Don't get me wrong, I like this book. It has some great explorative exercises that have appropriately challenged my students. For example, look at this exercise to help students derive the 30-60-90 triangle relationships. Take an equilateral triangle, its altitude, and the Pythagorean Theorem to find out the special relationships between the shorter leg, longer leg, and hypotenuse. Great.
Here's where I start to beat my head against the wall. The book uses diagrams that simply shouldn't be used, especially in the context of 30-60-90 triangles. Look closely...
That's right, the 30 degree angle is opposite the longer (drawn) leg for questions 1, 3, and 4. My students get bothered by this contradiction. I do too. I have no problem admitting this to them. I'm honest with them saying, "I know guys. It goes against everything we strive to do in here. I encourage you guys to attend to precision and check for reasonableness. Yet, I give you this. I'm sorry. It says at the top 'not drawn to scale', but they should be drawn to scale. Right guys?!"
I think this about sums it up. Students will come up and ask about the dimensions they've solved for and whether or not they're reasonable. I'm proud of my students for making sense of their answers and checking for reasonableness. I know something is a skew when my response to those students is,
"I never assume those things are drawn to scale."I feel rotten saying this to students. I feel like I've just provided them with a worthless and menial task. I've let them down. I feel dirty. Mr. Stadel's quality control group hasn't done their job. What message are we sending students? Do they think we're out to trick them? Do the directions read, "Find the mistakes?" They should. It's times like these that force me to (gladly) keep a closer eye on the content I provide my students with. Don't just throw some triangles at them with random angles and units. Make sure they're reasonable.
Have you ever felt this way? Have you ever been caught in this situation? What did you do? How do we avoid these situations again? How do we demand better quality content from publishers? How do we make sure we provide our students with content that matches the CCSS and Mathematical Practices? Maybe you're okay with these types of diagrams, so please explain why. I want to hear from you all on this.
nOt tO sCAlE,
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There are a few instances when I'll let the note "not drawn to scale" stand for a different reason, like when a person is drawn 8 times taller than her bike or a sketch of me being taller than you, but to use that note with 30-60-90 triangles is insane. Not cool to purposely deceive kids.
ReplyDeleteThanks, Andrew.
A sketch of you taller than me. I chuckled out loud with that one. Definitely now drawn to scale with that example. You're right, not cool to deceive kids.
DeleteI can see why you would be frustrated Andrew, there's a world of difference between 'not drawn to scale so you can't just measure it' and 'we don't care if the diagram makes sense'.
ReplyDeleteI find that some students learn NOT to trust diagrams, because they constantly see that they are not allowed to use what the diagram suggests. This of course backfires when they need to draw a diagram to help them with a problem!
Thanks Mr. S!
Exactly, I want to see my students avoid drawing (sketching) diagrams when solving a question.
DeleteYeah, that turns my stomach quite a bit.
ReplyDeleteI'll admit, I used to completely ignore drawing things to scale and it always drove my students crazy. I've since realized the error in my ways and am making an effort to be more accurate. And of course, students are less confused and more confident in the checking their answers now that I'm being more careful. In addition, I also have a square centimeter, square inch, square foot, and square meter hanging in my room to even better help with the visualization and estimation. Great post.
ReplyDeleteWe're in this together man! It's all about making the effort to give quality representations to our students, especially if we expect them to give it back to us. No double-standards, right? Thanks for the comment. First thing I'm doing after Spring Break is installing some examples (hopefully student-created) of square units and cubic units like you mentioned. That's a fantastic idea and wish I could go back in time to provide this to past Mr. Stadel's classroom and students.
DeleteEver since I've put the cubic units in my room, I've had a lot of students solve problems and say, "Woah, that's a lot." More proof that relevance and context are massively important in math classrooms. To most students, eating 2 cubic feet of ice cream sounds easy (and healthy). However, when they see it they quickly change their minds and jaws drop, and bets are made and students throw up. All in the name of math!
DeleteHere's brandtahedron's second post that didn't get published:
DeleteEver since I've put the cubic units in my room, I've had a lot of students solve problems and say, "Woah, that's a lot." More proof that relevance and context are massively important in math classrooms. To most students, eating 2 cubic feet of ice cream sounds easy (and healthy). However, when they see it they quickly change their minds and jaws drop, and bets are made and students throw up. All in the name of math!
I am one of the most anal-retentive people when it comes to things being drawn to scale. Maybe that comes from my engineering days when I had to draw cross-sections of semiconductors.
ReplyDeleteMy advice to anyone doing the same is to create drawings in Microsoft Excel where you can use to grid. I always do this for things like Pythagorean Theorem or similar figures. Nathan
I hear ya. Microsoft PowerPoint or Apple's Keynote can make you some pretty accurate drawings as well. That's just to name a couple.
DeleteI think that this is actually an opportunity in disguise. Math practice 3 requires students to critique the reasoning of others. Sometimes that can be challenging because they feel uncomfortable pointing out the errors in the teacher's work or other students' work. If you have students articulate what is wrong and how they would fix it, it will be great practice because in life there are mistakes everywhere from the cashier who can't calculate the correct percent off to the person trying to overcharge you for a product.
ReplyDeleteGreat points, especially the connection to MP 3. In this case, the students did articulate the error with the page. However, I think we need to tread lightly here and do this in moderation. If I were to move forward to next year. Phase 1: Help students be aware of errors that could exist. Phase 2: Create tasks (in moderation) that challenge students to find errors. Phase 3: Critique the reasoning of the task, example, teacher, diagram, etc. Does everything in this task make sense or follow logic? Explain.
DeleteThere needs to be a certain leve of confidence that both the teacher has in the content and the students have in the teacher to provide legitimate content. Students lose confidence in the teacher if the teacher is unaware of these errors. You're right, in life we need to beware of scams and people with subpar math skills. These errors are great opportunities to help students.
I think Michael Pershan's mathmistakes.org site will be a very useful tool in setting up errors without putting anyone on the spotlight.
DeleteTrue, you don't want students' do doubt the teacher but I also believe that creating an environment where everyone can make mistakes safely is as important as error checking.
Also, I would frame what happened differently than the way you are stating it. You may have been unaware that the error existed, but you noticed it once you saw it. That is at least one level above seeing the error and not understanding the content enough to know it is an error. That obviously still happens.
You are doing an exceptional job. If you go by the tried and true rule of "try to improve 10% each year", you are good to go until 2035 based on the work you do in one year.