Warning! Are You Sabotaging Your Students?

Ok, maybe the headline was a little TOO dramatic.  But I started to think about this the other day when I began teaching telling time and counting money to my second graders.  In ten years, these little second graders are going to be seniors in high school!  So here I am teaching them to tell time on an analog clock.  And now, using the fake plastic money to learn how to count money.


But think about it?  In ten years, will there be ANY digital clocks around other than Big Ben?  Pretty much everywhere you look you see digital clocks.  I remember my younger brother always going to the digital clock in our house when he needed to know the time (and that was more than 30 years ago!).  My own kids do the same!  They did learn how to tell time on an analog clock. But because they weren't obligated or had the need to use an analog clock (other than the one in school), they literally started to forget how to tell time on it.  But are there still advantages of using an analog clock?

Uhg! I'M GETTING OLD! (when did that coin change?)

Let's talk about coins and money.  True story.  I had no idea the new nickel had Jefferson facing forward!  Since I have been mostly teaching third grade the past ten years, teaching second grade is new to me this year.  I haven't taught lessons on money in a LONG time.  And I hate carrying around change! So I never really paid attention to the difference in the coins.  Ok, I did know about the new quarters.


So here I am teaching my students how to count coins.  What's the biggest hang-up or obstacle so far?  Recognizing the coins!  So to my surprise, the new nickel has Thomas Jefferson looking forward instead of a profile. I guess I hadn't been paying attention to coins for a LONG time.  As I said before, I hate carrying around change. If I ever have any, I just put them in a box and collect it until I can take it to coin counter (free coin counting at my credit union, btw).   So when my students asked me what a coin was in their math book, I said to look at the resource chart that we made together or to look in their math journal.  Of course, it had the old nickel!  So a quick lesson on the new nickel.  But think about it.  In ten years, will we STILL be using coins or will we be a cashless society?

THE iGENs (they're in your classroom right now!)

What skills are we teaching today that will NOT be needed in ten years by this generation?  By the way, they are called the iGen generation.  Fitting name!  Unlike their older brothers and sisters who are the Millenials who GRADUALLY went from laptops to tablets to phones, this generation is glued to a portable screen (tablets and phones) full time.


They are growing up in a world we can not even imagine what it will be like in ten years. The first tablet, the iPad, was introduced in 2010...the year my second graders were born!  They've grown up around screens and swiping.   What skills will they need by the time they graduate from high school?  It's hard to predict the future, but there are trends we see coming.

In this THREE part post, let's examine some skills that may or may NOT be needed in ten years time (2028).  Part THREE will look at some recent skills that will be of critical importance in any future that comes (think digital citizenship).


As I explained above, the only analog clocks we see anywhere nowadays seem to be in classrooms and grandfather clocks (oh yeah, they might have Roman numerals, too!).  Yes, I still have one in my classroom.  But you know what, the school Chromebooks also display the time digitally.  Guess which one the students prefer to use?  Is it a losing battle?  Do we choose between analog or digital?  If you're proficient at telling time using an analog clock, the switch to a digital clock is no big thing.


But think about all that went into learning how to tell time on an analog clock.  The movement of the hands on a clock vs numbers just changing on a display.  You could actually SEE the progression of time.  Granted that on a digital clock, you can usually use a stopwatch function to see the passage of time. But looking at an analog clock allowed us to do mental math (counting on or counting backward).

How many more minutes or hours do I have to wait?  You could visualize the hands moving on that clock.  It is very hard to imagine or visualize the numbers on a digital display changing to figure out how much time is left of how long I'll have to wait.  That involves adding and subtracting in your head, while with an analog clock, it's a matter of just counting on or counting backward.


But in ten more years, will there be mostly analog or digital clocks on display in public areas?  What about on screens?  Probably more digital than analog displays.  Digital just takes up less space and is easier to change or adjust.  So if students looking at a digital display can mentally add or subtract, then that's probably just as skillful as counting forward or backward on an analog clock.


