Tag: math teaching

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Challenges

It’s May.

There’s a month left of school. We just went through a round of testing. And we’re all a little fried.

And so, I was sitting in my office a few classes ago, with a topic (functions),  no particular lesson plan inspiration, and a textbook with a nice series of activities that checked all the boxes (mixed question/task types, at the right level, moving smoothly up Blooms…) I’ll be honest,  “Class, please complete the exercises on pages __ to ___” was a tempting plan.

Except

It’s May. The students are fried, they’ve worked hard all year and focusing on a textbook for a big chunk of class now was just not going to work.  They’re good, they’d try, but who could blame them for losing interest?

But was a nice series of activities, and I’m fried enough myself that I didn’t want to waste a perfectly good set of learning materials.

 

Solution: Envelopes.  Scissors. Some re-framing.

 

I made copies and started cutting and sorting. Those vocab words? A tiny matching activity, in an envelope. A big #1 on the front.  That exercise identifying functions? Cut apart and stuck in an envelope, it’s a card sort. With a big #2 on the front. That other exercise? I just stuck it in an envelope. Add a graph, ask for a sentence, envelope, envelope.

With a few additions and some cutting and editing, I soon had a few copies each of six enveloped tasks. I called them challenges and handed out the envelopes with that big #1 on the front. I told my students when they had completed the first challenge, they could exchange it for #2.

And it worked.

The mystery of the envelopes, the challenge, the sense of accomplishment and progress as they exchanged one envelope for the next added enough interest that they worked hard and stayed engaged the whole class.

In May.

When we’re all fried.   (Did I mention that fact?)

 

I love that they worked hard and learned.

But, I also love that I got a new tool in my teacher-kit.

Cut it up, put it in envelopes, set up a challenge – that’s a transferable skill-set for the next time I’m contemplating “please complete the exercises on page…”

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Lesson Sketch: Collective Notes

Problem: Students passively copy down whatever I write on the board as notes, and/or flail about and miss (or mis-organize) key information.

Topic: Any.

But, use judiciously. I save it for days when I need to deliver some specific, important content (For my class, for example, the steps to solve 2-step equations)

I usually try to keep lecture-y time to a minimum, and this makes the lecture more active, but also really sage-on-the-stage central and takes a bigger chunk of classtime.  In general, I like it best as a second class, after some more exploratory activity to introduce the concept.

 

Process:

  1. If your students struggle as much as mine, you might give them a lightly structured guide for their note taking. And/or talk about good note taking: types of information, purposes, ways of organizing etc.

Here is a generic note-taking organizer (as google doc or PDF)

  1. Teacher presentation (model solving some type of problem, explain some piece of content etc.) Be extra clear/slow/thorough – but resist the urge to write note-like things on the board.
  2. Stop periodically, ask students to look at their notes and share one ‘note’ with a neighbor (think/pair/share)
  3. Repeat until content is covered.

 

  1. Then, in groups, students review their notes and choose a few points they want to contribute to the collective.
  2. Groups share their selected points while teacher records, probing for information to fill in gaps or misunderstandings, until the class has a complete set of notes on the topic (I’m grateful for a computer/projector set up, but I’m confident you could make this work with whatever tech you have)
  3. Make a copy of the notes for each student and distribute.

 

Why:

– By my count, students will have run each piece of content through their brains approximately 11 times using five different modes (no really, I counted, I think it’s 11 if they’re actively participating: listen and write; then read, choose, speak, listen during the pair share; then read, choose, speak, listen and read again making the collective) That’s a lot of processing for an activity that looks a lot like a lecture.

– Note taking is a skill. It’s one my students need, but we don’t have a lot of time to teach.  This way, they get some support from the organizer, they get to see the best ideas of their peers, and watch the organization of the teacher transcribing it.  They’ve also had a couple rounds of reviewing their work and identifying the best items – I hope this helps them produce more of these kind of items when they’re on their own for note taking.

– Also, note taking, they’re not good at it on their own (yet!). This way I knew everyone got a solid set of notes to use as a reference.

– The group dynamics. The least visible, maybe the most important. They’re collaborating to create the thing together, as a group, where everyone contributes.   And, when they tell me what to write, they own it. That final, printed official looking reference on two step equations, they made that that, not me. It’s a nice reminder of their knowledge and capacity.

