Nov 122015
 

I have been in Elementary school classrooms this semester observing my learners teach lessons. They are amazing, and the UTeach model of teacher education is one with which I am completely on board. My learners will have spent so much time in the classroom being observed and getting feedback that they will have no choice but to be amazing teachers. Add in the fact that my math teachers will only be taught to use interactive and engaging methods like the 5E model, and you have a home run.

BUT, as I have been in 3rd through 5th grade classrooms, I have noticed a very disturbing trend. Like this board I saw in a 5th grade classroom.

2015-11-02 12.20.48 (2)

Notice that the objective here was to “Reacquaint yourself” with the math terms by designing a city. OMG. Seriously. This was in the 5th freaking grade. No wonder geometry is such a difficult class to teach in HS, the learners are bored stiff and resentful the teacher is lecturing them on something they have spent time on already.

Next up, a 4th grade classroom. The terms I heard LEARNERS using today were; expression, equation, identity, and inverse.

No joke. 4th grade. The learners were using the terms correctly, and identifying the difference between an expression and equation and using inverses to construct identities while solving equations.

This was not a Gifted and Talented classroom, this was an at risk, high needs, pretty normal, typical classroom.

If I were to summarize what I have learned this semester as a teacher of teachers, it is this. High School teachers, we need to seriously up our game. We need to realize that the reason our learners look bored and apathetic is because we are rehashing what they already know.

We are NOT connecting to what they already know (even if we think we are.)

We are NOT challenging them to reach for deeper understanding (even if we think we are.)

And, we are NOT realizing the learners are entering our classrooms with a great deal of prior math experience and love. Connect with it. Pull it out. Create engagement.

My eyes are open, and it scares me to death what I have done in the past to my learners. The CCSS standards are working. The shifts in mathematics education are working. We must be leaders and take advantage of it.

Go spend time in elementary school classrooms. It will shock you what the learners are doing today. What are we doing?

Jun 022015
 

In Algebra 2, I start with my 3 rules. They really are not “my” rules they are just restatements of the multiplicative identity, additive identity and balancing equation. I believe how I use them to set the tone and stage for the entire year is different, however.

On the second day of class (the first day I usually do a problem solving activity, cell phones, or other type of activity) I introduce the “rules”.

2014-08-10 15.43.10 You can zoom in, or you can grab the files from here. I had a long discussion with several teachers about the wording, and Meg Craig made up the files once we settled on the phrasing. Can they be tweaked more? Absolutely. I would love to improve them. Leave the suggestions in the comments.

Back to how I use them, and why I think I use them differently. I will say with 100% assurance that I use these rules differently than I have seen others, and I absolutely use them differently than I used to.

This grew out of the frustration of having a nice, simple equation to solve like, y=5x+2, and needing to solve it for x. I really wondered why learners would mess up the math on such a simple equation so frequently. And don’t even get me started if the equation looked like v=ba-d, because that was impossible.

And I realized that although I was teaching the idea of inverses and identities, I was not connecting learners with or building the idea that these things are used.

So, I turned to the “SADMEP” idea. (this link is a google search for SADMEP. That is sad, huh.) But as I worked with this a year or two, I realized that the SA always made a zero, and the DM always made a 1, so I added that to my SADMEP poster (sadly, there are no pictures of this, I threw them all away several years ago).

Which lead me to the idea that it is NOT subtraction or addition that is important, it is the ZERO! Same thing with the ONE, those are the important ideas. Those are the identities. Why do we subtract 2 from both sides of the equation above? Because 2 – 2 = 0. No magic. We can actually subtract ANY number, but we chose to subtract 2 because that is the most convenient way to reach zero in one step.

And then I stumbled upon another magic word that goes hand in hand here; convenient. Why do we chose the values we chose to add, subtract, multiply, or divide? Because those values are convenient ways to make a zero or one the fastest.

Back to the ugly equation above: v=ba-d. Solve it for ‘a’. Add ‘d’. Why? because -d+d=0. Do we care what d is, or represents? No, we know how zero works. Same goes with multiplying by 1/b.

Then, show this video: http://www.youtube.com/watch?v=seUU2bZtfgM up to the point where he goes into transcendental numbers (approximately minute 6).