In the meantime though, I still think it's important to continue teaching time on an analog clock because it is just more than just learning how to tell time.  It's using mental math and visualizing, which are still essential math skills no matter what century you live in.  Other math skills can be strengthened or taught through an analog clock:  counting by fives, understanding quarters or fourths, and the concept of am and pm.


What I do think we should stop teaching are all the VARIOUS ways of saying the same time.  You know...half past, quarter to, 10 'till.  Does anyone other than our grandparents even use that terminology today?  Ok, they might hear it more often in different parts of the country, and they will definitely read it in older stories, but do we get our "bang for the buck" when we take so much time to teach these archaic phrases?  I'm happy with my second graders just knowing it's "two-thirty" rather than "half past 2."  The only term that should be taught is o'clock!


In Part Two, let's examine some more skills such as counting money, reference skills (library skills and dictionary skills), cursive writing and keyboarding.

What do you think so far?  Do you feel you teach skills you know won't be relevant in ten years? Or maybe they won't be relevant, but are necessary to develop OTHER skills?  Share your thoughts in the comment section below!

Frustrated That Students Don't Know the Multiplication Facts? Part 3 The Derivative Strategies

In the final part of this three-part series on multiplication fluency, let's explore the Derivative Strategies.  These are the strategies that are taught AFTER the student has demonstrated proficiency in the Foundation Strategies (see the previous post for an explanation of the Foundation Strategies).  

The Derivative Strategies are the final steps that will eventually lead to multiplication fluency as well as, a better conceptual understanding of multiplication.



These are the strategies that continue to build that conceptual understanding of multiplication while building fluency over time.  The first strategy to teach is the halving and doubling strategy.  Why?

This strategy works with even factors.  So, if a student has learned and has fluency with the facts for the two times table, then he or she is ready to use this strategy to learn the facts for the four multiplication table.

Watch this example of a student using the halving and doubling strategy.

As you can see, knowing the 2s helps with learning the 4s because two is half of four.  The same strategy can be applied to learning the eight times table (though other strategies can be used as well).


This is the second strategy to teach students to use.  Once the 2s and 4s are learned by doubling and halving, students can move on to learning the three times table by learning to add a group.  If the student knows the 2s timetable, then teach the student to use this strategy to learn the 3s.

Watch this video of a student using the Add a Group Strategy.

THE DERIVATIVE STRATEGIES:  The Distributive Property Strategy

Of all the Properties of Multiplication, this is one of the hardest to teach.  Conceptually, students need to understand decomposing a number, partial products, and writing equations.  Not an easy task. Obviously, students need to be taught explicitly the Distributive Property of Multiplication.   I use this resource to teach the Distributive Property every year.  But whatever resources you use, once students are familiar with it, they can then practice the strategy for multiplying by seven and eight.

Before teaching this strategy, students really should be proficient in the facts for five.  Why? It is easier to decompose a number when one of the factors is five.  Multiplying by five leads to products with zero and five, which when added to the other partial product is easier to do mentally.

Watch this video of a student using the Distributive Property of Multiplication.


There are several other strategies to teach which include subtracting a group, using a nearby square and the patterns for the nine times table.


There is a definite sequence of teaching the multiplication tables.  I've created a resource that has everything you need to teach these strategies over time.  

Click to see the full product preview!

The Multiplication Fluency Strategies Resources includes teaching posters, student templates, practice pages, and games. See below for examples of each!

Click to see the entire resource!

I've also included sample lesson plans for the Foundation Strategies and Derivative Strategies.  It also includes the teaching sequence for the multiplication tables and well as, explanations for each of the strategies.



Click to see the entire resource!

Check out my Facebook page as well, as I will be posting more information and teaching multiplication, fact fluency, and more common core math topics!

Want more ideas and tips for teaching multiplication?  Check out my Pinterest Board for Multiplication and Division.

What are you go to strategies for teaching multiplication fluency? 
Share them below in the comments!

Frustrated That Students Don't Know the Multiplication Facts? Part 2 The Foundational Strategies

In the first part of this post, I wrote that we need to teach mental math strategies for multiplication just as we do for addition and subtraction.   But first, students need to learn about the concept of multiplication through arrays, equal groups, and repeated addition.  Then it is time to start memorizing the multiplication facts.


