Education ∪ Math ∪ Technology

Month: September 2014 (page 1 of 1)

What do mathematics teachers need to know?

I’ve been thinking a lot recently about what knowledge is needed by mathematics teachers in order to be excellent teachers. It is clear to me that teachers of mathematics must know the mathematics they are to teach, but what else do they need to know?

Different types of knowledge
(Source: Content Knowledge for Teaching: What Makes it Special?)

I’m not the first person to wonder this. In a 2008 paper, Deborah Ball, Mark Thames, and Geoffrey Phelps describe a categorization of the mathematical knowledge needed for teaching into five broad types, described in more detail below.

Knowledge of mathematics

  • Common mathematical knowledge:

    Deborah Ball et al. describes this as “knowledge that we would expect a well-educated adult to know” (Ball, Content Knowledge for Teaching. What makes it special?, 2008). It is closely associated with the curriculum that is being taught and includes knowledge of when students are making mistakes and when there are errors in a textbook.

    For example, common mathematical knowledge includes knowing how to subtract a negative number from a positive number. It includes knowing how to multiply multi-digit numbers together. It includes knowing how to solve a quadratic equation by factoring. It includes all of the mathematical knowledge necessary to solve mathematical problems in at least one way.

    Note that teachers should at least know all of the common mathematical knowledge they are expected to teach as well as how that knowledge is connected to the mathematics students are likely to learn later in school. If I teach my students that an equals sign (=) means “find the answer” they are likely to find algebra confusing until they learn that the equals sign means “the two expressions on either side of this = sign are equal in size”.

    Calculating percents
    (Source: Finding a percentage)

    Notice in the example above, the author of this video is dividing 4 by 16. Instead of making sense of the calculation, he starts by following the algorithm which results in a non-sensical calculation. Someone who understood this division calculation with a stronger conceptional understanding might realize that since 16 is larger than 4, we can skip the first iteration of this division calculation and move straight away into 40 divided by 16.

    When teachers lack this mathematical knowledge, they often end up relying on tricks instead of mathematical understandings. In my own practice, I remember when I was first called upon to teach the Chi-squared test, which at the time I did not know. I read ahead in my textbook and figured out how to follow the mathematical recipe that produced the Chi-squared calculation well enough to muddle through it in class. However years later I finally connected the expected values to probability and realized that I had for many years missed an opportunity to help students make a connection between this test and other things we were learning that year.

  • Specialized mathematical knowledge:This is “knowledge beyond that expected of any well-educated adult but not yet requiring knowledge of students or knowledge of teaching” (Ball, 2008). For example, knowing some of the many different representations of positive and negative numbers is mathematical knowledge that one could not expect every well-educated adult to know, and does not require knowing students or teaching.

    An example of specialized mathematical knowledge is knowing how to represent subtracting a negative number from a positive one in a variety of different ways. Since different students are likely to find different models for understanding current topics more or less easy to grasp, it is incredibly useful to know a variety of different ways of representing and approaching each area of mathematics that is taught.

    Models for understanding subtracting negative integers
    Different models for understanding subtracting negative numbers

Knowledge of students:

  • Pedagogical content knowledge
    • Knowledge of the ways students understand the content:This is the knowledge of not just the ways to do mathematics but the ways in which students understand the mathematics. While some mistakes students make are the result of over-taxed memory or carelessness, not all mistakes students make can be explained in these ways.

      Multiplication mistake
      (Source: Math Mistakes)

      What mistake did this student make above? What does this mistake mean they were thinking when this did this calculation? How would you help this student? Answering all three of these questions requires an understanding of the ways student understand this mathematical content.

      Magdalene Lampert once told me the story of an interaction she had with a 5th grader. Near the very end of a class discussion, he said that 0.007 is negative. The class ended and Magdalene was left to try and figure out why the student thought that 0.007 is negative. If teachers have the common mathematical content knowledge they need, they should understand that 0.007 is not negative. If teachers have sufficient knowledge of the many different ways students might understand mathematics, then they will know that there is a fairly logical (although incorrect) train of thought that leads students to the mistaken understanding that 0.007 is negative. Any guesses as to what this student was thinking?

    • Knowledge of the ways to teach mathematics:This is the knowledge of how to teach mathematics so that it makes sense to someone else. It includes thinking about the sequencing of mathematics concepts as well as knowing rationals for why various pieces of mathematics are correct. Teachers need to make choices about what mathematical representations to use with students and which of those representations are likely to understood and misunderstood by students.