I reach for my physics and chemistry books right about now, and find some ugly equations. These in fact. This is the file I start with to get the learners thinking about solving.

And then I go nuts. Put the formula sheet for AP Physics under the elmo. Do the same with AP Chem. Pick one. Pick another. Solve for any variable. Then solve the same equation for a different variable. For every single function / formula, the only thing they can write to justify their steps are “blah because it makes a zero” and “bleep because it makes a one.”

And we discuss that every single problem I can possible give them is solved simply by using these three simple rules. I make a huge deal of this in the log unit because they learn a NEW way to make a 1 in that unit. That is exciting.

All year long, my learners are shouting out, “because it makes a one” when we are working with exponent procedures (note they are not exponent ‘rules’) because that is how math works. Why do square roots and squares “cancel?” Well, they don’t. 1/2 exponents raised to the power of 2 means 1/2 times 2 which equals 1.

That’s it. I use it all year long. I rarely write a radical symbol, just fractional exponents. It just makes sense.

This is a couple of days of work, and I really think pulling from physics and chem texts helps. I have never had such success with solving and literal equations (in fact, they stop thinking literals are any different) as I have had the last two years.

There is a reason they are framed so nicely at the front of my room. They matter, and they solve every problem we encounter.

 

Some of the CCSS standards this idea hits:

A.SSE.1-4

A.REI.1-4a

A.REI.5-7

 

Apr 012015
 

I always tell my speech and debate competitors that a good speech takes multiple drafts, and this speech is no different. After sleeping on it overnight, and re-reading it today I realized that my speech was fighting itself in the wording, so I rewrote some key sections.

I like this version much better.

I really did not expect to spend spring break doing political activity, but here I am anyway. I also was just asked if I would do an interview for another local news story. Wow, say yes to one thing and more activities pile on. At some point I need to put this aside and start reading for my classwork. I need to do that soon!

This is the text of the final version of the speech. It is better than the previous one, I believe.

For the record, my name is Glenn Waddell, Jr., and I am the department chair and teacher of AP Statistics and Algebra 2 Honors at North Valleys High School. Chair Woodbury and members of the committee, thank you for allowing me to address you today and explain why I oppose the sections of AB 303 that delete reference to the common core. I NEED the core standards to be an effective educator. Most importantly, my learners need the common core state standards.

I need the core standards because the prior standards had different “enhancements” in Washoe and Clark counties; which means that I could not collaborate with teachers in the southern part of the state, let alone elsewhere. Today, I work with teachers in other states as much as I collaborate within my building. The internet facilitates connections with math teachers, the sharing of lessons, and pooling of resources with teachers in Oklahoma and New York as easily as teacher across the hall.

My needs pale when compared to the needs of my learners, however. My learners need the common core for two reasons. First, high standards create engagement. The current standards provide this through the shifts, the practices, and the standards themselves. An example of how much can be accomplished with the standards is two weeks ago, my learners were working on the A.REI group; solving systems of equations algebraically & graphically. My learners had a graph of two functions with solutions that were easy to find one-way and impossible to find other ways. They worked for over 30 minutes individually and in groups before they finally gave up and asked me for help. The understanding we found was; there was no algebraic way to find the solution, and they refused to believe it. The mathematical practices served my learners well. They showed perseverance, appropriate use of tools, making arguments, regularity of structure, and critiquing the reasoning of others. This is the heart and soul of a successful math classroom. My learners need and deserve this high level of rigor.

Secondly, my learners need the standards because they are working. All learners need a solid foundation beginning in elementary school upon which to build future mathematics content, and math teachers in my school agree the learners coming up from middle school are better prepared for high school algebra. The standards are not the maximum, they are the minimum body of knowledge that learners must know. The standards create a foundation that is stronger, substantive, and more demanding than we had in the past. My learners need the core standards so they can build their foundation, and launch themselves to higher mathematics with confidence. My learners do not come into my room to be average, they come into my room to be awesome, and the core standards allow and encourage them to be awesome.

Thank you.

 

Mar 312015
 

Tomorrow I am speaking to the NV Legislature on the Assembly Bill 303 (pdf text) that would eliminate the end of course exams that I don’t like, but would also eliminate the Common Core State Standards from all NV schools.