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Constant Cumulative Review: Pt 4 Implementation Considerations

Read: Part 1: Start with Why , Part 2: Making Materials without Going Insane, Part 3: Putting it into Practice and/or Classroom Routines: You will metacognate for background

Consideration 1: Why I love it…

  • Having a routine buffers the chaos at the start of class, and buys me a few minutes to deal with whatever needs dealing with that day (a student back from an absence, a message from the counselor, laying out materials for an activity) while students are getting to the work of learning. If it were not this, I am now sold on some consistent ‘do now’ activity.
  • Retention – My students’ learning never gets more than a week or two from the top of their minds, and this really does seem to help them retain what they’ve learned. (yay!) This has the side benefit of dramatically reducing frustration, and boosting their sense of competence as math learners. (double and triple yay!)
  • It makes me feel better when a student doesn’t totally grasp something in class. We’re not moving on and leaving them in the dust, they’ll see and try it (and get to ask questions about it) again and again and again until they get it.
  • The metacognitive elements, but especially students taking ownership – correcting, commenting, choosing independent work
  • All the good brain science (more on that below)

 

Consideration 2: But, the time!

I love this system, but, nothing is without trade-off’s. And the big trade-off here is that I devote a lot of class time to this.

The whole routine takes about 30 minutes (out of a two hour class) It could be less, but a) the hazards of teaching adults with jobs/kids/broken down cars  include someone straggling in late b) I time it by watching my slowest students  c) we’re doing a lot during this time.

(Although, also, I imagine that a K12 class might need less time)

For me, the returns in the form of learning are high enough that I’d do it anyways, but I have also learned that we do get some of that time back at a few points:

-Those all-review days– before the test, at the end of the year, at the end of the term etc. – because we’re reviewing as we go, I can have a ‘real’ lesson on those days.

-Some of the time I’d spend on “Ok class, who remembers what we did last time…?”  *crickets* Because we’ve just done a review, I can get to the new stuff faster. (Having the review at the start of class is also a great activator)

-Some of the time I’d spend catching up lost students, or those who take longer, or who had it but forgot, or who just haven’t grasped that one. specific. thing.  They’ll get to work on whatever that stuck point is, as part of the regular flow of class, so we’re not doing (as much) time away from class to address their confusion.

 

Consideration 3: High impact practices

Yes, time intensive, but worth it to me in part because it folds in so many high impact learning practices.

Retrieval practice (remembering what you learned by quizzing etc. )

 (My open notes decision undercuts this a bit but, frankly, most students are not taking great notes so I’m not sure who much help they get and they need the encouragement to take any notes.)

(I’m currently experimenting with a ‘purer’ retrieval practice, adding a question at the end asking them to put away the notes and free write what they remember from the previous class. My template includes versions with and without this) 

  Spaced repetition (aka distributed practice, the opposite of cramming)

Metacognitive Reflection

Interleaving (Studying a mix of topics)

Differentiation/ Student Choice work

Activating Prior Knowledge

(I also have a hunch that this is a sort of informal exposure therapy for the math and text anxiety that run rampant through my classroom. Face a well-supported, low stakes version every week and eventually it gets normal enough to be a little less scary….? I don’t have the research, except a suggestion in this study, if you know more than me, please comment!)

 

 

Consideration 4: Adapting it

Spacing – Perhaps a monthly or bi weekly schedule makes more sense for your class, or maybe you do a (more extensive?) version as a transition between each unit?

Correcting – My classes are ungraded (except for the minor matter of the ultra high stakes high school equivalency exam) Perhaps having students correct themselves wouldn’t work in your school culture. Perhaps you grade them, but maybe pairs exchange papers to correct, or the class reviews results together, or you beat me to the technological punch and use a self-grading quiz program.

Timing – A half hour suits my class, but perhaps you reduce the number of questions, or cut back on the folder comment writing and student choice work to make it quicker. Or, extend it to an occasional full-period activity with more questions and review stations.

Structure – Perhaps this is not your starter, but your end of Friday routine. Perhaps the review sheets go home as homework.  Perhaps pairs or teams work together, instead of solo.  Maybe the process of reviewing is what matters and you drop the folder writing and/or independent work elements altogether.

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Constant Cumulative Review: Pt. 3 Putting it into Practice

Read: Part 1: Start with Why , Part 2: Making Materials without Going Insane and/or Classroom Routines: You will metacognate for background

(In brief – My classes start with a cumulative review every week, it’s a process, and I love it)

Once I had figured out how I could make materials, I had to figure out how they’d fit into my classes. In the end, we built a multi step routine that works for us.

