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"My students learning levels are too varying"

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"My students learning levels are too varying"

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Differentiation is key when working with different backgrounds. Using this method we can help foster independence and critical thinking skills so students at any level can work towards a common goal.

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This goes back to the traits this process develops. If you help them learn to think for themselves and be creative then you won't need to spoon feed them.

Eventually the baby will learn to use the spoon if you stop holding it for them, but it will probably be a little messy at first.

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This goes back to what we view is a valuable use of our time. Do we want to spend time memorizing steps without understanding the why, or would we rather spend the same amount of time helping students to develop a deep understanding of the concept. If they retain the deeper meaning then they won't need to spend as much time repeating the same steps over and over again.

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Employers and colleges are looking for certain skills in their applicants and this approach develops those traits needed to succeed.

These traits are: Critical thinking, collaboration, communication, and adaptability.
(Wathall, p. 229)

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It's not what you teach, it's how you teach it. This approach can be melded into the lesson plan and can be worked into the curriculum.

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There can be push back, but we are encouraged to keep trying until we master it. Once you see the benefits of deep conceptual learning your concerns "will no longer be an issue." (Wathall, p. 233)

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Procedural vs conceptual mathematics instruction.

"Mathematical procedures, k nown to mathematicians as algorithms (are) procedures (which) enable (us) to find answers to problems according to set rules."

The traditional procedural approach

I argue that mathematical education has historically focused on imparting procedural knowledge through a 'Procedures-first, Understanding-second' approach' where teachers have as their first priority to teach procedures to students and where understanding hopefully follows with sufficient practise of those procedures. Following on, it would be tempting to describe what a traditional Procedures-first, Understanding-second approach looks like in action. However, the procedures-first, understanding-second is more of a mindset - an attitude - than an approach. When we use the traditional procedures-first, understanding-second approach, we hold the teaching of procedures as our #1 priority with understanding following as a second priority.

The problem with presenting maths in a way that prioritises the teaching of procedures over understanding is that, by default, it results in students spending significant amounts of lesson time not understanding what they are working on.

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To say this is problematic is an understatement because when students don't understand the activities they are working through the experience is unpleasant and demoralising. It inhibits learning. Furthermore, it makes the role of the teacher unnecessarily difficult.

Mathematical understanding is the realm of conceptual knowledge. In ' Procedural vs conceptual knowledge in mathematics education' I propose that in order for students to acquire conceptual knowledge, the teaching approach needs to firstly bring conceptual understanding to students, before prioritising the teaching of

procedures . In other words, we need a conceptual approach that also teaches procedures rather than a procedurally-based approach which hopes that understanding will naturally occur as a result of students learning those procedures. We need to adopt an Understanding-first, Procedures-second mindset.

​An Understanding-first, Procedures-second approach - what it looks like

Mathematical understanding in students is fostered through the use of activities and strategies which maximise the occurrence of aha moments for students.

The activities used are inherently engaging because they require students to use their own logic and reasoning early on, rather than having to follow a procedure provided by the teacher.

The initial activities lay the foundation for the procedures that follow; the procedures are presented traditionally, however students are more able to make sense of them because they are familiar with the underlying mathematics.

The approach is somewhat student-centred, yet at the same time highly-structured and well-scaffolded, affording the teacher a sense of control, over the learning.

Collaboration between students is integral to the approach.

There is a strong focus on higher-order thinking, including metacognition.

Be careful of speaking in binary

For a long time, I've been advocating the adoption of a conceptual approach by mathematics teachers. However, the message was muddied by the following misinterpretation: 'A conceptual approach obviously means a lack of emphasis on the teaching of procedures'. I had never advocated NOT teaching procedures, but I had not included the word 'procedures' in the name. This misconception remains true today because I commonly receive comments to articles stating "I didn't realise there was an approach that dealt with both concepts and procedures."
More recently, I've made it clear to teachers that we are referring to a hybrid conceptual-procedural approach, or, as I've mentioned here, an Understanding-first, Procedures-second approach. In my view, however, it is still very much a conceptual approach because conceptual understanding is the initial focus.

The challenge of transitioning to a conceptual approach

For many teachers of mathematics, the task of transitioning to a (hybrid) conceptual approach is a challenging one, not because a (hybrid) conceptual approach is especially difficult to implement. Rather, simply because it is unfamiliar to teachers who have always presented mathematics via a traditional procedural approach (Understanding-first, Procedures-second).

Conceptually-based approaches to teaching: Four common misconceptions

Few would disagree that hands-on activities are difficult to manage in many high school classrooms. In an ideal world, all students would explore hands-on materials within a well-structured activity, in a highly-engaged, inquiring and self-directed manner. Misconception 1: Conceptually-based teaching is synonymous with hands-on activities (which are extremely difficult to manage with my students.)

