Where Is The Mathematical Thinking and Why It Is Desperately Needed

Teaching mathematics in school is complex and often misunderstood.

The paper titled Why Are Learning And Teaching Mathematics So Difficult by Alan H. Schoenfeld does a great job shedding some light on why this is the case.

Schoenfeld introduces his paper with an idea that was expressed decades ago by Hans Freudenthal that “school mathematics is the fossilized remains of real reasoning processes or real mathematical thinking“.

He argues that teaching and learning math is complex because of these three (3) big ideas:

• • • Thinking Mathematically – What does it mean to do mathematics?
    • The Learning Environment – Structuring classroom experiences to allow students to think mathematically.
    • Cultural Surround – How surroundings help or deter students from experiencing mathematical thinking.

Let’s unpack these ideas so we can position ourselves and our fellow mathematics teachers to serve students in the most effective manner we possibly can.

Thinking Mathematically

Most school mathematics teaches students to memorize procedures, calculations, definitions resulting from the true mathematical thinking required to develop those procedures, calculations, and definitions.

The reasoning processes have been fossilized and in a sense “packaged” or “canned” into bite size chunks of memorizable “math”.

Schoenfeld goes on to argue that the mathematics that is taught to school aged children omits most of what matters when engaging in real problem solving:

• • • access to resources including mathematical processes, content, and practices;
    • access to use Pólya-like problem solving strategies (George Pólya – the Godfather of problem-solving);
    • metacognitive performance, specifically self-monitoring skills; and,
    • mathematical beliefs.

When you look at those four (4) big ideas above on what matters in mathematics ask yourself how many of the four are you seeing in the math programs at your school or in your classroom?

Most mathematics taught in school is focused solely on this first item and often times the mathematics taught is focused primarily on the curriculum content or standards, while the mathematical practices or mathematical processes are pushed to the side almost as an afterthought or, brought to the forefront if and only if time allows.

Here at Make Math Moments through our podcast, free online courses, video blog posts, and problem-based units with full teacher support guides, we’ve made it our mission to help teachers in this area by strengthening their own classroom mathematics program in six (6) key areas:

• • • Classroom Pillars / Routines;
    • Mathematical Content Knowledge and Proficiencies;
    • Mindsets & Beliefs;
    • Professional Learning Plans;
    • Pedagogical Content Knowledge; and,
    • Classroom Resources & Environment.

We’ve highlighted three (3) of the six (6) areas above that direct connect to Schoenfeld’s first big idea around engaging students in true mathematical thinking in classrooms.

If we teachers are to truly learn what matters in mathematics classes and communicate that to our students and fellow colleagues we’ll need to help students access mathematics and develop fluent processes, embed students in real mathematical problem solving while giving them the meta cognitive skills they need to succeed and help them believe they all can be mathematicians.

Here are three (3) recommendations to help shift your practice towards one that focuses on what truly matters in school mathematics:

• • • Teaching Math Through A Hero’s Journey
    • How To Help Students With Mindset
    • Classroom Mathematics Program Assessment

By focusing on real problem solving in your classroom, you’ll be well on your way to checking Dr. Schoenfeld first reason why teaching and learning mathematics is hard off the list by eliminating the focus from memorizing the “fossilized remains of real mathematical thinking” to a mathematics learning community that promotes the making of conjectures, reasoning and proving.

The Learning Environment

“Mathematics classrooms are the primary locales in which people’s mathematics identities are shaped.” ~ Schoenfeld

Teaching and learning math is difficult because the learning environment that our students are placed in has many dynamic and important factors namely these 5 dimensions that Schoenfeld says need to be developed to offer students a robust understanding in mathematics.

• The Mathematics
• Cognitive Demand
• Equitable Access To Content
• Agency, Ownership & Identity
• Formative Assessment

The Mathematics

“The extent to which the classroom activity structures provide access to students to become knowledgeable, flexible, and resourceful thinkers” – Schoenfeld

Teaching math is difficult because it’s educators mission to provide experiences for students to build their own math proficiencies. However, it can be easy to dismiss the importance of our own mathematics content knowledge and strength across all five strands of mathematical proficiency.

