Counting on Day 1

I usually like to take the first day of a course to talk about some interesting and simple math that students might not get elsewhere.  In the past, I've talked about why we write numbers the way we do, interesting facts about large numbers, and a basic overview of math subdivisions (topology, analysis, algebra, statistics).  This year I was inspired by a blog post from Andrew Stadel about counting dots.

The link to my slideshow is here.  Feel free to use as is or save a copy and modify to your own needs.

I begin with having them do a math meditation that I learned from Dr. Edmund Harriss at Twitter Math Camp 2014.  I ask the students to do a visualization exercise (close your eyes if you need to, or stare of into space if that helps).  Begin with a blank canvas -- it could be any color -- then we are going to add a certain number of dots to the scene.  I then call out the numbers 1 through 5 (maybe a number every 10 - 20 seconds).  I pause at five and tell them that this is about where we run into the limits of being able to put dots anywhere.  They might think about starting to organize the dots as we progress.  Then I continue on in the same manner until about 15 to 20.

http://www.printable-flash-cards.com/wp-content/uploads/2008/12/numbers-dot-preview.gif

When I stop, I ask students to describe to me the picture in their head.  It might be a 4 by 5 rectangle.  It might be two 2 by 5 rectangles.  It might be two groups of 3 by 3 squares with 2 dots to the side.  One person from Calculus said, "Three dice with the six side up and one with the two." (D&D player anyone?)

I then proceed with the sideshow and tell them I want them to try to figure out how many dots are on each of the next slides, but they will only appear for 0.6 seconds.  We do a few of these and they love the challenge of trying to get it quickly.  I'll explain what they are doing is called "subitizing" and how many animals (including humans) have the ability to do it with small numbers, but only up to a point.  Imagine if 97 dots flashed up there.  Could you count them in half a second?

I also like to stop after a couple of them and ask them how they figure it out.  I don't know much about the subitizing literature, but my guess is that with small numbers of dots you can make shapes with them easily (three dots make a triangle or line, 4 into some kind of quadrilateral or rows, etc.) then count after the image disappears if necessary.  Making a decagon is quite a bit harder, though.  Even with seven dots, some students will say they saw a row of 3 and a group of 4 (like we chunk phone numbers).

I then show the Vsauce video which talks generally about numbers (including logarithmic number lines as well as subitizing).  We discuss any questions, comments, or other thoughts they may have after viewing and then go back to trying out our counting techniques.

In the next few slides, the dots are organized into easier to think about shapes.  Even if they don't get the number correct after it flashes on the screen, we talk about how they might find the number of dots without counting each and every one.  The students sub-organize the shapes into smaller shapes (eg this section was a 3 by 3 square, then one more to the side) and it's interesting to hear the different ways they group them.  The switch to blue squares is not too different.

Once we do a few slides of blue-square counting, I ask them what percentage of the whole shape is made from blue squares.  In the slides I made, I have it animate to move the squares around and see the same amount in a different way.  By moving a blue square to a white space in an organized way, it's easier to determine the percentage.

To finish, I tell the students I wanted to show them this stuff to highlight some interesting psychology behind our number sense, but also to motivate the need for math as a subject.

By organizing and moving things around in a meaningful way, it is easier to interpret what is quantitatively in front of us.  Especially with my algebra students, I tell them that our notation and algebraic manipulation will cause some growing pains, but once we get some techniques under our belt (like we did with chunking shapes into smaller pieces or moving blue squares to fill white holes), it will make the process easier.

Trusting in An Answer

One thing I've been noticing this year as I find "challenge" problems online or give sets to my own students is that the first thing I acknowledge is the trust.

Often, we ask students to "Prove that..." or "Solve" in problems.  All well and good in a classroom setting, but in a more "real-world" situation, the first thing that needs to be assessed is whether that problem CAN be solved.  I think my own problem-solving strategies are sometimes directed by knowing that there is an answer before I even begin the work.

I trust that my teacher or problem-giver is giving me the opportunity to solve the problem when it is posed.  That is, if I am not allowed a calculator, I trust that it CAN be solved without a calculator and that my answer will most likely be a whole number or at least a simple fraction.  If I am in a 6th grade classroom then my problem will be "6th grade appropriate" (ie should be solved without calculus or other advanced methods).

