Tag Archives: online learning


Here’s a draft syllabus for March’s online course.

This is the first draft and will not be the official one. Participants will get the official one shortly before the course begins.

In the meantime, have a look and let me know what you’re wondering about.

It’s going to be a good old time.


The Mathematics in School Curriculum: Functions

Pilot spring 2012


Christopher Danielson
Twitter: @Trianglemancsd
blog: https://christopherdanielson.wordpress.com

Course Goals

The goals of this course are to broaden participants’ knowledge of (a) curricular approaches to function relevant to the middle school, and (b) the ideas behind the formal mathematical idea of function.

An important assumption behind the content of this course is that item (b) above encompasses both formal/logical components and psychological components pertaining to how both sophisticated and naïve learners think about these ideas.

Course Format

This course takes place entirely online with no requirements for synchronous participation. Regular, daily participation will be essential but time of day for this participation is at participants’ discretion and convenience. See Principles of the course below.

This course will run using the learning management system Canvas from Instructure.

Course materials

All materials will be provided as downloads or links through Canvas.

Principles of the course

  • Full participation will mean agreeing to spend about an hour a day for the duration of the course. The “hour” is an average and is at your convenience. Course activities will include working through mathematics tasks, reading articles, seeking resources and participating in asynchronous online discussions.
  • But full participation is not just about seat time. It is about committing to learning, and to supporting the learning of your classmates.
  • We are here to learn; this will require critical examination of what we think we already know. We cannot be possessive of old ideas—we need to be ready to expand them, to let go of them when necessary, and to welcome new ones.
  • We should seek to appear curious, not smart.
  • We all bring expertise; we should seek to share ours, and to take advantage of that of others.
  • This is not a pedagogy course. We will examine mathematics and curriculum quite closely, but implications for teaching are not the direct product of our activity. Conceptual insight is. Instructional implications will follow. These may require long term fermentation before ripening.
  • We should base our arguments and claims on evidence.
  • We should ask honest questions, and lots of them.
  • Discussions are not ever closed. Continue to contribute to old discussions as we move forward; it would be lovely to have each discussion be a record of our developing thinking.

Work load

Approximately one hour per day for the duration of the course is expected. The “hour” is an average and is at a participant’s convenience.

Course Grade

This course is ungraded and not for college or graduate credit.

All participants adhering to the principles of the coruse above and completing all assignments will be issued a certificate for clock hours towards relicensure. Participants requiring additional documentation of their participation should email the instructor with necessary details.

Summary of activities

Introductory activities: Reading principles of the course, introducing ourselves and exploring the online platform.

Discussion: What is a function? Participants will discuss their own understanding of functions, the ways that they and their students think about functions, and the relevance of these ideas to middle school curriculum.

Tasks: Participants will work a number of paper-and-pencil mathematics tasks involving function ideas. These tasks either come directly from elementary and middle school curricula, or are adapted from them. Sources include Everyday Mathematics, Connected Mathematics and Mathalicious.

Reading: Vinner, S. (1992). The function concept as a prototype for problems in mathematics learning. In E. Dubinsky & G. Harel (Eds.) The concept of function: Aspects of epistemology and pedagogy. Mathematical Association of America.

Discussion: Participants will work to integrate the ideas from the initial discussion with those in the tasks and the reading by considering the question, What images do you carry around pertaining to function? together with the implications of these images.

Task: Participants consider functions graphed in polar coordinates. They begin with a game from Connected Mathematics to develop polar coordinates, and move to simple (i.e. constant and linear) functions.

Create: Participants create a product for public sharing. This may take any number of forms, including (but not limited to):

  • a blog post reflecting on experiences as a learner and/or implications for instruction,
  • a lesson plan (for any audience),
  • an interpretive dance,
  • a work of visual art,
  • etc.

The exact form of the product is not important. The important thing is that it adhere to the spirit of the assignment, which encompasses these two criteria: (1) it should be made public (i.e. shared beyond the course participants), and (2) it should incorporate one or more ideas of the course pertaining to function.

To complete the course, the product—or a link to, or a photograph or other description of the product—must be submitted through Canvas.


Christopher dipping toe in water of online education

UPDATE: The course described below has filled and a substantial waiting list has been closed. I will post updates to this blog as plans for future versions of the course become more clear.

This coming March, I am going to be offering an online course.