Regardless of the curriculum resource provided by your school or district (or lack thereof), if we ourselves do not have a high level of mathematical proficiency, we will not be able to deliver a program that develops the mathematical proficiency of our students.

The only thing more challenging than developing an effective mathematics program is first understanding ourselves what it means to be mathematically proficient.

While schools or districts may work hard to secure updated textbooks, manipulatives, or other classroom resources with the intent to improve student achievement in mathematics, a common lack of clarity around what it means to be mathematically proficient will hinder the positive effects these tools and resources may have.

Make Math Moments helps you understand what it means to be mathematically proficient through our weekly free math podcast, annual free math virtual conference, online platform of mathematics professional development resources including live mathematics webinars and self-paced math courses, as well as over 60 problem based units of study with full facilitator guides to help you gain a clear vision of what it means to develop mathematical proficiency for all students and adults and at any age.

Once we strengthen our own mathematical proficiency we teachers will be more confident, and better prepared to give mathematically rich experiences to our students.

Cognitive Demand

“Meaningful learning occurs when students are stretched. If there is no challenge in the tasks they are working there is no real learning.” ~ Schoenfeld

A major pedagogical issue is that teachers reduce cognitive demand to make it easier on students, especially when teachers see students frustrated.

As math teachers, it is in our nature to “help” students and see them succeed, so it seems natural to step in when they are struggling and help them to the finish line.

By “helping” too much we’ve been robbing our students of the ability to attach real meaningful value to their learning and allow them to feel ownership in that learning.

If we are going to “help” and not let cognitive demand diminish we have to change our definition of “help”.

“The idea, consistent with Lev Vygotsky’s “zone of proximal development” or ZPD (Vygotsky, 1986), is that learning takes place most effectively when students work actively in the areas that stretch them – where students can build on what they know, expanding their current understanding.” ~ Schoenfeld

Students need to engage in a productive struggle to feel that the mathematics they are learning is meaningful to them.

Here are a few ways with the necessary supports to ensure that you are helping more students engage in productive struggle when teaching mathematics:

• How To Teach Math Through A Hero’s Journey;
• Using 5 Practices For Orchestrating Mathematical Discussions and Keeping Students Engaged; and,
• What you can do now to plan and implement elements of productive struggle.

Equitable Access To Content
Agency, Ownership & Identity

“It may be the case that there are ample, rich mathematical opportunities in a classroom. But who gets to take advantage of them?” ~ Schoenfeld

Teaching and learning math is difficult because many teachers, of course, have good intentions to give aid to their students, however fail to think or rethink which practices they use, mostly out of habit, exclude certain groups of students.

Differentiated instruction, while good intentioned, has backfired in some cases and “may systematically sideline students and deprive them of meaningful mathematical experiences” (Schoenfeld).

While Ontario makes great efforts to include a wider range of cultures in their mathematics curriculum (see the new Ontario Elementary Mathematics Curriculum as well as the new, MTH1W destreamed grade 9 mathematics course) and puts a focus on access, equity, inclusion and mathematical identity, there is a very long journey ahead before the effective implementation of these ideas will be observed in Ontario mathematics classrooms.

In order to design learning experiences for our students, mathematics teachers need to learn as they attempt to more intentionally integrate these ideas in their mathematics practice. Here are some supports that will help you in ensuring that you can address these important pieces of the mathematics teaching and learning puzzle:

• The characteristics of a culturally responsive educator;
• Why discipline needs to be revisited;
• How being a “warm demander” can allow you to expect more and get more from your students;
• How teachers can connect students’ cultures, languages, and life experiences with what they learn in school; and,
• What an Anti-oppressive Mathematics Education Looks Like [Join the Academy to Watch]

Formative Assessment

Ask educators what they believe the term formative assessment means and you’ll receive a variety of different answers – another reason teaching and learning math is difficult.

Here’s what you’ll typically here:
Quizzes, mini-tests, exit tickets, discussions, conversations, assignments, observations.

What do you notice about the terms?

“…it appears educators more often use the term formative assessment to refer to a particular kind of instrument than a process to improve instruction.” (Embedded Formative Assessment, Wiliam, p 40).

