Cutting the holiday pie

A good opportunity for exploration and problem-solving for your students.

With one straight cut, you can cut a pie into two pieces. Cutting it again and having the second cut cross the first cut gives you four pieces. With three cuts, you might get up to seven pieces.  What is the maximum number of pieces you can get with 6 straight cuts?

Your cuts do not need to produce equal pieces.  Once students identify the solution, can they keep going with more cuts?  Is there a pattern that begins to emerge?

Photo by Alex Loup on Unsplash

What Does the Data Tell Us?

In a recent conversation with some representatives from a major nationwide business company, we were discussing the needs in elementary math education to help lead students to be productive workers in tomorrow's business world.

The big question they said that students need to learn to answer is, "What does the data tell us?"  We also discussed the importance of probability and statistics,  but in the end, they said that they are looking for workers who can answer this question.

What would this look like in an elementary classroom?   I think it just reframes our questioning.  In many of our activities, we already expose students to real-world information.  We just need to be sure to be more intentional in our questioning in order to get them to look at data.

For example, look at this picture.  I took it thinking that it would be a good one for students to determine what was the best deal. What does it tell us?  What are some possible reasons a person would be willing to pay more for 3 Peep trees when they can get 9 for such a better deal?  

Noticing and wondering is a classroom routine that really benefits our students.  When they notice and wonder, you can ask them what the picture tells them.  Make them infer from the data that they have.  Numberless graphs are a good method of providing data that the students have to make sense of.  Here is a good example of a 2nd grade lesson regarding them.  You can find lots of examples and ideas for using them by reading some of the posts listed here.

What do you notice about this data?  What do you wonder?

We want our students to be able to compute and do basic math, but not at the expense of good math thinking and discussion.  These are the skills that will carry them into the future--not only in their career, but also in their roles as consumers and citizens.

How can you incorporate statistics and probability into your classroom (whether it is a K classroom or a 5th classroom) by framing your questions and your students' thinking around the question, "What does the data tell us?"

Continuing Thoughts on Math Facts

Some of the biggest concerns I get from teachers revolve around math facts.  Oftentimes, when they hear my response, they think that I am saying that math fact fluency is not important.  That is not true.  I do believe math fact fluency is important; I just don't believe that most of our class time should be spent on rote math fact practice.

This is a topic of much discussion among math educators, and books continue to be written addressing the ways to build true math fluency.  Graham Fletcher and Tracy Zager are piloting a math fluency kit that should be available soon, and I can't wait to see it in action!

I have been recently reading No More Math Fact Frenzy by Davenport, Henry, Clements, and Sarama.    It reinforces the ideas that I continue to communicate to teachers.  Rote memorization is not a method for students to best learn their math facts.  It does not lead to a true understanding or flexibility of number that defines fluency.

Some ideas to consider:

Count.  Choral counting is a great way to build number fluency.  Counting forwards, backwards, by different multiples....This not only allows students to think about the strings of numbers, but it helps to build a mental number line for students which is so helpful for future success with mathematics.  Learn more about choral counting here.

Make your fact practice visual. Create structure that students are familiar with, and have them see the fact rather than just memorize the fact.  For example:

Seeing 7 x 6 this way helps students to visualize how knowing 7 x 5 can help them get the answer for 7 x 6.  Using subitizing is still important as students get older!

Rekenrek and ten frame visuals are also great for addition and subtraction facts.  There is nothing wrong with students using fingers for a while either.  They are a built-in tool. We hope that they eventually gain the confidence to know the sums and differences without their fingers, but if using their fingers helps it to make sense, let them!

Use number talks.  Asking students to use dot cards to make sense of math facts is important because it allows them to decompose numbers in different ways and to hear different people's strategies.  The above image could be an example of a dot card number strategy for older students, but more simple dot cards for primary will also allow for students to see the fact.  Traditional number talks with numbers written horizontally also build fact fluency through exposure to multiple strategies.  For example:

When students discuss the way that they solved a problem like this, they gain a better understanding of number.  Maybe one student added 7 + 7 and then added one more.  Another decomposed 7 into 2 and 5 so they could make 10 + 5.  Maybe another started at 7 and counted up to 8.  Number talks give students opportunities to make sense of problems in ways that make sense to them, but they also give you the opportunity to make connections between the strategy they used and those that their classmates used.  The goal is for students to not only think more flexibly but also to look for a more efficient method.

Make connections:  Help students to see the connection between operations.  How does addition help us to do subtraction?  How is multiplication related to addition?  How are subtraction and division related?  Not only asking these questions, but having students explore with manipulatives and discover these relationships will help students to have a better conceptual understanding of the relationship between operations.

Play games:  Games that practice math facts are always good not only because students get the opportunity to use their facts, but also because they get to practice important social skills like taking turns, good sportsmanship, and taking care of materials.  The best games include students having to use strategy besides knowledge of facts.  Encourage parents to play games at home with their children.  This is one of the best ways for parents to help their children become better mathematical thinkers.

Make it real:  Find as many examples as you can in the real world to help students see how the operations are used.  Adding the chairs at one table to the chairs at the other table, finding an array in a display of student work, and talking through the math of your lunch count are all real world ways for the students to make more sense of their math.

Trying some or all of these ideas should help students to gain more flexibility and fluency with their math facts.  I believe it is better use of time than pages of math facts, repeated rote practice on a computer program, or timed tests.  Knowing your math facts fluently does free up some brain space as you work through more complicated mathematical concepts, but conceptual understanding is much more important than rapid fire!

Reminder of math fact progessions:
Kindergarten:  Fluency of +/- facts within 5
1st grade: Fluency of +/- facts within 10
2nd grade: Fluency of +/- facts within 20
3rd grade: Fluency of x and / facts within 100