      In my previous example, knowing the different models of understanding the mathematics is specialized mathematical knowledge, knowing which models to choose and how to sequence the models is mathematical knowledge for teaching.

      Multiplication models
      Different ways of representing 12 times 6

      These are a few of the different models for understanding multiplication. Which of these models do you think work best for initially introducing multiplication? Which of these models is most efficient? How are these models connected to each other?

      Tree x Tree = Line
      (Source: Jason Zimba, NCTM 2014)

      When teachers do not know how to teach a particular idea to students in ways that make sense to students, they may rely on mnemonics or other non-mathematical ways for students to remember mathematical content. If you look at the example above, it is clear that whoever created these diagrams is hoping that these will help students remember the math facts for 3×3 and 3×4. Wouldn’t it be better though for students to understand at least these simple multiplication facts based on pictures of what they represent or at least be memorizing the words that represent this relationship?

      This area of knowledge also includes how to ask questions in class that get all students thinking, how to give students feedback that helps them move forward, and how to choose instructional activities that support students learning mathematics.

    • Knowledge of the curriculum:Teachers also need to know the scope and range of the mathematics they are teaching. They need to know the standards that are appropriate for their particular course as well as what those standards mean. They need to know what resources they have available as well as how useful those resources are for their teaching. They need access to curricular resources and because of the limited time teachers typically have available to plan, they need to know where to find the resources they need without spending an enormous amount of time searching.

There is other knowledge teachers need to know outside of the knowledge related to their content area.

Knowledge of students and schools

  • Knowledge of students’ cultures and backgrounds:While this is not typically included in the lists of knowledge required for teaching, it is pretty clear to me that it is necessary. The ignorance that led this group of NYC teachers to not only wear these t-shirts but to also pose for a photo is a clear sign that there is a body of knowledge about students’ culture and teachers’ impact on it that exists and that not all teachers know.
  • Knowledge of the emotional needs of students:Teachers are also responsible for understanding the emotional needs of the children under their care. These needs change over time and are different for different children. Teachers need to pay attention to the status needs of their students and to understand how status impacts their classroom instruction.

    Who can be grouped with whom? Who looks like they need more support today? Which children look like they are being abused or neglected? Who is likely to be excited by today’s lesson?

  • Knowledge of the rules and procedures related to teaching:Every school has procedures teachers are expected to follow and every country and state has laws teachers are expected to obey. While knowledge of these rules and procedures is completely insufficient to be able to teach, a lack of knowledge of these rules and procedures has led many educators to disaster.


From my own experience, I learned very little of this before I started teaching, aside from the knowledge of most of the mathematics I was employed to teach. During the course of my career as I attempted to make sense of what students understood and planned lessons to build student mathematical knowledge, I slowly built up my understanding of these areas of knowledge.

Teacher knowledge
(Source: Embedded Formative Assessment)

The graph above shows a summary of research on the correlation between how many years of service literacy and numeracy teachers have compared to how much their students learn. Early in their careers, both teachers of literacy and numeracy struggle to use the knowledge they have to successfully teach students. One way of interpreting the graph above is that as they progress through their careers, both teachers of literacy and numeracy increase their knowledge of how to teach, and their students benefit.

What of the knowledge above can teachers begin to learn before they start teaching? How can we ensure that every teacher of mathematics learns enough of this content during the course of their career in order to be able to teach? Also, how does this post relate to teachers of other content areas or elementary school teachers?

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching what makes it special?. Journal of teacher education59(5), 389-407.

Wiliam, D. (2011). Embedded formative assessment.

20 things every teacher should do


The objective of this list is to identify a core set of teacher practices that every teacher should do. There are some things missing from my list. For example, I have not highlighted the need for every teacher to be aware of issues around the social status of their students, their students emotional needs, or have a certain level of cultural awareness. I think these things are important, but I’m not sure how to describe them in a sentence. If you have any suggestions…

I’m also fully aware that these sentences are probably too brief to be a complete picture of the complexity of teacher practice. There are entire books on giving good feedback and understanding formative assessment. I do not view this list as a simple check-list but rather as a starting place for teachers and teacher-educators to ask questions about effective teaching practice.

I’m sharing this graphic with a Creative Commons Attribution, Non-commercial, Share-alike license. You can make a copy to modify it here if you like, provided you make any changes from the original obvious, and provide appropriate attribution. Also, here is a PDF version.

