Math is Visual

Our students need access to math visually as much as possible.   This can come in different ways.  Drawing, manipulatives and online components are some places to start.

I spend a lot of time in classrooms working around word problems.  One of the things I find myself constantly coming back to with students is "Think like a reader:" use schema, make predictions, infer, VISUALIZE...that is what good thinkers do.  Asking students to draw what they are seeing happening in the problem is an important step.  That drawing can be a true drawing, a number line, a tape diagram, or anything else that helps them make sense.  Are they thinking through the problem or are they just pulling the numbers and computing without making sense? 

I also use numberless word problems a lot. Numberless word problems give students the opportunity to see the problem as a story and to make sense of it without worrying about the numbers.  Once you have practiced this strategy in class, it is a strategy they can refer to independently when they come across a word problem that confuses them.

Manipulatives are important, too.  CRA instruction helps us to build the visual into the child's mind by using tools to represent the math.  This is important for all students and provides equity in instruction.

There are many websites with visual representations.  One of the best is Math Visuals.  Berkeley Everett has created a treasure trove of visuals for the K-5 classroom.  His short videos to accompany skills like counting and computing are a fabulous resource, but they are just the beginning.  He brings visual supports to understanding mathematical properties and making connections through visuals, too.  In addition, his Math Flips cards are well worth the time to cut out and use with all of your students.

Many other awesome websites exist, too, that are very visual.  Math is Visual, Fraction Talks, Same or Different, Number Talk Images, and Slow Reveal Graphs are just a few of the visual resources that may help you and your students make more sense of mathematical concepts.

Visual math is for all students.  We have to be careful not to assume they can see it in their heads.  We have to help them see it in their heads...Visuals help students to make sense.  They help students to look for patterns.  Visuals help students to answer the question, "What does the data tell us?"

In what ways can you provide your students with more visuals?  This is an important area for all of us to grow.

Counting Connections

I love choral counting! This is confirmed every time I participate in one in a classroom.  It doesn't matter if it is a Kindergarten or 5th grade.  Choral counting gets everyone thinking--including the teacher!

One piece to choral counting that I have found to be important is the connections that can be made between different choral counts.

For example, this was an eye-opening choral count for some 2nd graders.  We began with the confidence builder of counting by 5s.  We broke it down by looking for patterns.  A lot of good talk and discussion here, but it really kicked up a notch when I challenged them to count by 5s off-decade.  We began slowly, but then the students were able to idenify patterns and get going.  We looked for patterns in this count, too, and then I pulled the two counts up side by side.  The ahas! that the students had were quite exciting!  So important for students to make these connections.

I did another one recently in a 5th grade classroom where we counted by 21s.

Students stepped up to the challenge of counting by 21s and looking for patterns.  But the real excitement came when I asked them to do another count by 2.1.  (I had copied the first count onto a clean page before we had found its patterns.) Students kind of gasped, but then I pulled up the clean screen of our previous count.  They counted by 2.1 as I added decimal points into the numbers.  When we were done, we made connections between the two counts and place value.  We also discussed what would have happened if I had asked them to count by 210.  These place value concepts were things that they "knew," but when we put it on paper, it really clicked for them!

Because I am in and out of classrooms, I did these "double-counts" in one session making the choral count a little longer than usual.  I certainly think that it would work for a teacher to do one one day and the second one the next day.  The importance is the time taken to connect the two counts, and following-up in later days with similar counts in order to help cement the counts into students' minds.

Beginning with choral counting is your first step.  Give one a try!  Once you get past that hurdle, you will feel more comfortable moving into a double-count!

An exploration for 4th grade

A site that I often frequent for good math inquiry problems is http://www.inquirymaths.org/.



This is a good inquiry I found there for 4th (and 5th) graders.

 24 x 21 = 42 x 12

The link gives good directions and discusses the importance of student conjectures and exploration.  But it leads me to another idea:  How much more valuable is solving this problem and thinking about it deeply than doing a page full of double-digit multiplication?   We sometimes practice the way we were taught, but many times, there are way better ways.  Students, in this case, solve two multiplication problems, look for patterns, develop conjectures, and then work to test those conjectures.  More math problems are being practiced while students are doing some important noticing and thinking.  So much better than a sheet full of double-digit problems.  

Sorts

Sorts are a great way for students to make sense of things around them.  It could be images or numbers or objects or words....the point is to get them thinking about a subject and have them make sense of it in some way.

Here are some ideas for sorts.  Don't think that these are just for Littles.  Older students enjoy them.  They offer a non-threatening way for students to look at things, and they offer us good insight into our students' thinking.  Sorting is very mathematical even when it is not about numbers.  It is about looking at things carefully, finding patterns, and making sense--that's math!

SORT:
pattern blocks
doors
coins
book characters
drain covers
shapes--2D and 3D
numbers (odd/even) (prime/composite) (square/not square) (multiples)
vehicles
words
patterns
emojis
food
volume/area/perimeter
shoes
states
types of graphs (no numbers or titles are necessary)
expressions
angles

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