that will help them solve a problem.  Here is a great summary of a few of these strategies.

Patient Problem Solving

In the end, its simple.  Help students think so that they can solve their own problems.  Don’t help them solve specific problems.  Long term, that’s not helping. If we scaffold each step, they will never be patient enough to think through something on their own.  On the other hand, if we scaffold thinking strategies, they will take the time to reason through their thoughts and work with others to make sense of the mathematics that we’re trying to teach.