For us, formative assessment is about both the instruments and the process of learning. Formative assessment is any strategy, tool, and process that allows us to learn about our students’ thinking and informs us on how to improve learning for a student. The other critical part must be WHILE the learning is happening.

Teachers must engage in learning activities that put themselves in positions to actively learn and gather information about student understanding.

Teaching through problem based lessons and actively utilizing the elements from Margaret Smith and Mary K. Stein’s work The Five Practices For Orchestrating Productive Mathematics Discussions with intentionality will position the teacher to access a gold mine of student understanding data they can use to push student learning forward.

Assessment needs to be about valuing growth and not attaching grades to students so they can be sorted.
You can continue to learn about how Make Math Moments supports teachers in a comprehensive course on Assessment in Mathematics.

Growing A Robust Mathematics Program Like A Strong Tree

The five (5) Dimensions to Teaching For Robust Understanding referenced above are embedded in the six (6) areas, we here at Make Math Moments believe are needed to strengthen for an effective mathematics program:

• Classroom Pillars / Routines;
• Mathematical Content Knowledge and Proficiencies;
• Mindsets & Beliefs;
• Professional Learning Plans;
• Pedagogical Content Knowledge; and,
• Classroom Resources & Environment.

If you are interested in addressing the three (3) big ideas that Alan Schoenfeld highlights as the real reasons why teaching and learning mathematics is so hard, then consider taking our 12-minutes mathematics program assessment screener to see which areas your current classroom is excelling in and which you might focus on next. We’ll email you a customized report that gives you actionable items and resources to improve your math instruction in your classroom to ensure you are heading in the right direction.

The Cultural Surround

“We have allowed this to happen to ourselves…we have, in effect, been committing an act of unthinking, unilateral educational disarmament.” – Schoenfeld

Teaching and learning math is difficult because of these barriers that surround the mathematics classroom:

• Curriculum & Testing & The Math Wars
• How Well-Positioned Are Our Educators To Teach Robust Mathematics?
• Cultural Inequity Barriers.
• How Can Leaders Grow Their Mathematical Programs?

Curriculum, Testing & The Math Wars

Prior to 2002, each state in the USA had its own education department that specified the curriculum, standards, and examinations. Textbooks played a significant role in shaping the curriculum, as school districts were often required to use textbooks approved by their state’s education department to receive reimbursement for textbook costs. This resulted in a lack of diversity in curricula, despite the apparent freedom of school districts to choose their instructional materials.

In 1989, the National Council of Teachers of Mathematics (NCTM) published the Standards, which aimed to promote problem-solving and a deeper understanding of mathematics. However, the implementation of these standards faced challenges. The National Science Foundation (NSF) had to support the development of textbooks aligned with the Standards because commercial publishers were unwilling to invest in the expensive production of new materials. Although the NSF-supported curricula performed as well as traditional curricula in terms of mathematical achievement, they did better on assessments of conceptual understanding and problem-solving.

The Standards faced resistance and controversy, with critics arguing that they reduced rigor and promoted “fuzzy math.” The politicization of the reform efforts led to a lack of progress in curricula and teaching practices in the West. Furthermore, even though the reform curricula were adopted by major publishing houses, there was insufficient professional development support for teachers to effectively implement them. Sound familiar? The “Math Wars” are still here. Back to basics or learn through problem solving?

Back To Basics vs. Problem-Based Learning A False Dichotomy

As mathematics educators avoid falling into the false dichotomy of “back to basics” versus “problem-based learning” approaches.
This artificial divide suggests that educators must choose between focusing solely on procedural skills and memorization or solely on problem-solving and conceptual understanding.

In reality, effective mathematics instruction requires a balanced and integrated approach that combines both elements. Students need a solid foundation in foundational mathematical concepts, procedures, and fluency, which are essential for tackling complex problems. At the same time, they also need opportunities to engage in authentic problem-solving experiences that foster critical thinking, creativity, and the application of mathematical knowledge. By embracing both aspects, teachers can create a rich and comprehensive learning environment that enables students to develop a deep understanding of mathematics and become proficient problem solvers.

