I feel quite fortunate that this past Tuesday and Wednesday, I (along with Associate Superintendent, Limary Gutierrez, Gabilan Principal, Abbie Arbrun, and Instructional Technology Coordinator, Carrie Sebora) was able to spend two days with Jo Boaler and Cathy Williams at the Mathematics Leadership Summit held at Stanford University. All of us came away with many “revelations” as to what we need to make happen in our schools to advance the mathematics instruction provided to our students. As we learned, the instructional block should be very clear on what math looks like in a classroom – students should experience math in the classroom in the same way they’ll experience it in the real world: in groups, making mistakes, explaining their thinking, and in a way that makes sense to them.

In addition, the point was hammered home quite early in the two-day presentation that we must emphasize “visual mathematics”. The new knowledge showing the visual processing of mathematical ideas, most likely explains the many research studies indicating that the teachers who emphasize visual mathematics and who use well-chosen manipulatives encourage higher achievement for students, not only in elementary school, but also in middle school, high school, and college. In fact, despite the prevalence of the idea that drawing, visualizing, or working with models is something only for young children, we learned that some of the most interesting and high-level mathematics is predominantly visual.

Finally, teachers should help students develop math facts, not by emphasizing facts for the sake of facts or using ‘timed tests’ but by encouraging students to use, work with, and explore numbers. As students work on meaningful number activities, they will commit math facts to heart at the same time as understanding numbers and math. They will enjoy and learn important mathematics rather than memorize, dread, and fear mathematics. One fantastic activity to develop number sense is Number Talks, one of the best methods for teaching number sense and math facts at the same time. Number Talks – a regularly scheduled activity at Rose Ferrero – was developed by Ruth Parker and Kathy Richardson. This is an ideal short teaching activity that teachers can start lessons with. It involves posing an abstract math problem such as 18 x 5 and asking students to solve the problem mentally. The teacher then collects the different methods and looks at why they work. For example, a teacher may pose 18 x 5 and find that students solve the problem in these different ways:

Students love to give their different strategies and are usually completely engaged and fascinated by the different methods that emerge. Students learn mental math, they have opportunities to memorize math facts, and they also develop conceptual understanding of numbers and of the arithmetic properties that are critical to success in algebra and beyond. There are some great books, one by Cathy Humphreys and Ruth Parker (in press) and another by Sherry Parish (2014), that illustrate many different number talks to work on with secondary and elementary students, respectively.

Research tells us that the best mathematics classrooms are those in which students learn number facts and number sense through engaging activities that focus on mathematical understanding rather than rote memorization. The following five activities have been chosen to illustrate this principle.

For addition facts, there is Snap It. This is an activity that children can work on in groups. Each child makes a train of connecting cubes of a specified number. On the signal “Snap,” children break their trains into two parts and hold one hand behind their back. Children take turns going around the circle showing their remaining cubes. The other children work out the full number combination. For example, if I have 8 cubes in my number train, I could snap it and put 3 behind my back. I would show my group the remaining 5 cubes and they should be able to say that three are missing and that 5 and 3 make 8. In How Many Are Hiding? each child has the same number of cubes and a cup. They take turns hiding some of their cubes in the cup and showing the leftovers. Other children work out the answer to the question “How many are hiding,” and say the full number combination. Example: I have 10 cubes and I decide to hide 4 in my cup. My group can see that I only have 6 cubes. Students should be able to say that I’m hiding 4 cubes and that 6 and 4 make 10.

One of the best activities to learn multiplication facts is How Close to 100? This game is played in partners. Two children share a blank 100 grid. The first partner rolls two number dice. The numbers that come up are the numbers the child uses to make an array on the 100 grid. They can put the array anywhere on the grid, but the goal is to fill up the grid to get it as full as possible. After the player draws the array on the grid, she writes in the number sentence that describes the grid. The game ends when both players have rolled the dice and cannot put any more arrays on the grid. How close to 100 can you get? In Pepperoni Pizza, children roll a dice twice. The first roll tells them how many pizzas to draw. The second roll tells them how many pepperonis to put on EACH pizza. Then they write the number sentence that will help them answer the question, “How many pepperonis in all?” For example, I roll a dice and get 4 so I draw 4 big pizzas. I roll again and I get 3, so I put three pepperonis on each pizza. Then I write 4 x 3 = 12 and that tells me that there are 12 pepperonis in all.

Many teachers (and parents) use ‘flash cards’ as a way of encouraging the learning of math facts. These usually include 2 unhelpful practices – memorization without understanding and time pressure. In our Math Cards activity, we have used the structure of cards, which children like, but we have moved the emphasis to number sense and the understanding of multiplication. The aim of the activity is to match cards with the same numerical answer, shown through different representations. Lay all the cards down on a table and ask children to take turns picking them; pick as many as they find with the same answer (shown through any representation). For example, 9 and 4 can be shown with an area model, sets of objects such as dominoes, and the number sentence. When students match the cards, they should explain how they know that the different cards are equivalent. This activity encourages an understanding of multiplication as well as rehearsal of math facts.

The activities given above are illustrations of tasks in which students learn math facts at the same time as working on something they enjoy, rather than something they fear. Moreover, the different activities also focus on the understanding of addition and multiplication, rather than blind memorization and this is critically important. As educators we all share the goal of encouraging powerful mathematics learners who think carefully about mathematics as well as use numbers with fluency. But teachers and curriculum writers are often unable to access important research, and this has meant that unproductive and counter-productive classroom practices continue. Students are “damaged” when they are exposed to practices that often accompany the teaching of math facts – speed pressure, timed testing, and blind memorization. High achieving students use number sense, and it is critical that lower achieving students, instead of working on drill and memorization, also learn to use numbers flexibly and conceptually. Memorization and timed testing stand in the way of number sense, giving students the impression that sense making is not important. We need to urgently reorient our teaching of early number and number sense in our mathematics teaching in our schools. If we do not, then failure will escalate. When we emphasize memorization and testing in the name of fluency, we are harming children, we are risking the future of our ever-quantitative society, and we are threatening the discipline of mathematics. We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it.

1). Teachers: Please remember to use the What, Why, & How regarding your Learning Targets … explaining to students What we are going to learn, Why we are going to learn this, and How the students will know when they have learned it.

2). Teachers: Whenever the opportunity arises, model your thinking in front of your students using “I” statements, and try to include some metacognition (thinking about our thinking) …. at minimum, it would involve using a “because, why, or how”. In this way, your students will soon learn how an expert thinker (you) begins to tackle questions/problems that they encounter.

3). Teachers: Let’s take to heart a line from above … that in our math classrooms, students experience math in the same way they’ll experience it in the real world: in groups, making mistakes, explaining their thinking, and in a way that makes sense to them.