Education ∪ Math ∪ Technology

Month: July 2014 (page 1 of 1)

Grand Challenge for NCTM

NCTM recently asked for Grand Challenges that are ambitious but feasible, positively impacting many people, and which should capture the public interest. Here is my grand challenge:

  • Develop a comprehensive, national professional development model that supports the high quality mathematics instruction they have been promoting for many years.


Here’s what I think that could look like:

  • We start by norming between a fairly large team of mathematics educators a core set of high quality mathematics teaching practices are, and what they look like in a real classroom.
  • We then carefully study (to ensure that the practices work) and implement these practices in model classrooms where the educators who have previously normed continue to study their practice while continuing to discuss and collaborate with the original group of educators.
  • These classrooms should be substantially open to the public (perhaps through video cameras, one-way glass, etc…) so that other educators, parents, and policy makers could come and visit the classrooms.
  • Once each original educator has established clear evidence (from evidence of student learning) that they are able to consistently and reliably use the set of core practices (along with whatever other practices they have developed), they start norming their practices with another group of educators.
  • This process continues until we have created a substantially similar core set of instructional practices that fairly large group of mathematics teachers consistently use.
  • Two of the core practices we would establish are the teacher as researcher into their own practices and the collaboration with other educators to study each others practices to see what works.


There are some other core practices that I think that every mathematics teacher should do, but the most important core practices are to study your own practice and to collaborate with others in doing the same. Once this is established, then at least when each of us is experimenting with a different practice it will be easier for us to see the connection between what we do and student learning.

Here are some other ideas I think are also worth pursuing.

  • Develop teacher training models at scale where teachers spend at least six months in close apprenticeship with an experienced classroom teacher with time to reflect on their emerging practice during this time. Follow this up with at least two years of additional support.
  • Support many (one in every major city) smaller TMC-like mini-conferences with the overall aim of building local communities of mathematics teachers. Continue support of these groups by creating online space for these educators to continue their conversations and share resources with each other.
  • Actively support the use of online professional development to increase the number of mathematics educators connecting with each other.



Why is it so hard to change math education?

Imagine you are asked to learn about something, and the only way someone can help you understand it is with words, because there are too few examples of it around to actually see it for yourself. You think you know what it is they are talking about, but you keep getting confused because your image of what it is seems so much different than what the other person is describing.

It gets worse because most of the other people you talk to haven’t really seen it before either, and are relying on a story they read about it once before in a book or occasionally on their attempts to tell the story to other people. They sometimes contradict each other, and then other people come in and start telling a totally different story. In fact you aren’t listening to just one story, but many different stories all at once.

When you were growing up the story you were told was pretty different. None of your friends or family knows this new story. In fact, you were never told this story in school or even university. You don’t even really understand why you are being asked to listen to this story because as far as you are concerned, the story you had growing up was a perfectly nice story. Why come up with a new story?

You try and tell the story yourself, but it turns out the story is so different from any stories you have ever heard that it is hard to remember all of it at the same time. You make a lot of mistakes telling the story and feel discouraged and decide you should just stick with your old story. It’s a lot easier to do, and virtually everyone you know seems to value it a lot. Gradually you stop trying to tell the story, and stick with the older story that you undersntand very well.


This description pretty much exactly summarizes a reason why I think math education is so hard to change. The narrative around the changes necessary is often just too different than people’s personal experiences of learning mathematics.



Categorizing Student Strategies

For the last two years, the project I am currently working with has been asking teachers in many different schools to use common initial and final assessment tasks. The tasks themselves have been drawn from the library of MARS tasks available through the Math Shell project, as well as other very similar tasks curated by the Silicon Valley Math Initiative.

Here is a sample question from a MARS task with an actual student response. The shaded in circles below represent the scoring decisions made by the teacher who scored this task.

Individual student work


This summer I have been tasked with rethinking how we use our common beginning of unit formative assessments in our project. The purposes of our common assessments are to:

  • provide teachers with information so they use it to help plan,
  • provide students with rich mathematics tasks to introduce them to the mathematics for that unit,
  • provide our program staff with information on the aggregate needs of students in the project.

We recently had the senior advisors to our project give us some feedback, and much of the feedback around our assessment model fell right in line with feedback we got from teachers through-out the year; the information the teachers were getting wasn’t very useful, and the tasks were often too hard for students, particularly at the beginning of the unit.

The first thing we are considering is providing more options for initial tasks for teachers to use, rather than specifying a particular assessment task for each unit (although for the early units, this may be less necessary). This, along with some guidance as to the emphases for each task and unit, may help teachers choose tasks which provide more access to more of their students.

The next thing we are exploring is using a completely different scoring system. In the past, teachers went through the assessment for each student, and according to a rubric, assigned a point value (usually 0, 1, or 2) to each scoring decision, and then totaled these for each student to produce a score on the assessment. The main problem with this scoring system is that it tends to focus teachers on what students got right or wrong, and not what they did to attempt to solve the problem. Both focii have some use when deciding what to do next with students, but the first operates from a deficit model (what did they do wrong) and the second operates from a building-on-strengths (what do they know how to do) model.

I took a look at a sample of 30 students’ work on this task, and decided that I could roughly group each students’ solution for each question under the categories of “algebraic”, “division”, “multiplication”, “addition”, and “other” strategy. I then took two sample classrooms of students and analyzed each students’ work on each question, categorizing according to the above set of strategies. It was pretty clear to me that in one classroom the students were attempting to use addition more often than in the other, and were struggling to successfully use arithmetic to solve the problems, whereas in the other class, most students had very few issues with arithmetic. I then recorded this information in a spreadsheet, along with the student answers, and generated some summaries of the distribution of strategies attempted as shown below.

Summary of student data

One assumption I made when even thinking about categorizing student strategies instead of scoring them for accuracy is that students will likely use the strategy to solve a problem which seems most appropriate to them, and that by extension, if they do not use a more efficient or more accurate strategy, it is because they don’t really understand why it works. In both of these classrooms, students tended to use addition to solve the first problem, but in one classroom virtually no students ever used anything beyond addition to solve any of the problems, and in the other classroom, students used more sophisticated multiplication strategies, and a few students even used algebraic approaches.

I tested this approach with two of my colleagues, who are also mathematics instructional specialists, and after categorizing the student responses, they both were able to come up with ideas on how they might approach the upcoming unit based on the student responses, and did not find the amount of time to categorize the responses to be much different than it would have been if they were scoring the responses.

I’d love some feedback on this process before we try and implement it in the 32 schools in our project next year. Has anyone focused on categorizing or summarizing student types of student responses on an assessment task in this way before? Does this process seem like it would be useful for you as a teacher? Do you have any questions about this approach?