You are baffled by the worksheet too. You are about to click the Share button.
But here are five reasons that’s a bad idea.
Almost invariably, the parent who shares the worksheet cites a degree as a credential for the critique. “I have a Bachelor of Science in electronics engineering,” wrote the parent in the recent “Letter to Jack”.
The math you need to know to be an electronics engineer is different from the math you need to know to be a math teacher. I am quite certain that electronics engineers use math that I have not studied, and similarly I use mathematical ideas that they have not studied.
When my teacher friends watch the following video of my son Griffin, they tend to see that a number line would be the right thing to draw to capture his thinking, and they tend to know what is coming when he goes to solve the problem on paper. They tend to describe my son’s work as demonstrating competence or proficiency with subtraction, but suffering mechanical errors when solving with paper and pencil.
Non-teachers who view this video are less likely to see a connection to the number line, and they tend to consider Griffin’s knowledge of subtraction to be weak.
The difference is that teachers have a different kind of mathematical knowledge from electrical engineers. Not necessarily more or less knowledge—different knowledge. This is because different mathematical knowledge is required to do their job.
Deborah Ball refers to this different knowledge as mathematical knowledge for teaching. It is what mathematics teachers know who are more successful in their work. This knowledge includes common errors with standard algorithms, as well as their sources [start at about 2:30 in this video]. It includes common correct, alternative ways of performing and showing computations.
Moral of the story: You may not want to look to an electrical engineer as your primary resource for the current state of math teaching.
Homework time can be stressful. This is not new to Common Core.
Parents are trying simultaneously to be helpers and enforcers. When a child does not understand what appears to be something simple, tempers can flare.
We parents are not at our most rational at these times, and this may prevent us from fully understanding the goal of the task.
When Frustrated Parent wrote to Jack, he committed two important errors of misinterpretation: (1) He assumed that Jack’s method was being taught as a preferred algorithm for subtraction, and (2) He assumed that something unfamiliar to him must be complicated.
These are totally understandable. I do not hold Frustrated Parent in contempt for his frustration.
I am simply asking the rest of us to resist sharing without asking critical questions.
The strategy shown on this particular worksheet is counting back.
This is how many people count change: You gave me $20.00 for a $3.18 item, so you get $17 minus $0.18 in change…that’s $16.90 minus $0.08…$16.82.
The number line Jack drew captures this thinking.
The number line could have been improved by (1) arrows on the arches, and (2) smaller jumps to suggest that Jack is counting by a small number (as it is, the relative sizes of the pictured jumps on the number line suggest Jack is counting by 10s or 20s to the left of 127—as Frustrated Parent notes in a later Facebook post).
This worksheet was not about getting students to use the number line as an algorithm. It was about having students try to understand the thinking of someone else.
This may have been a bad worksheet—but not for the reasons cited when people share it on Facebook.
[For the record, in this case Jack’s error was forgetting to subtract the 10 in 316. He counted back three hundred, then six ones. The result is that his answer (indicated on the left-hand end of the number line) is too big by 10. He gets 121. The correct answer is 111.]
There is nothing in the Common Core State Standards that requires students to use number lines to perform multi-digit subtraction. In fact, standard 4.NBT.B.4 requires students to “Fluently add and subtract multi-digit whole numbers using the standard algorithm”.
The standard algorithm, of course, is what the frustrated parent suggests that Jack use.
While parents who share these worksheets in frustration will make claims such as the old way worked for me, the research evidence is quite strong that the old ways did not work.
Changes being made to American mathematics teaching are (1) very slow to take root, and (2) based on years of American and international research on student learning.
The frustration and anger of a parent who is struggling to help their struggling child is completely understandable. Any parent who claims they have never been frustrated at homework time is living in (a) a fantasy world, (b) denial, or (c) both.
But when we widely share the product of others’ frustration online, we amplify the anger. Ultimately, classroom teachers are the targets of this anger, as they are the public face of the education system. As a group, teachers work very hard with limited resources. They are called upon to equalize the inequities our society creates, and to offer not just equal educational opportunities, but equal educational outcomes to all children.
Now—more than ever—teachers need our support, not our scorn.
If you are Frustrated Parent, you can write a level-headed note to the teacher. It might look something like this:
Dear Ms. Crabapple,
I worked with my child on this problem tonight. Neither of us could figure out what is going on with the number line. You can see the work we did together, but we did not know how to write an explanation to Jack. We are confused. Please help.