What else would you include? What do you disagree with? Do you have any clarifying questions about what I mean by anything on this list?


Could you build a nuclear power plant given the right reward?

From Facebook:

“my daughter and I were brought in to talk about her ” learning disabilities ” and how she was not applying herself . They talked about punishments and incentives . After listening to everyone I asked . ” If I asked you to build a nuclear power plant could you do it ?” They all answered ” no ” So I ask well what if i took a way all your free time at work , and did not allow you to go to the ofice party because you could not build it ? Could you build it then ?” Again the answer came back No . So I ask ” Okay then how about if I promise you a huge reward , could you build it then ?” Again they all answered no . So I say ” what if I read you a manual on how to build a nuclear power plant , gave you people who were educated and worked building nuclear power plants to show you how to build one , gave you every tool , all the materials to build one , could you build one now ” one said no still , but the rest said maybe . well I said to them my kid cant do this work if you punish her for not doing it ,or promise her a pizza party. She will only do this work if you put her with people who can help her understand . The world is a power plant to her .” [sic] ~ Adriene Kimiko Pauley

Could you build a nuclear power plant given the right reward? Or would you need support just to be able to get started?



I was not born to teach

I was not born a technology person. 

I am a person who does not accept that he does not know how to do something related to technology and explores how he does it until he figures it out. I am a person who will occasionally search to the fifteenth page of a Google search until I find the right piece of knowledge that helps me figure out my problem. I am a person who believes that I can eventually figure out any technological problem I am given, even if the answer sometimes is to find someone with more expertise. I am a person a who has been exploring the use of technology since I was inspired by the gift of my first personal computer at eight years old.

I was not born a mathematician.

I am a person who has been exploring numbers since before he started school, supported by a home environment rich in numbers and geometry. I am a person whose parents never told me that math was hard. I am a person who found solace in exploring ideas when he found out that kids can be cruel. I am a person who had a pair of exceptional mathematics teachers from grade four to grade six who taught me that mathematical ideas weren’t just meant to be written down and forgotten later but discussed and explored. I am a person who has studied mathematics for nearly 35 years.

I was not born to teach.

I am a person who often dreaded going to school as a young child either because I was bored or bullied. I am a person who struggled to make friends at school and never really understood other people. I am a person who decided to take musical theatre just to be part of the cool crowd in middle school. I am a person who needed to fill a block in eleventh grade and got pushed into peer tutoring by his guidance counsellor. I am a person who learned that he loved it when he could help other people understand. I am a person who worked hard every moment he was in his teaching program to the point of being called driven by his friends. I am a person who nearly gave up teaching in his first year because of how hard it was. I am a person who only survived his first year of teaching because he found out other people in his school were struggling as much as he was. I am a person who has dedicated a large portion of his life to learning more and more about the relationship between teaching and learning for the last twelve years of my life.

When we see someone who seems to do something well, we have the tendency to assume that they have special talent, rather than they had a set of experiences that prepared them to be successful.



Sharing individualized comments with students with Autocrat

Much of the research on formative assessment suggests that grades are not effective as feedback because they do not provide students with actionable information they can use to move their learning forward. Comments and questions are much more useful to students when grades are not included.

Unfortunately, if you have 180 students, providing individualized feedback to each student regularly is time-consuming and difficult, maybe even impossible. Grades are much easier to produce. 

One partial solution to this is to produce a list of descriptive comments and select the comments you want for each student from this list. Since students often require the same feedback, this provides enough individualization that is still helpful for the student without being over-whelming for teachers to actually do. One way to do this is to create a 1 or 2 page list of the comments and then check-off the comments that actually apply to the individual student, and then give each student their individually selected comments. Unfortunately, while not as time-consuming for teachers, this leaves students having to read a bunch of comments or questions that do not actually apply to them. (Aside: One interesting activity here would be to provide students with the complete list of possible comments and ask them to figure out which comments probably apply to their work.)

It is possible to do both; provide students with an uncluttered list of comments that apply to their work and not have it take an enormous amount of time. The best part of the process I’ve come up is that you get to keep a copy of the comments you actually gave to students for further reference.

This process uses Google Spreadsheets and the Autocrat add-on from New Visions to turn a template for comments into individual comment documents for each student. If you are using a non-Google Apps for Education account for this process, you should probably not include identifying student information in your personal Google account and just print these documents to share with students.

