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Curve Stitchings & Card Shufflings

Here’s something non-pedagogy related and rather brief and who knows. Here goes.

Fun Fact!

1) The solution to this problem (in short: how many perfect shuffles until a 52 card deck is back in its original order)

AND

2) the creation of this picture:

blah

Are the same thing:

Waning Moon

So… I’m just gonna let that sit out there in the open with very little explanation. Maybe you’re curious. Maybe not. I certainly am.

Related and very cool: http://www.openprocessing.org/sketch/192276

and: http://tube.geogebra.org/student/m992965

Two other sets of related questions:

1) Will an n card deck always take n perfect shuffles before it’s back in its original order?
2) If each of the n dots in the circle are really n nails in a piece of circular wood, can I create this design with a single thread of string?

Also, this post couldn’t have happened without Dan Anderson. All this math is his fault.

Idea: Intervention Table

The other day, MissCalcul8 asked me on twitter about ideas for setting up an intervention table. The exact text was: “Any ideas for setting up an intervention table? Mostly for students who don’t even know how to begin.” Twitter’s great for networking, but I’m far too verbose to fit my thoughts into 140 characters. So here are some thoughts on the idea of creating an ‘Intervention Table’.

When I’ve seen others try something like this – a designated intervention station in the classroom – I’ve seen two variations. One type of station is designed for students to use during the lesson as a signal to the teacher that they’re not understanding and need some help. Another variation is a designated station designed to be used during an assignment (such as bellwork or homework time or stations). There are some practical things to consider with these, but also lurking the background is creating a culture where advocating for yourself doesn’t have a negative stigma, and being careful to frame either of these interventions as an opportunity rather than a punishment.

Ideas for an Intervention Table to be used during a lesson

I once worked with a teacher who had a designated desk near the front of the room labeled the “Help Desk”. The idea was that if a student was struggling during a lesson – there was a step they didn’t understand or they couldn’t make it through a class problem – they could move to this designated desk with the promise that they’ll get help sometime soon from the teacher. This desk was right next to the teacher’s desk and had a direct line-of-sight to the board.

When presenting this idea to students, he framed it by giving a speech to his class that was something along the lines of: “If you’re paying attention to a lesson and you feel like you’re not sure what’s going on, feel free to come sit in the Help Desk. It’s near the front of the room so you can see easily, it lets me know that you need help so I’ll make sure to check in with you, and once you feel like you’ve got it then you can move back to your seat”. In this way, students were encouraged to advocate for themselves that they need help with the promise that the teacher will give them a little extra attention while they’re sitting there to help make sure they understand. It also lets the teacher know immediately that there’s a student who doesn’t understand in a way that doesn’t directly interrupt the lesson (related: Red, Yellow, and Green Cup Stoplights).

If I were to try this (which, in writing this, I don’t know why I’m not trying this), I’d want to add a ‘Tutor Desk’ next to the ‘Help Desk’. Both desks would be left empty during the start of the lesson and, If someone needs help, they’re free to move to the Help Desk to get some extra help. But also, students who feel like they know what’s going on can sit in the Tutor Desk as a signal that they’re available to help people too. I know that I have at least 2-3 students in each class who genuinely like helping others out, so offering them a subtle way to do so could be something that’s very appealing to them. I think I’d also find some subtle positive reinforcement to encourage students to act as tutors and to encourage students to ask for help (something along these lines).

Ideas for an Intervention Table to be used during an assignment

A Digression: Suppose you’re in a unit on Solving Systems of Equations. Suppose you’re on the topic of solving via substitution. As a scaffold to get students used to the concept of replacing variables with expressions, you’ve got a bunch of problems with the variable already isolated – for example: y = -2x + 8 and y = 5x – 22. Let’s say that, in working through these problems, you discover that a student understands that they’re supposed to set these two equations equal to each other (ie: -2x + 8 = 5x – 22), but then has no idea how to do the remaining algebraic steps to solve for x.

This digression also tends to happen during graphing lessons (can set up the expression but can’t plot the points using x and y), or during coordinate geometry (can plug into the formulas but can’t evaluate the integers) or during polynomial operations (knows they need to combine like terms but can’t evaluate the integers & get the right signs).

