kitchen table math, the sequel: "Practical Differentiation Strategies in Math"

Wednesday, December 2, 2009

"Practical Differentiation Strategies in Math"

I recently had the opportunity to attend an professional development seminar for elementary school math teachers. The topic was Practical Differentiation Strategies in Math.

It was a dinner-and-talk event. Several schools sent several teachers, and occasionally an administrator, to socialize and the listen to a 90 minute presentation. There were somewhere around 3 dozen people in attendance.

The talk was given by Liz Stamson, a math specialist with Math Solutions. You can see a related set of slides for a talk by her here. Those slides are for the webinar at this link to differentiation.

Stamson's talk was more specifically geared to engaging the math teachers. She spent most of her time talking about creating lesson plans and assignments that met the goals of differentiated instruction.

Stamson's talk assumed the full-inclusion classroom. She began the talk by saying that teachers were already needing to address students that could have vastly different "entry points" into the math curriculum. No mention was made of whether this was good or bad, whether grouping by ability was a help or hindrance. Full inclusion was a fact for teachers, not a point of argument.

Given that, she said, differentiation (that was the phrase, not differentiated instruction) was a way to help teachers engage all of their students. She stressed that differentiation could occur in content, process, and product. Her main ideas in how to differentiate content were to create "open tasks": questions that didn't have set answers, but had a multitude of answers. For example:

"what is the perimeter of a rectangle with length 10 and width 3", is a "closed" task. It has one answer. Such closed tasks, with one solution, and no critical thinking, does not lend itself to a differentiated classroom.

Instead, she promoted "open tasks" and "choice." What she meant by "open tasks" were problems that don't have one answer. By "choice" she meant allow the students to choose which problems on a problem set they must do. For example, instead of asking the above perimeter question, one could ask "describe a rectangle whose perimeter is 20." Such a task "opens" the problem to more creative thinking, allowing the more advanced students to find their own solutions (or to find a set of solutions), while still allowing the minimally skilled students to contribute. She said she'd offered the following "assignment" to her students, she said : "Do all the problems whose answers are even." This promotes critical thinking, she said, as they must investigate all of the problems to learn which ones fit the criteria.

She illustrated an open task with the "Which Does Not Belong" problem.

She gave the following sets:
2, 5, 6, 10
9, 16, 25, 43
cat, hat, bat, that

She asked the teachers to "work" the following problems. Teachers were seated together by grade (rather than by school), and were asked to collaborate. For each set, remove one item. Find a rule by which all of the remaining elements belong to the set, but the removed one does not. Repeat for all items.

The teachers gave a variety of answers.
Someone said that with the 2 removed, the remaining numbers' words were in alphabetical order. One said that with the 5 removed, the remaining numbers could all be written by 3 letter words. Another said all the remaining numbers were even. With the 10 removed, the remaining numbers were single digits said someone. With the 6 removed, the numbers formed a set of divisors of 10 (or alternately formed the number sentence 2 * 5 = 10.)

Ms. Stamson used this variety of answers to illustrate her point: all answers were valuable, no one had to get the "wrong" answer, even if some students didn't see immediately what the others did. This was practical differentiation--the more complicated patterns were found by the more advanced students, the less in depth by the students with less skills, but all could satisfy the assignment. The later grades could use factors, fractions, or other mathematical terms more appropriate to their lessons, while the younger grades could do this assignment with words or pictures if necessary.

She encouraged the teachers to create their own sets during this talk, stopping for several minutes.

Differentiation of process and product seemed to mean giving out different problems/homework/quizzes to different children. She advocated designing assessments/quizzes with several problems whose values were weighted, and requiring the student to complete a certain total value of problems, but allowing the students to pick which to do. The simplest problems were worth 1 point, say, and the hardest 10, and 15 points were needed for completion of the assignment. She suggested that such choice allowed all student to build confidence. She said also that it allowed a teacher to be "surprised" by a student who did difficult problems when the teacher expected to only do the easiest problems (as well as surprised when a good student did only the easiest). She also offered that the best students would challenge themselves and do all of the problems, so this provided a way to meet their needs for more or harder work.