Sincerely,
Frustrated Parent
If you run across the work of another Frustrated Parent online, please consider asking someone about it before sharing it as evidence of the decline of American mathematics education. Some possibilities:
Ask a teacher friend. You probably have at least one on Facebook.
Ask on Twitter. There are many eager-to-help math teachers who follow the hashtags #mtbos and #mathchat—sincere questions asked on those hashtags will get sincere answers and offers of help.
Ask on a website. The Mathematics Educators stack exchange is a new resource for people to ask and answer questions related to teaching and learning mathematics. Anyone is welcome to ask a question, anyone can answer, and everyone votes on the quality of the answers so that you can easily find the best ones.
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Note: Things got far beyond my ability to curate in the comments, so I needed to turn comments off. I would be more than happy to take up the dialogue on Twitter or through a pingback to your blog. You can also contact me if you wish to discuss further—Hit the About/Contact link at the top of the page.
Second note: I will curate and organize the major threads of the comment discussion in the coming days. In the meantime, I have sequestered the existing comments as the discussion threatened to overwhelm the point of the initial post. I have not deleted them.
]]>Posted without comment.
Thanks to David Radcliffe for passing it along on Twitter.
]]>Ugh what a mess.
Please share the annotated version widely.
I’ll say what I have to say (comments closed) and move on. If you wish to discuss further, hit me up on Twitter or pingback to the blog. Want to talk in private? Click the About/Contact link up top.
Also, Justin Aion—middle school teacher extraordinaire—wrote up his views on the matter. You can read them over in his house.
Here goes…
Dear Jack,
You only subtracted 306 from 427, not 316. You need to subtract another 10 to get the correct answer of 111.
Sincerely,
Helpful student
I cannot say whether this was the right task for this child at this time because I do not know the child, the teacher or the classroom.
I can say the following:
So what?
The Common Core State Standards do require students to use number lines more than is common practice in many present elementary curricula. When well executed, these number lines provide support for kids to express their mental math strategies.
No one is advocating that children need to draw a number line to compute multi-digit subtraction problems that they can quickly execute in other ways.
But the “Frustrated Parent” who signed that letter, and the many people with whom that letter resonated, seem not to understand that they themselves think the way Jack is trying to in this task.
Here is the test of that.
What is 1001 minus 2?
You had better not be getting out paper and pencil for this. As an adult “with extensive study in differential equations,” you had better be able to do it as quickly as my 9-year old.
He knows with certainty that 1001 minus 2 is 999. But he does not know how to get the algorithm to make that happen.
If I have to choose one of those two—(1) Know the correct answer with certainty based on the values of the numbers involved, and (2) Get the correct answer using a particular algorithm, but needing paper and pencil to solve this and similar problems—I choose (1) every time.
But we don’t have to choose. We need to work on both.
That’s not Common Core.
That’s common sense.
[Comments closed]
This friend asked by email:
For clarification: are you arguing that the sorts of problems that you point to will help students better understand calculus, or that these sorts of problems will help students do better in their calculus classes?
I was pretty sure that you were making the first argument, but not the second.
My reply, which I stand by, is this:
That these two things are different from each other is a pretty damning critique of the whole affair, is it not?
You know what will help them do well in their calculus classes? Memorizing about 20 of these:
That is…
Details and discussion coming soon.
In the meantime, see Kristin’s related post.
Full disclosure. Griffin was paid a sum of $0.25 for his performance.
In defining Bobs, Stacys, and the like, did you run into situations where your definition admitted shapes the students didn’t actually want. For example, once you defined a Stacey as a hexagon with three congruent acute angles, did you draw some other Stacys and have students blurt out “wait, that’s not a Stacy — that’s not what I meant!” If not, how did you privilege the *definition* over some sense of “I know a Stacy when I see one.” Is it because their definitions were based on some property they liked about one example, rather than trying to say what was the defining quality of some *group* of hexagons?
Yes, yes, yes.
This instructional sequence is all about moving through the van Hiele levels.
An important learning goal for these lessons is for students to separate what it looks like from its mathematical properties. That means we need to talk about this very issue.
Early in the process, a central challenge is identifying precisely the property the student had in mind. This requires a use of language that can be unfamiliar and strange. Does “has a right angle” include rectangles, which have several right angles? Or did you mean “has exactly one right angle”? That sort of work comes first.
Then we look for other shapes that have this property. If those early-discovered shapes violate the spirit of the original intent, the student may object and will be invited to revise. If she wants to add another property, then I will usually suggest that this is a second class of shapes, and that the one she was really after is the combination of these two classes.
