Fluency Fallacy #3:
Representations Are Strategies.
When we work with students on fluency, we often ask, “What strategy did you use and why?” Sometimes students will give a representation to show their solution. For example, if I ask a student to solve 39 + 12 these are some of the solutions I may get from students:
· Michael: I added 30 + 10 and then added 9 + 2
· Beth: I used a number line
· Cheryl: I used my 120 chart
· Robert: I pretended 40 + 12 and took one away
Strategies explain how students think about numbers. Looking at the examples above, Michael and Robert are sharing strategies. Beth and Cheryl are using tools (visuals or manipulatives) to solve. They did not explain what they did with a representation. As teachers we need to make sure students understand the difference between a strategy and a representation.
Representations are tools used to support student thinking and reasoning. They should represent or explain their thinking (their strategy). They may use a tool for support, but we do not know what they did with the tool unless you ask probing questions. For example, “Cheryl, how did you use the 120 chart?”
As teachers we need to explain to students the different between tools and strategies. See an example chart below:
Helpful Hints:
1. Share a chart with the difference between tools and strategies and illustrate the meaning of each.
2. Question students with precise language (example: “What strategy did you use?”
3. Ask what tool you used to help you solve using that strategy.
The Truth About Fallacy #3
Remember that representations are not strategies. They support a strategy, but we need to remember to ask those probing questions when working with fluency such as, “What strategy did you use? How does your tool or representation support your strategy?” Asking the appropriate questions is the key to diving deeper into understanding student thinking.
Fluency Fallacy #4:
Strategies and Algorithms Are the Same Thing.
Some think that strategies and algorithms are the same thing, but they are not. See the examples below in the chart:
Be careful not to turn strategies into algorithms. Students may know the algorithm, but can they explain how? For example, if a teacher poses 8 x 20 and a student solves using an algorithm, ask, “How did you solve 8 x 20? What strategy did you use?” Get students to explain (example: I know that half of 20 is 10 and 8 x 10 is 80, then I doubled 80. 80 + 80 = 160).
The Truth About Fallacy #4
Strategies and algorithms are not the same. Strategies require students to explain their reasoning and thinking while algorithms show the steps needed to solve a problem. Both are important but be cautious when explaining each method. Continue reading below to find out why the Standard Algorithm is not always the best choice to use.
Fluency Fallacy #5:
Once Learned, the Standard Algorithm is the Best Choice.
Growing up we were all probably taught the traditional algorithms for math. We learned that we needed to use these from textbooks and worksheets to master a skill and solve various types of problems. But this is not always the best choice.
Be cautious when using algorithms.
1. Make sure students are ready to use the algorithm (accuracy).
2. Teach students to use an algorithm only when it makes sense (efficiency & flexibility).
Have conversations with students to discuss when it is appropriate to use an algorithm. See example below:
The Truth About Fallacy #5
Standard algorithms can be used but think about what you are teaching and ask yourself if this is the best practice to use. Remember that algorithms should not replace strategies.
(Source: Figuring Out Fluency 2021 by J. Bay-Williams & J. SanGiovanni)
Comments