## In this eight-part series, we are unpacking and exploring each of **NCTM’s essential ***Principles to Actions* Mathematics Teaching Practices. Note: While each blog provides depth and detail for each teaching practice, please be aware that each one of the teaching practices should not be implemented in isolation. They need to be understood and used in a purposeful and impactful way with each other.

Today’s blog focuses on the second of NCTM’s eight research-based essential Mathematics Teaching Practices: implementing tasks that promote reasoning and problem solving.

### Mathematics Teaching Practice #2 - Implementing Tasks That Promote Reasoning and Problem Solving

The second Mathematics Teaching Practice is an essential stepping-stone for understanding and transferring mathematical concepts and understanding. For students to successfully engage in deep-level tasks that allow for reasoning and problem solving, their mathematics classrooms cannot limit their thinking to pure memorization and carrying out computations with little or no understanding. Students need to explore and investigate mathematical reasoning with confidence and understanding.

In order to implement tasks that promote reasoning and problem solving, teachers and students must aim to do the following.

**Teachers must:**

- Provide opportunities for students to engage in deep-level learning.
- Motivate students’ learning of mathematics through opportunities for exploring and solving problems that build on and extend their current mathematical understanding.
- Choose and develop tasks that provide multiple entry points for problems to be solved.
- Pose tasks on a regular basis that require a high level of cognitive demand.
- Know their students to find the “Goldilocks Zone” – not too easy, not too hard (or boring!)
- Support students in exploring tasks that connect to real-world mathematics.
- Encourage students to use procedures and strategies to make connections which can then be applied to solving tasks.
- Value procedural and conceptual thinking. The goal is flexible and transferable thinking from one concept to another.

**Students must:**

- Understand that learning can be difficult but embrace the challenge by persevering in solving problems.
- Move into deep-level learning at the right time (after moving through sufficient surface level) to reason through tasks.
- Use goals to understand where they are, where they need to go, and how they are going to get there through goal setting and clarity of task/understanding.
- Draw upon and make connections with prior understanding and ideas when provided with challenging and deeper-level tasks.
- Use tools and representations as needed to support thinking and problem solving.
- Use a variety of solution approaches to solve problems.
- Move from surface understandings to deeper understandings by defining, describing, discussing, and justifying to one another.
- Work to make sense out of tasks and persevere in solving problems, and if the problems are too challenging, be able to return to the surface-level knowledge to gain further understanding and skills in solving the tasks.
- Value procedural and conceptual thinking. The goal is flexible and transferable thinking from one concept to another.

### Research Behind Deep-Level Learning - Reasoning and Problem Solving in Mathematics

Deep learning is hard. It requires a cognitive demand that can be challenging. It often requires prior learning and the ability to compare and contrast, apply previous knowledge, and make connections. Unless the right conditions are in place to learn how to reason, many students are left at the surface and procedural level of mathematical understanding that relies primarily on using memory alone. Or, on the flip side, students are not ready to go deep, as they have limited skills and or knowledge to move into deeper levels of understanding, reasoning, and problem solving.

If students get into deep-level learning too quickly, they can lose confidence and self-efficacy about their mathematical ability. For this reason, we need to support all students to become engaged in high-level thinking and ensure that all students have the opportunity to experience reasoning and problem solving throughout a sequence of learning, not just those who are considered to be high achievers in mathematics.

When multiple procedures are required, there is a need to understand the surface level, yet find the right balance to move from surface to deep so that students can solve and understand more complex mathematical problems. Hattie (2012) defines this as the “Goldilocks Zone,” where the learning is not too easy and not too hard (or tedious) – it is just right for all learners!

Stein and Lane (1996) suggest that student learning has the greatest impact in classrooms where the learning tasks consistently encourage high-level student thinking and reasoning and the least impact in classrooms where the tasks are routinely procedural in nature. It is well documented that deeper-level mathematical tasks are often perceived to be more complex and difficult to deliver in the classroom and that tasks that are procedural with fewer cognitive demands are prioritized in place of deeper-level instruction (Boaler, 1997).

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#### The Importance of Using a Taxonomy

A taxonomy that is used effectively can encourage reasoning and access to the mathematics through multiple entry points that have supported students through the important learning and understandings being developed prior to deeper-level learning. This includes the use of different representations and tools that foster problem solving through a variety of solution strategies that move through different levels of learning. Furthermore, teachers can use a learning taxonomy to support the appropriate use and development of discourse and thinking.

A taxonomy can be used to support success criteria that moves from surface to deep level as the sequence of learning progresses. For example, using ‘I can,’ success criteria statements move students through the learning as the cognition progresses.

