Measuring Pedagogical Content Knowledge for Mathematics




photo-1pck5ut-300x225In a previous blog post I discussed the concept of pedagogical content knowledge for mathematics.  In this post I look how it has been measured.

Many would agree intuitively with the importance of both content and pedagogical knowledge for teachers. However, scholarly evidence for the existence of PCK separate from mathematical content knowledge and its effect on learning outcomes is thin (Hill et al., 2008).  Depaepe et al. (2013) identified six main research lines in PCK: ‘(1) the nature of PCK, (2) the relationship between PCK and content knowledge, (3) the relationship between PCK and instructional practice, (4) the relationship between PCK and students’ learning outcomes, (5) the relationship between PCK and personal characteristics, and (6) the development of PCK.’  Measuring PCK is complicated. It’s hard to distinguish content from pedagogical knowledge and determine their respective effects on student learning.  Research have used both quantitative and qualitative approaches to investigate PCK with mathematics teachers.

Qualitative studies have tended to take a situative view on PCK, something that makes sense only related to classroom practice (Depaepe et al., 2013). These studies rely on case studies, classroom observations, meeting observations, document analysis and interviews, usually during a relatively short period.   Longer-term qualitative studies that investigate the relation between teacher knowledge, quality of instruction and learning outcomes have the advantage they can track evolutions and tensions between theory and practice, but are rare.  An excellent (20 years old!) ethnographic paper from Eisenhart (1993) brings to life Ms Daniels based on months on interviews and observations at the school and teacher training institute.   Unfortunately, the current dominance of ‘evidence-based’ studies, often narrowly interpreted as quasi-experimental and experimental research, crowds out this kind of valuable in-depth studies.  These studies have confirmed the existence of pedagogical content knowledge independent of content knowledge.  A teacher’s repertoire of teaching strategies and alternative mathematical representations is largely dependent on the breadth and depth of their conceptual understanding of the subject.

Most Quantitative research is based on the Shulman’s original cognitive conception of PCK as a property of teachers that can be acquired and applied independently from the classroom context (Depaepe et al., 2013).  Several large-scale studies have sought to ‘prove’ the existence of PCK for mathematics as a separate construct from subject content knowledge using factor analysis.

Hill et al. (2008) used multiple-choice questioning to look for separate dimensions of  content and pedagogical knowledge.  Questions were situated in teaching practice probing teachers for representations they would use to explain certain topics, how they would respond to a student’s confusion or what sequence of examples they would use to teach a certain topic. Questionnaires were complemented by interviews to get more insight in teachers’ beliefs and reasoning (Hill et al., 2008). Several papers contain a useful sample of survey questions.

Item Response Theory (IRT) is used by several authors to assess the validity of these surveys to discriminate between subjects at various ability levels.  IRT quantifies how well a test discriminates between teachers with various levels of PCK.  Test Information Curves (TIC) depict the amount of information yielded by the test at any ability level.  In Hill et al. (2008) a majority of questions with a below-average difficulty level resulted in a test that discriminated well between teachers with low and average levels of PCK, but less well between teachers with good and very good PCK.


Test Information Curve from Hill et al. (2008)

 The amount of information decreases rather steadily as the ability level differs from that corresponding to the maximum of the Information Curve. Thus, ability is estimated with some precision near the centre of the ability scale, but as the ability level approaches the extremes of the scale, the accuracy of the test decreases significantly.

When evaluating their survey, Hill et al. (2008) found that teachers relied not only on PCK for mathematics knowledge for solving the questions, but also on subject content knowledge and even test-taking skills.  They used cognitive interviews for additional validity analysis, in which they asked teachers to explain why they had chosen a certain answer.  Secondly, their multiple-choice questions suffered from the fact that few teachers selected outright wrong answers, but differed in the detail of explanations of students’ problems they could give during the interviews.  The researchers found following kinds of interview items to discriminate quite well:

  • Assessing student productions for the level of student understanding they reflect
  • Use of computational strategies by students
  • Reasons for misconceptions or procedural errors

Baumert et al. (2010) analysed teachers’ regular tasks and tests, coding the type of task, level of argumentation required and alignment with the curriculum as indicators for PCK.  They complemented this with students’ ratings on teachers’ quality of adaptive explanations, responses to questions, pacing and teacher-student interaction.  Data from examinations and PISA numeracy tests were used to assess students’ learning outcomes.

Ball et al. (2001) discuss the concept of place value for multiplying numbers as a typical example of questions they used in their survey.  They found that teachers could accurately perform the algorithm – as would numerically literate non-teachers – , but often failed to provide conceptual grounding of the rule, and struggled to come up with sensible reactions to frequently occurring student mistakes.  Many teachers  using ‘pseudo-explanations’ focusing on the ‘trick’ rather than the underlying concept.  Ball et al. (2001) discuss similar examples in teachers’ knowledge of division (e.g. division of fractions), rational numbers (e.g. fractions of rational numbers) and geometry (e.g. relation between perimeter and area for rectangles).