When a teacher asked whether such choices were really practical, she admitted that such assignments might not be practical for daily practice, but suggested simply offering different students different problems. "Do you actually suggest assigning different problems to each student?" one teacher asked. Yes, she said: just hand out a worksheet, and highlight problems 3,4, and 7 for one child, and 1, 8, 9 for another. "If the culture of your classroom celebrates differences, then it's natural. We are all different, so of course we all have different strengths, and we all do different practice problems."

I could not tell from my vantage point, seated with a few lower grade teachers, if the room generally viewed the talk favorably. I assume the answer was yes, as who would attend such an evening if they were not already so inclined? The teachers at my table were quite dutiful at doing her assignments to us during dinner, and took very seriously everything she was saying. There were few questions from the audience, though. I couldn't tell, but perhaps that's just the style of talk, or the general behavior of a Minnesotan audience.


Anonymous said...

I have recently attended a PD workshop (Labeled "Brain Based Learning and Differentiated Instruction" ) that went exactly as you describe... Except it was for high school math and science teachers and included a few power point slides on brain anatomy.
But the idea is the same. Differentiation is a given. Dumbing down is a given. (Presenter actually stressed to us that "research shows -students cannot learn more than 10 vocabulary words A WEEK!" And we'd better create differentiated assignments - such as drawing a picture or making a poster... And group work, of course.To keep students engaged. And get them out of the seats every 10 minutes or so.
Oh my...
(I had to keep my mouth shut, unfortunately - it's my first year in this school, and I need this job. So quitely read a magazine under the table. The supervizors watched the teachers during the workshop and reminded everyone at the end that they want to see "the strategies" implemented.)


RMD said...

it's stunning to me how no proof is offered that the given technique/strategy/curriculum actually works, and yet schools are willing to adopt it


Eowyn said...

I just want to know when we're supposed to prepare all these personalized worksheets.

Cranberry said...

"She suggested that such choice allowed all student to build confidence."

Ah. "Confidence, if not competence" as a slogan. Do you think parents would find that compelling?

"She also offered that the best students would challenge themselves and do all of the problems, so this provided a way to meet their needs for more or harder work."

In what parallel universe would this happen? In which the "best" students were all masochists? Also, remember the bias towards group work. I suppose this means the best students will do everyone else's work, because they actually care about grades. However, repeating problems you already understand and have mastered doesn't increase your understanding of mathematics nearly as much as moving on to the next topic.

Also, I fail to see what was so "mathematical" about the number of letters required to spell the numbers. Numerology is higher math, in comparison.

Niels Henrik Abel said...

These are grade school kids, right? So in another ten years or so, they'll be headed for remedial math at your local college. Oh great.

On the other hand, there's plenty of silver linings if you have a Kumon (et al) franchise.

SteveH said...

Allison, thank you for the detailed account. This describes our schools. It's a given. It's not as if differentiated instruction was shown to be a good technique and then our schools decided to implement full inclusion. It's the other way around.

Teachers are supposed to have just one worksheet and then somehow figure out 5 or more levels for the different kids? Can you imagine when one of these sheets goes home and the parents see the level that the teacher has selected for their child?

How do you do this in the context of an existing curriculum like Everyday Math? EM uses spiraling to handle differentiated instruction. I think the idea is that the teacher just has to throw some enrichment at the kids who have already figured out the material when they spiral through it again. You can't do this with mixed ability groupings. Isn't that one of the main ideas of full inclusion? It's interesting to see that some of the solutions to differentiated instruction involve separating kids by level.

Much of differentiated instruction comes down to self-selection. Kids will never get carefully leveled assignments from the teacher or be properly challenged. For spiraling, the teacher doesn't even have to try and figure out how much to push. The spiral will take care of it all. Or not.

It's all quite astonishing. Where is the critical thinking?

Allison said...


Your statement "Also, I fail to see what was so "mathematical" about the number of letters required to spell the numbers. Numerology is higher math, in comparison." could be interpreted to suggest that the other answers were mathematical. But they were not.

There was no math present at all. At best, the assignments as presented were math appreciation. Randomly finding patterns or noticing relationships is not mathematics. Parlor tricks are not math. That someone can play with numbers and find several possible patterns in no way increases their mastery of arithmetic.

Allison said...

