Tonight a parent of one of my 6th graders e-mailed me about his math homework. I love it when parents contact me to discuss their student’s progress. Parental involvement has such a powerful effect on student success. In the process of crafting a response, I was forced to really grapple with what the student’s misunderstanding was. I think the result is interesting:
Alex* did both worksheets today, however his sense on how you’ve taught him to convert whole numbers into fractions appears to be different than what I know (keeping whole number as numerator and making the denominator 1). I think he must have been confusing whole number conversions with something else you were teaching but he was obstinate. Thus I think he had trouble with the generic rectangle calculations and I could not help him and we lost patience with each other.
Perhaps you can revisit whole number conversions with him as he would not believe me. I’ll have him do the three problems once he gets the concept correctly — or that you confirm that in this case I am smarter than a 6th grader….and if I am not, I am so sorry. It is probably better I write for a living.
*Not his real name
And then here is my response:
Don’t worry, you’re certainly smarter than a 6th grader. Certainly more wise, too, because 6th graders have a problem not realizing what they don’t know.
I haven’t taught students that they can write any whole number as a fraction over 1 (e.g., 4 = 4/1). As you realized, they sometimes don’t “get” why it works and end up getting more confused. Currently we’ve just been reasoning through multiplicatively via repeated addition: for example, if you had 4 times 2/3, that just means 2/3 + 2/3 + 2/3 + 2/3. To go further, drawing a picture of 2/3 four times and then counting the pieces to recognize you’d have 8/3. A lot of students are still struggling with the idea, and want to say the answer is 8/12 (because they multiply top and bottom by 4). But then you can present them with the question: Aren’t 2/3 and 8/12 equivalent fractions? This creates the dissonance for them to see that the answer should be more than what they started with, not the same. Put another way, multiplying by 4 changes the number of pieces you have (the numerator), but not the size of the pieces (the denominator).
He’s okay right where he is–a lot of students are still there, since we only just learned how to multiply fractions. We’ll be doing more work multiplying fractions by both fractions and whole numbers via the mixed number generic rectangle problems. I think with concepts like this, it’s important to foster both a continual emphasis on why the answer makes sense in addition to repeated drill, which gives students several times to make mistakes / correct them and notice the patterns themselves.
If he wants, we could do some problems together at lunch. I often have a few kids up for extra practice / more individual instruction.
What do you think the student’s struggle was with? What questions would you have asked him to help him resolve his misunderstanding?
The school year is off to a strong start, and for the first time since I entered the classroom two years ago, I feel like a real teacher. I understand how to present a persona that promotes an organized, safe classroom. No matter how much I may want to crack a joke or do something goofy, it behooves me to secure the authority necessary to make things run well. It’s that tricky blend of “warm and strict” (from Teach Like a Champion
) that I knew was ideal. A perfect cocktail that my first-year self had no idea how to brew.I also understand the importance of routines and procedures and how to teach them. Teach for America mentioned them, but I don’t feel like I really learned them. There’s so much going on in a classroom that experienced teachers forget all they’re doing. And to an untrained first year, these things are invisible
. It’s like discussing thick/thin contrast on the rounded letters of a font. Most people have, at best, an unconscious recognition of these things.
But knowing how important the start of a class is and making it priority number one at the beginning of a year. It’s the first thing you do with students every period. If you set the tone well at the beginning, they fill in the blanks and end where they expect themselves to be. People are predictors, constantly imagining how things should be. As mentioned in Steve Pavlina’s recent post, the essence of frustration is when our predictions don’t match reality. So teaching kids how things should begin is half the battle.
I’ve recognized a few things that work really well with my 6th graders:
- Having them line up outside, tell them a preview of what they will do when they come in, and making sure it’s purposeful and strictly timed does wonders for those first 5 minutes. In my class they have 4 things to do:
- Take out their HW and HW trackers (a weekly sheet that I stamp daily, with a rotating quote, and space for parent communication)
- Write down tonight’s HW in their planners (keep this short, because 6th graders are notoriously slow writers)
- Check their answers against the posted HW solutions OR Complete the Do Now OR set up a new entry in their ISN (Interactive Student Notebook)
- Review your work with your teammate, & see whether he or she is ready for class.
- If it’s a minute into class and I’m still waiting for students to take out HW trackers, I know it’s because they’re unfocused. We line back up outside, I reiterate that they weren’t meeting my expectations, and then I send them in with the assurance that they can do a better job. Some days I’ll use a timer and report their start-up time, challenging them to improve tomorrow. (Maybe I should be more consistent about this?)
- Students know the HW tracker is a big deal. They’re responsible for it for the whole week. (I print it on bright yellow paper, to tilt the odds in their favor!) If they lose the tracker, they lose credit for the HW. (Our math department’s policy is that late HW is not accepted, because the point is that it’s daily spaced practice.) The other piece is that I only grade HW on effort:
- 2 – Attempted Everything
- 1 – Partial Effort
- 0 – Little to No Effort
- I stamp every student’s tracker daily. By personally coming by and checking, I create a personal obligation. I want all of my students to do HW, I believe it’s valuable for them, and I’m disappointed when they didn’t do it. Telling them I know they can do better tomorrow and regretfully having to stamp that 0 kills me inside. But I’ve noticed that HW completion is higher than it’s ever been in my past two years.
- Finally, I have these chimes for the classroom. I hate countdowns; they feel so authoritarian. I’m not really a “clap twice if you can hear me” kind of guy; I think those sorts of calls to order often release too much energy. But the kids seem to like the soothing tone (I certainly do), and I can hit them quickly or slowly. I taught the class that my expectation is that by the time the third chime sounds, everyone is silent. I gave more space between strikes early on, to build success. Now I can hit the first two quickly and pause an instant for the third chime, until the room is silent. They always see themselves quieting just in time, and then I compliment them and move on quickly.
- As a side note, my first year I borrowed something from the 7th grade teacher. She would whisper “Good morning, class” And they would whisper back, “Math is life.” Then she would speak it loudly and they would echo in turn. I tried to do that with my classroom motto, “Strength in Numbers,” but I didn’t pull it off as well with my 8th graders.
So that’s the start of my class. Then we’re off into the day’s lesson. And each day keeps getting better!
Next post, I’ll talk about the management strategies I’ve found successful during class.