The newly revised California Mathematics Framework (CMF) is full of research-informed recommendations for how teachers should update their instructional strategies to be more aligned with current understandings of how students best learn mathematics. Unfortunately, the hard work of the CMF authors will be for naught unless we properly support teachers to carry out the changes called for. Specifically, we need to begin by supporting teachers to increase their Mathematical Knowledge for Teaching (MKT).
Mathematical Knowledge for Teaching is crucial for math teachers because it goes beyond simply knowing math—it involves understanding how to teach it effectively. According to a recent talk by Michael Kirst (professor emeritus of education at Stanford University and appointed four times as president of the California State Board of Education) developing the MKT of teachers is one of the most important essential aspects of improving student achievement in mathematics.
Essentially…if we want improved student outcomes, we need to focus on building teacher capacity in the classroom.
Mathematical Knowledge for Teaching is a framework that describes the specific types of mathematical knowledge that teachers need to effectively teach mathematics. It was developed by Deborah Ball and colleagues at the University of Michigan and extends beyond general mathematical knowledge to include the specialized understanding necessary for instruction.
Components of MKT
MKT is typically divided into two broad categories:
1. Subject Matter Knowledge (Knowing the Math Itself). This refers to a teacher’s deep understanding of mathematics, including:
- Common Content Knowledge (CCK): The math that any well-educated person should know (e.g., solving an equation, calculating area).
- Specialized Content Knowledge (SCK): Math knowledge unique to teaching, such as explaining why a division algorithm works or diagnosing student misconceptions.
- Horizon Content Knowledge: Understanding how mathematical ideas connect across grade levels (e.g., knowing how early fraction concepts lead to algebraic reasoning).
2. Pedagogical Content Knowledge (Knowing How to Teach Math). This refers to knowledge that combines mathematical understanding with effective teaching strategies, including:
- Knowledge of Content and Teaching (KCT): Knowing different ways to present a concept, choosing the best representations, and designing lesson sequences.
- Knowledge of Content and Students (KCS): Understanding how students think about math, predicting common errors, and knowing how to respond to misconceptions.
- Knowledge of Content and Curriculum: Understanding how concepts fit into broader learning progressions and standards.
MKT is the teacher’s ability is twofold: 1) to understand how the 1’s are the same or different, and 2) then explain it to students in a manner that builds conceptual understanding.
Mathematical Knowledge for Teaching (MKT) plays out in multiple ways in a math classroom. Here are some detailed examples that illustrate how MKT helps teachers make informed instructional decisions:
1. Anticipating Student Misconceptions (Specialized Content Knowledge)
A teacher is introducing the concept of division with fractions, using the problem: \(\frac{3}{4} \div \frac{1}{2}\)
A student incorrectly reasons that since division makes numbers smaller, the answer should be less than \(\frac{3}{4}\).
A teacher with strong MKT:
- Recognizes that students often struggle with the meaning of fraction division.
- Uses multiple representations (visual models, number lines, and real-world contexts) to show why the quotient is actually greater than \(\frac{3}{4}\).
- Asks guiding questions like, “How many halves are in three-fourths?” to help students make sense of the operation conceptually.
2. Choosing the Right Instructional Representation (Knowledge of Content and Teaching)
When teaching multiplication of negative numbers, a teacher wants students to understand why \((−2)×(−3)=6\). A teacher with strong MKT:
- Avoids just stating the rule “two negatives make a positive” without explanation.
- Uses a real-world example, such as repeated changes in temperature:
If the temperature drops 2 degrees per hour for 3 hours, that’s \((-2) \times 3 = -6\).
If we reverse the direction (negative time), we must reverse the result, making it positive 6. - Introduces a number line model or pattern approach to justify the rule in different ways.
3. Understanding and Responding to Student Thinking (Knowledge of Content and Students)
A student solving \(35\times 19\) uses the strategy
$$(35×20)−(35×1)$$
A teacher with strong MKT:
- Recognizes this as the distributive property in action.
- Asks the student to explain their reasoning to the class, validating their thinking.
- Encourages other students to share different methods to deepen their understanding of multiplication properties.
4. Differentiating Instruction Based on Student Needs (Horizontally and Vertically Aligned Knowledge)
A teacher working on proportional reasoning notices some students struggle with unit rates. A teacher with strong MKT:
- Connects the concept to prior knowledge, such as equivalent fractions.
- Provides scaffolding using a double number line or ratio table.
- Helps students see the connection between unit rates and slope in algebra to prepare them for future learning.
5. Designing Effective Assessments (Knowledge of Content and Assessment)
A teacher designing a quiz on solving equations chooses problems carefully:
- Instead of just asking students to solve \(3x + 5 = 20\), they include a multiple-choice question where students identify which step is incorrect in a student’s work.
- This assesses not only whether students can solve but whether they understand the process conceptually.
- They include an open-ended question: “Write and solve an equation that represents a real-world scenario.”
6. Building Student Mathematical Discourse (Knowledge of Content and Teaching)
During a lesson on geometric transformations, a teacher notices students struggling with reflections. A teacher with strong MKT:
- Encourages student talk by asking: “How does the reflection affect the coordinates?”
- Has students use patty paper or digital tools to explore reflections physically.
- Facilitates a discussion where students compare reflections across different axes.
Why MKT Matters
Teachers with strong MKT can:
✅ Explain mathematical ideas in multiple ways.
✅ Anticipate student misunderstandings.
✅ Choose effective instructional strategies.
✅ Assess student thinking and adjust lessons accordingly.
Here’s why MKT is essential:
- Deepens Content Knowledge – Teachers need to know math conceptually, not just procedurally. MKT helps them anticipate student misconceptions and provide multiple representations of mathematical ideas.
- Enhances Pedagogical Strategies – MKT enables teachers to choose the best instructional methods for different concepts, ensuring students engage in meaningful learning rather than rote memorization.
- Supports Student Thinking – Teachers with strong MKT can analyze student work, ask targeted questions, and guide discussions that promote mathematical reasoning and sense-making.
- Facilitates Differentiation – A teacher with MKT can modify instruction to meet the diverse needs of students, including multilingual learners and those with special needs.
- Strengthens Assessment Practices – MKT helps teachers design effective formative and summative assessments that align with learning goals and provide insight into student understanding.
- Builds Confidence and Engagement – When teachers have strong MKT, they feel more confident in their instruction, which translates into more engaging and effective math lessons for students.
Conclusion
MKT allows teachers to move beyond procedural instruction and instead create rich, conceptually driven learning experiences. By understanding what to teach, how students think about it, and the best ways to support their learning, math teachers can make their classrooms more engaging, inclusive, and effective.
If your district in contemplating adopting a new math textbook, then properly supporting teachers should be part of the process! Specifically, we need to support teachers increase their MKT in order to take full advantage of the new curriculum your district is adopting.
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