--it's stunning to me how no proof is offered that the given technique/strategy/curriculum actually works, and yet schools are willing to adopt it

Works at what, RMD? Works at what?

These techniques work very well at their goal: to give everyone the ability to answer the question correctly, and to help the teacher feel better about the full inclusion classroom she has.

Allison said...

Examples of the worksheets can be found in the slides at the link.

Steve, I don't see any difficulty in using this inside a curriculum like EM. Again, we're talking about
a 15 problem worksheet, where 5 problems are perimeter problems for rectangles, 5 are area problems for rectangles, 4 are word problems that involve drawing rectangles, and 1 is drawing that floorplan of your house that I posted a few weeks ago.

Then 3 days later, when EM switches gears, you again have such a worksheet on counting by 3s: 5 problems of which are add a number to a sequence, 5 problems of which are to remove the incorrect number from the sequences, 2 of which are word problems to guess something, and 1 of which is to write down a question whose answer is a sequence of 3s.

Allison said...

re: "WHEN" does a teacher do this?

The answer as given was to encourage collaboration of the teachers at a school, so that the workload was not overwhelming. Since the basic premise for the questions was independent of knowledge, teachers in other grades were just as likely to come up with an idea for a given grade as any other.

Eowyn said...

Re Allison's comment on collaboration:
This works as long as you have other teachers teaching the same material. (Which you probably do in a grade school.)

What if you're the only teacher teaching the post algebra 2 courses? All three of them?

And as a more general question, at what point can one stop differentiating? Because I know college instructors don't. The math profs I deal with set fixed parameters on what skills need to be mastered when to pass their courses.

Anonymous said...

There are two things that bother me with this kind of thing and I'm speaking more generally here than just this particular example.

The first problem is this and I'll pose it as a question. If your neighbor tossed a rock through your window every day would you...

a. Ask him to stop doing that

b. Go to a class to improve your window repair skills

All differentiation is fixing the wrong problem. Why do we accept sending kids with enormous dispersion to a single class room? It's because we have no structure to do otherwise!

The second problem is that lots of times these things are trying to instill 'critical thinking' into a topic which is basically a simple definition. Perimeter, for example, is nothing more than a definition. If you want to think critically about it then it needs to be one component of a more complex domain. So you might ask about the relationship between perimeter and area in a regular octagon. That requires critical thinking! To think it through you have to know the facts of area and perimeter.

You can't think critically about perimeter, by itself, anymore than you can think critically about the meaning of the word pot.

farmwifetwo said...

There's a teacher on another bb I play on that hates the fact that I'm unimpressed with the school system... shocked, eh??

I mentioned once about the lack of feedback to parents and children, citing the example below w/ my son and our afterschooling...

Her answer:
"There was no need for it, children knew what they could and could not do".

This was the topic we were discussing at the time...
Hence we as parents should "trust them" b/c they know more than we ever will... (cough, choke, eye-roll)

farmwifetwo said...

What are we teaching permimeter and area in Gr 3, YET.... still haven't taught children to add, sub, mult, div.... Ohhhhh... but they are to bring a calculator to school starting Gr 3.... sigh...

Cranberry said...

Allison, I was thinking of arranging the numbers into sets. It's a very low level task, though, and to perform it in a group doesn't increase its complexity. It's also perilously close to the Sesame Street exercise, "which of these things is not like the other." Also, there are right and wrong answers, of course. There are answers which are beyond the concerns of the math classroom, such as, "these words all rhyme." For some grade levels, the concept of odd and even numbers is new, as might be the concept of multiples.

"Why do we accept sending kids with enormous dispersion to a single class room? It's because we have no structure to do otherwise! "

I don't know why the public schools have accepted and promoted extreme inclusion. I don't think it serves any child in the classroom well. The worst answer is extreme inclusion coupled with group work on problems, or coupled with extended class discussions of one (simple) problem. In that case, only a few children actually do the work. From my eldest child's descriptions of group work in math, with a heterogeneous group, the best case is for one child to do all the work, and the others to approve. The most frustrating case is to spend time debating the right answer--and to lose, because one's outvoted, so the group chooses the wrong answer. That is, of course, if you're fortunate enough to end up with a group which can read, understand, and accept the instructions.