What usually happens is that some students are good at memorizing and can recall them on demand.  But most will learn a multiplication table and then start forgetting it soon thereafter.  Yes, there are ways to improve memorization skills and I do teach my students those tips.  Memorization is part of being fluent in the facts.  As we probably already know, we will have many students who will battle with memorizing all the multiplication facts.


But what if there was a different way to address multiplication fluency?  How? By teaching students multiplication mental math strategies that will help them arrive at the product more efficiently WHEN or IF memory recall fails.

These mental math strategies are first practiced with paper and pencil, but eventually (just like the addition strategies) they begin to use them mentally with more efficiency.  There are two categories of strategies:  Foundational Strategies and Derivative Strategies.

In part 2, let's explore the Foundation Strategies.


The Foundation Strategies must be taught first as they are the foundation for the later Derivative Strategies.

The Foundation Strategies involve basic mental math skills such as skip counting, finding a pattern, knowing the Identity and Zero Properties of Multiplication and learning by memory the all-important square numbers.  Here's a chart comparing the Foundation Strategies for both Addition and Multiplication.


If a student uses these strategies, the student will know about 50 facts on a multiplication chart.  That is a great start to eventually learning the rest and being fluent in the computation of multiplication facts.  The student must have these skills under control in order to advance to the next level of strategies, the Derivative Strategies.

Let's take a look at how the Foundation Strategies can be taught.


Skip counting is one of those mental math skills that begins early in kindergarten.  By second grade children should be able to count by 2s, 5s, and 10s.  Counting by 2s, 5s, and 10s will enable students to quickly learn the multiplication facts for those tables.  


Skip counting can be done with songs (https://youtu.be/SCBwSSDk9Mg) are a great way to motivate kids and keep them engaged while learning.  Counting dimes and nickels is another way to motivate students to learn to count by 5s and 10s.  A game for learning to count by 2s is thinking of things that come in pairs.  How many pairs of eyes are in our class? How many shoes?  There are lots of possibilities!  Games, flash cards, multiples strips, and bookmarks also help with learning to count by 2s, 5s, and 10s.


Teaching and learning the Properties of Multiplication lay the foundation later for Algebra and more advanced math.  So it is important that we explicitly teach these building block properties.  Start with the Identity Property and the Zero Property,

I like to teach these two properties in a fun way by making Zero a Hero and One a Bum.  Zero can obliterate any factor to produce a product of zero while One is a bum because he's lazy and produces a product equivalent to the factor.


Finally, the products of squares (2 x 2, 3 x 3, 4 x 4, etc.) should be learned and memorized.  The square numbers run diagonally along a multiplication chart.  The resulting line of squares produces a mirror image of the other products (3 x 4 = 12 and 4 x 3 = 12). That is why it's important to memorize these square number products.  It reduces the number of facts to learn!

To teach the square numbers is to make cards with all the square factors (1 x 1, 2 x 2, 3 x 3, etc) on one card.  Then on other cards write all the products of those squares.  Then hand out one card to each student.  On a signal, they are to quietly get up and find their partner to match the square factors to the square product.  When they've found their match, the pair stands back to back.  You can practice this multiple times in a 5 - 10 minute period by giving students a different card each time.  You can even time the class to see how fast all the pairs can partner up.


Students should practice the Foundation Strategies until they are proficient because they will be needed later to develop the Derivative Strategies.

In Part 3, we'll take a look at the Derivative Strategies including video of strategy use in action!

Frustrated that Students Don't Know The Multiplication Facts?

Have you've encountered the student who doesn't study the multiplication facts? Or the student who learns one multiplication table and then seemingly forgets most of the facts? Or the student who studies but has a hard time recalling the facts?


I'm sure you have!  Multiplication is introduced in second grade as counting groups by a certain number (2, 5 and 10).  It is not until third grade that most students are formally introduced to the concept of multiplication through equal groups, arrays, area model, etc.  And for the most part, most students "get" multiplication as a concept.

