Are your students struggling with basic mathematics fact recall? Memorization of basic mathematics facts is a basic principle in Classical Christian classrooms. This presentation will discuss basic facts’ instruction and how it looks in the classical classroom. It will also provide insight on how to improve recall without increasing anxiety in students. The researcher will share the stages students go through when memorizing facts and review the research that overwhelming supports memorization of basic facts. Finally, the researcher will provide guidance on intervention when fact memorization becomes difficult for students. You will walk away better informed about basic math facts and better equipped to teach for memorization in your classroom.
Cristina Dube
Dr. Cristina Dube is in her 7th year as the grammar school mathematics specialist at the Geneva School of Boerne. She holds a Doctorate degree in Curriculum and Instruction from Liberty University and has been involved in mathematics education for over 30 years. She is the author of The Three R's of Mathematics published in The Journal in 2019. The results of her dissertation, Self-Efficacy Score Differences Between Supported, Unsupported, Departmentalized, and Non-Departmentalized Classical Christian Elementary Mathematics Teachers, will be published soon. Prior to working at Geneva, she spent 14 years as an assessment specialist writing math textbooks and tests. She has a passion for mathematics education, classical education, and connecting mathematics to other disciplines.
As grammar school teachers, we have to teach many subjects with a limited time to prepare for each of them. is session’s objective is to equip you with some tools for teaching math. is seminar will o er ideas on how to teach the good, true and beautiful in math, as well as speci c suggestions on how to integrate math with other subjects. You will leave with several very practical tips: a list of warm-up activities, wise time activities and easy ways to challenge your students.
Hana Rodgers
Hana has a master’s degree from the Czech Republic in English, social sciences and education. She loves sharing her love for God with her 3rd-graders and is thankful for the opportunity to pursue the mission, vision and values of The Cambridge School in San Diego. She is grateful for the opportunity to point her students daily to the truth of God’s Word and to form the minds of her students. Hana is passionate about creating wonder and
a love of learning in her students and is always eager to improve in this area. She enjoys teaching about the Creator, diagramming, thinking of new ways to integrate the many subjects she gets to teach, creating innovative ways to bring history to life for her students, helping her students enjoy math and much more.
Many discussions of mathematics from a classical Christian perspective focus on presenting math as true, good and beautiful. While this is an integral conversation to bring into the classroom, it’s an incomplete picture. Students would leave our schools ultimately unchanged in how they practice and understand mathematics. This presentation will challenge educators on how to complete the sentence “Math is ” with language that considers the practical experience. How do we understand not only the philosophy of mathematics, but the practice from a Christian perspective? How do the practices and liturgies of the math classroom impact the affections of students? We will end by offering some practical examples that can be implemented in your own classroom.
Josh Wilkerson
Josh has a bachelor's degree in
mathematics from Texas A&M, a
master's degree in historical theology
from Dallas Th eological Seminary and a doctoral degree in math education from Texas State University. He has taught math at Regents School of Austin for the past seven years and serves as the Chair of the Mathematics Department for the logic and rhetoric schools. In his spare time, he runs the website www.GodandMath.com, which is devoted to the integration of math and Christian faith. Josh also serves on the board for the Association of Christians in the Mathematical Sciences.
Geneva School transitioned to a conceptually-based, problem-solving focused mathematics curriculum for upper grades. We implemented Math in Focus for lower grades eight years prior. This curriculum uses collaboration to foster learning and focuses on students making sense of mathematics for themselves. We will share lessons learned from the process, feedback from parents, implementation challenges, obstacles and success stories. Come learn from our experiences and see how you might make a similar transition within your school.
Janet Andreasen
Janet teaches prospective and
practicing math teachers at the
University of Central Florida. Her
research interests include examining
mathematical knowledge for teaching
and using technology to foster learning
of mathematical concepts. She received a bachelor’s degree in biomedical engineering from the University of Miami and both a master’s and doctoral degree in mathematics education from the University of Central Florida.
Kevin Clark
Kevin serves as Academic Dean of
Th e Geneva School, where he has been
a member of the rhetoric faculty for 14
years. Dr. Clark is a founding Fellow
of SCL’s Alcuin Fellowship and speaks
regularly at SCL and Alcuin retreats and
conferences.
Christine Miller
Christine joined Th e Geneva School
faculty in 2006 and teaches mathematics to dialectic and rhetoric students. She received her bachelor's degree in computer engineering from the University of Central Florida and
worked as an engineer in the Central
Florida area for seven years. In its
inaugural year, Christine was the upper
school winner of Th e Geneva School
2012 Paideia Award for Excellence in
Teaching.