The first step is to create a Google Spreadsheet and enable the Autocrat add-on. For more information on the Autocrat add-on, see this page. I also recommend creating a folder for each assignment to keep all of the comment documents in. You may also find that this process plays well with Doctupus if you are already using it.

The next step is to enter in names and email addresses for each of your students as in this sample spreadsheet. If you want to avoid storing student information in a Google Spreadsheet, you could use pseudonyms at this stage and enter your own email address, and then print the comments once they are created. Otherwise, you can choose either to print the comments or share them with students electronically.

Example of initial data entered into the spreadsheet


Now, in this spreadsheet create another tab. Into this new sheet, enter letters from A to Z as necessary into column 1, and enter comments, questions, or other feedback into column 2. These comments should be relatively generic but apply to specific common issues you have noticed in the student work. Remember that every one of your students, even the ones who did very well on the assignment, need constructive feedback to move their learning forward. The comments shown below are associated with with this mathematics task.

Step 2: Adding comments to a separate sheet


Now it is time to associate these comments with specific students. Create a number of columns equal to the maximum number of comments you think you will give to any individual student. The title of these columns doesn’t matter, but I used Replace 1, Replace 2, Replace 3, etc… to help remind me that these comments are not the ones I intend to share with students. Instead, in these columns place the letters associated with the comments you want to give students instead. You may find it helpful to print the sheet of comments and their associated letters from step 2 to help enter comments when you look at student work. Notice that I have left some of the cells below blank since I only have 2 or 3 comments for those students.

Step 3

The next step makes those letters useful for students. Instead of sharing the letters with students, you want to share the comments. Here is where I used a Google formula to take the letters and substitute them for comments in another cell. The actual formula itself will depend on the name you used for the comment sheet you created in step 2 and the number of comments intend to share with each student.

Step 4

The formula I used is =if(isblank(E2), “”, vlookup(E2,Comments!$A:$B,2)) which you can copy and paste into the top-left most cell of your comment columns as shown above, and then edit to meet your needs. Basically what this formula does is check to see if the associated cell (E2 in this case) is empty, and if it is not empty, it looks up the comment in the sheet you created in step 2 that is next to the letter entered. If you need to edit this formula, E2 refers to the top-left most cell where you entered the letters corresponding to the comments, and Comments! refers to the name of sheet with the comments in it that you entered in step 2, and finally $A:$B refers to the two columns from the Comments sheet where you entered the letters and comments. Once you have the formula entered into this cell, you can drag it over and down to fill all of the relevant comments, as shown in the brief video below.

You will also need a template document like this one. Autocrat will take this template document you create, make a copy of it as either a PDF or Google Doc for each student, and replace the variables entered into the template (which look like <<Comment 1>> in this sample) with the comment information you entered into the spreadsheet.

Now you have all of the preparation work done that you need to run Autocrat. Launch Autocrat from the menu above (if you have not added Autocrat to the spreadsheet, you will have to do this now).

Step 6


Click on New Merge Job in the new Autocrat sidebar.


Click on Drive in order to select the Google Doc template you created and then give the merge job a name.

Confirm that the <<tag>> and Sheet header information is correctly set. Here Autocrat has attempted to match the headers of this spreadsheet with the variables you entered into the Google Doc template.

Click Save, and then enter the naming format each document to be shared with students should have. You may also want to open up the advanced settings here and select a folder for these documents to be created in.

Now you should be able to click on the Run Merge button on the right. Autocrat will share a copy of the template with each email address entered into the email column with the comments entered instead of the variables. You may want to click on Preview before running the merge, just to see what the documents will actually look like before they are created. Once the merge is done, the documents will exist, be shared with students (assuming you entered their email addresses and not your own), and there will be links to the documents in this spreadsheet.

Merge done


Here’s what a folder of these comment documents looks like. At this point you can download these documents (you can download all of the files in this folder as a zip archive, and then once you extract them from the zip, print them out for students – you can do batch printing in both Windows and Mac OS) and then print them out for students or students can access the documents themselves online.

The good news is that once you have set up this process once, replicating it again is fairly easy. The part that will always take a while to do is creating the list of feedback comments and/or questions for students. For more examples of good feedback comments and questions, check out the feedback pages of the Classroom Challenge lessons from the Mathematics Assessment Project.

Hopefully this process will make it a bit easier to give students individual feedback in the form of comments. Let me know if you have any questions.