Category Archives: Curriculum

The hexagons are here! [#nctmnola]

Forgive the delay. Here are pdf files of the hexagons we built for use in my hierarchy of hexagons lessons. You should be able to open and edit them in Adobe Illustrator. Consider them CC-BY-SA.

Set 1 (pdf)

Set 2 (pdf)

Shout out to former students Jen Carlson, Nadaa Hassan and Brenna Magnuson for collaborating on these.
Creative Commons License
Hexagons by Christopher Danielson is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Update: Below is the current complete set, with added hexagons from former students Ruth Pieper, Brandon Schwab and Mona Yusuf.

hexagons.complete.set

The latest “Common Core” worksheet

You have seen this on Facebook.

Original (Click to enlarge)

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…

The intended answer

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

The purpose of this task

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:

  • Analyzing errors is a useful way to encourage metacognition, which means thinking about your thinking. This is an important part of training our minds.
  • The number line here is a representation of a certain kind of thinking—counting back. The number line is not the algorithm. The number line records Jack’s thinking. He counted back from 427 by hundreds. Then he counted back by ones. He skipped the tens. We can see this error because he recorded his thinking with a number line.
  • Coincidentally, the calculation in question requires no regrouping (borrowing) in the standard algorithm, so the problem appears deceptively simple in its simplified version.
  • This task is intended to help students connect the steps of the standard (simplified) algorithm with reasoning that is based on the values of the numbers involved. Why count back by three big jumps? Because you are subtracting 300-something. Why count back by six small jumps? Because you are subtracting something-something-6. Wait! What happened to the 1 in the tens place? Oops. Jack forgot it. That’s his mistake.

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.

The Common Core State Standards dictate teaching the standard algorithms for all four arithmetic operations.

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.

A task

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]

Brief thoughts on being ready for calculus

A smart friend (whose permission I have not asked) read an article of mine that will be published in Mathematics Teaching in the Middle School sometime soon. The article is based on my NCTM talk last spring, titled “They’ll Need It for Calculus”.

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:

 

“I can teach them more math than they have done in their entire lives”

Here are two conversations for which I have no patience at all…

  1. “They should know X”
  2. “You should teach X in manner Y.”

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.

 

Decimals and curriculum (Common Core) [TDI 6]

The Decimal Institute is winding down. This week, I have a short post outlining the relationship between our discussion these past weeks and the Common Core State Standards (with links). Then next week we will wrap up with a summary of what I have learned and an invitation to participants to share their own learning.

Screen shot 2013-11-04 at 9.27.08 PM

The Common Core State Standards build decimals from the intersection of fraction and place value knowledge. Fractions are studied at third grade and fourth grade before decimals are introduced in fourth:

Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

One of the issues we have been wrestling with in the Institute has been how much decimals are like whole numbers and how much they are like fractions. In light of this conversation, I found the following statements about comparisons interesting.

  • CCSS.Math.Content.1.NBT.B.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
  • CCSS.Math.Content.2.NBT.A.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
  • CCSS.Math.Content.4.NBT.A.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

These all refer to comparisons of whole numbers—at grades 1, 2 and 4. Comparisons of decimals appear at grades 4 and 5. For example:

  • CCSS.Math.Content.4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. [emphasis added]

The phrase, Recognize that comparisons are valid only when the two decimals refer to the same whole, struck me as odd. If I am comparing 0.21 to 0.5, I need to make the whole clear, but if I compare 21 to 5, I do not?

This seems to be an overcommitment to decimals being like fractions rather than like whole numbers. Or not enough of a commitment to the ambiguity of whole numbers.

In any case, the treatment of decimals in the Common Core State Standards is probably one of the major challenges for U.S. elementary teachers, who may be accustomed to curriculum materials that emphasize the place value similarities of decimals to whole numbers rather than the partitioning similarities to fractions.

I will provide some examples of pre-Common Core U.S. curriculum in the Canvas discussion to support this claim. Join us over there, won’t you?

Non-U.S. teachers, please share with us your observations about how these standards relate to curricular progressions you are using. An international perspective will be quite useful to all of us.

And please start thinking about what you can do in the coming weeks to share/demonstrate/document/extend your learning from our time together. Consider it your tuition to the Institute.

Van Hiele levels in my geometry instruction

This mathagogy video has generated some behind-the-scenes discussion in recent months.

This week, I got a note from Chris Hill with some questions. I wrote him a long email in response. Here it is:

First a disclaimer.

I have never taught high school geometry.

I taught middle school, including some geometry there, have written some middle school geometry tasks, and I now work with future elementary teachers. That last is the context in which I developed my hexagon thing. The sequence I describe in the video is the end result of several years worth of trying to work out a way to get my students to geometric proof.

Of course there is a lot more involved in the process than can be conveyed by a short video. Have you seen my blog posts on the sequence?

And now your questions:

Do students develop the ability to move up the levels through the entire year, or on each unit?

I think of the van Hiele levels more as an organizing principle for instruction than as a principled diagnostic for students. But the levels do serve both purposes.