If a school used this sort of scattershot approach on a consistent basis, can the teachers be said to be teaching? There's an assumption built into this assignment, that the best students will be the most compliant. I know a fair number of really intelligent people, and none of them are wild about wasting effort on simple problems. They're more likely to come home and complain to their parents that the math is too easy, boring, and disorganized. And they'd be right.

Cranberry said...

"Presenter actually stressed to us that "research shows -students cannot learn more than 10 vocabulary words A WEEK!"

So, how do college entrants develop the necessary 11,000 to 14,000 root words? 10 words per week x 36 weeks = 360 words per year?

If students do not come from an educated household, which encourages vocabulary growth, such prescribed rates of vocabulary instruction would mean that schools choose not to prepare their students for college. By definition.

Katharine Beals said...

Allison, Thanks for this post, which gives me a much better idea about what educators mean by "differentiated instruction" or "differentiation." In particular, I hadn't fully appreciated its connection to "open-ended" assignments.

Besides all the concerns raised above, another big problem with this model is that it drastically shortchanges under-achievers, which it seems to assume simply don't exist. I know many math whizzes who are lazy and unmotivated enough in class to do the bare minimum asked of them.

Returning to my point in an earlier post, I suspect that an underachieving child perceived by his/her teacher to be of (upper) middle class background and of Northern European extraction will tend to earn very low grades in this system (which, in many cities, will severely limit his or her opportunities for a decent high school education, to the extent that such opportunities exist at all).

farmwifetwo said...

"So, how do college entrants develop the necessary 11,000 to 14,000 root words? 10 words per week x 36 weeks = 360 words per year?"

My son's Gr 5 class is doing 4 words/wk. The Gr 3 class is doing 10.

lgm said...

Very interesting and thanks for posting.

The bigger problem here is that students choose to do nothing, not something when the work is insultingly below their abilities, as it is in full inclusion.

>>She also offered that the best students would challenge themselves and do all of the problems, so this provided a way to meet their needs for more or harder work.

Ah, she buys into the 'busywork=challenge' equation. The braniacs don't. I guess they are supposed to go elsewhere since they can't accelerate or do independent study in the 'fully' included classroom.

Cranberry said...

I think that if the assignment goes home, parents in the know will make certain that it's all filled out, even if they or a tutor have to step in. I was much more naive when I was younger. I now do believe that in certain households, the work gets done--by someone.

This (if my cynical take is true) automatically gives the "best students" crown to the children with the most aggressive parents. It also tilts the field toward the most affluent in the group, as tutors are expensive, as is the time to do your child's homework. (Which, to be clear, should never happen.)

Katharine Beals is correct, however, that grading practices have consequences for high school placement. The teachers don't intend it, but certain practices or strongly held beliefs do tilt the field toward the students whose parents have firm control over their academic work, especially if homework is graded.

And, group work, if graded, will decrease the class average of the naturally academic kids, even if they're extroverts. Pair an A kid with a B kid with a D kid, and the average product won't be an A project--unless the A kid does all the work.

Finally, I think I'm perceiving a terrible trend toward academic summer camps, in which kids take algebra at camp, before it's covered in school. There's nothing wrong with taking courses in the summer from natural curiosity. There is quite a bit wrong with taking the courses in order to game the class placement. You will naturally get better grades in a course you're taking for the second time. Homework grades will nudge you before the child of like ability who--more fool he--didn't take the class before the class.

kcab said...

The thing about differentiation is, it's better than NOTHING. In the hands of a dedicated, smart teacher, it's a whole lot better than nothing. Unfortunately, that is too often the only other choice which is available.

Though I suppose one could argue that getting nothing might be better, since that would push parents like me to pull their kids out entirely. That's what will likely happen next year in my family. I do agree with the objections above regarding best students "challenging" themselves by doing all the problems.

cranberry - regarding academic summer camps - they seem like a natural consequence of schools which don't teach. My older child is taking algebra this year and I can't see why anyone thinks that should be the name of the course. I expect to have to cover the material with her separately, most likely through an on-line course since she doesn't like to be taught by me.

farmwifetwo said...