Success in mathematics requires a variety of skills, all of which are perfectly situated within classical Christian schools. Classical Christian educators can use God’s Word to help students develop these skills. Recognizing the good information from the bad is a key objective sought by classical educators. The ability to see truth in a world full of untruths is imperative. The same skill applies to mathematics: Students must be able to recognize relationships in mathematics to be able to know how to proceed. This recognition, the first R, will help students get started on one of the toughest parts of mathematics at any age, problem-solving. The second R, retrieval, is a basic tenet of classical education: Students must memorize their basic facts and be able to retrieve the facts quickly. Finally, the third R, resolve, can be taught both biblically and through developing students’ mindsets. Classical Christian educators have at their disposal biblical truths in developing students’ resolve.
RECOGNITION
Developing students who have good number sense is critical in mathematics. Nguyen et al. discovered that early numeracy ability in preschool is a strong predictor of fifth-grade mathematics achievement scores (2016). What is number sense? It is a group of skills that allows people to work with numbers. Witzel, Ferguson and Mink discuss five components of number sense: magnitude comparisons, strategic counting, retrieval of basic arithmetic facts, word problems and numerical recognition (2012). The authors go on to discuss three methods for improving number sense. First, the authors support constructivist claims that children construct their knowledge through manipulating concrete materials. Second, they discuss how proficiency of skill should not just include algorithms, but also the meaning behind the algorithms. Finally, the third key element of the theory, the importance of making language connections, is offered to integrate math to everyday life. The first R, recognition, is best accomplished through these language connections. In her article on this idea of language in mathematics, Susperreguy emphasizes the importance of math talk, specifically the use of language comparisons. The use of math talk that includes cardinality and counting is ubiquitous in homes. What is missing, according to Susperreguy, is the use of comparisons: more than, less than, parts and wholes (2016). It is the recognition of parts and wholes in problems that unlocks mathematical understanding. Knowing that two parts are given in a problem allows the solver to add, no matter what the numerals are that are being added. Knowing that a whole and a part are given allows the solver to subtract. Taking time to recognize the information in the problem is key. This is best understood in the context it is required: problem-solving.
Nicholas needed to distribute 5 ¼ bags of grass seed on a lawn. He distributed 3 ½ bags in the morning. What is the total amount of seed he still needs to distribute before running out?
These rational adjectives (5 ¼ and 3 ½) can sometimes cause angst for students, and students then struggle with knowing which operation to choose. What if, on the other hand, the problem had no fractions in it?
Nicholas needed to distribute 5 bags of grass seed on a lawn. He distributed 3 bags in the morning. What is the total amount of seed he still needs to distribute before running out?
The problem becomes much easier for upper elementary students and they can immediately recognize that a whole and a part are known and that they need to subtract. By using the language of parts and wholes, students recognize the relationship and are on their way to solving. Knowing the operation required to solve problems eliminates one of the most common errors: choosing the wrong operation (Ferrucci, Yeap, & Carter, 2003). The same idea applies to multiplication and division, the only difference being that the parts are equal parts and that if you have equal parts and know the number of parts, you multiply to determine the product.
When we encourage students to seek truth in the relationships and carefully work through problems, we encourage the biblical virtue of carefulness. Phillip Dow writes in his book Virtuous Minds that, “Those who are intellectually careful earnestly want to know the truth; thus, they are reasonable and consistently careful that they do not overlook important details and habitually avoid hasty conclusions based on limited evidence.” When we teach students to take time to discover parts and wholes in mathematical problems, we are teaching this carefulness. And finally, from John 8:32, “And you will know the truth, and the truth will set you free.” God’s truth illuminates the need for seeking truth.
RETRIEVAL
Retrieval of basic mathematics facts is a hot topic in education. Classical educators, however, have consistently held students responsible for memorizing basic facts. The importance of quick retrieval of basic facts cannot be overemphasized. A study by Calderon-Tena and Carerino in the Journal of Science and Mathematics Education in 2016 supports this return to holding students accountable for memorizing their basic facts – something classical educators never left. The researchers found that longterm retrieval skills became a better predictor of both mathematics calculation and mathematics problem-solving as age and grade increased.
