(This is the last post in a two-part series on this topic. You can see Part One here.)
Susan Hobart asked this week’s “question-of-the-week":
What strategies help math facts stick besides the old “drill the skill” and, if someone is not proficient at addition facts, can learning multiplication help or confuse?
Responses in Part One came from educators Anne Collins, Sue O’Connell, Alexandra Mattis and José Luis Vilson. Today, Leslie Texas, Tammy Jones and Denise Flick share their thoughts, as do a number of readers.
You might also be interested in a ten-minute conversation that I had with Anne and Sue on my BAM! Radio Show.
Response From Denise Flick
Denise has taught grades one thru seven, and numeracy intervention and enrichment. She is now a district learning coordinator. An instructor of Math Methods and Aboriginal Issues Courses for the University of British Columbia, Canada. Her thesis study was on teacher and student math anxiety:
Every September teachers vow their students will complete the school year having memorized addition and subtraction facts. Yet, come June many students have not. What is the problem? Hard working teachers are providing planned repetitive opportunities for practice. Computer programs, classroom games, card games, songs, chants, flashcards and Mad Minutes have not done the trick? For many, the above practices are ineffective for the development of fact automaticity. Unfortunately the above game and drill activities may be tremendously effective nurturing negative math disposition and anxiety.
The very fact that so many students in grades four, five, six, and beyond do not know their math facts should be proof that drill is not effective. Students should not be expected to memorize number facts that they do not understand. They certainly should not be expected to complete series of facts under timed conditions. If a student is struggling with automaticity, the reason for the student`s struggle must be found. Questions need to be asked.
•Is the student able to demonstrate an understanding of addition and subtraction using manipulatives, pictures, words, and/or numbers?
•Does the student have multiple strategies for solving number facts? (Counting or touching are not efficient and/or counter productive to sense making.)
Remember that strategies only work if students have number sense on which to build otherwise, strategies are just “tricks”. Instead of the old drill and kill, provide practice in the use and selection of strategies once they have been student developed through teacher facilitation.
On the road to automaticity of the basic facts, students pass through several phases:
•strategy development and use
Students who struggle with number fact automaticity often believe that there are hundreds to be learned. Students who commit facts to memory are able to apply many strategies to any number sentence.
Here are some potential teaching strategies to consider:
Provide opportunities for un-timed practice of the facts. Students should be instructed to answer questions for which they are sure of the answer. (Practice does not make perfect, practice makes permanent. Never ask a student to answer a math fact if they might do so incorrectly.)
Next have students circle the questions for which they have a strategy. Circulate and have students share their strategies with you and with other students.
Change it up a bit. Ask students “If the answer is 16, what might the fact be?”
Check to see if students are able to partition 16 into all of its possibilities and organize the possibilities to ensure they have all.
For more ideas and information to help with math fact facilitation please visit What About Those Math Facts.
Response From Leslie Texas & Tammy Jones
Collectively Leslie Texas and Tammy Jones have almost 40 years of classroom experience in teaching mathematics in elementary, middle, and high school. One of the things that set Leslie and Tammy apart as consultants is their work with classroom teachers, modeling and offering continued support throughout the year to build capacity at the building and district levels.
Their book series, Strategies for Common Core mathematics: Implementing the Standards for Mathematical Practice, is available from Routledge:
So how does one get from practice to proficiency? First, there is nothing wrong with drill and there are several different approaches that can be used in this context.
Through the use of games, students develop fluency while practicing in a non-threating environment. Students can practice and build confidence in problem solving and procedural mathematics through interactive game play. Students extend critical thinking skills while building strategies for successfully completing the game. NCTM (The National Council for Teachers of Mathematics) has the Product Game on their Illuminations Website. Shodor also has several interactive activities such as their Arithmetic Four that can be used and is easily leveled to meet the needs of individual students.
In our book series, “Strategies for Common Core mathematics: Implementing the Standards for Mathematical Practice” (Eye On Education, Texas & Jones, 2013) we have a section devoted to the use of interactive games and how they support several of the CCSSM Standards for Mathematical Practice. One such game is the ABC Sum Race. Check it out here.
The question is, “How can I develop mathematical thinkers and provide an opportunity for computational proficiency?” (Texas & Jones, 2013)
In order to gain mastery, they need to practice the skills. This means working math problems in order to become computationally proficient.
Expertise is generated in practice but implemented through process. The Standards for Mathematical Practice are how the student engages with the mathematical content to develop both procedural fluency and conceptual understanding. It is for this reason that we recommend teachers focus on the Standards for Mathematical Practice since they serve as the door to accessing the content. You will notice that one of the ABC Sum Race task cards employs building arrays. Through the use of multiple representations and through being involved in a kinesthetic activity, students experience and see the mathematics.
The processes are how students gain proficiencies in the content that allow them to develop the practices that ultimately become sound habits.
You can see a larger version of this infographic at Routledge’s website.
Responses From Readers
I teach a high-level version of Algebra 1 to some pretty bright kids. We drill for skill, but we always move forward. Every day we begin with a review that covers past skills because not everyone learns at the same rate. And, the fiddler who practices most plays best.
When I have strugglers I always check their birth date because it is the math that trips up the young ones. As long as they are age appropriate in their grade they should not be left behind. The slower kids can often catch on to the next level even without having mastered the earlier skill.
That doesn’t mean the unmastered skill should be overlooked it just means that if the student can gain a little confidence in another skill he can be caught up in a past skill.
I have always used 2, 3, or more dice with kids working on addition facts. Keep throwing the dice and having the student call out the sums. Their speed will pick up pretty quickly.
One way is to focus on estimation and explaining your reasoning before working out a problem. Using 10’s is a good anchor. For example, if you don’t remember the answer to 4x8, do 4x10 , then subtract 2 four times and get 32. it’s even easier to do that with addition and subtraction and only a little harder with long division. Don’t try to divide 879 by 46; divide 800 by 50 in your head and you’ll be pretty close to the right answer. What I am trying to say is that you can use 10 or multiples of 10 to get a rough answer for almost any problem. For an exact answer, use the single digit you excluded to get the additional amount. (I could illustrate this better than I can explain it.)
Thanks to Leslie, Tammy and Denise, and to readers, for their contributions!
Please feel free to leave a comment your reactions to the topic or directly to anything that has been said in this post. As I mentioned earlier, I’ll be including readers’ comments in Part Two.
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