Can I complain for a second on how difficult it is to give a 3 minute speech? OMG! My first draft was around 8 minutes long, and I finally have it down to 3 minutes on the dot. Below is the text of my speech. If you have any suggestions, I am open to tweaking or rewriting. I leave tomorrow at 2 pm for Carson City!

There may also be an opportunity to be on a local PBS channel show about this bill as well. Who would have guessed that I would have spent this year’s spring break in political advocacy? Not this guy, that is for sure.

For the record, my name is Glenn Waddell, Jr., and I am the department chair and teacher of AP Statistics and Algebra 2 Honors at North Valleys High School. Chair Woodbury and members of the committee, thank you for allowing me to address you today and explain why I oppose the sections of AB 303 that delete reference to the common core. I NEED the core standards to be an effective educator. Most importantly, my learners need the common core state standards.

I need the core standards because the prior standards  had different “enhancements” in Washoe and Clark counties; which means that I could not even collaborate with teachers in the southern part of the state, let alone elsewhere. Today, I work with teachers in other states as much as I collaborate within my building. The internet allows me to connect with math teachers from across the United States and share lessons and pool resources with teachers in Oklahoma and New York as easily as I can with the teacher across the hall.

My needs pale when compared to the needs of my learners, however. My learners need the common core for two reasons. First, my learners need a solid foundation beginning in elementary school upon which to build future mathematics content. The current standards provide this through the shifts, the mathematical practices, and the standards themselves. An example of how much can be accomplished with the standards is two weeks ago, my learners were working on the A.REI group; solving systems of equations algebraically & graphically. My learners had a graph of two functions with solutions that were easy to find one-way and impossible to find other ways. They persevered for over 30 minutes individually and in groups before they finally gave up and asked me for help. The understanding we found was; there was no algebraic way to find the solution, and they refused to believe it. The mathematical practices served my learners well. They showed perseverance, appropriate use of tools as well as making arguments, regularity of structure and critiquing the reasoning of others. This is the heart and soul of a successful math classroom. My learners need and deserve this high level of rigor.

The second reason my learners’ need the standards are because the core standards are not the maximum, they are the minimum body of knowledge that learners must know. The core standards raised the bar tremendously from prior standards, and in so doing created a foundation that is stronger, substantive, and more demanding than we had in the past. My learners need the core standards so they can build their foundation, and upon this foundation launch themselves to higher mathematics with confidence. My learners do not come into my room to be average, they come into my room to be awesome, and the core standards allow and encourage them to be awesome.

Thank you.

Any suggestions? Comments?

Mar 072014
 

My goals:

  1. Construct a consistent vocabulary of problems that can begin in Algebra 1 and extend through to Calculus, Statistics, and all courses in between.
  2. The problems must have the potential to be engaging to learners.
  3. The problems must hit at least 4 of the eight Mathematical Practices & high school math standards (CCSS).

My idea started with this idea for Algebra 2: Model the escape velocity of a rocket on the Moon and the Earth. ( PDF and Word DOCX) This ended up being a far more difficult task than I expected, mainly because the learners did not connect the idea of writing the equation of a line with the fact we had a function in front of us.

I Desmosed the project for a visual display, and we spent another day discussing it and achieved the goal. [Is it okay to use the name as a verb? I don’t care, I am doing it anyway.]  It turned out great in the end, but it made me start thinking hard about how to connect Algebra 1 through Calc and Stats and make the ideas more real, more understandable, and more connected.

From there came the idea of using an “off the shelf” structure in a new or different manner to extend the lessons. Enter http://graphingstories.com . Dan Meyer started the Graphing Stories with a long time ago, and they are awesome. But they also fit the idea of using the video / graph combination to write the equations of lines and finding area under the curves.