Recent studies often start from teaching practice in analysing the role of knowledge.  Even teachers with strong PCK (as based on surveys) may, for a variety of reasons, not use all this knowledge when teaching  (Eisenhart, 1993).  Rowland and colleagues (2005) observed and videotaped 24 lessons of teacher trainees.  Significant moments in the lesson that seemed to be informed by mathematical content or pedagogical knowledge were coded. Codes were classified and led to the development of the ‘knowledge quartet’.  They illustrate the framework using a grade 8 lesson on subtraction from a hypothetical student called Naomi.  The framework looks promising as a guide for discussions after lesson observations.  Its focus on the mathematical aspects of lessons, rather than on general pedagogy was positively perceived by mentors and students (Rowland et al., 2005).

Various interpretations of PCK exist and it’s important to make clear which definition of PCK is used or which components are included.  A more cognitive interpretation as devised by Shulman has the advantage that it can be clearly defined, but in that case it is only one (hardly distinguishable) factor of many that affects instructional quality. A more situative approach tends to imply a wider definition of PCK beyond the scope of content knowledge, including affective and contextual factors. This may widen PCK so much that it means ‘everything that makes a good teacher’.

Few studies on measuring PCK have been done in developing countries. In their systematic review, Depaepe et al. (2013) found only one study of PCK that included an African country (Botswana, in Blömeke et al., 2008).  In Cambodia we used surveys with multiple-choice questions and lesson observations to assess teacher trainers’ PCK.  Some lessons learned are:

  • Language is a major barrier, as questions and answers were translated between English and Khmer, complicating assessing conceptual understanding and further probing during interviews and coding during lesson observations.
  • Response bias is an issue in surveys and lesson observations.  Teacher trainers tend to respond what they think the researcher likes or what they think will bring them most benefit in the future. Due to administrative requirements lesson observations are usually announced beforehand, resulting in teacher trainers applying the techniques you want them to apply for the occasion.  This makes that the picture you get is the optimal achievement rather than the average achievement.
  • The initial test we used was based on items from the TIMSS survey. However, most questions were too difficult for teacher trainers, resulting in low ability of the test to discriminate between teacher trainers’ PCK. Recent teacher graduates have much stronger content and teaching skills though.  An IRT analysis would have been helpful here to devise a valid and reliable test.
  • The small population of teacher trainers and the crowded donor landscape makes it hard to devise an experimental study. A more ethnographic approach that also investigates how PCK that is learned during teacher training is applied or fails to be applied in schools seems more useful to me.  However, care should be taken to include a variety of characters, school settings and ages in this fast-changing society.

Finally, PCK seems most useful to me as a theoretical framework to underpin sensible professional development. To be discussed in a next post.

Selected references

  • Ball, D.L., Lubienski, S.T. and Mewborn, D.S. (2001) ‘Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge’, 4th ed. In Richardson, V. (ed.), Handbook of research on teaching, Washington, DC, American Educational Research Association, pp. 433–456, [online] Available from: (Accessed 12 September 2013).
  • Hill, H.C., Ball, D.L. and Schilling, S.G. (2008) ‘Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers’ Topic-Specific Knowledge of Students’, Journal for Research in Mathematics Education, (4), p. 372.
  • Rowland, T., Huckstep, P. and Thwaites, A. (2005) ‘Elementary Teachers’ Mathematics Subject Knowledge: The Knowledge Quartet and the Case of Naomi’, Journal of Mathematics Teacher Education, 8(3), pp. 255–281.
  • Depaepe, F., Verschaffel, L. and Kelchtermans, G. (2013) ‘Pedagogical content knowledge: A systematic review of the way in which the concept has pervaded mathematics educational research’, Teaching and Teacher Education, 34, pp. 12–25.

What knowledge do teachers need to teach? Pedagogical Content Knowledge for Mathematics

What knowledge do mathematics teachers need in order to teach successfully?   In a series of four blog posts I want to summarize some of the research on the topic. The first blog post looks at the concept of pedagogical content knowledge of mathematics.  The second will discuss research attempts to measure teachers’ knowledge and link it to students’ learning outcomes.  In the third one, I write about the implications for teachers’ professional development.  In the final blog post, I will relate the first three blog posts to the South African context.