  1. This is my ‘do now’. The copies are waiting on the front table when students arrive, and so early or late they know to start when they get to class. It’s open notes (to encourage note taking and using), and question-asking is allowed (so confusion gets cleared up).
  1. When students are done they correct it themselves with an answer key. And  I’m making (slow/uphill) progress at teaching them to use the key and/or questions as a tool to understand the ones they missed on the first try (instead of marking a despairing X and moving on).
  1. Each student has a (plain manila file) folder with a log stapled inside, where they file the corrected review sheets + their scores + a comment (my metacognition!). I keep the folders in the classroom so they don’t get lost, and so I can review periodically and see how students are progressing)

(I have ambitions of these folders working/feeling more like portfolios, but so far they remain just collection points)

Here’s a version of the score/comment log that I use

  1. As they finish the folders, my students know there are milk crates full of materials – and that their task is to grab something they need to brush up, and work independently until the next activity starts. (Some need more reminding of this knowledge than others, but in general, they’ve just had a good reminder of what math needs some practice)

 

While students are arriving/settling/working on this review, I’m also collecting and checking homework folders and finishing any set up for the rest of the lesson.

 

Essential elements:

Students doing most of the work.  I answer questions and attend to those who are stuck, or off-course, or need a nudge to work, but after the first couple of rounds most of my students can handle most of this process.

Making it routine. My students can do much of the work because they’ve done it before. The first weeks of the term, I spend more time directing the process, but they get into the routine pretty quickly. The routine also saves my brain – I don’t have to decide how we’re starting class, or figure out how to do review because I’ve already set that up. I’m grateful to put that brain power to some other planning question.

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Constant Cumulative Review: Pt. 2 Making Materials without Going Insane

Read: Part 1: Start with Why and/or Classroom Routines: You will metacognate for background

When my student asked for cumulative reviews the first, highest barrier was figuring out how I could possibly make custom materials, at a scale that would be useful, without going insane or putting in more hours than I had to give.

There was a learning curve, but with practice, and a system, it now fits into my weekly prep routine without too much fuss.

 

Essential Supplies: One of those accordion folders with multiple pockets, scissors, glue stick, paper.

(I have dreams of one day getting organized enough to do this electronically, but I’m not there yet)

The Process:

Round 1:  New school year, I stock the first section of my accordion folder with a few worksheets worth of review skills that students should know coming in. I cut out 1 or two problems from 3 or 4 sheets and glue stick them to a piece of scrap paper.   I make copies and answer keys, the class does them, then moves on to learn some new math. At the end of our learning, I put some of the leftover materials in a new pocket of the folder.

Round 2: One or two problems from 2 or three review skills, plus a problem or two of the newly learned skill. Glue stick. Copies. Answer key. Class. New math. Extra materials to the accordion folder. (Each unit or major skill gets a pocket in my folder)

Round 3: A few review problems, one or two of the skill we learned two weeks ago, and one or two more of the skill we learned last week.  Glue stick. Copies. Answer key. Class. New math, extras to the accordion folder.

Rounds 4 to n: 6 or 8 or 10 (if I’m feeling ambitious, mostly 8ish) problems cut from increasingly scrappy looking worksheets.

As the year goes on and we’ve learned more than 6 or 8 or 10 skills, I select quickly, without getting too bogged down. There’s always the last thing we learned, a selection of previous skills, and usually some basic number sense item that I want to drive into their brains through the power of repeated practice.

Most weeks, it takes me about half an hour to make review sheets, answer keys  and copies for two levels of classes. (My classes change just enough each year that I haven’t found it worth it to try to reuse them from year to year, but maybe you’re consistent enough to pull it off)

I’ve created a template (although you hardly need it) 

 

The essential elements:

Organization. For me, the accordion folder (perhaps for you another easy-organization binder-ish alternative)  If I had to find 6 or 8 or 10 types of math questions every week, I wouldn’t do it, but I can cut and glue 6 or 8 or 10.

Letting go of any perfectionist impulses. I move quickly and don’t agonize too much over which items to include (or how DIY it looks). This is the power of doing it regularly – anything I don’t cover sufficiently this week, will certainly be covered in other weeks.

 

Stay tuned for Part 3: Putting it into Practice and Part 4: Implementation Considerations

 

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Constant Cumulative Review (pt.1)

As I’ve mentioned, my two primary math classes start every week with a cumulative review. It is a bigger commitment than most of the practices I describe here, but I think the impact is worth it.