We shouldn't have to pull them out to get success. We shouldn't have to have conversations like I did with the Spec Ed dept about alternative placements - who btw returned my call within 24hrs, WOW!!! - and then write emails wrt that phone call to the school.

But we do... and the call went well, so let's see where we go from here.

Cranberry you are correct, all you have to do is see the beautiful science/social studies projects that children present to know exactly who did the work... it wasn't the child...

Cranberry said...

Kcab- I agree that the growth of private providers of academic services is linked to what is, or what is not, happening in the public school classrooms.

I think that a few teachers can do differentiation well. I think there's a natural limit on the span in students' preparation. Beyond a certain range, (guess) of 18 months in ability/preparation, it becomes a way of preventing parental complaints (in my opinion.)

The kids at the upper end of the performance range are bored, and resentful. Their parents may pull them out (as we have done). The kids at the lower end of the range are in worse shape, though. What do the teachers expect them to be able to do by the end of the year? How can you measure progress, if the easiest approach to any problem is always available. There does seem to be a trend towards direct instruction (small d, i--i.e., non-progressive teaching methods) mandated in IEPs. Whom does this method of instruction benefit? Who learns more under this plan than in a tracked system?

At least in a tracked system, the parents of the kids on the lower end could determine what the teacher or school expects of their children, and they can agitate for higher standards. I don't know if that's possible in such a loosy-goosy setup. "Every answer is right." "No one fails." How does that translate to high expectations?

kcab said...

cranberry - yes, I agree, the range possible to cover is probably less than 2 years either side. Obviously a problem since the students readiness is likely to cover a wider range, especially by upper elementary.

I won't argue against ability grouping, would much prefer that to be used.

SteveH said...

I keep realizing that not much of what I've said is new. You could probably go back years onto the original KTM blog and find that I say the same things. It's almost as if I'm in a constant state of disbelief. How can I achieve anything constructive if our schools are on another planet? There is no process, and in our area, little or no choice.

Laura said...

Doesn't the solution to the teaching problem have to be the same as the solution for when a child is misbehaving in the classroom or struggling to get work done? The right reinforcement scheme? I mean, I understand and relate to the frustration, but ultimately, it doesn't matter if teachers or administrators or even parents *want* to do the right thing--it just matters that they are given the tools they need to do it, are properly motivated to do it, and that it becomes self-reinforcing.

There must be some way to break the larger goals discussed on KTM into smaller components and find ways to reinforce teachers/administration for implementing those components.

Allison said...


Yes, it's important to reward the behavior we want, not the behavior we don't.

What rewards can we create? What positive reinforcement can happen?

How can we apply the "what gets measured, gets improved" to school boards, administrators, teachers, etc.?

The first bit is to notice that parents in their role as parents are the least able to create these reinforcing structures, because parents take too many years to recognize what's going wrong, and have or will age out of the system before the can create a critical mass of structured opposition. Their children could easily become victims of their actions as well--not just in the sense of a backlash, but in the sense of "we're going to experiment"; that's pretty risky--who wants to take the risk that some actions MIGHT improve results for their kids, but MIGHT make them worse?

They can't create a phalanx, either, because they have no way to pit teacher against administrator; they can't demonize anyone because their children are in the mix.

PalisadesK has talked about this a lot; she thinks the place to put effort into change is at the teacher level, by teachers, so to speak.

I have decided something similar, and am currently creating a math institute to teach some actual mathematics to the middle school math teachers in my area--Wu is going to give a conference for a week, and I will do the rest. I'm only working with schools whose viewpoints are already in line with my own, but the idea is that I'm making those schools a lot stronger. Other people may be able to create different methods for positive reinforcement.

But always remember that the education industry has so much money and so much inertia that it's positively reinforcing failure. The more parents shout "save our schools!" the more the current educational worldview can co opt that concern and turn it into money for their own failing schools and their own failing policies. I don't think anything can fix that, but then again, perhaps a coming hyperinflation that wipes out all middle class entitlements will have an upside after all.

SteveH said...

" am currently creating a math institute to teach some actual mathematics to the middle school math teachers in my area"

Great! I look forward to hearing more details sometime. It's nice to know that there are some schools in your area that share your viewpoint. I wish I could say the same in our area. By the way, do these schools use any form of full inclusion? How close are they to your viewpoint? Will you be dealing with schools that use Everyday Math?