The introduction of the No Child Left Behind (NCLB) Act in 2002 further complicated the landscape of mathematics education. High-stakes assessments became the focus, leading to “teaching to the test” and neglecting aspects of mathematics that were not assessed. This resulted in a stripped-down curriculum that deprived students of a deeper understanding of mathematical concepts and problem-solving skills. Cheating scandals also emerged as a consequence of the intense pressure to meet test score expectations.

A decade later, the Race to the Top (RTT) initiative attempted to address the flaws of NCLB by promoting comprehensive reform plans and the adoption of common standards and assessments. The Common Core State Standards for Mathematics (CCSSM) were introduced, aiming to reduce incoherence between state standards. However, the implementation of the CCSSM posed challenges as existing textbooks did not align with the new standards, leading to a scramble to develop new curricula. The detailed nature of the CCSSM also resulted in a focus on covering the standards rather than emphasizing big ideas and conceptual understanding.

Overall, despite some progress in promoting problem-solving and conceptual understanding, mathematics education still faces significant barriers related to curriculum and testing. The lack of diversity in curricula, the emphasis on standardized assessments have hindered the development of effective teaching and learning practices in mathematics.

How Well Positioned Are Our Teachers To Teach Robust Mathematics?

When the pandemic forced schools to shift to online instruction, teachers faced the daunting task of adapting quickly without adequate preparation. This highlighted the hierarchy that often places teachers in challenging situations, expected to excel despite the obstacles. Let’s explore the undervalued perception of teaching expertise and the challenges faced by teachers in terms of professional development, curriculum support, and equity in mathematics education. By recognizing these issues, we can work towards empowering teachers and creating meaningful learning experiences for students.

Valuing Teaching Expertise

Despite extensive research on teaching, much of it remains academic and inaccessible to teachers. Teacher preparation programs often provide limited content knowledge and pedagogical training, leaving beginning teachers to grapple with classroom management and instructional strategies. This lack of support contributes to the high attrition rate among new teachers in the US, resulting in a majority lacking essential mathematical thinking and problem-solving skills.

Challenges in Professional Development

While researchers and professional developers understand effective instructional practices, scaling up these practices to reach a large number of teachers poses challenges. Many teachers rely on their own experiences or informal learning from colleagues, hindering the development of effective instructional techniques. This leads to insufficient support and limited growth in pedagogical approaches.

Inadequate Curriculum Support

Current curricula often fail to provide problem-solving opportunities and culturally relevant mathematical content. This lack of alignment makes it challenging for teachers to create robust learning environments that foster critical thinking and engage students. Biases and patterns of inequity can also go unnoticed, perpetuating disparities within the classroom.

The challenges faced by teachers in terms of content knowledge, pedagogical skills, and access to resources are significant barriers to effective mathematics instruction. Recognizing and addressing these issues is crucial to empower teachers and enhance student learning outcomes.

To support teachers, there must be a collective effort to improve teacher preparation programs, provide meaningful professional development opportunities, and develop curricula that promote problem-solving and equity so that a teacher’s mathematical tree can grow strong and wide.

Cultural Inequity Barriers

Systemic Racism and Educational Inequalities

The tragic deaths of George Floyd, Breonna Taylor, Trayvon Martin, Sandra Bland, Ahmaud Arbery, and others brought attention to the deep-rooted systemic racism in American society. While the Brown v. Board of Education decision officially ended racially segregated schooling in the US, it did not eliminate the structural inequalities that persist. Schools serving students of colour often face higher teacher turnover rates, limited resources, and a lower quality of education. The COVID-19 pandemic further exacerbated these inequalities, as students from resource-rich families had greater access to technology and support for remote learning, while those from under-resourced districts faced significant challenges. These disparities emphasize the urgent need for equity in education.

Structural Inequalities and Biases in Mathematics Education

The educational system is not immune to the structural inequalities and biases present in society. Teachers often lack resources and support, impeding their ability to provide high-quality instruction. Moreover, societal biases can impact student identities and opportunities for meaningful engagement with mathematics. Creating a mathematically rich learning environment that supports every student and meets high standards while addressing these challenges is a complex task. However, existing curricula and testing often fail to promote sensemaking, metacognitive competencies, and productive beliefs, hindering effective learning and teaching of mathematics as discussed above.