But there eventually comes a time when the math will get more complicated (and maybe harder).  So as a mathematician, you want to be more efficient.  Learning by memory all the multiplication facts will make you a more efficient mathematician when the math does get more complicated.


So now we are back to the students and memorizing the multiplication tables.  There are lots of tips and tricks and strategies for memorizing the multiplication facts.  I've even developed resources for my students that give them tips and strategies for remembering the tables.

But the problem with memory is that it needs to be recalled easily if it is to be useful.  So beyond teaching memorization tips and strategies for the multiplication tables, what else can we do so that students learn those facts to be more efficient if they can NOT recall from memory?


Teach them strategies for fluency!  Let me show you some mental math strategies that you can teach your students to use with memory is not enough!  But first, what exactly is "fluency?"


The Common core DOES specify that memorization of the multiplication tables is an expectation.  But that does not mean that meaningless memorization is the route to go.  We want students to understand multiplication conceptually as well as recalling facts.  In the Common Core, there are various terms used:  know from memory, be fluent in, demonstrate fluency, etc.  It even states that "by the end of Grade 3, know from memory all the products of two one-digit numbers."

Now, let us clear up some terms.  To know from memory means to recall from memory a fact.  It is remembering.


Fluency is more complicated. Fluency is NOT the instant spouting of memorized facts.  Fluency is the combination of recalling from memory OR using patterns and strategies when memory is not enough which then leads the student to use mental math strategies for multiplication.


The mental math strategies used to teach multiplication fluency can be grouped into two categories:  Foundational Strategies and Derivative Strategies.

The Foundation Strategies are the first strategies to teach and use the reliable mental math strategies of counting by a number (also known as skip counting), knowing the square numbers, and knowing the Zero and Identity Properties of Multiplication.

The Derivative Strategies build on the Foundation Strategies and teach halving and doubling, adding or subtracting a group, using a nearby square, the patterns for the nines multiplication table, and the Commutative and Distributive Properties of Multiplication.


The strategies are similar in use to the addition strategies you would teach students learning to add: decomposing a number, doubles, doubles plus one, adding one more, etc.  They are mental math strategies that are explicitly taught and practiced with pencil and paper first before becoming part of the mental math repertoire for adding.  The same applies to these multiplication strategies.  They are explicitly taught and practiced with pencil and paper until the student can use the strategy in mental math.


Are you eager to see the rest?  Then download the FREE Guide to Achieving Multiplication Fluency!  It's a 15-page guide that explains all the strategies, some suggestions for teaching the strategies and how to assess strategy use.  It's FREE and you can download it HERE or click on the image below!

Click HERE to get this FREE GUIDE.

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In Part 2 of this 3 part blog series, come back to learn more about how to teach the Foundation Strategies.  In Part 3, we'll take a look at the Derivative Strategies.

In the meantime, check out the latest resource in my Teachers Pay Teachers store!  Made explicitly for teaching multiplication fluency strategies!


Overwhelmed? Self Care is Important and Vital to Success!

I recently had the opportunity to chat with Rachel Davis of Elite Edupreneurs on her podcast featuring Teachers Pay Teachers sellers.  We discussed many topics about being a seller on Teachers Pay Teachers.  One important aspect of being a seller is knowing when to take care of yourself.


When you're a TpT seller, there is a lot of stress involved.  A seller has to know how to create high-quality resources, learn marketing techniques, learn copyright and trademarks, and run a business while trying to balance all that with the rest of your life.

As a single dad of two boys (13 and 10), it's not always easy to balance raising children, working full time as an elementary teacher and running a store on Teachers Pay Teachers.  But I try.  Join Rachel and me on the podcast now! (It's free! Just click on the link).


5 Must Have Presents That Will Make a Teacher Happy!

It's that time of year again when you know your students are wondering what gift to get you this year.  I've been teaching for 31 years, and I've received plenty of beautiful gifts that have been given with gratitude, from the heart and from an interesting point of view of what a teacher might like.  Yes, that means I've gotten some very "interesting" presents over the years. But I've appreciated every single one!  Even the one which was a bag of coal! 