"The more parents shout 'save our schools!' the more the current educational worldview can co opt that concern and turn it into money for their own failing schools and their own failing policies."

In our area, all issues are translated into money. It's not a matter of doing more with the same amount of money. That would imply that they are doing something wrong. All problems are explained away as a need for more money.

Around here, my hope is that a few charter schools might show the way. This is unlikely to happen. We have very few charter schools in our state and the only ones that are approved are for kids that the regular schools have failed. The only semi-rigorous charter schools are high schools and most of their kids are just struggling to recover from a bad K-8 education.

Even the fancy private schools use Everyday Math. They can get away with it because the kids get so much support at home or with tutors. At the private school my son went to, the headmaster always pointed to the tony prep schools the students went to as the ultimate proof of what they do.

Our state requires that all of our 7th and 8th grade teachers must be certified in the areas that they teach. I believe that this was one of the reasons that our school finally got rid of CMP and offered a real pre-algebra and algebra sequence. The problem now is the non-linear jump that Wu talks about when kids have to make the leap from K-6 to the rigors of algebra.

K-6 math is the big philosophical nut to crack. How do you get them to understand the critical importance of timely mastery of the basics? Although our 7th and 8th grade teachers know something about math, and even though they see kids poorly prepared in the basics, nothing seems to drive changes back to the lower grades. They don't want to rock the pedagogical boat.

If schools are like-minded, then I can see how teacher education will work. But for schools that have other ideas of education, how can anything work? Maybe if they see what is being accomplished at the school down the road, that will have an impact, but in our area, there are no schools down the road. Maybe we will have to look far down the road to MN.

Laura said...

That sounds terrific, Allison. The schools involved are very lucky to have such a great resource made available to them.

I wish I had access to something like that for K-6 math. I am supposed to start looking for a job soon (not that there's any great danger of getting one in this area), and I am very worried about the idea of someone with my (unfortunately) low level of math knowledge trying to supplement Investigations and make sure all the students are getting what they need.

More generally, though, I do wonder if it would be possible for parents to create some kind of incentive for teachers--some kind of fund that could dole out money for things teachers want (for the classroom, not as individuals) in exchange for them being willing to implement some kind of assessment system for particular skills. Probably not very realistic, but I do wonder.

Laura said...

I've been thinking about it some more, and what about creating some kind of program like the Accelerated Reader program, though not specifically for accelerated students. Students could get points for passing computer-based assessments, and schools could have rewards for individual students, and then schools could submit all of their points as a group, or something like that. Points might not be just for how high students score, but could also be rewarded for rate of progress, or something along those lines.

California Teacher said...

Allison said:
"I don't think anything can fix that, but then again, perhaps a coming hyperinflation that wipes out all middle class entitlements will have an upside after all."

That is an intriguing comment, but I don't understand your point. Can you elaborate on what you mean by "middle class entitlements", what problems they cause (relative to schooling I assume), and what would happen if/when they are wiped out?

I am asking because I teach in a school that serves a fairly even mix of upper-middle class and working poor. We are in a semi-rural suburb (California) with a large population of immigrants from Mexico who work in local agricultural and service jobs. About 55% of the students are low-income immigrants and the rest are middle/ upper-middle class, mostly white, with a sprinkling of other ethnicities.

Our school offers a Spanish Immersion strand that attracts the upper middle-class families. They will mix with the Latinos as a benefit to their own children in acquiring Spanish, but will not place their kids in the regular English strand because they perceive that mixing with the Latinos in a regular English class would be disadvantageous, due to the "inclusion" issues which are being discussed on this thread.

Hence, back to my original inquiry.
Thanks for the response...

Lisa said...

Oh my, and this passes for professional development? It explains why my kids still in PS do the whole group's project and why my HIGH SCHOOLERS are still making posters for heaven's sake. (These are straight A students taking AP classes.) I'm sure that method would not have worked for me. I would have found the bare minimum of work to get the grade I wanted and gone back to whatever novel I was reading. What about the bright but bored kids? I guess they don't matter. They'll so fine on the state tests anyway.