*identify*problems and helps us

*solve*them. Beyond global issues, everyday living involves financial literacy, interpreting data, and basic computation—all elements of mathematics.

*now*so that students can apply them

*later*to authentic learning. The impact of this approach is that far too few adults feel confident in mathematics and far too few develop competence in mathematics (Aguilar, 2021; Hart & Ganley, 2019; U.S. Department of Education Institute of Education Sciences, 2020). To fix this, we need to reimagine mathematics learning experiences with an eye on the future. That means extinguishing some long-standing cultural practices in mathematics classrooms, and replacing them with practices that reflect the reasoning and decision-making that define mathematics as a discipline. This reimagined approach increases

*competence*, because students are actually doing mathematics (which goes well beyond remembering processes) and

*confidence*, because confidence comes from realizing you can figure out a problem without needing someone to show you how to do it.

**STEM Disciplinary Practices in the Classroom**

*The Case for STEM Education: Challenges and Opportunities*, Rodger Bybee explains, “The purpose of STEM education is to develop the content and practices that characterize the respective STEM disciplines” (p. 4). These disciplinary practices are best described in the Standards of Mathematical Practice (CCSSO, 2010) and the Science and Engineering Practices (NGSS Lead States, 2013) as listed in Figure 1.

*do*as they determine things such as how to predict global warming trends, analyze traffic patterns or airline flight schedules, assess safe bridges, and understand countless other problems. Yet to become competent and confident in these disciplines, the practices should also define what

*students*do on a daily basis in math and science classes. In a nutshell, students should be thinking and making decisions as STEM professionals do, whether students are solving real-life problems or learning to subtract whole numbers.

- Students choose
*how*to solve a problem. - Students choose representations and tools to support their reasoning.
- Students notice patterns and structures and use them to solve problems.
- Students justify their processes and their answers.
- Students reflect on their answers and decide if they make sense.

*doing*mathematics (i.e., engaging in any of the mathematical practices)

*,*they are just

*learning about*mathematics (i.e., this is how to solve for

*x*). Not only is the

*learning-about*approach a missed opportunity to develop disciplinary practices, but it also misses the mark in terms of developing mathematical fluency.

**What Is Mathematical Fluency?**

*flexibility*(it was chosen based on the numbers in the problem),

*efficiency*(could be done mentally and more quickly than other options), and

*accuracy*(the answer is correct). Figure 2 illustrates these three fluency components, along with actions that reflect the components.

*learn about*mathematics, they:

- Don’t understand mathematics (
*Why are we crossing out and regrouping?*). - Develop anxiety (
*I might forget the steps, I might be wrong, that will be embarrassing*). - Don’t feel like they are good at mathematics (
*I’m making mistakes, other students are faster than me*).

- They don’t like mathematics (
*I don’t like this feeling of being wrong, and I see other students are faster, and therefore smarter, than I am*). - They don’t want to pursue mathematics (
*I am not going to take math my senior year of high school and/or in college*). - They don’t pursue STEM professions (
*I wanted to become a ___ but I had to take too many math courses*).

#### Let’s extinguish the statement 'I was never good at math,' which actually means, 'I was not good at remembering processes that had no apparent purpose.'

**Mathematics Experiences to Extinguish**

**1.**

*Memorizing***.**Memorization is not effective for learning and retention. In 2010, the Program for International Student Assessment (PISA) assessed the mathematical learning strategies of more than 250,000 15-year-old students in 41 countries, looking for correlations with achievement (OECD, 2010). They found that “memorization/rehearsal strategies are almost universally negative, suggesting that memorization is an ineffective strategy for learning mathematics” (p. 99). Memorizing says to students, “No need to make sense of math.” Having students memorize basic facts is particularly damaging because they lose out on the opportunity to develop essential number relationships that will support their fluency beyond basic facts (Bay-Williams & Kling, 2019).