But what gift would REALLY make a teacher happy?  What would a teacher write down on her or his list for Santa?  So I've come up with a list of 5 MUST HAVE presents I know all teachers want.  Maybe we don't ask for these outright, but in the back of our minds, we do wish for these!


Respect.  Like Aretha Franklin says...R-E-S...P-E-C-T, that is what a teacher needs!  After 31 years of teaching, I have noticed a dramatic drop in respect for teachers from all parts of society.  From some administrators, from MOST politicians, from parents and from students.  It used to be that a teacher was considered a wise and fair sage.  Now, all that has changed.


Sometimes we are treated as if we don't know what we are doing (as in politicians telling us how and what to teach).  Sometimes we are treated as if we are devils (as in parents not believing us when we report misbehaviors by their children).  Sometimes we are treated as a bothersome itch (as in students who keep on talking while we try to teach or ignore our teaching altogether).

Of course, there are many wonderful parents, students, and administrators who do go out of their way and treat us respectfully.  But what we really want is that respect from everyone.

To the politicians out there:  we know what we're doing!  Let us teach!  We have the teaching degree, you do not.  To the parents who never believe us when we discuss their child's behavior:  you're doing your child more damage than you can imagine!  To the students who seem to think we're interrupting their day with our teaching:  I will still try my hardest to teach you, but remember that you reap what you sow!


Less stress.   You would think after 31 years of teaching, stress would not be an issue.  But it is.  Yes, I have stress in all parts of my life, but in the past 10 years teaching has become even more stressful with mandates by politicians, demands by parents and the ever-increasing needs of needier students.


Most people think that teachers just sit at a desk all day and tell students what to do.  Nope.  I don't even have a regular desk.  I'm always moving around.  I'm teaching.  I'm working with a group of students.  I'm working with one student.  The point being?  I'm working here!  And planning for all those lessons, keeping up with the grading, and the myriad other teaching-related duties a teacher has, the stress can be demanding.


A simple thank you.  As teachers, we get a silent thank you every time a student learns to do something new or improves.  Every time a student lights up with that proverbial light bulb is a thank you.  But, I'd like to think that I have one of the most essential jobs in the world which is shaping the future! So I believe society should be a more thankful we teachers are so dedicated!

Wouldn't it be nice if teachers were given ticker-tape parades as a thank you?  Wouldn't it be nice if teachers were given a Nobel Prize as a thank you?  Wouldn't it be nice if teachers were rewarded with multi-million dollar contracts?  Shouldn't teachers also get signing bonuses?


The point being is that teachers need to be thanked more often and in more ways for all we do.  I can't say all 100% of teachers will go above and beyond their duties, but probably 99.99999999% will.  Which other professions can say that!


Better pay!  You've heard this one before.  If you want to keep attracting the best and brightest to the teaching profession, you have to draw them with a good salary, good benefits, and good working conditions.   Many teachers now begin their careers saddled with student debt.  No one is going to want to be a teacher if they realize that on a teacher's salary, they'd still have to live at home with their parents because they would not be able to afford to do so on their own.

For those of us who are veteran teachers, we have gone years without pay raises while the cost of living has continued to go up.  I can tell you (and prove to you with receipts) that I spend about $1,000 - $2,000 per year of my own salary on supplemental materials, school supplies and other resources for my class.  So over the 31 years of teaching, I probably have spent close to a starting teacher's salary!  We're not asking to double our pay.  We're asking to be paid a decent wage to support our families!


A coffee mug!  In my 31 years of teaching, I have gotten a coffee mug EVERY SINGLE YEAR.  I've said that when I retire, I will open a coffee shop featuring all these mugs.  So keep them coming!


What has been your favorite present from a student?  

If you could choose one present as a teacher, what would it be?

Let me know in the comments below!

Happy Holidays!

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3 Tried and True Ways on How to Teach Multiplication

The gateway to learning algebra and higher forms of mathematics is multiplication.  It is critically important that elementary school children learn the concept of multiplication as well as, just learning the multiplication facts.  I am a big believer in using manipulatives and concrete objects to teach concepts to elementary school children. When introducing the concept of multiplication or when my students are practicing with the idea, we use manipulatives such as the foam tiles that are included in the math program.  You can use anything:  beans, counters, buttons, etc. 


