1 Philosophy of Mathematics Education
1.1 Embodied Arithmetic
George Lakoff’s theory of embodied arithmetic offers a fascinating perspective on how humans understand and interact with numbers. This theory suggests that our ability to comprehend and manipulate arithmetic concepts is deeply rooted in our physical experience of the world. One of the most compelling pieces of evidence supporting this theory comes from the phenomenon of subitizing, the ability to instantly recognize the number of objects in a small set without counting. Remarkably, this ability is observed in infants, indicating that our capacity to grasp basic numerical concepts is innate and closely tied to our sensory and perceptual experiences.
Recent research into infant cognition supports Lakoff’s theory by showing that even very young children exhibit an understanding of basic arithmetic operations. A study by Simon, Hespos, and Rochat (1995)1 investigated whether infants as young as five months old can comprehend simple addition and subtraction using a violation-of-expectation paradigm. Infants were shown scenarios where objects were either added to or removed from behind a screen. When the final number of objects did not match the expected outcome, infants showed surprise by looking longer at the display. This reaction suggests that infants possess an intuitive sense of numerical consistency, which aligns with the idea that arithmetic understanding is grounded in embodied experiences.
Moreover, the study introduced conditions where objects changed identity, making the outcomes physically impossible but arithmetically possible. Interestingly, infants seemed less concerned with changes in object identity and more focused on numerical consistency, indicating that their primary expectation was based on the number of objects rather than their specific characteristics. This finding emphasizes that numerical comprehension in infants is not merely a response to visual or physical cues but involves a more abstract understanding of quantity. Such behavior supports the notion that our arithmetic cognition is not solely learned through explicit teaching but is also rooted in our natural, embodied interactions with the environment.
These findings bolster Lakoff’s theory by illustrating that our earliest encounters with the physical world shape our understanding of numbers. The infants’ sensitivity to numerical outcomes, even in the absence of explicit counting or symbolic number use, shows that basic arithmetic concepts are intertwined with physical experiences. By engaging with the world through touch, sight, and manipulation of objects, humans develop an intuitive grasp of numbers and arithmetic that precedes formal education. This embodied approach to arithmetic highlights the deep connection between our cognitive abilities and our sensory and motor experiences, suggesting that our understanding of mathematics is fundamentally grounded in the way we interact with the physical world.
Furthermore, according to Lakoff, this embodiment of arithmetic extends beyond simple number recognition. It influences how we understand and perform more complex mathematical operations, suggesting that our bodies and physical experiences play a critical role in shaping our cognitive processes. This theory challenges traditional views of mathematics as a purely abstract domain, proposing instead that our understanding of numbers is intimately connected to how we interact with the world around us.
The implications of Lakoff’s theory are profound, suggesting that learning and teaching mathematics could benefit from approaches that emphasize physical interaction and experience. By acknowledging the role of embodiment in arithmetic, educators can develop strategies that align with our innate capacities for understanding numbers, potentially making mathematical concepts more accessible and intuitive for students.2
1.2 Large Group Socratic Teaching
Large-group Socratic teaching, often referred to as the Socratic method or Socratic dialogue, is a form of teaching based on asking and answering questions to stimulate critical thinking and to illuminate ideas. Some education theorists refer to LGC as simply teaching via Guided Inquiry. While traditionally associated with higher education, its adaptation to the elementary school classroom offers unique opportunities and challenges. The methodology in this context is tailored to young learners’ developmental stages, encouraging engagement, critical thinking, and deeper understanding of content. This book is written having Large Group Socratic teaching in mind, and using it as a guiding philosophy.3 Here’s how LGS can be structured and applied in an elementary setting:
1.2.1 Preparation
Teacher Training: Educators need to be well-versed in Socratic questioning techniques and how to manage discussions among younger students. Typically students in US classrooms expect a lecturing approach from their teachers, and do not easily accept the level of participation that is demanded in LGS style of teaching. However, in the context of elementary school a teacher could set-up expectations easily.
Teachers prepare a series of open-ended questions that encourage students to think deeply and articulate their understanding. Thus, a more intense and careful lesson planning is required here, but in the long-run these questions will become part of the pedagogical arsenal of the teacher.
1.2.2 Setting Expectations
Classroom Environment: A supportive atmosphere that encourages risk-taking and respects diverse opinions is essential. The teacher encourages students to participate by asking yes/no question: hang gestures could be used to represent ‘yes’ or ‘no’. Students learn the importance of listening, turn-taking, and how to respectfully agree or disagree with peers. You can ask, “Jose could you pick a quiet hand,” or “Jessica, please call on someone to answer this question, look for the quietest hand.”
1.2.3 Implementation
Ultimately, however, teachers will have to ask more critical questions to drive the lesson, prompting students to think critically. Teachers guide the dialogue with follow-up questions, encouraging students to delve deeper or consider alternative perspectives. Students are also encouraged to ask their own questions, fostering a sense of ownership and active engagement in the learning process.
