Just a quick note here for the folks who are (a) not on Twitter and (b) not following Talking Math with Your Kids.
I created a shapes book for all ages. The digital version is free for now. Details are in this post over at TMWYK.
Just a quick note here for the folks who are (a) not on Twitter and (b) not following Talking Math with Your Kids.
I created a shapes book for all ages. The digital version is free for now. Details are in this post over at TMWYK.
You should seriously go check out Polygraph. Four versions of a delightful and challenging game:
The hexagons will be familiar to long-time readers of this blog.
I have run the parabolas version in College Algebra, and the hexagons version in my Ed Tech course. It was a huge hit both times—lots of conversation happened both electronically and out loud in the classroom. It’s a ton of fun.
I am especially pleased with the rational functions version. It makes for challenging work—even among the mathematically astute Team Desmos in recent trial runs.
Read the Desmos blog post on the matter if you like.
Back in the spring when the Letter to Jack was a hot item, I took to Twitter to wonder why there was no Common Core Math for Dummies. One thing led to another, I proposed it to Wiley and now you can expect it in the spring.
Audience is parents, and this may appear in the title (Common Core Math for Parents For Dummies is the working title). It goes for the big picture in each of the grade levels, K—8.
The For Dummies format is pretty rigid but there will be no mistaking authorship. A few sample section headings (and the grades where they will appear) to whet your appetite:
1st grade. Saying bye-bye to key words
1st grade. Understanding the importance of ten
2nd grade. Why units matter
2nd grade. Place value
2nd grade. More about place value
2nd grade. Seriously. Place value.
4th grade. Multiplication: What is it and why not just memorize the facts?
5th grade. Standard algorithms: Doing things “the old-fashioned way”?
6th grade. Dividing fractions—More fun than you’d think!
6th grade. Area: It all goes back to rectangles
8th grade. Congruence and similarity: Two kinds of sameness
Catch you all later. I have some writing to do!
I’ll keep you posted.
Posted in News
(NOTE: If you arrived here via Facebook, welcome. You may also be interested in the perspective of a middle school teacher, Mr. Aion. Click on through for more.)
You are browsing Facebook on a Sunday evening. Someone has shared a baffling piece of math homework that was sent home with their child. Accompanying commentary bemoans the current state and future trajectory of mathematics teaching and learning, and lays blame for this at the feet of the Common Core State Standards.
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.
(NOTE: Frustrated Parent is in fact male. A quick Google search will turn up his interview on Fox News and his public Facebook post about this letter.)
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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The New York Times published a piece on “Common Core homework” in July. I wrote a response to it that clarifies a critique in the article about dots. I invite you to read there for more information.
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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 in News
You have seen this on Facebook.
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]
Posted in Curriculum, News, Opinion, Talking math with your kids
Tagged algorithms, ccss, common core, subtraction
I want to use this space to make a pitch for a conference session.
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.
Last summer, the super-smart, super-creative team at Desmos (in partnership with Dan Meyer, who may or may not be one of the Desmos elves) released a lovely lesson titled “Penny Circle“. It’s great stuff and you should play around with it if you haven’t already.
The structure of that activity, the graphic design, the idea that a teacher dashboard can give rich and interesting information about student thinking (not just red/yellow/green based on answers to multiple choice questions)—all of it lovely.
And—in my usual style—I had a few smaller critiques.
What sometimes happens when smart, creative people hear constructive critiques is they invite the authors of the critique to contribute.
Sometimes this is referred to as Put your money where your mouth is. So late last fall, I was invited to do this very thing.
I have been working with Team Desmos and Dan Meyer on Function Carnival. Today we release it to the world. Click through for some awesome graphing fun!
It was a ton of fun to make. I was delighted to have the opportunity to offer my sharp eye for pedagogy and task design, and to argue over the finer details of these with creative and talented folks.
Then let us know what we got right and what we got wrong (comments, twitter, About/Contact page).
Because I just might get the chance to work on the next cool thing they’re gonna build.