In brief, my system is a short mixed review sheet that I make each week. I pick problems from a folder full of worksheets from the whole year, so that it is a constantly expanding cumulative review. Students know to work on the review when they first get to class, then correct it themselves and log their results. If they have extra time they work independently on any weak points the review sheet turned up.

I’ll be back with more details, in three parts

  • How to stay sane while making weekly cumulative reviews
  • Review sheets in action
  • Implementation considerations

But for today, start with why. 

We do it because…

  1. Retaining what you’ve learned is maddeningly difficult when you only have a few hours of math a week, and you spend the rest of your time thinking about other pressing non math things.

And because I got tired of students working hard in class to learn something, only to get to an end of term review or high stakes test (or the next weeks’ class) and realize they had forgotten it. This is discouraging for everyone.

 

  1. The brain science/pedagogy: recall practice and interleaving, low stakes formative assessments, the way it set up student-lead differentiated review.  All good, research-backed pedagogical reasons.

 

  1. But really, because a student asked for it.

This student, D–, was one of my sweetest, favorite students. The kind who cheerfully worked away at a subject that baffled her, who kept at it despite setbacks, who’s going to make a great early childhood educator someday, and who paid attention to her own learning enough to know that she didn’t always retain what she had studied.

So, when she asked for review materials that combined of all of the math we’d covered, I paid attention. It was a good request, but not one I could easily answer.

The textbooks had a cumulative review at the end, but we hadn’t done every unit, so that wasn’t so helpful. And I didn’t have a custom one ready for her. Making a custom one would require digging back through all of the units we’d done over the last six months, and that would take time (which is always limited)  I think/hope we did a review day soon after, but mostly I kept thinking about her request.

  1. And I do it because systems solve things.  (personal philosophies…)

No, I didn’t have a stack of perfectly aligned cumulative reviews when she asked for them, but by September, I had figured out a system. We could have them going forward.

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Lesson Sketch: Proportions Project

Lesson Sketch:  Not a lesson plan, but a sketch, an idea, adaptable to your context and content

 

Theory: If you’re going to do math you need numbers. And they might as well be about something. And that something might as well be how learning happens.

DSC_1495

Practice: Tonight’s lesson on ratios and proportions.

  1. Conduct teeny (anonymous, paper) survey about big learning topics.

For my class,  three yes/no/maybe questions: learning preferences (aka, learning style*), do you know yours?; math anxiety, do you have it?; studying independently, do you do it?

(Here’s a version, if you’d like to borrow it) 

2. Three questions, become 3 student groups. Groups convert data to ratios, then ratios to proportions as they estimate the number of students in the school who do or don’t have math anxiety, study independently etc. Statistics become posters as they present their findings. (Or not, if like me, you run out of time for posters)

 

Expansion possibilities:

-Offer resources on learning preferences/styles, anxiety etc. when students complete math project.

-More statistics! Sample vs. population, survey methods etc.

 

Easily adapted to: Making bar/pie graphs, percents, simple statistics.

 

 

*Yes, I know the research that learning styles aren’t a thing. But for students who’ve always experienced school as a disempowering struggle, thinking about their preferences and the different ways one might learn is still a useful conversation.

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About

 

About Me

I was that girl. Smart, good in school, and somewhere around sine waves and that teacher I didn’t like but had two years in a row, I decided I wasn’t a math person. I was a social sciences person, I was going to save the world, and I didn’t see how calculus was going to help with that.

Years later, a volunteer gig in a GED class showed me there was joy to be found in teaching adults and the intellectual and interpersonal challenge of convincing them that they were math people. Or at least, capable math learners.

One career change later, I teach math, but really I think about brains and learning, about anxiety, efficacy and metacognition. In short, about the human side of math class

 

About my class

My students left school as teenagers, before that they mostly went to schools that struggled and where they struggled. They’re back as adults, and balancing work, kids, family, life and school. They’re determined (and awesome, in my humble opinion) but they’re not necessarily convinced they can learn math.

My job is to convince them otherwise (and then help them learn enough math to get a high school credential). I have two about hours a week with each group of students, a math education that stopped in high school, and faith in them.

In the process, they’ve taught me about second chances, doing hard things (and how fun it is to achieve them), and more about teaching and learning and what it takes than I learned in grad school.

 

This blog is about the human side of math class. How students learn to be math learners, and what they teach their teacher in the process.  Welcome.