These situations are probably the motivation behind this idea of a designated ‘intervention table’ – a place for students to work on the underlying skills that they need before they can continue with the current class content. These situations were also the motivation for my Wall of Remediation, which I still use in my own classroom. Part of addressing these issues is deciding how you feel about the following statement: If a student can’t do the underlying steps to a problem (integers, algebra, graphing, etc), then there’s no point in having them keep trying these problems that are above their head. In other words, instead of a student learning integer arithmetic and algebra and systems (and, depending on your unit, graphing systems) all at the same time, why not just reduce it down to integers and then build back up to the other stuff?

Adopting this mentality is scary because, at times, it means throwing away your objective for the day. On the other hand, its relieving for the student since it means they can focus on the real roadblock to their learning and feel like they’re making tangible progress. It also means that you, the teacher, need to be really prepared for when these roadblocks manifest. You need to be ready right away with a worksheet or set of notes or something to be able to give to the student to say “Try these – just these – then we’ll build back up to what we’re working on today”.

So – with all of this in mind, if I were to have a designated intervention table, it would have:

—A collection of worksheets with isolated skills that are also curated and checked by me. Here are most of the ones that I use currently. I also get worksheets from worksheetworks.com (I like that their answer keys show step-by-step solutions) and rarely from Kuta. I would also have an Answer Binder for students to check their answers immediately. When possible, I would have an answer bank for students to check their answers even more immediately (something like this). I let students turn in any remedial assignments like these for points to make up past homework assignments, but I always pick the assignments they complete.

—Number lines. Grids for multiplying numbers or polynomials. Positive & Negative tiles. Graphing squares with the numbers written on the sides. Whatever other tangible representations that are usually used by elementary and middle school teachers to give concrete representations to the things that high school teachers have made abstract. These are things I can grab easily and quickly show a student how to use to solve problems, using it as a temporary scaffold that eventually gets removed. If I can find good procedural guides for how to solve problems, I’d have those too (I’m thinking of things like multiplying fractions or graphing coordinates) for students to look at and reference.

—Whatever notes or definitions or formulas they need for the lessons, even if they’re ‘supposed to know it’. For example, I recently stole Sam Shah’s folder system for organizing myself, but I’ve been added notes and definitions to the center of the folders for students to reference:

2015-04-04 09.26.09

This could also be included at an ‘intervention table’ – not necessarily printed versions of your lesson, but the bare minimum important diagrams/definitions/notes/etc that students can reference quickly. This goes a long way in giving students that entry into a problem, especially those with high absence rates.

Working at the Intervention Table

So – let’s say you’ve set up your intervention table and you’ve got a student sitting there with a basic assignment to work on. Let’s say that assignment is on basic 2-step equations and you’re working on the problem below:

blah

In working with this student, you feel torn – you want to provide meaningful help that is worth their time, but you’ve also got a class full of students who are also working on an assignment and would like your feedback. The issue is: working meaningfully with this student could take at least 10 minutes and means addressing all of the misconceptions in this one problem, such as: which number do you start with? why that number and not the others? my teacher always told me to start with the number on the left. my teacher always told me to start with the positive number. my teacher always told me to start with the smallest number. why did you add the numbers instead of subtract? why is the answer negative and not positive? why are you dividing by -5 – why not 16? why is the answer positive? am I done? how do I start the next one?

The Tension: You’re trying to help this student in a meaningful way that they will remember for next time, but you also need to bounce around to everyone else in the room to help them with their assignments and for just basic classroom management sanity. If you spend a long time working with this intervention student and honestly addressing all of their questions, you’ll feel obligated to get up and walk around once you’ve done only one problem in depth – but the student at the table will probably need some reassurance before they can work independently and so, once left alone, will not have the confidence to do a problem completely. On the flip side, you could try to rush through a problem in order to get back to your classroom, but then the misconceptions and underlying questions are never really addressed, so the student can’t transfer this ‘band-aid’ fix to any of the other problems, and usually won’t remember the first step and can’t get started.