Of course, this is not the case in higher mathematics where the main question can often be, "Is this solvable?"  Certainly less so in every day life when we are faced with innumerable unsolvable problems.  I'm not suggesting we give students unsolvable problems, but it might be interesting to mention this to students that we are in an artificial, safe environment in this classroom where these problems have already been solved and I (the teacher) am giving them to you knowing you should be able to complete them.

My Favorite Unit Circle Question

In Precal, I love the development of the unit circle.  It brings together so many ideas and concepts they've learned from algebra 1 and geometry and makes sometimes unexpected connections.

Two years ago I developed this question that I think really helps students see how the circle comes together.  It originally stemmed from students memorizing THE Unit Circle with it's 16 (or 17 if you count 0 and 2pi differently) special angles and coordinate pairs (although they often mixed up the coordinates since the "patterns" weren't quite what they thought).  Later they would begin to believe that the circle only held those 16 angles and other angles just popped out of the calculator.

So, the question goes like this:

1. Draw a right triangle in Geogebra.  Use it to measure the lengths of all three sides and the angles.
1. Using the ideas from similar triangles, shrink or expand your triangle so that it will fit within the unit circle. [This means to divide each side by the hypotenuse, but we'll have gone over the idea of similar triangles and ratios before a quiz.]
2. This triangle can be flipped and flopped to fit into the unit circle in 8 different ways. Draw a picture with all 8 triangle orientations. [It looks like a double-winged butterfly with all right angles against the x-axis and hypotenuses radiating (pun intended) from the origin.]  
3. Find all associated 8 angles and their coordinate pairs based on your triangle.

For example:

This would correspond to the angles:

1. 40º (0.766, 0.643)
2. 50º (0.643, 0.766)
3. 130º (-0.643, 0.766) [Got this from 180º - 50º]
4. 140º (-0.766, 0.643)
5. 220º (-0.766, -0.643) [180º + 40º]
6. 230º (-0.643, -0.766)
7. 310º (0.643, -0.766) [360º - 50º]
8. 320º (0.766, -0.643)

I think this question really helps students see the utility of the unit circle.  And how we could use it to figure out any of the 6 trig functions for any angle if we were precise enough with a compass, protractor, and a ruler.

My SBG Letter

This may or may not be of any interest to you, but this is what I hand out to my students along with their syllabus to discuss my version of Standards Based Grading.

Standards Based Assessment

Philosophy

            This class will be graded on a system known as “standards based assessment.”  Grades will be assigned for content knowledge based on understanding of the main standards needed in the course.  Thus, the grade for the class will accurately reflect the student’s understanding of the material from the class.

            This system has many positives for the student, teacher, and the student’s family.  Grades will be posted based on the concepts.  Seeing a 74% on the Chapter 4 test in the gradebook is not informative to any of the participating parties.  The student may have a wonderful grasp of three quarters of the material while missing only a single key concept or he could barely have a working knowledge of the entire chapter’s material.  On the other hand, an 80% on Trig Graphs is much more informative to all the stakeholders to show where the student may need to improve his understanding.

            The main goal of the class is to teach students the mathematical material for the course.  Thus, a student’s grade should reflect his knowledge and understanding of that material without being clouded by completion or participation grades, extra credit for material not related to concepts from class, or behavioral issues.  Any behavioral issues, nonparticipation, or incomplete homework will certainly be addressed, but will not be directly reflected in the grade for the class.

            Since mathematics often requires a good understanding of a topic before being able to work well with the following topic and because I wish for all students to understand all of the topics from class, I will give students the opportunity to retake assessments over concepts covered in class.  No graded assignment will be dismissed since every assignment is given for a purpose.  Retaking an assessment can help students more accurately show their understanding of the material and helps students to learn the material before getting too far ahead in the curriculum.  If a student has a bad day or does not fully understand the topic on the day of the quiz, he will have the opportunity to show his improved understanding at a later time.  I care less about when the material is fully grasped than the idea that it actually is understood at some point.  That being said, grades and understanding of concepts can be time sensitive, so students will need to complete all retakes in a timely manner.

            The new grade will completely replace the original grade.  Thus, students should never give up on the class, no matter how low his grade gets.  As long as he is keeping up with the material in class and working on reassessing the material for which his understanding has improved, an F in the class can quickly become an A as zeroes are replaced with better scores.  The grade is constantly in flux and should be viewed as such.  Until the final grading period, parents should concentrate on the individual grades to see what their child should work to improve rather than the overall grade for the course.