Clearly, many
high school students have become removed from that ideal world! Therefore, releasing hands-on materials en-mass to your average high school mathematics class is likely to be ineffective in bringing about conceptual understanding. The late Grant Wiggins excellently explored this point in his Experiential Learning article by expanding on the statement ' Just because it's hands-on doesn't mean it's minds-on' :

The belief that conceptually-based teaching is synonymous with hands-on activities is, in my view, a misconception.

A quality, conceptual approach can incorporate some use of hands-on materials as part of well-scaffolded, teacher-led activities infused with multiple leading questions. However, student interactions with the materials may be best governed by the teacher in accordance with the capacity for the particular group of students for learning. In other words, the activities are essentially student-centred yet with a level of teacher direction. For example, here are 3 Conceptually-Based Maths Activities that Don't Require Hands-on Materials.

While it is true that many teachers have lost important lesson-time experimenting with conceptual-based teaching, it is often not the approach that is time-consuming, but rather the way the approach is being implemented in the classroom. When conceptual-based teaching is implemented poorly, students are likely to obtain little insight from the activities presented. However, when the approach is implemented well, Misconception 2: Conceptually-based teaching is too time-consuming to be realistically implemented. conceptually-based teaching has actually been seen to save time in the classroom.

This misconception also ties into the previous one, for hands-on activities that are not well-scaffolded and that lack teacher direction can definitely be more time-consuming in the classroom.

. A conceptually-based approach to teaching mathematics requires a very different pedagogy to a traditional procedural approach. One of the main differences is that a conceptual approach requires a degree of student-centred-ness, which is a mode of operating foreign to many teachers. Misconception 3: Conceptually-based teaching is too difficult to manage.
This misconception, I suspect, stems from a fear of the unknown
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The uninformed view of the student-centred approach is that the teacher loses his or her sense of control. However, in the highly-structured student-centred approach advocated here, the opposite is true - teachers can feel like they have more control over the learning.

In my experience, once high school mathematics teachers have been guided through the transition to (an Understanding-first, Procedures-second) conceptual approach they are impressed by the improved engagement and understanding of their students and by how easy the conceptual approach is to manage.

Misconception 4: I'll lose my role as teacher, and therefore also my enjoyment that comes with that role.
This is absolutely a misconception and stems from the idea that teaching conceptually and assuming an 'active facilitator role' means the teacher becomes passive and has less opportunity to 'work the group with charisma'. Arguably, a student-centred, conceptual approach offers more avenues for a teacher's charisma to shine - there are more chances to connect with students one-on-one and in small groups - although the charisma will be expressed somewhat differently.

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Below are links to two free tutorials which fit nicely into an Understanding-first, Procedures-second mindset. Note that the approaches covered in the tutorials are about much more than the specific topic in focus - i.e. avoid thinking "I know how to teach that topic, therefore I don't need the tutorial' (!!)
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Click the links to gain detailed info. Then sign up if you are keen.

We'd love your thoughts at the bottom of the page. (Your email address will not be required)

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Procedural vs conceptual mathematics instruction. "Mathematical procedures, k nown to mathematicians as algorithms (are) procedures (which) enable (us) to find answers to problems according to set rules." I argue that mathematical education has historically focused on imparting procedural knowledge through a 'Procedures-first, Understanding-second' approach' where teachers have as their first priority to teach procedures to students and where understanding hopefully follows with sufficient practise of those procedures.

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"Adopting a growth mindset will make this journey smoother..."

While the road may be tough, we as educators also need to have a growth mindset about our curriculum and believe that "with effort everyone can do math when instruction is focused on conceptual understanding."
(Wathall, p. 228)

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Put the students in the center of the lesson

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Put the students in the center of the lesson

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In traditional math classrooms, teachers present lessons, students work through problems individually or in groups, or perhaps they're asked questions or take a quiz to demonstrate understanding, and then teachers correct or affirm answers.

On the surface, it's a concise, logical teaching model where the teacher "starts with an instructional objective and then designs a lesson with the goal of students demonstrating proficiency," write educators Sam Rhodes and Christopher R. Gareis for ASCD's In Service. But it's an approach they say tends to "relegate equity to an afterthought, inadvertently positioning many students as passive observers of mathematics." Over time, this passive positioning impacts students' math identity-especially kids from diverse backgrounds who might already struggle to connect classroom learning to their own life experiences-building a "sense of disinterest, inadequacy, and disenfranchisement," note Rhodes and Gareis.

A more equitable model of math instruction, they believe, begins with student identities firmly in mind during the lesson planning phase, with the teacher thinking first about students' beliefs around math-for example, considering their comfort level with language, or how they view themselves academically and as mathematicians. "We cannot leave considerations of student identities to the end," write Rhodes and Gareis, an assistant professor of elementary math education at Georgia Southern University and a professor of educational leadership at William and Mary, respectively. "Rather we need to consider what dispositional outcomes we intend for students," and then intentionally design curriculum backward, keeping "the aims of equity and sense of self in mind" so that more kids begin to see themselves as competent mathematical thinkers.

