This is going to be a super quick post and I would LOVE some feedback here and/or over on the twitter (where you can find me @mrdardy)
I am one of three teachers of Geometry at my small school and I am also the chair of our department. I feel that the three of us ought to have pretty similar policies to make life feel a little more unified and fair to our students. I have opted in the past to keep most of Geometry calculator free. I feel that this is one of the last opportunities to try and help firm up some number sense and some self reliance with minor calculations. I also encourage my kiddos to leave answers like the square root of 160 as a final answer or even something like ‘the sum of the first 98 natural numbers would be the number of handshakes’ to help offset any anxiety about calculations beating my students down. I also place very little emphasis, pointwise, on arithmetic mistakes. One of my colleagues pretty vigorously disagrees and feels that having a calculator by their side eases pressure, and is simply a more realistic way to approach life for her students. I find myself questioning my decision here since I do not restrict calculator usage in general in my other classes. I do, however, worry a bit about all of this since I am hearing pretty consistently from recent alums that they head off to college and are not permitted to use calculators in their freshman classes. My recent Calculus students report this very consistently, they take freshman Calculus at college without a calculator. I know that there are all sorts of reasonable arguments that we should not make high school decisions based on college realities. I also know that I am hearing back from a small group of students.
So, I guess all of this rambling is really about one thing – Give me advice! Let me know how you approach this question. Does it depend on the level of the class? Is it a departmental decision? A school or district policy? Am I simply holding on to some quaint idea that mental arithmetic really matters? I fear that I am not being coherent or consistent in how I think about this issue. HELP!
7 thoughts on “Questioning Myself”
I try to let students choose when possible. When to use tech is probably different for each student. What I want is for them to be able to evaluate if it’s helping or not.
Grading, I don’t pay much attention to computation errors. My transition to SBG went through just grading each problem with an A, A/B, B, etc… based on understanding shown. Correct answer, no thinking was usually a B. Good math with some mistakes could easily show A understanding.
Just as students choices can be different, so can teachers’. If you trust your colleagues to be fair (and they can explain how they are), maybe they need to be able to choose, too.
Thanks for the measured response. I worry that life already feels so arbitrary and unfair to some of our students that any perceived disadvantage due to different policies would be detrimental to our department and our school. Perhaps, I need to be more trusting that people can roll with changes and as long as individual class policies are well explained, then the need to compare and look for consistency is greatly diminished.
How does allowing your students to use a calculator help or hinder their ability to master important Geometry concepts? For some students, knowing the calculator is an option may relieve some anxiety, and then they’ll find out they don’t really need it anyway. If building number sense is a goal, start every class with a Number Talk to give kids the opportunity to hear others. Even students who are good a computation will benefit. Too often, students and teachers rely on calculators as an answer-getting tool, rather than a learning/understanding tool, which is why I think there is sometimes the urge to ban them. Used appropriately, calculators (and dynamic geometry software) can really help students learn more mathematics and at a deeper level of understanding. The reality is that students can use calculators in most high stakes testing situations, so there is also a need to help them develop a sense of when a calculator is helpful and when it’s not, as well as how to use it efficiently (SMP 5).
I do think that consistency within a grade level or course team is important. Without it, we create equity issues for students.
P.S. Thank you for not making students simplify radicals.
I do not believe that it helps or hinders on Geometry concepts, what I do find is that it is often a crutch on figuring out ideas and a hindrance in showing their thought process on paper. I can fight that by being super clear about what I expect in terms of work shown to support answers. I have chosen to fight it by eliminating that tool and, as I tried to express above, I am not at all confident in this decision.
Thanks for chiming in and helping me to clarify my muddy thoughts.
I vote calculator yes, number talks also yes. Encourage them to try each problem without a calculator then use a calculator to check, but don’t stress it much. Allow students to demonstrate their no-calculator methods in class and have a “Calculator Free!” sticker for those who can go for it if you want. But allow calculators and, more, assign problems where a calculator is helpful (so many realistic geometry modeling problems need calculators and rounding to be useful).
I skipped simplifying radicals one year and now I do it again (though I tell them it is only required in a final answer if simplest radical form is explicitly requested). Too many cases where students didn’t realize 6/sqrt(2) was a congruent side to 3sqrt(2) in a large complex problem.
David – interesting idea for the calculator free choice. Knowing my students, perhaps tossing them a bone in the form of a bonus point or something like that might be an interesting experiment although I tend to shy away from any bonus conversations. This might be an interesting experiment to try. I tend toward questions where the calculator might be a meaningless tool. For example on my last test (the first of the year in Geom) I asked about midpoint, slope, and distance for a line passing through the points (x, 6) and (x+4, -6) I feel that a calculator would make almost no difference to any student here (unless the difference between 6 and -6 is an issue) These kinds of questions make me feel, at least marginally, better about not having calculators at their desks.
With the example of 6/sqrt(2) compared to 3 sqrt(2), what other context clues could have steered them to realizing that they were congruent? I wonder how many of our precalc students would quickly recognize this equivalence.
Thanks for jumping in on the comment thread!
I dont think anybody, including me, would quickly notice their equivalence. Which is why I’ve gone back to having them essentially always rationalize denominators now. There is value in having a single standard representation for a quantity, mostly in finding equivalence, which matters quite a bit in geometry.