Number sense

One of the things that I notice about my students is that they often lack number sense, by which I mean an intuition for numbers and arithmetic operations.

For example, an Algebra 1 student came by after school today to ask for some help. (That in itself was great and made me happy.) At one point, this student was struggling with how to find the solution to “-8+y=2”. We tried lots of different approaches: using the blank-out method where you replace y with a box, using the number line, etc. The student was throwing out guesses, but she wasn’t guessing wildly even though her guesses were way off–she really was trying to guess the correct answer. Here’s what I mean.

The picture above shows all of the guesses that she made. She would give a guess for “y” then I would write out what happens when we add her guess to -8 so we could see if it equaled 2 or not. I purposely wrote all of the guesses in numerical order, not in the order that she guessed. That allowed her to clearly see all the guesses she had made and I was hoping the visual pattern of the numbers would help her see the answer. Two things surprised me: (1) that it took her so many guesses before she got the right answer, and (2) the guesses that she produced were seemingly random: she would guess “y=5” then “y=-8” then “y=-2″ then y=0”. However she didn’t seem to be guessing wildly because she was trying quite hard to get it right and it was clear to her that I wasn’t going to just give her the answer (which is what students are trying to coax out of you when they are guessing wildly).

Certainly we could have just jumped to the algebraic method of adding “+8” to both sides of the equation to solve for y, but it seemed a much more valuable thing to help her develop more number sense, in particular, the intuition that negative numbers are less than positive numbers and that adding a lesser number means you’ll get a lesser answer.

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Today was a good day overall. Went for a walk outside with Geometry class (because they were starting to get ornery) and I happened to see some nice shadows cast on the ground by some trees so I snapped some photos and when we got back to class I pointed out the nice similar triangles in the photo (foreshadowing of what we will do in class next week).

Most of my Algebra 1 students are beginning to show mastery of some key skills and concepts. I feel like we are making slow, steady progress.

New students in classes

Two new students got added to my classes today. One in Geometry, one in Algebra 1. Part of me starts to feel inconvenienced by having to get these students up to speed, but then I just imagine what it must feel like for them.

One of the two students told me she hates math and is only back at school because a judge ordered her to go back to school.

In other news, it seems that students are becoming more used to my seat arrangement system. I have an Excel spreadsheet that randomly generates seating charts and we use that to shuffle seats every three weeks. I explain that not everyone is going to be happy with the seating arrangement but that we do this because I think it’s important for students to learn how to work well with each other. The level of complaining that I get from students seems to be decreasing. Today we only had about one minute of complaining from my Algebra 1 class. I’m learning to be content with micro bits of progress.

Student in wrong class

One of my students told me yesterday that she thinks she is in the wrong class. She’s a ninth grader, says she passed Algebra 1 but she’s in my Algebra 1 class now. I’m sad that she managed to slip through the cracks until this point. I vaguely remember her telling me that she already passed Algebra 1 and I told her to speak to the school counselor, but that was months ago and I forgot to follow up on that comment.

I’ve arranged an appointment between this student and the school counselor tomorrow morning. If she is able to switch to Geometry, I will require her to stay after school for extra help until she catches up with the class. Will post an update later on what happens.

UPDATE: Either the student was genuinely confused or she was lying. The counselor looked up her grades and she failed Algebra 1 last year in 8th grade. <sigh>

Back to school

Today was the first day back to school after the winter break. I was anxious about going back to school over the last week and it got to the point where my digestive system was off and I was experiencing weird back pain. Anxiety before the first day of school is pretty normal for me, but usually it doesn’t bother me so much because I know that I’ll get so happy to see my students that I’ll forget I’m back at work. Perhaps the anxiety this past week was due to the absence of this mitigating effect. But still, after walking on campus today it was nice to see some of my high school students; others… I can only pray for a bigger heart.

But I didn’t let that anxiety show. Teaching is part performance anyway. Since students have been away from school for three weeks, I started each period by reviewing some classroom rules and those myths about mathematics.

Today, I was again reminded by how students’ motivation is strongly linked to their beliefs about whether they will be successful at the task set before them. Since the tasks today were relatively simple tasks to help students review what they’ve learned, all of my students got to work with no complaints.

I saw another instance of this principle while helping a student with Algebra 1 after school. This student is not one of my students and was in a heightened emotional state today. Her worked involved graphing linear equations. She had learned an algorithm for graphing lines, but it was not clear that she really understood what she was doing. For example, to graph the line y=5x-2, she knew that she had to “start at -2” then “go up 5 and over 1” but she didn’t know whether the y-intercept of -2 meant to start at (0,-2) or (-2,0). I’ve worked with this student enough to know that it is better to let her try, give lots of encouragement and choose judiciously when to gently offer corrections. She was motivated to graph lines, perhaps because she knew the algorithm enough to be successful at the task. Once I started to probe more about whether (0,-2) or (-2,0) fit the equation, she lost the motivation to work. With this student, I’ve learned that it will take time for her to develop enough confidence to move out of her comfort zone.