Here’s How I Navigate That Tension:

First, we’re never solving just one problem – we’re always solving at least 8. And, especially at the beginning, we’re not doing one problem to completion – we’re doing five or six problems one step at a time. The trick for me is to break this problem into pieces that are general enough to be applied to most problems and can be applied quickly to several problems. Once we do the step for the first problem, I make them repeat just that step for several more (allowing me to do a quick pass around the room). Along the way, I’m checking for understanding on just that step before moving on to the next one. Here’s a pretty much word-for-word account of what I would do with this struggling algebra student to navigate this tension:

1) “In an equation, the most important thing is the equal sign. Find the equal sign and put your finger on it. Now draw a line through the equal sign, splitting your problem into two pieces. Now do this for the next 6 problems” (Walk Around – come back – check work)

2) “Which side is your x on? If it’s on the left side, write an L. If it’s on the right side, write an R. Now do this for the next 6 problems” (Walk around – come back – check work)

3) “We need to get the x by itself, so I’m going to look at the other number with the x and that is still on the same side as the x. Find that number an underline it. Now do this for the next 6 problems” (Walk around – come back – check work)

4) “We need to find the opposite of this number so they can zero out. What number is the opposite? Write it underneath on both sides. Now do this for the next 6 problems” (Walk around – come back – check work)

5) “These terms zero out and we’re left with _____. On the other side, we need to do some math. What do you get when you combine these numbers?” (At this point, if they’re struggling with integers, we stop the algebra and start working just with integers). “Good – now do this for the next 6 problems” (Walk around – come back – check work)

6) “Now we need to get the x by itself. What does it mean when a number is next to a variable? And what’s the opposite operation? So what do you think we should divide by? Just on this side? What happened on the side with the variable? And on the other side? Good – now do this for the next 6 problems” (Walk around – come back – check work)

7) “Great! Now do these last 2 from start to finish” (This is important – that you always save a few problems to do completely on their own from start to finish).

Some things that are done intentionally: Each step has something tangible for the students to write/draw/circle/etc for me to check later. Each step is broken up so its manageable, but also lets me check for all of those tiny misconceptions that can crop up. And, by doing several problems at once, I can see a specific misconception that I may miss if I do only one problem at a time (for example: if the first 5 problems all have the variable on the left side, It’ll be a while before I uncover misconceptions a student may have about variables on the right side of the equal sign – but, if we’re solving several problems at once, I’m more likely to notice and ask questions about the one problem with a variable in a ‘weird’ spot). Last intentional thing: the decision of which number to work with first in solving that problem is not a trivial decision – I purposefully add scaffolds to help make that decision more concrete and logical rather than a series of special cases that feels closer to memorization than algorithmic problem solving.

A lot of times I come up with these strategies on the spot as I’m trying to navigate this tension between meaningful help that applies to several problems, while also managing my classroom. My guiding principles are: have them do something tangible, have them break decisions into smaller pieces, try to isolate the steps where I know most misconceptions can occur.

A Completely Valid Point: But the student still doesn’t really know what they’re doing or why they’re doing it! You’ve just given them a procedure to follow to get the answer!

Response: Yep. If we’re still talking about a single student in the middle of a class who needs a very targeted intervention, then yes – that’s exactly what I’ve done. It’s not perfect, but it’s how I’ve reconciled the cost-benefit game of these moments. There’s the benefit of trying to explain the conceptual underpinnings of algebra with balance scales or developing a real-world analogy, and then there’s the challenge of competing with their attention span, my resources at that moment, their motivation in that moment, and the time I have with them – all of these lead me to conclude: it’s not realistic that I can fix years of conceptual misunderstandings in a small moment that takes place in the middle of another classroom lesson. If I’m lucky, a student who suddenly gets the procedure will start asking “but why does this work?”, which can lead to that conceptual conversation, but it doesn’t always happen.

There is a place for these conceptual conversations though – its either a more in-depth tutoring session, or a dedicated intervention class (which is what I usually teach). This is when I try to build that conceptual framework and hold them accountable for it – but the middle of a lesson on a totally different subject is not the time or place for that.

So…. there are lots and lots of thoughts and ideas and opinions. Thanks for reading.