How Can Leaders of Mathematics Grow Their Math Programs & Overcome Barriers

Clearly teaching and learning mathematics is difficult. As educational leaders we play a vital role in shaping and improving math programs within our school districts. With the right mindset, knowledge, and guidance, leaders can help to clear a “well-lit” pathway to effectively strengthen math programs and drive student success.

Setting a Clear Vision and Objectives

To initiate growth and improvement, leaders must establish a clear vision for their math program. This vision acts as the trunk of the tree, providing direction and purpose. By collaborating with district stakeholders, leaders can define what they want math education to look like within their schools. This involves identifying priorities and setting measurable objectives that align with the overall vision.

Using Data to Drive Instructional Decision-Making

Data plays a critical role in evaluating the effectiveness of instructional strategies. While many districts rely solely on end-of-year standardized test scores, leaders can adopt a more proactive approach. By leveraging data to influence teacher practices, leaders can make informed decisions and measure progress throughout the academic year. This approach avoids the disconnect between action items implemented and student outcomes assessed only at the end of the year.

Implementing Measurable Tracking Systems

To ensure accountability and progress, leaders must establish measurable tracking systems. These systems enable leaders to track the impact of specific actions taken by teachers and monitor the alignment of classroom practices with the set objectives. By directly measuring the results of implemented strategies, leaders can assess their effectiveness and make data-driven adjustments when necessary. Measurable tracking systems provide an organized structure for accountability and help ensure that goals are met.

Providing Professional Development and Support

Educational leaders should prioritize professional development and support for teachers that helps achieve their goals and vision of mathematics. By offering targeted training sessions, coaching, and resources, leaders can empower teachers to implement evidence-based instructional strategies and build their own content knowledge. This could include promoting building thinking classrooms, encouraging math talks, emphasizing productive struggle, and focusing on effective use of mathematical models and flexible, fluent strategies. Ongoing professional development ensures that teachers have the necessary tools and knowledge to drive student success.

Conducting Needs Analysis

To effectively prioritize areas for growth, leaders should conduct a comprehensive needs analysis. This analysis identifies the specific areas that require attention and resources. By understanding the unique needs of their district, leaders can tailor their efforts and allocate resources where they are most needed. This ensures a targeted approach and maximizes the impact of interventions.

At Make Math Moments we encourage you to take our 12-minute assessment screener to receive a free, customized improvement plan to shape and grow the 6 parts necessary for a strong, healthy, and balanced mathematics program.
Our report should be your first step to addressing the barriers that surround mathematics classrooms.

Summary

Teaching and learning mathematics is undeniably challenging due to various factors that shape the learning environment. Schoenfeld’s five dimensions—mathematics, cognitive demand, equitable access to content, agency, ownership, and identity, and formative assessment—highlight the complexity involved in fostering robust mathematical understanding.

Developing teachers’ mathematical proficiency is essential for creating effective mathematics programs. Without a deep understanding of mathematics content and proficiency across different strands, educators may struggle to deliver impactful instruction.

Teaching mathematics is challenging due to the diverse experiences students encounter in their classrooms, the instruction they receive, their self-perception as math learners, and the assessment practices used. The barriers presented by curriculum and testing, the need to support teachers’ expertise, and the importance of addressing cultural inequities all contribute to the complexities of mathematics education.

To overcome these barriers and enhance mathematics programs, leaders must value teaching expertise, provide meaningful professional development, and develop curricula that promote problem-solving and equity.

By working collaboratively, educators can create an environment that fosters deep mathematical understanding, supports all students, and prepares them for success in mathematics and beyond.

Make Math Moments offers professional learning resources, including courses, problem-based units of study, and a podcast, to support teachers and leaders in mathematics in developing mathematical proficiency for themselves, fellow educators, and their students. Let’s grow your mathematics program like a strong, healthy, balanced tree.

To assess the areas for improvement in your mathematics program, take our 12-minute screener. We will provide you with a customized report containing actionable items and resources to enhance your mathematics program. Together, we can continue to improve teaching and learning in mathematics and create a positive impact on teachers’ and students’ mathematical proficiency and growth.