It will not be Massively Open. It will not be a MOOC.

Here’s the plan:

I want to gather a medium-sized group of middle-school math teachers to study the mathematics that lurks just beneath the surface of quality middle school math curriculum materials.

I’m talking about big ideas. I’m talking about what if? questions.

I’m talking about starting with the familiar and ending up with the strange. We will stretch our minds to consider new corners of the mathematical world.

If you want Common Core implementation help, go pay your $400 to NCTM. This won’t be that.

If you want machine-scored, lecture-based online instruction, go join a Coursera MOOC. This won’t be that, either.

If you want credit towards your Master’s Degree or other contract-based lane change, talk with your nearest graduate institution. Because this won’t be that.

This will be a community of 20 smart teachers learning with and from each other in a structured environment.

This will be a set of semi-synchronous activities and discussions designed to advance our own understandings of the foundational mathematical idea of function. Eventually, the course will expand to other topics, such as inversesymmetrychange and operation. But this first version will just be function.

What you’ll get:

An instructor (me) with over 15 years of professional development experience, 18 years of teaching experience, an open mind, a whole mess of math knowledge, and a deep curiosity about ideas and ways of thinking.

Also, you’ll get:

  1. Smarter than you are now,
  2. More connected to other smart math teachers, and
  3. A beautiful, suitable-for-framing certificate to assist in relicensure in your state/province/district/etc.

The details:

Course title: The Mathematics in School Curriculum.

Dates: I need 20 middle-school math teachers who are interested in spending about an hour a day thinking about functions for the two weeks March 17—30, 2013.

Platform: The course will take place on Canvas, an Instructional Management System (IMS) developed by Instructure. I have used this system in place of my institution’s adopted IMS (Desire2Learn, or D2L) and have been delighted with its design—especially the way it supports discussion and sharing of resources. Your Canvas account will be free of charge.

Content: We will pilot a functions unit—the first of what will eventually be five units embedded in a larger course. The goals of the pilot will be to broaden our knowledge of (a) curricular approaches to function relevant to the middle school, and (b) the ideas behind the formal mathematical function.

Cost: This pilot will be free. Eventually, I will charge a reasonable fee in compensation for my time and effort (each of which I imagine will be substantial).

Commitment: In signing up for the course, I will ask for your commitment to full participation. We will be looking to build community, and that won’t happen if we don’t commit to the effort together.

Full participation will mean agreeing to spend about an hour a day for the duration of the course. The “hour” is an average and is at your convenience. Course activities will include working through mathematics tasks, reading articles, seeking resources and participating in asynchronous online discussions.

But full participation won’t just be about seat time. It will be about committing to learning, and to supporting the learning of your classmates.

How to sign up: By filling out a Google form. I will cap the pilot course at 20 participants. First come, first served. I know that people’s lives are hectic and variable, so I’ll maintain a 10-person waiting list before ending registrations altogether.

Click here to register for the course.

What questions do you have? Send them my way.

I hope to catch up with you online in March. We’re going to have a ton of fun.

We were promised jetpacks

We were promised jetpacks and we got algorithmically-generated online skill practice instead.

This was in my InBox from a college textbook publisher today:

Enhanced Web Assign is unique to Cengage Learning.  What makes it enhanced?  We offer rich tutorial content, algorithmically generated problems (each student gets a different set of numbers to practice the problems), comprehensive grade book, and unique pedagogical features such as


Unique pedagogical features, huh? Read, watch, practice instead of watch, practice? Ugh.

Help! My parent and my teacher are both apps

People outside of the education profession look at Khan Academy and they see brilliance because Khan conforms closely to American cultural scripts of teaching. Teaching is telling and Khan tells in a friendly, seemingly competent way (which is actually incompetent in some important and non-obvious ways, but more on that in the coming weeks).

This is the drill: Tell students some stuff; ask them some questions to see whether they remember what you told them. See those first two headings on the Khan Academy landing page?

With thanks to Michael Pershan (@mpershan) for noticing this on Khan Academy.

Watch and practice.

People outside of the education profession look at iPad apps and online schools and see efficiency because these-again-closely follow the script of teaching and learning in this country.