Here are four things to keep in mind when designing student-centered math lessons.

Develop a Clear Mission Statement

Consider creating a mission statement that articulates the intentions of your school's math curriculum and communicates "what teaching mathematics should look and feel like in the school," write Rhodes and Gareis. This is a simple way to "codify the beliefs and identities that [teachers] aim to foster in students."

The mission statement might highlight the importance of creating a community of learners "who are seen as doers of mathematics," for example, and set a goal of giving every student the chance to "develop and communicate deeper understandings of mathematics through flexible thinking, reasoning, and problem-solving."

Connect to Students' Experience

Children are naturally drawn to explore the math around them. "From young ages, we quantify, recognize patterns, and question the equivalence of things, even before we have those words for it," say Rhodes and Gareis. "As we grow, these informal learning opportunities are intrinsically tied to home and cultural experiences and identities." Drawing on these experiences in the classroom can be a powerful way for teachers to "create mathematical understandings that are inherently connected to the lives of their students."

While clearly not everything in the math curriculum can directly relate to students' life experiences, it's important to plan for lessons that include more connection points for kids in your classroom-similar to how a thoughtfully assembled classroom library would include a rich variety of options that reflect students' diverse tastes, cultural backgrounds, reading levels, and specific interests.

In his seventh-grade math classroom, Kwame Sarfo-Mensah plans a unit where students investigate an issue of interest. It's an effort to help students "make sense of the world in which we live," he says, and in the process, connect them more deeply to mathematics. He starts the unit with a survey to gauge students' areas of interest. One year, the responses led to a three-week project examining the intersection between law enforcement and communities of color in Boston.

Sarfo-Mensah helped students come up with a focus question and brainstorm the different math-related data points needed to investigate it-statistics, graphical representations, geometric diagrams, and functional relationships-and he made sure to align the work with the appropriate academic standards. He gave students three options for their final product, providing "multiple access points for diverse learners," he writes.

Allow for Multiple Solving Pathways

In vibrant math classrooms, teachers often "show different ways to solve the same problem and encourage students to come up with their own creative ways to solve them," writes Matthew Beyranevand, a K-12 math and science department coordinator for Massachusetts Public Schools. "The more strategies and approaches that students are exposed to, the deeper their conceptual understanding of the topic becomes."

After students solve a problem using a single method, encourage them to brainstorm alternative solving pathways, then discuss the various options as a class. It's a subtle shift that puts emphasis on developing critical thinking and encourages students to embrace asking questions and sharing strategies as a way to make sense of complex material. "Whereas a focus on [right or wrong] answers results in judgments of correctness, a focus on thinking builds and refines understandings from what students know and understand," write Rhodes and Gareis.

Encourage Productive Struggle

Problem-solving is an integral component of math, and allowing students to struggle productively as they attempt to solve complex problems "sends the message that the teacher believes students are capable of doing and creating mathematics," write Rhodes and Gareis.

High school math teacher Solenne Abaziou, in an effort to build up her students' problem-solving skills and stamina, assigns weekly open-ended math tasks called problem solvers-problems like Dice in a Corner and Snowmen Buttons. "Students often struggle with persistence-they're uncomfortable with the idea of trying a solution if they're not confident that it will yield the desired results, which leads them to refuse to take risks," Abaziou writes. "Helping students get past this fear will give them a big advantage in math and in many other areas of daily life."

A good problem solver "has a low floor and high ceiling," Abaziou notes. "The skills needed to tackle the problem should be minimal, to allow weaker students to engage with it, but it should have several levels of complexity, to challenge high-flying students." As students engage with the problem, they should be "confused at the beginning, which encourages them to struggle until they get on a path that will likely lead them to the solution." It's only by working through that initial frustration that students begin to build "problem-solving resilience," she writes.

All students are capable of doing math, Rhodes and Gareis insist. "We believe that diversity of thought enhances understandings of mathematics for all students, and we believe that allowing student voices and experiences to shine in mathematics classrooms is a crucial step towards rehumanizing the subject," they conclude.

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4 Ways to Build Student-Centered Math Lessons

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In traditional math classrooms, teachers present lessons, students work through problems individually or in groups, or perhaps they're asked questions or take a quiz to demonstrate understanding, and then teachers correct or affirm answers.

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Engage them with your opening, bring variety and flexibility during, then reflect upon the now and build for the later in the closing.

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Help students better understand math by presenting multiple examples, encouraging collaboration on alternative solutions, and framing the class with a clear agenda and effective summary.

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We need to help our students understand that it is ok to not get something right away.

The book uses the quote, "not yet", but I've always said, "I don't know, but I will find out".

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These align closely with the 8 practices we are more familiar with. There is a need for conceptual understanding the both practices address that need.