1.2.4 Reflective Closure
The discussion ends with a summary of the main ideas explored, emphasizing the connections made and insights gained. Students reflect on what they learned and how they participated, fostering self-assessment skills.
Teachers must adapt questions to suit the varying cognitive levels within a large group. Keeping a large group of young students engaged requires dynamic facilitation skills and sometimes incorporating multimedia or tangible objects to anchor the discussion.
Finally, implementing large-group Socratic teaching in the elementary school classroom demands
significant preparation and skill on the part of the teacher, but the rewards can be substantial. It promotes an inquisitive mindset, critical thinking, and a collaborative learning environment. To be effective, it should be tailored to the developmental level of the students and integrated thoughtfully into the broader curriculum.
1.3 Liping Ma’s Knowing and Teaching Elementary Mathematics4
Liping Ma’s book presents a compelling examination of the differences in mathematical understanding between Chinese and U.S. elementary school teachers, suggesting a significant factor in the superior performance of Chinese students in international mathematics assessments. Despite having fewer years of formal education, Chinese teachers begin their careers with a more profound understanding of elementary mathematics than their U.S. counterparts, a gap that widens over the course of their teaching careers. This discrepancy is attributed not only to the teachers’ initial grasp of mathematics but also to the systemic support for the ongoing development of their mathematical knowledge in China.
Ma’s book discusses several illustrative scenarios that highlight the gaps in U.S. teachers’ mathematical understanding as observed by the author during her studies and professional observations in the U.S. These include teachers’ struggles to correctly apply mathematical concepts in teaching scenarios and their lack of noticing mathematical mistakes made by students or student teachers.
Ma’s research utilized interviews from the Teacher Education and Learning to Teach Study (TELT) to compare the mathematical understanding of Chinese and U.S. teachers. She found that U.S. teachers, despite having more advanced mathematical training, lacked a comprehensive understanding of the mathematics taught in elementary schools—a gap that was not evident among Chinese teachers.
Liping Ma delves into specific mathematical topics like subtraction, multiplication, division by fractions, and the relationship between area and perimeter, showcasing how Chinese teachers’ deep understanding of these subjects contributes to effective teaching practices. Ma introduces the concept of Profound Understanding of Fundamental Mathematics (PUFM), arguing that a teacher with PUFM not only knows how to compute correctly but also understands and can teach the conceptual structure, basic principles, and attitudes of mathematics inherent in elementary mathematics.
The latter part of the book investigates how Chinese teachers develop such profound understanding and discusses the systemic and cultural factors that support this development. Ma contrasts these with the conditions in the United States, which she argues are less conducive to fostering a deep understanding of mathematics among elementary teachers. She concludes with suggestions for changes in teacher preparation, support, and research in mathematics education in the U.S. to help close this gap.
1.4 The California Common Core Mathematics Education Standards
The California elementary mathematics education standards serve as a comprehensive framework designed to guide educators in the systematic teaching of mathematical concepts from kindergarten through sixth grade. These standards are organized into several key sections, each addressing different mathematical strands such as Number Sense, Algebra and Functions, Measurement and Geometry, Statistics, Data Analysis and Probability, and Mathematical Reasoning. This organization not only facilitates a structured approach to mathematics education but also underscores the importance of a well-rounded mathematical foundation for all students.
The significance of the California elementary mathematics standards lies in their role in ensuring consistency and depth in the mathematical education provided across the state. By setting clear expectations for what students should know and be able to do at each grade level, the standards help to ensure that all students, regardless of their background or the school they attend, have access to a high-quality mathematics education. This is critical for preparing students not just for academic success in higher grades, but also for practical problem-solving and analytical thinking skills required in daily life and future careers.
The standards are organized according to each grade level. For example, Grade 2 begins with a summary of objectives:5
In grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.
This is followed by more in-depth elaboration on each of the (1) - (4) objectives. Note that the current book only covers standards (1), and (2):
Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including comparing. Students understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens + 3 ones).
Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds.
In the Content Standards, on p. 20 you can also see a more in-depth description of the standards (1), and (2) above, and their corresponding code 2.0A, and 2.NBT. Note the OA stands for “Operations, and Algebra,” while NBT stands for “Number, Baste Ten”.
Teachers are encouraged to use these standards as a roadmap for planning their instruction throughout the academic year. The standards provide a clear outline of the content to be covered, allowing teachers to design lessons and activities that not only meet these benchmarks but also engage students in meaningful learning experiences. Furthermore, the standards are meant to be integrated with pedagogical strategies that promote understanding, application, and critical thinking, rather than rote memorization. Note, for example, the standard 2.0A1:
All of this pertains to our goal of teaching via Guided Inquiry, and teaching in a manner that ensures conceptual understanding instead of procedural.
Each section of the standards is clearly labeled by grade level and mathematical domain, making it simple for teachers to identify the expectations for their specific grade. Additionally, the standards are available in various formats, including online databases6 and printed documents, which are easily accessible and searchable. This accessibility ensures that teachers can quickly reference the standards to confirm the grade-level appropriateness of their content, align their instruction with state expectations, and identify areas for individual or group advancement.