Here is what I mean…I do think that most of my students (high school graduates; some young, some older; all students at my community college) come at a pretty easy to observe Level 1. They know names and properties of shapes but cannot talk much about how these properties relate to each other. That, I think, is pretty standard for entering high school geometry students too.

I do not think that they become Level 3 geometers over the course of our work.

Instead, I use the van Hiele levels as a way to organize instruction. So we start where they are with some noticing and naming of properties. Then we move to a tremendous amount of level 2 work. I probably describe some of that in the blog posts, and I should probably write up more of that.

Over the course of several class sessions (we meet for 80 minutes twice a week), I push a little bit harder, and then harder still, on the arguments students are making. Eventually, through this modeling and through my explicit talk with them about it, they begin to be more critical of their own arguments, and eventually we reach a question that sort of requires proof; it seems true, but is non-obvious, and it has arisen from the questions we have been asking (at level 2) about how properties relate to each other. Creating and organizing the hierarchy of hexagons is often a useful part of this sequence.

And then we move on. If we had the whole semester (or whole year!) to spend on geometry, I would imagine that level 2 work would be an important introduction in each unit. This is Serra’s conjecturing phase in the Dan Meyer comment.

Is it reasonable to expect every high school geometry student to get to the top two levels within a unit (like circles)?

I don’t think the top level (level 4, rigor) is a reasonable goal for every student. Sure, you can do Taxicab Geometry and notice that “circles” there look like Euclidan squares, but I am not convinced that this has a lasting effect of rigor on students.

I do think having all high school students (for whom future elementary teachers are my proxy) use their ideas about relationships among properties of shapes to build a more formal argument that a proposition must be true is reasonable. Different students will be capable of different levels of sophistication in terms of the propositions they argue, and in terms of the arguments they make, but they can all do some level 3 work for sure.

If I try to talk about the levels explicitly, do you think high school students will be insulted, not care or would they feel validated when they take a long time to get to a higher level?

This is a good question. I debate this a lot in my own teaching. How much of the motivation and justification should I provide my students for what we are doing? I tend to talk about things in that spirit—we are doing X today because it will address needs that I know we have; here is how I see this activity getting us to our goals. And I try to provide specific evidence and the source of that evidence. A research basis is a good source of evidence, as is student work on quizzes and things kids say in whole group discussions.

I would not spend time telling students what van Hiele level they are at. And I might not use the actual language of the van Hiele levels with kids (but then again I might).

Rather, I might say something like this:

Our goal is to make formal mathematical arguments (proofs) in this course. Mathematical arguments take a different form, and have a different standard than a scientific argument or an argument you may have with your mom. But that is a goal, not our beginning place.

I know from research I have read, from my experience in previous years, and by observation of your work and the ways you talk about geometry that an important part of getting to that goal will be exploring properties of shapes and how those properties relate to each other. So we will do a lot of that work.

We will argue with each other. And over time, I will help you to make these arguments more mathematical. Eventually, you will be able to make such a great mathematical argument that we can call it a proof. Some of these proofs will be developed and written by the class collectively, and I will ask you to try do write some on your own. But it will be a while before we get there.

Then, when we do get an example of a solid mathematical argument (whether because the class developed it or because I provide it), I make sure to stop, notice it and to have a conversation with the class about how that argument is different from the other work we have been doing.

Do you know of any standard assessments that I can use to track my students’ progress with these levels?

No. I ask my students to make short arguments on quizzes, and try to pose questions that admit both informal and more formal arguments. But I don’t know of good vH level instrumentation. It may exist. But I don’t know about it.

Ready for Calculus

I’m working on a paper that will be submitted for publication. It mostly records and expands on my NCTM presentation in Denver a while back.

need.it.for.calc.08

Along the way, I decided to a bit of looking into the claims I have made about the standard view of what it means. I did it the lazy way. Google search on “Ready for Calculus”. 

See for yourself. (Click on the images to be taken to original sources on the web.)

Exhibit A

Screen shot 2013-06-26 at 4.11.13 PM

exhibit B

Screen shot 2013-06-26 at 4.12.32 PM

Exhibit C

Screen shot 2013-06-26 at 4.15.57 PM

 

Exhibit D

I did find one that had a different nature

The most important precalculus concept is the notion of a functional relationship between two variable quantities. This relationship may take many forms: linear functions, power functions, exponential functions, logarithmic functions, trig functions, polynomial functions, rational functions… Functions from these basic families may be combined, transformed, and inverted to produce still more functional forms. Functions also appear in various representations: formulas, graphs, data sets… You will have to be familiar with the basic families of functions, and all of their representations, in order to succeed in your study of calculus. The concept of function underlies everything that calculus considers.

This was nice, and if you’re at all interested in this topic, you should go read the essay in its entirety. (Don’t worry, it’s short.)

Conclusion

And now let’s all imagine how a community college developmental math program that took Exhibit D more seriously than Exhibits A—C would be different from the present-day state of affairs.