With that in mind, I offer the following Desmos files:

File 1:

  1. This uses the Graphing Story of water being poured into a graduated cylinder to create the graph. I took some points from the graph on screen, and wrote a function that goes through the point (0, 0) because we know it was empty at time 0.
  2. Notice that the line does not go through exactly all 4 points! That allows for discussion of variability and observation skills.
  3. I also used the (h, k) form to write the function f(x) because it is the easiest way to show the line.
  4. What does the slope MEAN?  A standard AP Statistics interpretation is: As the time increases by 1 second, the water increases by 40.67ml.
  5. Next, find the area under the curve. Move the slider for “b” to the right and you see the area highlighted.  Okay, standard triangle, ½ b*h, and you get 5205.33 ml*sec. ??? What does that even mean?
    1. It is called “absement” and it is the time-integral of displacement. Yes, we don’t need to discuss that for Algebra 1, but as teachers we should know it.
    2. The area is the sum of all the instantaneous moments of water before. With the Desmosed file, you can see and clearly communicate what it means. It means that you are adding up the area of the little triangle when b=1 with the larger triangle when b=1.5, and then with b=2, etc. Except the area is the sum of the instantaneous areas, not the discrete areas.

Notice that this one lesson required the learner to interpret a real life action, pouring water, into a graph, and then find the slope and write the equation of a line, and then interpret the slope, and then find the area under the curve.

These are all essential skills of the Calculus learner, done at the Algebra 1 level!

A second one.

  1. Now we are removing cups from a scale.  There are actually several questions that the video brought to my mind, like is this really a continuous line, or should it be more discrete? Time is continuous, but the weights really are stepped.  But, I left it as is though because I wanted to not change it from what the video shows. That is a larger conversation in class.
  2. We now have a negative slope to calculate, which does not really make a huge difference for interpreting the slope: As the time increases by 1 second, the weight of the cups decreases by 3 grams.
  3. The fact the line only hits 1 point absolutely creates some conversation about which point to pick, variability, ect.
  4. The area gets fun, however.
  5. Notice that the FULL area is still a triangle. However, if you move the “b” slider across, you notice the partial areas, the area at 5 seconds, 8 seconds, etc, are trapezoids! Now the learner can be challenged and pushed to incorporate some extra questions of find the area of trapezoids.
  6. We still are doing and absement calculation and not a displacement calculation.

Finally, the Desmosed Lunar Modeling I started with:

It is far more complex and involved, but that is why it is an Algebra 2 lesson and not an Algebra 1 lesson.

Feb 242014
 

I am trying to come up with ways to connect the ideas of Statistics and Calculus down through the curriculum in all levels, even into Algebra 1. I think I have a way to do it that makes partial sense and can be done with reasonable effectiveness, but I have an interpretation problem.  If anyone can see a way out of it at the end, please let me know in the comments.

First off, I started with http://graphingstories.com and selected the water volume exercise. My reasons are focused and simple. 1. It is linear, so it works well in the algebra 1 course during the first semester. 2. Finding the slope of the line is straightforward, and requires some transfer of skills, but not a lot of transfer. 3. The area part of it is just a triangle, so the math is not complex.

I am trying to build from simple through the more complex, so starting off simple is helping me wrap my brain around it.

So, I start with this video:

watervolume

You end up with a graph that looks like this (click to enlarge):

watervolumepic

Great, now for some math. Find the rate of change of the line: Best guess: (610ml – 0ml)/(15 sec-0sec) = 40.67 ml/sec.

What does that mean? As the seconds increase by 1, the water volume increases by 40.67 ml.

The equation of the line is milliliters= 40.67 ml/sec * seconds + 0.

So far so good. The interpretation of everything so far makes complete sense. It is attainable for algebra 1 or geometry, and has meaning in context of the video.

So I want to inject a little geometry into the problem and I find the area under the curve. No big deal, it is a triangle, so it is 1/2*610 ml * 15sec = 4575 ml*sec.

What does that even mean? Does it have a meaning? Am I doing something that should not be done? What does a value of 4575 ml*sec even mean? It is not the total volume, that is 610 ml. It is not the sum from 1 to 610, that is 186,355 according to wolframalpha.

In calculus, we find the areas under curves all the time, but they are specific types of curves.  We find the area and it is the displacement, or it is the total distance. But does this area have a context or meaning?

I really would like to figure out a way to make sense of this, but if there is not a way, then I will have to go back to the drawing board.