So your shape is special because it has all sides the same length AND it has at least one right angle. Let’s do this…let’s call hexagons with all sides the same length equilateral and let’s give a name to hexagons with at least one right angle…Who has a name for us?
A “sally”? Good. From now on, a sally is a hexagon with at least one right angle.
Now this shape is both equilateral and a sally. Let’s give this special category of shape a new name.
Et cetera.
That’s OK on day 1. Our goal on day 1 of the lesson sequence is to have a set of between five and eight named classes of shapes that have interesting interrelationships.
I need to be the judge of when we have enough, and whether they are of sufficiently interesting variety (I have screwed this up before). I reserve the right to add in some properties that I know will be interesting. For example, I make sure concave and equilateral make it into the mix somehow—either by student introduction or by my own.
After day 1, we need to move away from what things look like. We will be operating only on the properties as we have defined them. If we accidentally left ambiguities, we can plug those holes as clarifications. But these definitions cannot otherwise change. It is important to notice that shapes with very different appearances can share important properties.
It is important to notice that mathematical properties of shapes behave differently from the look of a shape.
For example, this semester we defined a class of hexagons this way.
A windmill is a hexagon with three acute angles and three angles greater than 180°.
The iconic windmill is this one.
Much later, as we were in the process of trying to decide whether a windmill can be a utah (a utah is defined as a hexagon with two sets of three parallel sides), we happened upon this most un-windmillish object.
In trying to push the limits of windmill-ness, we started to consider shapes that had the right properties but that looked nothing like the original shape.
For what it’s worth, we have produced a proof—which some of us can reproduce and others cannot—that a windmill is never a utah. This proof depends on the Dani principle, which states that when two sides of a polygon are parallel and separated by a single side, then the two angles formed sum to 180°. This happens twice in a utah, which accounts for 4 angles less than 180°, leaving (at most) two possible angles to be greater that 180°, and thus a utah is not a windmill.
It should further be noted that the Dani principle needs amending to admit the possibility of angles that sum to 360° (when one of them is greater than 180°) or 540° (when they are both greater than 180°). And it should be yet further noted that standards for proof are socially negotiated in all areas of mathematics, and that we had become quite confident that these cases would not come up with a utah, thus the weak version of the Dani principle was good enough for our work.
Finally, if you have made it this far, you may be interested in the speculative origins of the hierarchy of hexagons. It has been a long road.
I will be presenting the hierarchy of hexagons at NCTM in New Orleans. Friday morning at 8:00 a.m. See you there?
This has made the rounds on the Internet, and it has angered lots of folks in education. And rightly so. Because there is no learning going on in that video.
But those teachers are being trained to deliver that sort of instruction to students in classrooms. Go ahead and search EDI or whole-brain teaching. You’ll see these very techniques being promoted as good practice.
So, is it how people learn, or is it not?
Tip of the cap to David Wees for reminding me that the parallels are not necessarily obvious.
]]>Conversation number 1 is the reason they are students, and the reason you are a teacher. What they should know is not relevant here; only what they do know is. So let’s factor that into our instruction.
But I need to focus on conversation number 2 today. EdSurge reposted an article today that struck me the wrong way. An excerpt:
Of course, the problem is deeper than a handful of students who accidentally say ironically stupid things. The problem is that American high school students are taught something named “math” for four years which is not even close to math.
Pretty sweeping generalization here. But I don’t disagree with the basic premise, which is that we aren’t doing the job of bringing mathematics to students (and students to mathematics) that we should be doing. I do disagree that the K-12 system is the only place this problem exists, but let’s get back to the matter at hand.
I fear my rant may disguise my true intentions: the problem is not the content. Geometry and calculus and algebra are very fine subjects of mathematics. The problem is that they’re taught in a way that strips out all the math and leaves a vapid husk of an education.
Now things are starting to spin a little bit out of control. Vapid husk of an education? Wow.
And the solution?
[I]f you give me an hour with a group of disillusioned but otherwise motivated high school students, I can teach them more mathematics than they have ever done in their entire lives. I can give them a dose of critical thinking and problem solving like no algebra problem can.
Child, please.
I teach at the college level these days, so I am accustomed to this sort of bravado. I try (perhaps unsuccessfully) to avoid it in my own writing because it is (a) unproductive, and (b) false.
But my beef isn’t so much with the author (although…) No, my beef is with EdSurge.
Why not feature the vibrant work that is going on in K—12 math education?
