Practical Thoughts on Differentiation in the Secondary Math Classroom

“There might be a student that doesn’t need modeling, but there is always one who does. So why wouldn’t I provide that first?”

I heard this recently on a favorite site. It sounded reasonable, and was spoken by a teacher acutely interested in her students’ success. She was talking about the “I do, we do, you do” strategy of gradual release, making sure that her students fully understood the expectations of the task she set before them. At times it may seem like the perfect fit, but far too often in secondary math classrooms we overuse this well intentioned, time tested lesson plan.

One powerful reason NOT to provide that first is that the modeling robs students that don’t need it of a chance to be creative, problem solve, and use strategic thinking – all higher depth of knowledge ways of processing their learning.  When we provide scaffolds for all it is easy to reduce everything to a dok level 1 – “Watch and do exactly as I do.” For more on depth of knowledge, see Robert Kaplinsky’s work here.

Differentiation is an important topic in education. At the high school level many people find it challenging to truly differentiate in a 55 minute period.  But here is a simple opportunity that actually creates time in your period – time that students can engage in those higher depth of knowledge types of thinking.

How can that work? Several models are possible, but all revolve around reducing or eliminating the whole class modeling and instead planning timely hints appropriate to various sticking points in the process.  Start with thinking about hints for your highest kids. What is the least you can say? What question could you ask to spark their thinking? The next set of hints could be more directed, potentially less open, and may encourage them toward a specific method. For the most helpful hints, basically just do your “I do” – either in writing or by giving then a link to a short video they watch on their phone. Label the hints 1A, 1B, 1C, 2A, 2B, 2C . ..etc. 1 represents the earliest sticking point – how to start.  A, B, and C represent the level of the hint. Students should be encouraged to take the least help possible. It is amazing how quickly your highest kids will adapt to that suggestion . . they are inspired by the challenge. But kids who need more help have it readily available.  

Double the effectiveness of this strategy by putting the students on vertical non-permanent surfaces. Definitely worthy of a blog post all it’s own, but simple enough to implement tomorrow. For a look at this strategy and the research behind it, check out Laura Wheeler’s blog here. A quick summary of the fuller body of work, “Building Thinking Classrooms” by Peter Liljedahl, can be found here.

Struggling with the mismatch between some hypothetical classroom ideal where every student is motivated just because they want to learn and grow, and the reality of grade-chasing, point-grubbing, homework-copying in your classroom?*  Long term, the research above can really make a difference, but to help in the meantime, some teachers use a pointing system for their hints.  All groups start with 10 points (or more if it is a pretty hard task with lots of sticking points).  C hints cost 1 point, B hints 2 points, A hints 3 points.  Points that aren’t used are extra credit on the assignment, and students can spend into debt – making their highest possible score 95/100 for example. Remind struggling students that 95, or 86, or whatever they work their way down to is so much better than the 50% for an incomplete assignment, or a 0 for not turning it in. And to begin to change that culture, be sure you celebrate successful completions equally – let students have joy in completing the challenge no matter how many scaffolds were used along the way.

Every day and lesson is different, and occasionally a bit of “I do, we do, you do” may be just the perfect amount of scaffolding/guided release for the specific task your students are working on.  Optimize those moments by listening to your students thinking along the way. But too often in a diverse regular classroom, we mis-serve both the top and bottom segments of our classes by doing all the higher level thinking for them. This model of differentiation is one way to support ALL learners in those classes.

 
*Ready to try something different with grading? Matt Vaudrey has a nice post about his experiments with a variation of Standards Based Grading  here.  And there are a thousand other great ideas out there. Look here for what came up when I googled: MTBOS on homework.

It truly is a wonderful time to be a math teacher.

 

Over-scaffolding: the loving art of loosening our grip

Parenting is an ever-changing job.

In the youngest years we are required to give our children instruction and advice on how to do every minute life task: how to angle their foot to put on a shoe, how to walk safely in a parking lot, how to put their pee in the potty, how to pick up the edge of their long t-shirt before putting pee in the potty. Those things are eventually mastered and we smile as we watch our little ones handle them all by themselves. Then we move on to new tasks: how to pick out a matching outfit, how to make a bed, how to clear dishes, how to put those dishes in the dishwasher,  how to put the laundry in the hamper, how to take the laundry out to the washer and run it through. For most of my kids-at-home parenting years, these tasks did not seem to stick as well. They were learned, but never seemed to be owned. But when my children moved out, and had to do it themselves, that training finally came to fruition, and now they manage all of those things with ease and grace. I never have to call my 26 year old and ask him, “Did you wash your work clothes? Are you wearing clean underwear?” But they’ve been through the stage where they had to decide it was important to them to have clean dishes to eat from and clean clothes to wear to work. Clean clothes for a college chem lecture did not seem to be equally motivating at first, but happily they eventually began to see the advantages of not stinking to high heaven when sitting by an attractive co-ed, and they reassessed their prioritizations.