1.5 Exercises
1. For the following exercises please refer to the California Common Core Mathematics
- The Common Core Standards include specific objectives for understanding and practicing multiplication within elementary mathematics education. Locate the standard that details the objectives for multiplication for Grade 3. What is the specific code for this standard, and how does it describe the introduction of multiplication to students?
- Fractions are a crucial part of elementary mathematics education. Find the standard that outlines the objectives for teaching fractions in Grade 4. What is the code for this standard, and what key concepts should students understand about fractions at this grade level according to the Common Core Standards?
- The concept of place value is fundamental in elementary mathematics. Locate the standard that specifies the objectives for place value understanding in Grade 2. What is the specific code for this standard, and what are the essential place value concepts Grade 2 students are expected to grasp?
2. For the following exercises please refer to the discussion on Liping Ma’s book Knowing and Teaching Elementary Mathematics
- “Subtraction with Regrouping” is a foundational mathematical concept taught in elementary schools. According to Liping Ma, teaching this concept effectively requires more than instructing students on the procedural steps. Discuss how the concept “regrouping” as opposed to “borrowing” in subtraction can be introduced to students in a way that enhances their conceptual understanding.
3. Consider the following two summaries of lesson plans designed by two teachers on the subject of subtracting two-digit numbers such as 21 - 9 requiring regrouping:
Teacher 1: Whereas there is a number like 21−9, they would need to know that you cannot subtract
9 from 1, then in turn you have to borrow a 10 from the tens space, and when
you borrow that 1, it equals 10, you cross out the 2 that you had, you turn it into a
10, you now have 11−9, you do that subtraction problem then you have the 1 left and
you bring it down.7
Teacher 2: In order to subtract 9 from 21 we must begin by looking at the ones place of the minuend, and notice that there aren’t enough ones to take away 9. Thus, we must decompose from a higher value unit, the tens. We unbundle one of these two tens into 10 ones, giving us a total of 11 ones. We now take-away 9 ones from 11, obtaining 2. We must put the digit 2 in the ones place underneath the 9. Since we have no tens to subtract, we put the 1 remaining ten in the difference area. Thus, the difference is 12.
Compare and contrast the two different lesson plans on 21 - 9. In which ways is the first
teacher’s lesson plan insufficient? Identify a Common Core Standard that is supposed to be covered in this lesson plan, and in which ways is the second teacher’s plan superior in meeting this standard goal?
4. You are preparing a lesson plan for a 4th-grade math class on the topic of multiplication. Using the large-group Socratic teaching method, which of the following approaches best aligns with the methodology’s emphasis on guided inquiry? After you make your choice, write a brief paragraph providing your reason why this methodology is superior to the others.
Directly teaching the multiplication tables through lecture and then giving students a worksheet to complete independently.
Asking students to solve a real-world problem, such as finding the total number of apples in several baskets, by using multiplication. Guide the discussion with questions like “What operation can we use to find the total?” and “Why does multiplication help us here?”
Having students memorize multiplication tables first and then asking them to recite them in front of the class to check for understanding.
Providing students with a list of multiplication problems to solve at home and discussing the answers in the next class.
5. How might George Lakoff’s theory of embodied arithmetic influence a 1st-grade teacher’s approach to teaching number concepts, and what practical strategies could be employed in the classroom to align with this theory?
6. In several studies involving infants the violation-of-expectation paradigm was used to demonstrate infants’ understanding of simple arithmetic. These studies can be found in Simon, Hespos, and Rochat (1995) Do infants understand simple arithmetic? A replication of Wynn8 and also Karen Wynn’s Addition and Subtraction by Human Infants.9 How does the idea that their expectations are violated help show that infants’ are capable of subitizing, and performing basic arithmetic of numbers 1 - 5, and why is this significant for early childhood education?
Simon, T. J., Hespos, S. J., & Rochat, P. (1995). Do infants understand simple arithmetic? A replication of Wynn (1992). Cognitive Development, 10(2), 253-269. https://doi.org/10.1016/0885-2014(95)90011-X↩︎
Lakoff, G., & Núñez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books. This book provides a comprehensive overview of Lakoff’s theory, including the significance of subitizing and the embodied nature of mathematical thought.↩︎
For a summary of teaching via Guided Inquiry, see: Y. Yumiati, Mery Noviyanti, “Abilities of Reasoning and Mathematics Representation on Guided Inquiry Learning”, Journal of Education and Learning (EduLearn)↩︎
See Ma, L., & Kessel, C. (2003). Knowing mathematics. Houghton Mifflin.↩︎
See the searchable California Contents Standards Database.↩︎
Ma, L., & Kessel, C. (2003). Knowing mathematics. Houghton Mifflin, p. 2.↩︎
https://www.sciencedirect.com/science/article/pii/088520149590011X?via%3Dihub↩︎
https://www.researchgate.net/publication/21647991_Addition_and_Subtraction_by_Human_Infants↩︎