Math Facts through Conceptual Understanding Emilie Schiff K-5 Math Specialist and Instructional Coach. Lake Orion Community Schools

Making the Case… Focus on number sense! Research indicates that early number sense predicts school success more than other measures of cognition like verbal, spatial or memory skills or reading ability. Jordoan, Kaplan, Locuniak, and Ramineni, 2007 Why are we talking about fluency and number sense

Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts By Jo Boaler Each take a paragraph. What stood out. Highlight, underline, mark up…Be prepared to share your M.I.P. Professor of Mathematics Education, co-founder youcubed with the help of Cathy Williams, co-founder youcubed, & Amanda Confer Stanford UNIVERSITY Mark It Up.

What is Mastery of Basic Facts: Addition and subtraction by the end of 2nd grade Multiplication and division by 3rd grade Fact fluency leads to other math fluent skills… Imbedded in the Common Core-Standards for all students- Quick response (3 seconds) w/o resorting to inefficient methods such as counting

Guided Intervention Fact mastery will not magically happen. “Time is a poor intervention.” Parents can help, by showing your children your active strategies, and your thinking aloud. Teachers who relied heavily on textbooks that focused on memorizing basic fact strategies had students with lower number-sense proficiency.

Developmental Nature of Basic Fact Mastery Counting Reasoning Mastery Teaching basic facts requires essential understanding that student’s progress through stages that eventually result in “just knowing”. Listed are the three phases: Phase 1: counting strategies Phase 2: reasoning strategies-using known information to logically determine an unknown combination. 4 + 7, student knows 3 +7 = 10 so just adds on 1 more. Phase 3: mastery producing answers efficiently, just knows it, 4 + 7 = 11.

Phase 1: Counting Strategies Counting strategies- using concrete objects verbal counting Example includes: 5 + 1 = count on starting at 5, 6 5 + 2 = 5, 6, 7 4 +7 = 7 then 8, 9, 10, 11

Phase 2: Reasoning Strategy Reasoning Strategy: using known information to logically determine an unknown combination 4 + 7, student knows 3 +7 = 10 so just adds on 1 more. 10 x 8 is 80 half of that is 5 x 8 is 40

Mastery Phase 3: Producing answers efficiently just knows it 4 + 7 = 11 Retrieval within 3 seconds Grade appropriate strategy Efficient In first grade knowing 5 + 2 , (5 plus 1, plus 1) more would be seven would not be grade appropriate at 5th grade Counting by 5’s is good to get to 20, but not 2000

Knowing Facts from Memory- “Passive Storage View” Thinking that students will learn their facts if they just practice enough What does this mean- 100 isolated addition facts 100 isolated multiplication facts Also having to memorize subtraction and division- well over 300 pieces of information to remember Not only having to memorize, but to keep practicing as well. Strong evidence that this method does not work well! Think of how many students simply don’t have their facts mastered. People may say, well I did it, but early research shows that those who mastered facts used some strategy to remember those facts. Inefficiency. There are too many facts to remember.

Effects of not knowing facts fluently… Lack of basic fact automaticity has been shown: Limit participation in math class discussions Impede successful problem solving Severely impair the development of the standard algorithms for multiple-digit addition and subtraction, long division and fractions Misapply facts and not seeing reasonableness of answers Inflexible thinking skills If students don’t make the correct relationships by themselves they will often use incorrect relationships. An example is 1 + 0 and 1 x 0 Inappropriate applications. Students misapply the facts and don’t check their work. Inflexibly. Students don’t learn flexible strategies for finding the sums (or products) and therefore continue to count by ones.

Struggling Learners and Students with Disabilities: Have difficulty memorizing so many isolated facts (but can be successful with strategies) Drill creates, in a majority of students, unnecessary anxiety Undermine student interest and confidence in mathematics Connecting to what students already know allows students to learn basic facts for life.