 

Maybe those early tasks are mastered with independence earlier because we let them go. When a problem arises – like pee along the front hem of a t-shirt – we can step in and provide some additional instructions and advice. We do not stand and watch our nine year olds put pee in the potty and critique their technique.  We provide scaffolds as needed when problems arise. And then we step back.

 

Stepping back has been the hard part of parenting. All of my early on the job training was at the level of minutiae, and detailed instruction with constant reminders were essential. But gradually, if we let them, they grow to be able to figure a few things out for themselves.  Maybe if I had let it happen my kids would have learned earlier that underwear with skid marks were not a good idea to wear to gym class and they better get a load through. Maybe if they had to scrape yesterday morning’s oatmeal off their bowl on a semi-regular basis, or their ice cream got served up in their dirty spaghetti bowl, timely cleaning of dishes would have seemed more important.

 

Thank goodness, in spite of me, they are making their way to being amazing and wonderful young men who can problem solve as things arise and handle life. I wish I could say I was as good at learning the parameters of my new role. “Did you call your landlord about that broken garbage disposal?” “Did you check reviews on that car repair shop?” “If you hang your clothes right when they come out, you won’t have to iron those dress shirts so much.” “Is that sweater dry clean only??? You really should check washing instructions before you buy. You can’t afford dry cleaning every week.”

 

When they are young, we tell our children what to do because we are applying our priorities to our life. It was a priority for me to get them out of diapers for my life, not just because eventually at 5 they’d be glad to not wear diapers to kindergarten.  But when they are adults, their actions and decisions have to come from their priorities. Fortunately life has a wonderful way of providing timely feedback to help us all make decisions around adjusting priorities. As parents, it’s important for us to learn how to get out of the way and let that feedback get to them.

 

Adulting is hard. . . and then it’s not. It’s up to us to give them the chance to get there.

 

So lately, I have been thinking about how this plays out in the classroom.

 

The current buzz word is “over-scaffolding”, which translates to helping more than students  really need. If we insist on owning the responsibility for all the thinking, they will eventually stop trying to think.  Instead we need to carefully structure activities that help only as much as they need.

This is extra tricky because students grow at differing rates. I used to think that was about ability, but now I think it is mostly because of the access or lack thereof to appropriate feedback. The student who waits in his group for others to decide on a solution path is never trying out his own ideas and learning what works and what doesn’t, or why it’s important to, for example, isolate the radical before squaring both sides of the equation.

 

Imagine if class begin with two or three simple equations on the board
equation 1

and you ask your students to work individually to play around with and figure out the answer. Many would use their natural number sense to guess and check, but a few would use their equation skills they are comfortable with and reason about what they might add to the new situation. It’s likely at least a few will come up with “ Well, if that equals five, then before the square root it must have been twenty-five.”

However they approached it, encourage them, still all by themselves, to check their answer and see how they did. Guess-and-checkers started with this, because they are still trying to make sense of the numbers.  But for many whose heads are full of procedures, they have let go of this sensemaking move, and only do it as the last step in a procedure if that is what their teacher requires.  So here we see the first of the difficulties we often cause for our students – we teach compliance over thoughtfulness. I am beginning to think the most important thing we can teach them is to seek and use feedback -especially feedback they seek out and create from their own efforts.

 

Next, gather answers. Listen to methods and value all of them. Form hypotheses together about how this might work.

Some guess-and-checkers will be be persuaded to the “square both sides” method, but others will continue to trust their better skill – number sense. That’s okay. Children learn at different rates. We need to give them the opportunities to learn. So up the ante with the next question. Give the higher students a chance to explore the edges of their thinking, and the slower ones a chance to clearly see the limitations of their method. Again, gather answers, listen to methods and what has been learned. Adjust hypotheses.

 

You might have to keep raising the level of difficulty several times before the strong-number-sense/poor-equation-sense students decide that maybe there is something to their classmates methods.  Here are a few ideas:

equation 2

Finally, when they are mostly won over, together compile the methods they have figured out with your careful guidance, using their own developing number sense and feedback-seeking skills.

 

Scaling parenting to 35 kids in one classroom is hard. But continuing to tell 100% of them exactly what to do is preventing them from growing up and owning their own learning. I challenge you to build a thinking classroom, one that respects the strengths of all your students and gently prods them to the next step in their development. Encourage them to try something and see if their answer makes sense, and to think for themselves about how to approach a problem.

When problems arise, as needed, give just enough help and then get out of the way.

Be watchful, be positive, and even prepare in advance some scaffolds they might need, but give them as needed.

 

And I encourage you to be patient with yourself in the process. Learning, real learning, takes feedback and adjustments. It’s messy. So you won’t be perfect the first time, but you’ll learn. And guess what – learning can be exciting and fun! And helping the children in your class, the children I am sure you love,  grow to an adulthood that doesn’t require the scaffolds of childhood, can be the most rewarding part of your job. It is why teaching is such a rich and rewarding profession. Let it be that for you.