Resources for Intervention Classes

I gave a presentation last week on strategies that I use in my dedicated intervention classes for high school math students. The feedback was pretty positive, and I thank the local Tucson folk who attended. One of my favorite moments was letting teachers imitate their worst students, then trying to teach a lesson to them – cell phones out and shouting out answers and heads on desk and unprepared for class and all of the best behaviors of our worst students. That was fun. However, I still ran out of time – so much to talk about.

Anyway – for any other interested parties, I’ve compiled the resources from my talk all in one place: http://www.schneiderisawesome.com/MEAD (the MEAD is from the Tucson Math Conference where I first gave the much-too-short version of this presentation). This collection of resources includes:

  • The presentation I gave
  • A collection of resources for how I teach basic Integers & Algebra
  • An article about Mindsets from Carol Dweck
  • Lots of resources for starting to implement SBG
  • Some resources & ideas for classroom engagement strategies
  • Some details of how my intervention classes are organized at my school from a placement & administrative perspective

Leading this workshop has made me want to do two things:

  • Have a video recording of the first week of my intervention classes when I do a lot of culture building and attitude reshaping and a little teaching.
  • Write a blog post detailing the process that me and my administration have gone through to implement our version of intervention. One of the things I could feel at this workshop is that a lot of teachers are being ‘thrown in’ to an ‘intervention’ class without clear details about what the needs of the students are or how to measure progress. This is a stark contrast to how our school has done intervention, so I’d like to share some of that – another time.

Anyway – for any other interested intervention folk, I hope those resources help. Hopefully I’ll find time to share more of my thoughts on intervention in the coming months.

Life in Intervention (or: I’m leading a PD session)

So – fun fact: over the last 2 years, I’ve fallen into a curious niche called ‘math intervention’. I teach an ever-changing curriculum where the only real overriding theme is ‘help students get better at math’. I get the students who’ve failed math their whole life and, if things go well, they start to get better.

Apparently, I’m doing a good enough that I was invited to present an hour-long session of my strategies and thoughts and tricks at a local Math Educator conference here in Tucson. The feedback was pretty good, but I know it could’ve been better – mainly: I wish I had more time, and I wish I had anticipated my audience a little better.

Well, soon I’ll have a chance to try and fix both of those issues: I’ll be presenting a 4-hour session on Teaching Math Intervention at the University of Arizona on March 7th. The link to register is here.

I’m designing the time for high-school math teachers who are currently teaching some kind of support or intervention class, which I guess is becoming more common than I had expected. I plan to talk about some curriculum things I do (like how I teach integers or basic algebra to students who ‘should’ know it already), the way I structure the class in terms of assignments and grading (spoiler alert: its SBG based), how I encourage the ‘growth mindset’, how I deal with difficult students, and how I work with my administration to find the right students for the class. Honestly, most of the things I plan on talking about were instigated on this blog, which has made planning for this a really interesting jaunt down memory lane.

So – if you or anyone you know are looking for strategies for some kind of intervention class, maybe this is something that’ll help them; again, the link to register is here. Also feel free to contact me with any questions – my email is somewhere on this blog.

Game Design & Task Design

Quick post here as I’m preparing for lots of other things:

Watch this clip from the movie ‘Indie Game’ – it’s game designer Edmund McMillen talking about how he designs the first few levels of a video game:

http://www.criticalcommons.org/Members/fearv/clips/indie-game-the-movie-edmund-mcmillen-discusses

Now can we talk about how amazing this clip is if you replace the phrase ‘level design’ with ‘task design’ or ‘activity design’ or ‘worksheet design’?

Related: Dan Meyer & his whole ‘Open Middle’ analogy between good game design and good task design

Another Reason for Twitter

Hey everyone,

I’m getting worse at keeping my blog updated… I’ve been wanting to add something to this for a while because I haven’t liked that the first post people see when they come here starts with ‘Shameless Promotion!’. That’s just tacky.

So – in an effort to move that from the top of my front page, I want to write about Twitter. In particular, I want to write about one person on twitter: Alexis Huicochea. This person is 90% of the reason I still use twitter today.

Alexis isn’t a teacher or educator. She’s not someone that I follow for professional development – that’s the other 10% of why I use twitter. I’ve never met her in person, nor have we ever had a conversation on twitter. Alexis works for the local newspaper in my city – she writes primarily about local education. I like reading her articles because they keep me informed about local educational news – things happening with districts and stuff. Most cities probably have this – that’s not a big deal.