But it is also essential to use the manipulatives with strategies that can transfer to paper and pencil models as well.  Though there are many ways to teach the concept of multiplication, I have always emphasized equal groups, arrays, and number lines.  Each one has its strengths and difficulties for students to use.


The first strategy is to teach equal groups.  After all, multiplication is the grouping of equal numbers of objects to quickly find a total.  Students must be explicitly taught, and they must practice forming, identifying and counting equal groups.  The student's desktops become whiteboards in my class. Students can be told to get 12 tiles and see what kind of equal groups can be formed (3 groups of 4, 4 groups of 3, 6 groups of 2, 2 groups of 6, 1 group of 12, 12 groups of 1).

Once grouped, students can draw circles around each group and then use skip counting.  Once students are able to form equal groups, I also give them a number that can not be formed into equal groups such as 13.  This is to emphasize the importance of having each group equal.  When groups are not equal, you can not multiply.

Here are some videos that illustrate the concept of equal groups.  They can be assigned to the students to watch individually or watch it as a class.  They are kid-friendly and offer good examples and explanations. If you're using Google Classroom, you can post them as a resource for students to use at home or in the classroom.  If you use EdPuzzle.com, these videos are excellent for prompting the students with questions so that viewing the video is just not a passive experience.

This particular video really emphasizes the ideas of groups, by using real-life examples of "group holders."


Once students have a solid understanding of equal groups, arrays can be introduced.  Arrays are to multiplication what ten frames are also.  Arrays provide a structured way to see groups making it easier to count totals and recognize quantities.  I use the same tiles to create arrays.  I like to start out a lesson on arrays by asking how groups could be arranged in a way to make them easier to count.  Eventually, someone discovers or builds an array.  Then we have a discussion of why arrays are easier to count.  It's also a great idea to show them real-life examples of how ordinary everyday objects are grouped in arrays.

Check out this video!

Arrays are also a useful tool for discovering the Commutative Property of Multiplication.  Just rotate an array 90 degrees, and you have a related fact!  If you have students eventually draw their own arrays, it is a good idea to use graph paper.  Graph paper will help the students keep their arrays from morphing into uncountable blobs!

Here are some more kid-friendly videos to demonstrate the use of arrays.

This particular video is very useful because it also prompts the students with questions making it more interactive.

Which kid doesn't know about MineCraft®!  Keep the motivation going with this MineCraft® themed explanation of arrays.


I always use this method last.  Why? Because though it looks straightforward to use, students make many mistakes when using it!  Sometimes students do not count enough spaces to jump or confuse jumps with how many to jump at once.  In either case, it requires careful teaching and making sure the students understand the steps involved in using a number line to multiply.  I've also thought about using an open number line to multiply as this may lead to less confusion counting the tick marks to jump.  An open number line requires the student to SKIP count by a certain number for each jump.  A marked number line requires a student to count the same amount of ticks each time.

I have number lines that are laminated, and the students put them on a marker board to use. Number lines do not lend themselves very well when using manipulatives.  But by this time, most understand the concept of equal groups.

Here are some videos that can be used for a review or for teaching how to use the number line to multiply.

This first one also points out to students common mistakes when using the number line!

Here's an example of using an open number line.


Once I have established the concept of using each of these strategies using manipulatives or, I want my students to start connecting multiplication expressions to go with equal groups, arrays, and number lines, I use a PowerPoint I created that explicitly explains how to write multiplication equations.  It's a three-part PowerPoint that teaches equal groups, arrays and number lines to multiply.  It comes with printables that are used along with the PowerPoint.


The printables help connect the manipulatives to writing multiplication expressions. The PowerPoint is animated and has sound to keep the students engaged.  Many questions are embedded into each slide to keep the students thinking about what is happening.  Presenter's Notes for the teacher also guide the teacher through the PowerPoint lessons and provide questions for stimulating mathematical thinking.




Take a look at the full PREVIEW HERE.

Click below to see all my 
multiplication resources!


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