**2.**

*Speed/timed activities***.**Children as young as 1st grade experience anxiety in math (Ramirez, Shaw, & Maloney, 2018). Extinguishing timed basic fact tests could go a long way in lessening that anxiety. Board races as well as apps and games that position the faster thinker as the winner must go. As Figure 1 emphasizes, automaticity with basic facts—not speed—is the goal. Pressure to recall a fact quickly doesn’t make a student automatic—it makes them anxious.

**3.**

*Teacher shows how to solve a problem, students replicate***.**An “I do, we do, you do” approach to mathematics is the opposite of the disciplinary practices approach in mathematics and science, both of which are inquiry-based fields. Familiarity with and understanding of algorithms is important, but weeks spent working on standard algorithms, at the exclusion of other methods, is malpractice. “I do, we do, you do” is the predominant way that math is learned in school, especially middle and high school. This will be difficult to extinguish, but if these days were pared down to, say, less than half, and the other days were designed in ways that prioritized the mathematical practices, the impact on student competence and confidence could be profound.

**Mathematics Experiences to Expand**

*How might a fluent person solve these problems?*

- 9 + 7
- 2.8 + 3.9
- 6(
*x*– 1) – 4(*x*– 1) = 10 [solve for*x*]

**1.**

**Reasoning strategies involve choice and reasoning. To solve for 9 + 7, for instance, there are several good options. One is “Making a Ten.” Move 1 from the 7 to the 9 to create a new expression 10 + 6. The answer is 16. This strategy is based on knowing how far away a number is from 10, which is a fundamental understanding in mathematics. So, in teaching strategies, you are developing strong**

*Teach students to use reasoning strategies.**conceptual*foundations. Can that strategy be used for 2.8 + 3.9? Yes, it can! Now, we are “Making a Whole.” Move 0.1 from 2.8 to 3.9 to create the expression 2.7 + 4. The answer is 6.7. Mathematics educators teach this strategy using manipulatives, visuals, and stories, as illustrated in Figure 3.

**2.**

**Many students think that once they learn the standard algorithm, they are supposed to always use it. Instead, standard algorithms should be thought of as an addition to a student’s existing repertoire of strategies (Bay-Williams & San Giovanni, 2021). Within that repertoire, a student’s job is to notice features of a problem to help them**

*Teach students to choose among strategies (and algorithms).**make an appropriate choice*to solve the problem efficiently.

*x*– 1) – 4(

*x*– 1) = 10. Rather than immediately “eliminating parenthesis first,” look at the features in this problem. The quantities inside the parenthesis are the same, so combining those two terms will be efficient. The new equation is 2(

*x*– 1) = 10. At this point, relational reasoning can be applied (what can

*x*be so that the quantity inside the parentheses equals 5? 6).

**3.**

**Practice should involve reasoning, not replicating! Practice can be motivating and meaningful, especially with games. For example, you can have students play “For Keeps” (see Figure 4 for instructions), saying aloud the reasoning strategy to find the sum. If they draw 5, 2, 9, 4, and make the expression 29 + 45, they can use the “Make a Tens” strategy. This game can be adapted to decimals (2.9 + 4.5) to practice the “Make a Whole” strategy. “For Keeps – Subtraction,” similarly, can be used to practice subtraction reasoning strategies.**

*Practice reasoning strategies.***4.**

**Math tests are graded based on correct answers. That is an accuracy assessment, not a fluency assessment. How are flexibility and efficiency assessed? Prompts can be added to tests that focus on efficiency and flexibility. For example, trade out the instruction, “Solve using ___ method” for one of these options:**

*Attend to and assess fluency.*- Choose your method for each problem.
- Use at least three different strategies in solving this set of problems.
- Select one problem and explain what strategy you used and why you chose it.

#### A key distinction in the reimagined classroom is student decision-making—students choose methods and reflect on their choices, changing them out as needed.

**“I Was Never Good at Math”**

### Math Fact Fluency

An indispensable guide for any educator who needs to teach basic mathematics facts from experts Jennifer Bay-Williams and Gina Kling.