Why not republish Fawn Nguyen’s brilliant reformulation of a crappy textbook problem?
Why not post Andy Schwen’s video of a kid talking about the relationship between slope and rate of change while working on Function Carnival?
Why not feature the work of people trying to bring real mathematics to young children? Moebius Noodles, Math in Your Feet, Talking Math with Your Kids, Math Munch—these are projects where people are working on a daily basis to help parents, teachers and caregivers to support meaningful mathematical thinking for children. No bravado. No blame. Just hard working, thoughtful people working to solve a problem.
Because there is a problem. For sure there is a problem.
But an hour with Professor Awesome isn’t going to solve it.
Oh dear. Class inclusion is giving some of my elementary kiddos fits. Do we have any post-Piagetian research on this?
— Christopher (@Trianglemancsd) February 7, 2014
I suspect some of it is linguistic and probably appears not just in their math work but in their writing too.
— Christopher (@Trianglemancsd) February 7, 2014
Things quickly got too nuanced for Twitter.
An example of something my students struggle with is answering a question such as, Is a square a rectangle?
This type of question asks about class inclusion. Is an element of a subset also an element of the larger set?
Many useful and interesting questions in geometry have to do with whether one class is a subset of another class. Do all isosceles triangles have a pair of congruent angles? Are all quadrilaterals formed by connecting midpoints of other quadrilaterals parallelograms? Are all Stacys concave?
I am trying to sort out the extent to which my students’ struggles with questions of this sort are linguistic, and the extent to which they are about struggles with the idea of class inclusion.
Justin suggested this wording, which I will investigate:
Is a square an example of a rectangle?
Or, more generally:
Is an X an example of a Y?
My suspicion is that this will be helpful for some students when asked in this direction. But I also suspect that asking it in the other direction will be problematic.
Is a rectangle an example of a square?
See, part of what I wonder about is whether class inclusion—and the fact that it doesn’t have to be symmetric—is at the heart of a particular kind of struggle in geometry, and whether this is also related to the ways students think about and use language.
I hope these three (and others) will weigh in here where we have more space to work than we do on Twitter. The ideas are really useful. If you’d like to follow the prior discussion, you can follow this link.
See, there is this thing called Twitter Math Camp. It is professional development by teachers, for teachers—nearly all of us connected through Twitter. It takes place this summer near Tulsa, OK.
I am presenting with Malke Rosenfeld. Our official description is copied below.
Malke and I have developed a really productive collaboration this year. You can browse both of our blogs to see the kinds of questions and learning this collaboration has developed for each of us.
Here is my pitch for our session…
We are planning a session that will force our groups (including ourselves) to wonder about the origins of mathematical knowledge. We will question our assumptions about terms such as concrete, hands-on and kinesthetic.
We will participate in mathematical activity both familiar and strange—all in the service of better understanding the relationship between the physical world and our mathematical minds.
We will dance.
We will make math.
We will laugh and possibly cry.
Below is an example of Malke’s work. When I participated in a workshop last summer, my head was spinning with math questions as a result. It’s great stuff and we will use it as a launching point for inquiry into our own classroom teaching.
So if you’re coming to Tulsa, please consider joining us for our three 2-hour morning sessions.
Of course you’ll miss out on other great people doing other great sessions. But you won’t regret it. I promise.
And if you choose a different session (perhaps because you’re leading one of them!), I have a hunch there will be after hours percussive dancing in public spaces. Come join in!
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This workshop is for anyone who uses, or is considering using, physical objects in math instruction at any grade level. This three-part session asks participants to actively engage with the following questions:
What role(s) do manipulatives play in learning mathematics?
What role does the body play in learning mathematics?
What does it mean to use manipulatives in a meaningful way? and
“How can we tell whether we are doing so?”
In the first session, we will pose these questions and brainstorm some initial answers as a way to frame the work ahead. Participants will then experience a ‘disruption of scale’ moving away from the more familiar activity of small hand-based tasks and toward the use of the whole body in math learning. At the base of this inquiry are the core lessons of the Math in Your Feet program.
In the second and third sessions, participants will engage with more familiar tasks using traditional math manipulatives. Each task will be chosen to highlight useful similarities and contrasts with the Math in Your Feet work, and to raise important questions about the assumptions we hold when we do “hands on” work in math classes.
The products of these sessions will be a more mindful approach to selecting manipulatives, a new appreciation for the body’s role in math learning, clearer shared language regarding “hands-on” inquiry for use in our professional relationships and activities, and public displays to engage other TMC attendees in the conversation.