Edit: The word for what I am looking for is called absement. Links to resources are:
http://nqtpi.blogspot.com/2012/09/the-area-under-distance-time-graph.html
http://thespectrumofriemannium.wordpress.com/tag/absement/
http://wearcam.org/absement/examples.htm
http://forums.xkcd.com/viewtopic.php?f=18&t=34744 [read all the way to the bottom]

And a pic that makes the idea of absement more clear: http://ow.ly/i/4HEwh

Jan 172014
 

I am soon to be embarking on a new direction / undertaking in my professional career, and in doing so will need to seriously delve into the realm of research and resources. Starting this summer, I am beginning a Ph.D. in Mathematics Education at the University of Reno, NV in the Math, Science, Technology and Society area of emphasis.

As a high school math teacher I need to be an expert in CCSSM. As a Ph.D. student I MUST, without question be an expert in CCSSM. To that end and because I have always been interested in research, I am going to compile lots of CCSSM resources and create a page here to house the ongoing collection.

The page can be found here:  http://blog.mrwaddell.net/ccssm-resources or at the top of the page.

 Posted by at 12:03 pm  Tagged with:
Jul 032013
 

This is a two part lesson, an experiment and simulation that together meets the CCSSM S.IC.B.4-6 and explains AP Stats concepts as well:

clip_image001

This exercise is one we did at the SilverState AP Institute with Josh Tabor, but the nice this is that if you do the first part it fits in Alg 2 nicely, but you can extend it to AP Stats as well.

It is LONG, with lots of pictures to explain how to use the applet so I will put the post after a break:

Continue reading »

May 232013
 

Tonight we had a discussion of “5 Practices for Orchestrating Productive Mathematics Discussions” (B&N, Amazon) on Twitter, and I wanted to post the Storify here. One caveat, one of the members, MrHodotNet is protected so he didn’t show up in the timeline. I copied and pasted his contributions at the end, but they are not in order.

 

The link to the Storify.com transcript is here.

The transcript can be read after the break.

Continue reading »

May 152013
 

The one problem with having a summer list is that I always want to add to it. I try not to, but things come up that I cannot say no to.

This is one of those things. It is a course taught by Stanford University professor Jo Boaler. If you don’t know about her, she has gone to bat for math teachers and taken some professional licks for it, and she is the author of the fantastic book “What’s math go to do with it.” (Amazon, B&N). I can not say how much I liked this book, and then to have the opportunity to take a class from her for free; well, that is too good to pass up.

The class is: EDUC115N: How to Learn Math.

Many of my Twitter PLC has already signed up for it, and I just sent the link out to my department and hopefully many of them will sign up too.

I hope many people will sign up for it so we can have some great conversations in the class.

Here are the topics the course will cover, which makes it even more inviting and exciting! See you there!

CONCEPTS

1. Knocking down the myths about math.
Math is not about speed, memorization or learning lots of rules. There is no such thing as “math people” and non-math people. Girls are equally capable of the highest achievement. This session will include interviews with students.

2. Math and Mindset.
Participants will be encouraged to develop a growth mindset, they will see evidence of how mindset changes students’ learning trajectories, and learn how it can be developed.

3. Teaching Math for a Growth Mindset.
This session will give strategies to teachers and parents for helping students develop a growth mindset and will include an interview with Carol Dweck.

4. Mistakes, Challenges & Persistence.
What is math persistence? Why are mistakes so important? How is math linked to creativity? This session will focus on the importance of mistakes, struggles and persistence.

5. Conceptual Learning. Part I. Number Sense.
Math is a conceptual subject– we will see evidence of the importance of conceptual thinking and participants will be given number problems that can be solved in many ways and represented visually.

6. Conceptual Learning. Part II. Connections, Representations, Questions.
In this session we will look at and solve math problems at many different grade levels and see the difference in approaching them procedurally and conceptually. Interviews with successful users of math in different, interesting jobs (film maker, inventor of self-driving cars etc) will show the importance of conceptual math.

7. Appreciating Algebra.
Participants will be asked to engage in problems illustrating the beautiful simplicity of a subject with which they may have had terrible experiences.

8. Going From This Course to a New Mathematical Future.
This session will review where you are, what you can do and the strategies you can use to be really successful.