The time that is devoted to fact retrieval tends to focus most on the initial counting stages and on the ubiquitous practice of timed tests. How to get effective practice at that middle stage will be the focus of this section, and brainbased research will help explain why it is important. In his 2014 book The Confident Student, Kanar discusses the three stages of memory: sensory memory, short-term memory and long-term memory. Sensory memory is the memory that takes in information. What a person sees, hears and touches all are taken in and sensed by the brain. If what the brain senses is attended to and processed, then it makes it into short-term memory. Short-term memory manipulates and processes information for about 30 seconds. Finally, if the information is rich enough and engaging enough, the information gets transferred into long-term memory. How does knowing this information help with basic fact retrieval? Simply put, attention matters. Students typically are first taught basic facts through a progression similar to the following: counting, adding zero, doubles, doubles +1, combinations of ten, make ten, doubles +2, +9, +4 in addition, then using addition facts to help retrieve subtraction facts (Purpura, Baroody, Eiland, & Reid, 2016). When they are taught these strategies, such as doubles plus one, teachers use effective manipulatives and visuals to first teach the meaning behind the basic facts. This follows cognitive learning theory first introduced by Jean Piaget and further developed by Jerome Bruner. Bruner significantly added to learning theory by stating that children first need to use concrete manipulatives to learn concepts, then transition into pictures of the objects, and finally transfer to abstract numerals to represent the number of objects. In math fact retrieval, they may count eight blocks, then add two blocks to work in the concrete stage. Next they might use a ten-frame to show pictures of blocks and visualize that 8 and 2 always make 10. Finally, they will write the equation 8 + 2 = 10 and work with numerals. This type of practice is in every mathematics curriculum in the United States, including those used by classical Christian schools. This is as it should be, for students who have good number sense and practice with rich strategies are more successful at transferring the information into long-term memory (Purpura, Baroody, Eiland, & Reid, 2016).
The question should follow then, why we have so many students who struggle with their retrieval of basic facts? The answer lies in what comes next in schools around the country. Students who initially practice retrieving their facts by spending time counting to retrieve them, such as 8 + 2 = 9, then 10, do not experience the same level of richness as students who associated their facts with known facts. Utilizing what is known in memory to learn unknown information is key to all of learning, but especially to basic fact retrieval. Students must be fair-minded enough to try new methods for retrieving facts. Dow speaks of the importance of fair-mindedness as well in developing students who have virtuous minds (2013). Fair-mindedness in mathematics is crucial to understanding the subject. Fair-mindedness comes into play in problem-solving, understanding relationships, and yes, in basic fact retrieval. Students who are retrieving their facts by counting as fast as they can should learn new retrieval routes, but in order to do so they must be fair-minded.
Classical Christian schools traditionally emphasize the importance of basic fact retrieval, and they should. I am not saying that basic fact retrieval is time wasted. On the contrary, research demonstrates that it is time well spent. What comes next, however, in many schools is the use of timed tests to retrieve basic facts before students are ready to be timed. Much research has shown that the overemphasis on timed tests at too early of an age results in math anxiety, something we all want to avoid for our students (Boaler, 2016).
Why not allow more practice for basic fact retrieval within the associative, strategic stage? This is no small task, and I do not mean to trivialize it. Most educators do not know what this looks like. What I am calling for is a change in both curriculum and instructional practices that still allow for accountability, a key component of classical Christian education. Students who are struggling with their fact retrieval do not need more timed tests or more manipulatives. Instead, they need more time associating, or deriving their facts. At our school, The Geneva School of Boerne, students are doing just that. If they show signs of counting or skip-counting while trying to retrieve their addition, subtraction, multiplication or division facts, they are given the tools to help them practice more in the deriving stage. We still require them to spend time retrieving their facts, and we hold them responsible for memorizing those facts. However, using standard flash cards can be just as detrimental to developing math anxiety as timed tests if pressure is placed on students to retrieve them quickly. Rather they should spend time altering their retrieval by associating the unknown fact to known facts. Students need rich practice to transfer information from short-term memory to long-term memory, as Kanar suggests (2014). They also must be fair-minded enough to try new methods to retrieve their facts if they have continuously built the counting pathway in their brain. The second R, retrieval of basic facts, is a key tool that students must possess.
RESOLVE
Finally, the third R, resolve, must be considered as an important characteristic for students to develop. Students who think they can solve math problems are the most successful. Self-efficacy, or beliefs about one’s abilities to accomplish goals, can influence activities people participate in (or not), the amount of effort they give to tasks and the persistence of effort and level of achievement reached (Boaler, 2016; Cerit, 2013). Self-efficacy is an area of study that needs to be further investigated in all teacher research studies, but specifically in the content area of mathematics.