So powerful is this cultural script that minor tweaks are seen as revolutionary. (Rewind the video to hear the same explanation again! As many times as you like!) So powerful is this script that our roles as parents can be misconstrued as preparing our children to be this type of student. From the comments on this blog last month:

Around 6-7, I think it is important for children to first internalize basic arithmetic equations as memorized, right-brain pattern recall. Once they do this, their minds are free to think about other aspects of the math problem in front of them. Once basic one-digit equations have been internalized, the next pattern needed is the simple process of stepping through more complex problems.

This was in response to my description of something I had done that had gotten my five-year old daughter to think, rather to respond in a rote way.

A parent talking with his child about mathematics gets redirected to a fact-drilling app.

I’ll make the analogy to literacy again. The equivalent would be a parent writing about how turned on his kid was by a story, and how his kid applied the ideas of that story to thinking about her lived experience. And then got pushback about the importance of phonics instruction to prepare a child for reading in first grade.

But we know that a lifelong love of reading is fostered by reading aloud with parents. (Of course, there are exceptions, blah blah blah.) And if you love reading—barring disability—you’ll learn to read as long as you are provided competent reading instruction.

We need to similarly foster a lifelong love of numeracy. And that does not start with math-drilling apps (Of course, there are exceptions, blah blah blah.) It starts as all things do, through play, conversation and wonder.

Let’s keep the focus, shall we?

Further reading:

Alfie Kohn (of course) on summer learning loss.

Will Richardson on online learning and apps.

Can you calculate your grades by hand?

In college, students and teachers have differing expectations of technology. Teachers typically expect one of two things from online technology:

  1. Increased content consumption by the student, and/or
  2. Decreased grading workload for the teacher.

Students typically expect one of two things also:

  1. Increased access to their teachers, and/or
  2. Increased access to updates on their grades.

There are outliers in both groups, of course. I make a gross generalization in order to make a point-there has been great attention to and investment in educational technology in recent years, but teachers and students are not in agreement on what purposes that technology should serve.

I have written recently about Sophia, an online social learning platform, that relates to the first expectation of teachers and students above.

But on my mind right now is students’ access to updates on their grades.

Every college subscribes to one or another Instructional Management System. Ours is Desire2Learn and it is a mess. (I have complained about it in writing before, and on the radio.)

So I do not post grades on D2L. My students are critical of this and I have wondered why.

I contend that their discontent goes deeper than their expectation of being hyperconnected and instantaneously updated.

I contend that teachers have used electronic gradebooks to make their grading schemes too complex for students to understand. I contend that students don’t expect to be able to figure out their own grades, so they look to D2L to figure those grades for them rather than looking at their scores on work that has been returned to them.

Consider an example.

My first semester at my current institution, I had a student whom I will call Aaron in my Math Center course. The Math Center is where we teach our developmental math courses. It is a carefully constructed machine in which each individual teacher has a narrowly defined role to play, and where there is little autonomy. In particular, the grading scheme is standardized across all sections: 60% tests, 20% final exam and 20% participation points.

Aaron was shooting for an A in the course. He had scored an 89 on the first test, a 65 on the second and he wanted to know what average he needed in order to get the A that was his goal. Our conversation began something like this…

Well, you are averaging 77% on the first two tests. There are five tests for 60% of the grade, so you have 77% of the 24% of the grade determined by these two tests. Let’s assume you get all of the participation points, so you have 100% of that 20% of the grade. So we need to figure out what percent you need of the remaining 36% of the grade that comes from the tests, and what percent you need of the 20% that is the final in order to get 90% or better in the course.

Even I was confused.

So we thought about it algebraically. If we let x be the average on the remaining tests and final exam, then we need to solve the following inequality:


But in order to solve this inequality, Aaron would already have to have passed the course in which he was enrolled.

In the courses I teach outside the math center, I take a different approach. The semester has 100 points. The weighting is built into the point values of each graded item. So if I want exams to be worth 60% of the grade, then I have 60 points to distribute across however many exams I am giving. At any moment in the semester, a student can simply add the points they have gotten, the total points, divide one by the other and consider the quotient as a percent.

I can figure these grades quite easily without my computer and I can answer a question like Aaron’s quickly and easily.

Can you say the same for your grading scheme?

If not, can you defend the complexity of your scheme? Does it serve to motivate, inspire or inform? Or does it serve to obfuscate and to place a barrier between performance and evaluation?

Computerized gradebooks allow us to create complex grading schemes. But that doesn’t mean we should.