 Many intuitively feel that a thorough understanding of content is necessary to be a good teacher. However, accurately describing what and how much knowledge a teacher should have to teach successfully, has eluded researchers and policy makers.  Nevertheless, the question matters. Ball et al. (1991) argue that insufficient insight in the knowledge it takes to teach well contributes to low numeracy levels and a lack of interest in maths with many people.  Questions such as “What is the probability that in a class of 25, two people will share a birthday? Or Is a square a rectangle? leave many baffled.

This is due to an excessive focus on procedures, rules of thumb and ‘drill and kill’ kinds of practice, instilled by a dated, but tenacious view of mathematics as a fixed body of knowledge rather than an system of human thought (Ball et al., 1991).  This view is reinforced by textbooks that list ‘hints’ instead of developing conceptual understanding and a wide but shallow curriculum. A similar diagnosis, in my opinion, can be made for other sciences.  Many feel that what makes a good teacher is tacit and consider teaching as an art, based n common sense with little need for professional learning.

Research on the relation between teacher knowledge and student learning took off in the 1960s. Quantitative studies used the number of certificates or mathematics courses taken as proxy variables for teachers’ knowledge.  These studies rejected a straightforward relation between more teacher mathematical knowledge and more student learning.  A (weak) positive relation for undergraduate courses was found, but with non-linear relations and threshold effects complicating matters (Ball et al., 1991).  For more advanced graduate courses the relation was absent and in some cases even negative (e.g. Belge, 1979; Monk, 1994).  This may be due to the increasing compression of knowledge in advanced courses, which can complicate the ‘unpacking’ of content necessary in teaching and with higher exposure to conventional teaching approaches in advanced courses (Ball et al., 1991).

In the 1980s, researchers set upon closer probing of mathematical knowledge, rather than using second-order indicators.   Lee Schulman (1986) conceptualized pedagogical content knowledge (PCK) as a unique domain of teacher knowledge covering various aspects of subject knowledge of mathematics that are relevant for teaching including:

  • Knowledge what concepts students develop in various stages of their development
  • Knowledge of common student misconceptions on mathematical concepts
  • Knowledge of curriculum, threshold concepts and in what order they are best taught.

With PCK Shulman intended to vindicate the central role of subject content knowledge in teaching quality, in addition to generic pedagogical knowledge. It offers a fine-grained conceptualization of the kind of content knowledge a teacher requires for teaching successfully. PCK relates to the ability of not only knowing the content, but the ability to enable others to know it.  A powerful example of PCK is given by Deborah Ball and colleagues on multiplication of decimals, worth quoting in full:

‘The teacher had to know more than how to multiply decimals correctly herself.  She had to understand why the algorithm for multiplying decimals works and what might be confusing about it for students.  She had to understand multiplication as iterated addition and as area, and she had to know representations for multiplication.  She had to be familiar with base-ten blocks and to know how to use them to make such ideas more visible to her students. Place value and the meaning of the places in a number were at play here as well.  She needed to see the connections between multiplication of whole numbers and multiplication of decimals in ways that enabled her to help her students make this extension.  She also needed to recognize where the children’s knowledge of multiplication of whole numbers might interfere with or obscure important aspects of multiplication of decimals.  And she needed to clearly understand and articulate why the rule for placing the decimal point in the answer – that one counts the number of decimals places in the numbers being multiplied and counts over that number of places from the right – works.   In addition, she needed an understanding of linear and area measurement and of how they could be used to model multiplication.  She even needed to anticipate that a fourth-grade students might ask why one does not do this magic when adding or subtracting decimals and to have in mind what she might say.’ (Ball et al., 1991, p.448)

Since Shulman introduced PCK, the concept has been refined.  Krauss et al. (2008) distinguish three dimensions of PCK: knowledge of mathematical tasks as instructional tools, knowledge and interpretation of students’ thinking, and knowledge of multiple representations and explanations of mathematical problems.  Ball et al. (1991) include a component of subject knowledge, called ‘horizon knowledge’, that includes insight in curriculum structure and how concepts are gradually introduced over grades. Hill et al. (2008) use the terms ‘common content knowledge’ and ‘specialised content knowledge’.  The former relates to mathematical knowledge that numerically literate non-teachers are likely to know.  The latter refers to specific knowledge for teaching, such as what kind of mistakes are typically made at what age, or what representations create powerful learning.


Shulman’s conceptualisation of PCK (from Depaepe et al., 2013) and Ball’s refinement

Not everyone finds Shulman’s PCK concept helpful . Margaret Eisenhart (1993) dismisses the distinction between subject and pedagogical content knowledge as fuzzy and prefers to use procedural and conceptual knowledge as components of teacher knowledge.  Cochran et al. (1993) coined the term pedagogical content knowing (PCKg) to stress PCK as a dynamic knowing-to-act’ that is inherently linked to and situated in the act of teaching within a particular context.  Shulman’s concept was a theoretical construct, which proved difficult to confirm with empirical data.  Tim Rowland  (et al., 2005) used grounded theory to develop an empirically based classification, the ‘knowledge quartet’ that distinguishes between foundation, transformation, connection and contingency.