Here’s the big deal: Alexis Live-Tweets Every School Board Meeting.

And they are fascinating. They are enlightening. They catch things that don’t make it into the newspapers. They are, at times, hilarious. Seriously – check out this gem right here.

There’s a lot going on these days in education policy – changes to statewide assessments, funding issues, unions & tenure – all sorts of stuff. The more I know about what’s happening in my district, the better. And I can’t think of a better way to get that info unfiltered and raw, which is what I want. I don’t want the boiled-down bullet points for the general public – I want to know what’s up for discussion and where people are leaning. I want to know what might be coming down the pipeline because these people affect me directly. I want to be informed damnit.

So – if you’re a teacher on twitter, I encourage you to find someone in your area who does this same thing. Maybe this is common and I’m late to the game. Or maybe this is the start of a trend that I hope starts to pick up – live-tweeting school board meetings so we all know what’s going on.

Actually – here’s a question for you, reader of this post: are there any journalists in your area that go to board meetings and tweet the results? Do any school districts in your area do this voluntarily? If not, why not? Why don’t more people do this? Can we get more people to do this?

Lastly, for the record, I live in Tucson, AZ and I follow Alexis because she live-tweets the Tucson Unified School Board (TUSD) meetings, which is one of the largest districts in the state. So if you live here too, you should give her a follow. Ironically, I don’t teach in this district anymore and no one live-tweets the school board meetings from my district. I wish someone would.

Shameless Promotion: Listen To Me Talk About Things!

Hello Everyone,

Thought I’d share: Adrian Pumphrey has started a neat little podcast called MathEd Out where he’s been interviewing lots of awesome people in Math Education. I’m on his most recent episode talking about all sorts of things – first-year teaching, Standards Based Grading, what my classroom looks like, Standards of Mathematical Practice, PARCC implementation, an amazing Math joke in the last few minutes, etc. It’s 40 minutes! Holy crap – how did that happen?

Anyway – if you’re interested, you can listen to that here: Mathy McMatherson on MathEd Out.

And, if you’re really interested, I talk a lot about first-year teaching and preservice teaching and feedback and SBG and lots of other things in the first episode of Infinite Tangents, a podcast by Ashli Black that started strong but has since petered out (which I think is because Ashli is now a big deal facilitator for Common Core Professional Development as part of the Illustrative Mathematics team). This was recorded in the Spring of my second year teaching as I was just beginning to seriously reflect on my implementation of SBG, leading to a whole series of posts that people seem to find useful.

Anyway – if you’re interested, you can listen to that here: Mathy McMatherson on Infinite Tangents.

Cheers – Mathy

Answer-Getting and: Mastery Quizzes

See Previously: Some Thoughts on Interventions & Answer-Getting

So here’s another thing I did a few times in my class this year with dramatic effect: Mastery Quizzes

A Mastery Quiz Is: A collection of skill-based questions designed to be completed in 10-15 minutes. The types of problems on the quiz should be very similar to the ones they’ve done in class – no surprises. For full effect, some problems should be procedurally difficult in the sense that they are multi-step and require you to be careful with all the little details of the problem. Think along the lines of “Perseveres in Solving Problems” for the remedial student. It is not multiple choice.

A Mastery Quiz Works Well With: Skills that have some kind of concrete or procedural foundation that, if students just relied on this foundation, they would get it right. Examples include:

  • Integers for students who know how to use the number line but try to take shortcuts and make little mistakes
  • Multiplication for students who can use the box model but get stuck with the standard algorithm and make little mistakes
  • Exponent Rules for students who understand the individual rules but try to do too many steps at once and make little mistakes
  • (Procedural Skill) for students who understand (the foundation grounded in a process or scaffold) but (do this bad thing) and make little mistakes

Here’s my Mastery Quiz for Integers:

Here comes the important one:

You Grade a Mastery Quiz By: Everyone Either Gets a 0% or 100%

Read more…

Answer-Getting and: Analyzing Student Mistakes

See previously: Some Thoughts on Interventions & Answer Getting

So here’s something I started doing in an effort to keep students more engaged in problems and less focused on the most direct route to getting to the answer: I started having them analyze their own mistakes. This isn’t new – Kelly O’Shea’s idea for The Mistake Game has been around for a long time and it was a definite inspiration for my explicit focus on mistakes. But, I didn’t think I could jump right into having my students create their own mistakes – here’s why:

  1. Generating mistakes requires you to be confident enough that you could solve the problem without a mistake in the first place. My students don’t start out having this confidence – I needed to work on building it first.
  2. Generating mistakes requires you to care about how you got your answer, not what the final answer ends up being. Its possible to generate two mistakes which, as the problem progresses, cancel each other out and give the final answer. As a teacher, I see this as an incorrect problem even though the final answer is correct. Students with the Answer-Getting Mindset will see this as correct because the final answer is correct – learning to see the problem as an entire body of work is something that I need to train them to do before purposefully making mistakes has any meaning to them.
  3. Generating mistakes requires me to ask a “how/why” question rather than a “what” question. Answer-Getting is all about what: “What is the answer?”, “What is the next step?”. How and Why questions focus on process: “How did you get from this step to this step?”, “Why are you allowed to do this?”, “Why can’t I do this instead?”, “How would you explain this process in words rather than numbers or symbols?”. In general, I want to be asking more how/why questions rather than what questions.

Analyzing Mistakes

At the start of the year, all of my students took a pretest that covered basic arithmetic, solving basic equations, and basic graphing skills. As the year progressed and I had begun to explicitly teach these skills and build confidence, I would eventually come to a lesson where we revisited these tests. To do this, I took lots and lots of pictures of these initial tests (without any student names showing):

mistake1 (600x800)mistake2 (600x800)2-Step Arithmetic 1

 

mistake7 (600x800)2-Step Arithmetic 42-Step Algebra 1

Students would walk into class and see a single picture on the board. They wonder whose mistake it is, which gives some social buy-in. Each person wants to be the first person to find the mistake, which gives some competitive buy-in. As class starts, I ask them: Where is the mistake? At first, students tell me the answer is wrong (which is true). Then they try to work it out themselves and give me the right answer – but, since we’re still in the middle of mastering these skills, these answers still aren’t very reliable. Usually there are multiple answers, so I let them debate for a while. Then I interrupt:

“You’re not answering my question. I asked ‘Where is the Mistake?’, not ‘What is the Answer?’. I don’t care what the answer is. I care where they made a mistake”

I let that sink in for a minute because, for someone with an Answer-Getting mindset, this stops them cold. A teacher just told them that they don’t care about an answer. This either makes this class new and intriguing or new and terrifying. Either way, we’re gonna work through this.

“I want to get to the point where I can circle the part of the problem where this student made a mistake. This means I need to look at each line of their work and ask myself ‘Do I understand what they did?’ and then ‘Was it the right step? Did they get the right number?'”

And we’re off. I guide them through looking at each step of the work. I ask them to describe that this person did from one line to the next, ask if this was an okay thing to do, then verify that they got the right numbers. We talk about how not showing our work can make it harder to find mistakes. I show them how to circle the individual step of the problem that is incorrect. We look at the next mistake, which is the same starting problem but with a different mistake somewhere in there. Isn’t it interesting how one problem can have so many different mistakes? Is this a mistake you might have made? Are there multiple mistakes?

For students who’ve built up walls around these problems in the form of dismissal and anger and solve-as-fast-as-I-can, they tend to give these problems a second-chance because they’re not actually being asked to solve the problem. This slight change in what I’m asking them to do is enough for them to engage with the work even if working out the problem itself would have been a motivational challenge. For students who’ve built up walls around these problems in the form of apathy and not-trying and fear of failure, I found that they start to quietly participate in these discussions because this type of question is new and doesn’t have any past stigmas of failure associated with it. Either way, I have students genuinely engaging with a set of problems and, hopefully, feeling successful as they do it.

Full Disclosure: I’m leaving out all the important work that happens in the classroom as we discuss these – the graceful handling of student pushback; the subtle encouraging and guiding of students thoughts; the questioning strategies and No Opt-Out mentality that I have; etc etc. There’s lots of other explicit and implicit things that need to happen, but they’re the things that differ from teacher to teacher.