Policies

• Students may retake only ONE section per day.
• Students must retake the entire section (even if it is more than one question)
• Students may retake each section two times (in addition to the original)
• To have the retake ready for the following day, students must sign up before 8pm (according to Google’s clock).
• Students should be able to show proof of work done to improve understanding since the original assessment to merit a retake
• Students may take it in during encore or before or after school.
• The new grade replaces the old grade entirely (whether it’s better or worse) to reflect the current understanding.
• Retake questions will assess the same concept, but may be drastically different in format or more difficult than the original.  Students should be prepared to demonstrate mastery of the topic, not just hope for an easier question.
• Students must take all the retakes for this semester before the date to be announced.

Procedure

The procedure for signing up for retakes will be done online.  Students should look at the teacher’s website for detailed instructions.

Our Little Baby All Grows Up

I lived for 11 years in Music City, Nashville, Tennessee.  I found myself putting music in a bunch of holes I found in my life (after breaking up a 5 year relationship, after leaving school, etc.).  Being there, when I was, I found a number of outlets to consume local, indie music.  In the early to mid 2000s, enjoying indie music was even cool, so there was the weird thing of being part of a popular movement whose whole point was to enjoy things that were not mainstream.

I feel like this is where we are with our online community of math teachers, affectionately known as the mathtwitterblogosphere (or MTBos).  We're the rising indie band who is a couple steps away from having Sony call to offer us a contract.  It's a little bit of a scary place.

Many of the things that define this community are directly related to the smallish size and "grass-roots" popularity that we enjoy.  Will we "sell out" if our little group grows too big?

On the one hand, we want our ideas and the group itself to be open to anyone and everyone who feels moved to join.  After all, what we do is for the kids.

On the other hand, it feels like we will lose a lot of the things I love about the group if it grows too big.

I was trying to tell my wife about my confliction with my vision of the future of this group.  Sarah asked if I wanted TMC to just be a little get together of my friends.  It hurt when she said it, because it sounds bad and exclusive, but I'll admit that I selfishly kind of do want that.   I have found some people that I would consider friends for life through this community and I am continuing to find more here and there, so I guess I shouldn't be worried about that part.  Whatever happens to the community at large, I can hope that people I really want/need to interact with will still be there for/with me.

Sam posted about the close-knit community we have.  How can we maintain that while growing much larger?  Will we split into groups (#geomchat, #algchat, #statschat, Northeast, West, Central, early adopters, newcomers, etc.)?  What happens when we're bigger than physical space will allow us to get together during the summer?  Would corporate sponsors at meetings help or hurt?

I guess what I'm saying is that change is hard.

Phrases that Annoy Me

There are a couple phrases that students use that really annoy me in lessons.

"So, is it just...?"

and

"So, do you always...?"

The main reasons they annoy me is because I can tell students are just looking for an algorithm rather than learning the background reasoning.  They want a formula they can plug into.  They don't want to think about it.

I mean, I get it if they're looking for a generalization and if they truly understand what is happening and are putting the pieces together to make a formula, then I'm all for it. 

The worst thing about the phrases, though, is that they shut out learning.  I will try to turn the questions back around and ask them, "You tell me.  Is it always...?"  Then they get frustrated that I didn't answer their "simple question" and solidify their, "Well, I guess I'm just going to fail this quiz" mentality.  If I answer, even if I follow it up with, "Yes, but here's WHY it works (or doesn't)," they shut off their brains after I say, "Yes." 

Perks of SBG

This deserves a longer post, but this post made me think I should post now.

There are at least 2 times where Standards Based Grading saves my butt with no real work.

1. Talking to parents.  So much responsibility goes to the students for this type of grading.  They have no reason to fail.  Help is always available  There are opportunities out the Wazoo for them to succeed.  So parents who try to point a finger have to just turn back around and look at their own kid.

2. Final exams.  As a teacher I get so sick of students talking about how in their chem class they can turn in a blank exam and still get a B.  They take the last 2 weeks of school to just be bodies in seats and not work at anything.  With SBG, the kids have to prove their knowledge on this final assessment and their grade can jump up or die fast with their performance on it.  So, even my A+ students who might be able to get a decent grade with half the exam blank in a traditional system know that they have to still put in the effort to show me that they understand the material and deserve the grade they are shooting for.