Foundation consists of the teacher’s theoretical knowledge and understanding of mathematics and beliefs about the nature of mathematics, including why and how it should be learned.  It’s called foundation because it determines the potential for the three other categories. These describe how foundation knowledge informs teaching decisions. Transformation refers to the capacity to transform it into powerful pedagogical forms, enabling others to learn.  Connection describes teachers’ ability to convey mathematics’ inherent coherence through well-chosen sequencing of topics, tasks and exercises within and between lessons.  Finally, contingency is the preparedness of the teacher to listen to student responses and readiness to suitably respond and even deviate from the set lesson agenda.

Foundation Awareness of purpose; identifying errors; overt subject knowledge; theoretical underpinning of pedagogy; use of terminology; use of textbook; reliance on procedures.
Transformation Choice of representations and explanations; choice of examples, teacher demonstrations
Connection Making connections between procedures; making connections between concepts; anticipation of complexity; decisions about sequencing; recognition about conceptual appropriateness.
Contingency Responding to children’s ideas; use of opportunities; deviation from agenda; swift and correct analysis of student errors and difficulties

Recent research efforts tend to focus more on the practice of teaching rather than teachers’ knowledge.  Strong content knowledge, or even strong PCK does not always translate into strong teaching, due to both teacher factors as environmental constraints.  In her case study of Ms. Daniels, Eisenhart (1993) splendidly describes the tensions between the focus on conceptual understanding in policy documents and the teacher training courses and elements at the personal level (limited conceptual understanding of students) and the school level (beliefs cooperating teachers, wide curriculum) that push teachers towards more procedural approaches.  Deborah Ball hits the nail on the head:

 ‘The pull toward neat, routinized instruction is very strong.  Teaching measurement by giving out formulas – l x w = some number of square units and l x w x h = some number of cubic units – may seem more efficient than hauling out containers, blocks and rulers and having students explore the different ways to answer questions of ‘how big’ or ‘how much’. With focused, bounded tasks, students get the right answers, and everyone can think they are successful.  The fact that these bounded tasks sometimes results in sixth graders who think that you measure water with rulers may, unfortunately go unnoticed’ (Ball et al., 1991).

In the next post I’ll discuss some of the approaches that have been used to measure teachers’ pedagogical content knowledge.  Comments and suggestions welcome!

Selected references:

Ball, D.L. (1990) ‘The mathematical understandings that prospective teachers bring to teacher education’, The elementary school journal, 90(4), pp. 449–466.

Ball, D.L., Lubienski, S.T. and Mewborn, D.S. (2001) ‘Research on teaching mathematics: The unsolved problem of teachers’ mathematical knowledge’, 4th ed. In Richardson, V. (ed.), Handbook of research on teaching, Washington, DC, American Educational Research Association, pp. 433–456, [online] Available from: (Accessed 12 September 2013).

Baumert, J., Kunter, M., Blum, W., Brunner, M., et al. (2010) ‘Teachers’ Mathematical Knowledge, Cognitive Activation in the Classroom, and Student Progress’, American Educational Research Journal, (1), p. 133.

Cochran, K. F., DeRuiter, J. A., & King, R. A. (1993) ‘Pedagogical content knowing: an integrative model for teacher preparation’, Journal of Teacher Education, 44, pp.263-272.

Depaepe, F., Verschaffel, L. and Kelchtermans, G. (2013) ‘Pedagogical content knowledge: A systematic review of the way in which the concept has pervaded mathematics educational research’, Teaching and Teacher Education, 34, pp. 12–25.

Eisenhart, M., Borko, H., Underhill, R., Brown, C., et al. (1993) ‘Conceptual knowledge falls through the cracks: Complexities of learning to teach mathematics for understanding’, Journal for Research in Mathematics Education, 24(1), pp. 8–40.

Hill, H.C., Ball, D.L. and Schilling, S.G. (2008) ‘Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers’ Topic-Specific Knowledge of Students’, Journal for Research in Mathematics Education, (4), p. 372.

Rowland, T., Huckstep, P. and Thwaites, A. (2005) ‘Elementary Teachers’ Mathematics Subject Knowledge: The Knowledge Quartet and the Case of Naomi’, Journal of Mathematics Teacher Education, 8(3), pp. 255–281.

Shulman, L. S. (1986) ‘Those who understand: Knowledge growth in teaching’, Educational Researcher, 15(2), pp.4- 31.