After doing this as a class, I would give students a problem set in two-columns. The left-column had a problem and a hand-written solution that intentionally had a mistake. The right-column had the same problem and a space for work. Students needed to find the mistake and circle it in the left column, then solve the problem correctly in the right column.

Push-Back Against Answer-Getting

  • Part of the Answer-Getting mindset is avoiding mistakes because they’re bad. Explicitly analyzing mistakes and getting exciting about them starts to remove a lot of that stigma and fear. This is all over Math & Science Education research – the idea of normalizing mistakes and internalizing it as growth versus verification – and this is how I managed to do that in my classroom.
  • Students get practice explaining their steps and discussing how to get to answers versus what is the answer. There are lots of students who make little mistakes leading up to answers, but are actually very capable of explaining their thinking out loud. For some students, this oral explanation needs to become part of their problem-solving toolkit. For these students, they start to realize that all their steps may be correct even if their final answer isn’t, which is an easier problem to fix and helps to build confidence.
  • When students get a problem incorrect, it starts to become normal for me to ask them to find their mistake. In doing this, students begin to see value in showing their work. And not just because the teacher said so – because it makes it easier to retrace their thought process and identify their mistakes. This is one of the first times in the class that I say to students “I think you should do this because it’s helpful to you, not because I’m being mean and want you to do extra work”, and they begin to believe me.
  • This is one of the first times where I’ve made it explicit that there’s more to math than just getting the right answer. Which, in terms of motivation and attitude towards math, may be something my students need to hear.

A Few Other Things

Here are all the Integer Mistakes.

Here are all the Algebra Mistakes.

Here is Michael Pershan’s Math Mistakes website, which has tons of other mistakes to peruse.

The same day I published these thoughts, Michael Pershan publishes his thoughts on How He Gives Back Quizzes. Embedded in this post is his method of displaying mistakes as a way to generate discussions as part of quiz feedback – if you enjoyed reading this, try reading his ideas too.

Even with all of these resources, I highly recommend finding a way to naturally generate your own student mistakes to analyze. The buy-in this creates from students – “Who’s is that? Is it mine? Is it _______ from 4th period?” is important, and the fact that we’re normalizing our own mistakes is important.

Some Thoughts on Interventions & Answer-Getting

A long while ago, Andrew Stadel posted a call for ideas on Intervention Strategies and the only person I could see who valiantly answered was Michael Pershan. Then, more recently, this topic showed up again here and here. If there’s one thing I’m good at, its noticing trends in the blogosphere.

In looking at these posts, I found myself wanting to write something about interventions because I live in the world of intervention. For the last year, I’ve only been teaching an intervention class called ‘Math Lab’ to a group of sophomores who have (literally) failed math most of their life. I’m part of a brand new school-wide math intervention process designed to increase math fluency and confidence. I’ve been helping to develop the curriculum and identify effective strategies for my particular demographic. My students have built up mental walls and self-handicapping strategies and a slew of negative coping mechanisms to deal with their distaste and distrust of mathematics. Their issues aren’t only their skill deficits and cognitive issues – it’s their underlying behavior and mindset that causes the most issues. These are my students.

I want to contribute somehow to this discussion of intervention, but this post has been in the ‘draft’ status for months because I’m not sure who my intended audience is. Teaching in a class designated as strictly intervention with a 15-student cap is not the typical classroom setting, which makes some of my best strategies less feasible to the typical classroom teacher (which may be you). At the same time, in talking to teachers in town and across the twitterverse, some kind of school-wide intervention model seems to be the new attractive thing for schools and districts looking to respond to low test scores and the incoming PARCC assessments, which might make these thoughts attractive to someone who may be facing an intervention class next year (which may also be you).

In trying to find the overlap between these two audiences, I realized what I’d like to do is share my experience combating something I’ve started calling the Answer-Getting mindset. Hopefully this is something the typical classroom-teacher can relate to, and it’s definitely something that an intervention teacher will face head-first in their own classroom. In thinking of all the things I consider intervention strategies – from affective techniques to teaching strategies to grading systems – the underlying theme is how all of these are designed to defeat this Answer-Getting mindset that is developed in the students with the most need.

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