When effectively using a taxonomy, mathematical tasks are viewed as placing *higher-level *cognitive demands on students as they are encouraged to engage in active inquiry and exploration or to use procedures in ways that are meaningfully connected with concepts or understanding. Tasks that encourage students to use procedures, formulas, or algorithms in ways that are not actively linked to meaning, or that consist primarily of memorization or the reproduction of previously memorized facts, are viewed as placing *lower-level *cognitive demands on students.

A taxonomy should not be perceived as a checklist, but instead as a framework and chance to move from procedural to conceptual understandings. This gives students the opportunity to transfer their understandings from one context to another as teachers move from placing lower-level cognitive demands on students to higher cognitive demands that require reasoning and problem solving.

##### The SOLO Taxonomy

The Structure of Observed Learning Outcomes (SOLO) Taxonomy was developed by Biggs & Collis in 1982.

**The SOLO taxonomy consists of two major categories with increasingly complex stages:**

- Surface (uni-structural and multi-structural responses)
- Deep (relational and extended abstract responses).

**Within the two categories, the SOLO taxonomy consists of four levels (Biggs & Collis, 1982):**

- One idea
- Multiple ideas
- Relating the ideas
- Extending the ideas

Biggs and Collis argue that the purpose of the Structured Observed Learning Outcome (SOLO) taxonomy is to balance deep knowledge with surface knowledge when preparing students for deep-level understandings and transferring these to new contexts.

Each level of the SOLO taxonomy increases the demand on the amount of working memory or attention span. At the surface levels, a student only needs to encode the given information and can use a recall strategy to provide an answer. At the deep levels, a student needs to think not only about more things at once, but also how those objects interrelate.

The SOLO taxonomy is used extensively throughout the **Big Ideas Math Series** to inform the progression of learning and design goals through the consistent implementation of learning targets and success criteria.

### How Can Teachers Effectively Promote Reasoning and Problem Solving?

As outlined in this blog and the prior blog, the use of learning intentions and success criteria are vital in providing instructional practice that offers students clear and purposeful goals and a foundation for what success looks like. The learning intentions should plainly explain what students need to understand and what they should be able to do. In addition, this also helps teachers plan learning activities.

Professor John Hattie’s research demonstrates that having clear learning intentions and success criteria coupled with the support of a taxonomy/framework that moves the students from procedural understandings to more complex deeper understandings help students self-regulate. When teachers set and communicate clear lesson goals to help students understand the success criteria, students know where they are, where they need to go, and how they are going to get there. They move from the procedural through to the conceptual. They cannot move to conceptual without the surface-level procedural.

You can find effective examples of learning intentions (also known as learning targets) and success criteria in the Big Ideas Math Series. These align to both the learning within the series and the teacher notes, providing teachers with an excellent starting point for their teaching strategies.

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### Key Takeaways

When promoting reasoning and problem solving, we suggest that teachers should consider using a taxonomy. Using a framework such as the SOLO Taxonomy can help teachers to ensure that tasks include lower and higher-level understanding, skills, and knowledge and that students are encouraged to engage in higher-level, problem-solving tasks that build up from lower-level cognitive skills. In order for students to learn mathematics with understanding, they must have opportunities to regularly engage in tasks that allow for reasoning and problem solving and make possible multiple entry points and varied solution strategies.

Remember, deep-level learning is hard, but we embrace challenge. Surface-level understandings are just as important to helping students to reason and problem solve as deep-level understandings.

Together with Big Ideas Learning and National Geographic Learning, I will continue to provide you with support as we navigate the complexities of 2020, whether this is online and remotely or back in the classroom. Despite the challenges, we are committed to supporting teachers in moving forward with high impact teaching and learning, starting with NCTM’s Mathematics Teaching Practices.

We will continue to share practical examples and connections to the NCTM evidence-based Mathematics

Teaching Strategies that will impact all classroom environments when used explicitly and intentionally.

Keep in touch, ask any questions or comment via twitter (@_sophie_murphy_) and through Big Ideas Learning and National Geographic Learning. Stay safe and well. I look forward to connecting with you all again soon.

*Biggs*, J., & *Collis*, K. (*1982*). *Evaluating the quality of learning The SOLO taxonomy*. New York Academic Press.

Hattie, J. (2012). *Visible learning for teachers: Maximizing impact on learning*. Routledge.

Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. *Journal for research in mathematics education*, 524-549.

Stein, M. K., & Smith, M. (1993). Practices for orchestrating productive mathematics discussions. *Reston, VA: National Council of Teachers of Mathematics*.