Education ∪ Math ∪ Technology

Year: 2014 (page 2 of 5)

Two different approaches to teaching

Here’s an approach to teaching about the relationships between the different forms of the equation of a line that is based on constructivism.

  1. Be clear with students what the objective for the lesson is today.
  2. Give students a tool for looking at the relationships between the different forms of the equation of a line.
  3. Ask them to write down what they notice and what they wonder after they have played around with the tool themselves. Circulate around the room to see what they write down so you can better make use of it in the whole group discussion.
  4. Next, as a group, discuss what students notice and what students wonder about the graph, or use a different talk protocol.
  5. As a group summarize what appear to be the most important observations made, and everyone (including the teacher) writes them down. If important points about these representations of a line are not brought up, either ask questions to prompt students to realize these points themselves or make a note to yourself to structure a task that will help draw out these points next time.
  6. Come back to this concept multiple times in a variety of different ways over the next few weeks, and even find ways to connect to this topic through-out the rest of the year.


Here’s an approach to this same lesson based on cognitive load theory.

  1. Be clear with students what the objective of the lesson is for today.
  2. Give them notes on the different forms of the equation of a line and some worked examples.
  3. Give students twenty or so practice problems to do themselves. Circulate around the room while students do this and give them timely, efficient, and useful feedback.
  4. Quiz students on their understanding of the concept. Use this to guide your planning in subsequent lessons.
  5. Review this concept with students at spaced intervals over the course of the rest of the year.


My recommendation is that if you are unclear on how either of these approaches is helpful for students, then you should try it at least once. I have in fact tried both of these approaches myself, and I’m quite clear on which approach helps students make better connections to other areas of mathematics, and which approach did not work.

Which approach is more likely to lead to an instrumental understanding of mathematics? Which approach is more likely to lead to a relational understanding of mathematics? For reference, here’s a comparison of instrumental to relational understanding.


This Kind of Teaching

I read an important passage in Elizabeth Green’s book “How to Build a Better Teacher” and decided that I had missed something important when reading about Magdalene Lampert’s teaching. Below is a summary of some of the important features of her teaching as I see them. The first tweet is the thing that I had misinterpreted as students talking to justify their reasoning, which is similar but not exactly the same as students talking to prove their reasoning is correct.

A catalog of mathematics education resources

I’m tired of having to search all over the place to find the-link-to-that-mathematics-resource-I-really-want-now-and-bookmarked-a-year-ago and so I created a spreadsheet to keep track of the various mathematics resources as I learn about them. Yes, I know I could have done this with Diigo or Delicious, but I prefer the portability and simplicity of a spreadsheet.

I’ve opened the spreadsheet up for editing and will curate it to ensure it only includes what I consider to be the highest quality resources. The spreadsheet contains the categories of websites, mobile apps, software, books, research, and organizations. If you have a suggestion for a resource you think I should include, please offer it below, or add it to the spreadsheet. Note that I have currently protected the first sheet (the list of websites) so that you can comment on it, but not edit it directly.

This is by no means complete and will continue to evolve over time as I learn about new resources.



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?

The Confirmation Bias Cycle

Educational research flow-chart


I’ve been working hard to read research carefully, both research with which I agree, and research with which I disagree. I still struggle with my tendency to overlook the flaws in research with which I agree, and to find fatal flaws in research with which I disagree.

This does not mean that I should ignore research; only that I continue to be careful to read all research with a critical eye, and discuss the findings with other people. My suspicion is that norming about what research means with people who have a wide variety of view-points might reduce my tendency toward personal bias.



Ineffective feedback and the Khan Academy

According to Grant Wiggins, there are seven essential elements of effective feedback.

Feedback should be:

  • goal-referenced,
  • tangible and transparent,
  • actionable,
  • user-friendly,
  • timely,
  • ongoing,
  • consistent.


I recently looked at the new Common Core Math section of the Khan Academy. I’d like to analyze it within this framework of feedback.

Here is a sample exercise from the Khan Academy, related to the Common Core grade 3 standard of 3.G.A.1. Not familiar with this standard? Just follow the previous link; the language is identical to what is written on the Khan Academy website.


Khan Academy - 3rd grade geometry problem


There are two places where goals are embedded within this question. The first is an implicit goal, described by the Common Core standard linked to this problem. The second goal is embedded within the problem; get five correct in a row.

An obvious issue here, given that the Khan Academy is mostly intended to be used independently by students (the definition of personalized learning the Khan Academy is apparently using), the language of the standard is probably not student-friendly, and in particular it does not lend itself well to be a goal for students.

Getting five questions correct in a row is clearly a goal that a 3rd grade student can understand whether or not they have achieved, although the question remains about whether five correct questions in a row means students understand the standard being assessed by these questions.


Tangible and transparent:

There are two main ways students get feedback from the Khan Academy exercises; they can check their answer, and they can ask for a hint. If students click on check answer, and their answer is wrong, the button they just clicked on shakes back and forth. Presumably this means the answer is not correct, but both of these feedback opportunities are opaque.  

The box shaking back and forth does not tell the learner WHAT they did incorrectly, nor is a shaking button even explicitly a signal that something is wrong.

Next, feedback that is tangible is something a student understands. Look at the series of four hints students are given when working on this problem.

Khan Academy hints

Now imagine you are a 3rd grade student, and you do not know what a right angle is, what is meant by 4 equal sides, or what a rhombus, rectangle, or square is. How does this feedback help you understand these key concepts or the connection between these concepts and the diagrams given. Finally, telling students that the correct answer is “none of these” is not helpful feedback because it does not respond to whatever thinking a student may have done about this problem, it just states something that should be accepted as fact.



Giving students the correct answer to a problem or telling them that they are wrong does not give them actionable feedback because it doesn’t give them any actions they can take in order to improve. A student who receives this feedback has no other alternative but to continue guessing until they are able to get the answers right. This does not lead to student understanding of mathematical concepts.



The key to user-friendly feedback is to give a small amount of feedback at once so that students have an opportunity to make use of the feedback and not be overwhelmed, but at the same time give them feedback that they know how to act on. Feedback that is constructed in language students may not know, or which requires them to use knowledge they do not have is not user-friendly. What evidence do the developers of the Khan Academy have of the effectiveness of the feedback they have included in their system?



What little feedback the Khan Academy exercises give, it is certainly timely.



Again, this feedback is ongoing. If students want another chance to try again, they just need to do another exercise.



The feedback offered by the Khan Academy is certainly consistent, in fact for any given exercise, it is likely virtually identical. This actually leads to problems, since some variance in the language used in feedback is often necessary in order for students to make sense of it.


This analysis of the feedback offered by the Khan Academy platform suggests that their mechanisms for feedback fail on four out of seven of the criteria shared by Grant Wiggins. One open question here; is it possible for a computerized feedback system to give feedback that passes all seven of these questions? Next question; what is learned from using a system that gives poor feedback?