Additional research on self-efficacy has been conducted recently by Carol Dweck (2006), who clearly shows the importance of students’ mindsets in her book, Mindset, by elucidating the difference between students who have a fixed mindset and those with a growth mindset. Those with fixed mindsets believe that they either have a talent, or do not. Those with growth mindsets, on the other hand, believe that if they work hard enough they can learn anything. Boaler has connected mindset research from Dweck to the area of mathematics in her book Mathematical Mindsets (2016). Students who have growth mindsets score higher on mathematics achievement tests. Teachers, according to Boaler, can encourage a growth mindset in their students in several ways. For example, the praise that teachers direct towards students is extremely influential. Praise suggesting a student is smart furthers the fixed mindset, whereas praise suggesting the student has worked hard furthers a growth mindset.
Classical Christian educators, however, have the best tool available to help develop students’ mindsets: God’s Word. We can first give examples of grit from the Bible. Moses took a long time to reach the promised land and faced great strife. Yet he persevered. We also know from 1 Peter 1:3-5 that we have a promise of hope and that this promise is not wishful thinking, but rather confidence in God’s faithfulness. A second way to inspire grit is to remind students of times when they were successful in the past. If you develop a relationship with students and know their past success stories, you will be better equipped to help them through challenges they encounter in the future. The third method for mindset development is to model it yourself as a leader. Students look up to their leaders who have grit and are honest about their struggles. We know that one way students establish their own self-efficacy is by watching it modeled by their peers. Tenacity, or resolve, is a virtuous trait that can be developed by reminding students that hard work pays off. Resolve, a virtuous trait, is worthy of being titled the third R in mathematics.
CONCLUSION
The three R’s in mathematics – recognition of relationships, retrieval of basic facts and resolve to work through difficult problems – can be developed by parents, teachers, coaches and mentors. Students need to be surrounded by people who show that they care and take time to help students develop these traits. The Christian virtues of carefulness, fair-mindedness and tenacity can help students develop the three Rs, which, in turn, will help them succeed in their mastery of mathematics.
All too often, math is taught as a set of irrelevant, isolated algorithms. In this workshop, we will explore how to combat this and learn how to teach math in alignment with classical thought by asking better questions. We will practice and use specific categories of inquiry about a mathematical concept, such as historical context, key figures, connection to the Quadrivium framework of mathematics and theological implications. By investigating math units using these categories and learning to show students how to do this for themselves, we can teach them to be better thinkers and mathematicians.
Jeff Chambless
Jeff Chambless has been teaching at Westminster School at Oak Mountain since 2011. Prior to that, he served as a youth minister. He likes to connect his degrees in mathematics, divinity and philosophy to student learning in the classroom. He has one wife, three children and one cat.
What does a Singapore Math classroom look like and sound like? Throughout this in-depth session, we will demonstrate effective Singapore instructional strategies, including questioning, mathematical discussions and writing in the math journal. You will view student samples and experience the importance of inquiry through the anchor tasks. Come join the conversation and learn this approach to teaching that allows students to master mathematical concepts in greater depth for deeper understanding and improved confidence.
Mo Gaffney
Dr. Mo Gaffney currently serves as Head of Lower School at The Covenant School in Charlottesville, Virginia. Before that, she was the Co-Director of the Central Virginia Writing Project and developed teachers of writing through the Summer Writing Institute at The University of Virginia. She has done extensive writing research in elementary schools and has presented her ndings at the NCTE national conference. Mo is an Adjunct Professor for
The University of Virginia, teaching courses in the Department of Curriculum and Instruction. She has led professional development workshops and has presented at Society of Classical Learning conferences on teacher evaluations, reading and writing connections, homework in schools and Singapore Math.
Susy WIlletts
Susy Willetts is the Math Coordinator for Pre-K through 6th-Grade students at The Covenant School in Charlottesville, Virginia. Susy has extensive teaching
and administrative experience with students of all kinds, including gifted and intellectually advanced students in independent specialty schools, struggling learners in public schools and students in Christian schools. Susy leads professional development workshops about Singapore Math Strategies. She
and her husband, Bo, have three teen sons and enjoy spending time together outdoors with their dogs and horses.
This session will introduce a curriculum and pedagogy for mathematics grounded in the classical Christian tradition. It will give special attention to 7th through 12th Grades (or pre-algebra through calculus), though many topics will be of interest to K-6 teachers. This classical approach, which is under active development for release through Classical Academic Press, will demonstrate the possibilities opened by thorough attention to the traditional categories of the Quadrivium, including 1) a pedagogy of puzzle, proof and play; 2) a curriculum of wonders; and 3) mathematics for the sake of wisdom and worship. Everybody will leave with a preliminary packet of new pedagogical models, a sheet of great math quotes and an overview of the classical math curriculum envisioned. Join us to consider how we can recover for students the wonder of an enchanted cosmos that God has spoken — or perhaps sung — into being.