Parent Supports Scaffold the Language, but not the mathematic strategies. Explain the vocabulary they don’t know, in simpler terms, but don’t “dumb down” the mathematics. Use Realia and Models Use Graphic Organizers Use gestures for together, take apart, groups etc.

Explicit Strategy Instruction: Strategies can be effective to learning math facts. Supports students thinking rather than give the students something new to remember. Key: Help students see possibilities and let them choose strategies that help them get to a solution without counting. Strong evidence indicates that these strategies can be effective to learning math facts. Supports students thinking rather than give the students something new to remember. Teachers who relied heavily on textbooks that focused on memorizing basic fact strategies had students with lower number-sense proficiency. Key: Help students see possibilities and let them choose strategies that help them get to a solution without counting.

Addition FACT Strategies Use Doubles Double Double Plus 1 Double Plus 2 Bridge to 10 All Facts Count On Count On 1 Count On Turnarounds Count On 2 Count On 3 Count On 0 Research shows that the most effective way for students to learn the basic facts is to arrange the facts into clusters. Each cluster is based on a thinking strategy that students can use to help them learn all the facts in that cluster. (Fuson, 2003, Thorton, 1990) For example use doubles is on of the addition clusters. Within that cluster are double facts and facts that involve doubling then adding 1 or 2.

Prepare… Before teaching strategies students need to know… Subitizing The ability to instantly recognize the total quantity of objects in a group without counting Small infants can subitize.. Trajectory state is developing states that by age 1-2 they should know 1, 2 and more Many of our children come to school without any formal experience that allow them to build on this subiizing idea.

1. Introduce Concept Through Real Objects Transitioning to Model Counting bottles Cubes in a cup Addition stories Count on 1 Cards

2. Reinforce Through Models Count on Cards Cube Trains Count on Cards with Numeral Cards Does This Make Sense? Moving from physical models to semi-abstract models to symbolic models (with numbers) These are models that will continue to work with kids throughout their math cubes Cube trains starts the work around arrays Does this make sense starts students on the track of CCSSM college/career ready by having to justify their answers “there is not a job out there that doesn’t require thinking! If it is automated, it is automated-we build machines to do the work…

3: Practice through Games Spin, Count On 1, Record Fact Reinforce with Count On Flash Cards Count On 1 Bingo Total and Expression Subitizing games/Other resources: Youcubed.org http://teachmath.openschoolnetwork.ca/grade-1/number- sense/subitizing/ https://www.pinterest.com/explore/subitizing/?lp=true Send home as homework for family engagement

Multiplication Strategies Use a Rule One’s Facts Zero’s Facts Build Down and Build Up Nine’s Facts Six’s Facts Last Facts Use Tens Five Facts Doubling Two’s Facts Four’s Facts Eight’s Facts

Multiplication Strategies Use Tens Five Facts Doubling Two’s Facts Four’s Facts Eight’s Facts Use a Rule One’s Facts Zero’s Facts Build Down and Build Up Nine’s Facts Six’s Facts Last Facts

What to Do When Teaching Basic Math Facts Ask students to self-monitor Focus on self-improvement Drill in short time segments Work on facts over time Involve families Make drill enjoyable Use technology Emphasize the importance of quick recall

What Not to Do When Teaching Basic Facts Don’t use lengthy timed test Don’t use public comparisons of mastery Don’t proceed through the facts in order from 0-9 Don’t work on all the facts at once Don’t move to memorization too soon Don’t use facts as a barrier to good mathematics Don’t use fact mastery as prerequisite for calculator use.

Did we? Identify fact fluency is a developmental process. Identify research based strategies that help develop fact fluency Understand strategies that are ineffective for fact fluency development Learn specific strategies that help develop fact fluency Hand out games and additional resources from Boalers article.

Emilie Schiff: Elementary Math Specialist Contact Information Emilie Schiff: Elementary Math Specialist Office (Main Office Hallway) is at Orion Oaks Phone: 248-393-0010 (x 4010) email: emilie.schiff@lok12.org