Do Students Really Know Their Math Facts?
What is fact fluency? Is it memorizing facts and being able to cite them quickly? I would agree that many students can accurately state or write math facts in a timely manner. Do those same students really understand what the facts mean and represent? This is often a topic for discussion about how to teach math facts and if they have the conceptual understanding of the facts.
I was fortunate to attend a session at the Wisconsin State Math Conference facilitated by Dr. DeAnn Huinker who is on the Board of Directors, National Council of Teachers of Mathematics. She is also a professor at the University of Wisconsin-Milwaukee as well as a co-author of many books including Principles to Actions (2014). Our district math vision and practices are grounded in the the eight mathematics teaching practices from this book published by NCTM. The session that connects to students knowing their math facts focused on the practice of, Use and Connect Mathematical Representations: Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.
The visual model below shows the importance of connections within and between the various representations.
National Council of Teachers of Mathematics, Inc. (2014). Principles to Actions: Ensuring mathematical success for all. Reston, VA.
By embedding the above into practice, students will be able to make connections and develop a deeper understanding of math facts. Here is an example of how to expand on what you may already be doing:
Find the answer:
Explain how you figured out the answer.
Draw a picture that shows what ___X___=___ means.
Write a story problem for ___X___=___.
If we ask a student to tell us what 3x4 equals, they may be able to tell us that it equals 12. We should not stop at that and assume that the student really knows what 3x4 equals in context. Asking more of them could tell a different story. Let’s take a look at a couple of student examples below.
By embedding the above into practice, students will be able to make connections and develop a deeper understanding of math facts. Here is an example of how to expand on what you may already be doing:
Find the answer:
Explain how you figured out the answer.
Draw a picture that shows what ___X___=___ means.
Write a story problem for ___X___=___.
If we ask a student to tell us what 3x4 equals, they may be able to tell us that it equals 12. We should not stop at that and assume that the student really knows what 3x4 equals in context. Asking more of them could tell a different story. Let’s take a look at a couple of student examples below.
Huinker, D., Dr. (2019). Developing Representational Competence in Our Students [Handout]. Board of Directors, National Council of Teachers of Mathematics.
Professor, University of Wisconsin-Milwaukee, WI.
If Olivia was just asked what 4x7=, we may think that she knows that fact. Having students explain their answer helps us know what strategy they are using. Olivia is skip counting here. Her picture could lend itself to teaching her a visual four groups of seven with manipulatives and then how to illustrate it. It is clear that she could also benefit from instruction on how to write a story problem for this fact.
Professor, University of Wisconsin-Milwaukee, WI.
If Olivia was just asked what 4x7=, we may think that she knows that fact. Having students explain their answer helps us know what strategy they are using. Olivia is skip counting here. Her picture could lend itself to teaching her a visual four groups of seven with manipulatives and then how to illustrate it. It is clear that she could also benefit from instruction on how to write a story problem for this fact.
Huinker, D., Dr. (2019). Developing Representational Competence in Our Students [Handout]. Board of Directors, National Council of Teachers of Mathematics Professor, University of Wisconsin-Milwaukee, WI.
Dylan also got the answer correct. He also used a skip counting type of strategy. He is ready for a more sophisticated strategy. Dylan’s picture shows twenty-eight total using tally marks. He is ready for learning that multiplication requires groups of objects. Dylan could also spend some time revising his story with four baskets and putting seven apples in each basket.How do these two examples change your thinking about teaching and learning facts? Asking students to draw and use visuals to support their thinking around facts provides insight into next steps for instruction. Having students connect the facts to real life stories not only tells what they are able to do, but it creates meaning of the math. A take-away for me in the session was that students need to make connections between the visual and contextual elements of math facts. I encourage you to give it a try and see what you learn about your students.
Citations:
Huinker, D., Dr. (2019, May). Developing Representational Competence in Our Students. Seminar conducted from the Wisconsin State Math Conference, Green Lake, WI.
National Council of Teachers of Mathematics, Inc. (2014). Principles to Actions: Ensuring mathematical success for all. Reston, VA.
Great post, Michelle - I love the diagrams of the connection between the various representations. Great examples, too. There are many aspects of fact fluency!
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