Ravi Jain
Ravi Jain graduated from Davidson College with a bachelor’s degree and interests in physics, ancient Greek and international political economies. He worked at various churches, received a master’s degree from Reformed Theological Seminary and later earned a graduate certificate in mathematics from the University of Central Florida. He began teaching calculus and physics at The Geneva School in 2003, where he has developed an integrated double-period class called The Scienti c Revolution. In this class, students read primary sources like Galileo and Newton in order to recapitulate the narrative of discovery while preserving the mathematical and scientific rigor expected of a college-level treatment. During his tenure there, he co-authored The Liberal Arts Tradition: A Philosophy of Christian Classical Education. He has given over 100 talks and workshops worldwide on topics related to education, mathematics and science. He has two young boys, Judah and Xavier. After the duties of the week have been discharged — usually by 8:53 on Saturday nights — he enjoys his few remaining hours with family, friends and his wife, Kelley Anne, whom he met in Japan.
Do you ever wonder why we teach specific representations in Grammar School? Do you wonder how you can connect your algebra curriculum to Grammar School mathematical knowledge? Come explore how areas of early mathematics connect to higher-level mathematics. We will explore multiplication, specifically, and will examine the connection between the early understanding and representation of whole numbers and the algebraic manipulations learned later in a student’s education.
Janet Andreasen
Dr. Janet B. Andreasen is an Associate Lecturer of mathematics education at the University of Central Florida (UCF). She is the Coordinator of Secondary Education and works with prospective and practicing mathematics teachers at the elementary, middle and high school levels. Dr. Andreasen’s research interests include examining mathematical knowledge for teaching and using technology to foster student learning of mathematical concepts. Prior to joining the faculty at UCF, Dr. Andreasen was a high school mathematics teacher. Dr. Andreasen has published books, book chapters and articles in state and national publications, and has conducted professional presentations throughout the United States. She is a member of the Association of Mathematics Teacher Educators, the National Council of Teachers of Mathematics and the Florida Council of Teachers of Mathematics.
Whether you are implementing Singapore Math or are simply looking to improve the math culture in your school, this session is for you. Changing the way math is taught requires more than just a change of textbooks. A successful transition requires the development of habits of mind in our teachers and students. This session will include how we became a model school with Math’s No Problem by developing instruction that promotes metacognition, exible thinking, and striving for accuracy. Come learn from our five-year journey of professional development, parent education, and student achievement.
Mo Gaffney
Mo Gaffney has served as Head of Lower School at The Covenant School in Charlo esville, VA, for the past six years. Before coming to The Covenant School, Mo taught primary grades in both public and private schools. She earned a BA in Early Childhood Education, an MEd in Elementary Education, and a doctorate in Curriculum and Instruction, all from the University of Virginia. While there., she taught writing and beginning teaching courses as well as conducted extensive research in elementary classrooms. She has led professional development workshops and has presented at national conferences on teacher evaluations, reading and writing connections, leisure in schools, and Singapore Math. This past fall The Covenant School hosted a successful Singapore Math conference, featuring Dr. Yeap Ban Har and Maths No Problem, winner of Publication of the Year in the United Kingdom. Mo and her husband, Je , have four young adult children.
Susy Willetts
Susy Willetts Society for Classical Learning • Summer Conference 2017
Susy Willetts serves as the Pre-K-6 Math Coordinator at The Covenant School in Charlottesville, VA. Before coming to The Covenant School, Susy has extensive experience teaching in independent specialty schools for gifted and intellectually advanced students, public schools, where she served struggling learners and Christian schools as a classroom teacher and administrator. She earned a BA in History from James Madison University with a concentration in secondary education and an MAT from Mary Baldwin University. This past fall she and Dr. Gaffney hosted a Singapore Math Conference at the The Covenant School featuring Dr. Yeap Ban Har and Math’s No Problem, winner of Publication of the Year, in the United Kingdom. Susy and her husband are raising their three teenage sons, three chickens and three dogs. Extensive research in elementary classrooms. She has led professional development workshops and has presented at national conferences on teacher evaluations, reading and writing connections, leisure in schools, and Singapore Math. This past fall The Covenant School hosted a successful Singapore Math conference, featuring Dr. Yeap Ban Har and Maths No Problem, winner of Publication of the Year in the United Kingdom. Mo and her husband, Jerry, have four young adult children.
