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Hairy Money: Trick or Strategy?

 Hairy Money!

As I near my 20th year in Education, I’ve learned a great many tricks designed to “help” students on their paths to successfully understanding math. Some of these “tricks” are simply that...a trick that happens to work. 

• A trick that often causes students to stray away from truly grasping math in the long run; or causing students to eventually hate math because every year they learn a new “trick”. 

• A trick that eventually expires and has no mathematical foundation. 

For more examples that include concepts from grades 2 through Algebra 1, download this free ebook - Nix the Tricks.

But when I ran across “hairy money”, at first I wondered if this cute approach to skip counting values was actually a trick or what we might call a Tiered strategy.

A tiered strategy is one that takes an approach to learning by encompassing some layer of instructional support [ie: number sense in this case]. A tiered strategy is often used during explicit and systematic instruction. In math (intervention) it’s coupled with manipulatives and graphic organizers. 

So for example, if I were teaching a 3rd grader to add 3-digit numbers with regrouping, I might show them how to line up their place value, then add beginning with their ones place. Next I might model how we regroup to the tens place and so on through the hundreds place. 

But if this 3rd grader struggles with this abstract algorithm, the most appropriate thing to do is intervene by scaffolding back to a more concrete approach to instruction within this concept. Explicit and systematic instruction would include pulling out place value charts (graphic organizer) and base ten blocks (ones units, ten rods and hundred flats,...even a thousand cube). I would then create 3 (no more than 4) steps walking the student through adding with regrouping beginning at the ones place. Teaching the child to rename ones larger than 9 as tens and ones (regrouping) using the models. 

Then I would wean the student away from the concrete and have them still use the place value chart (a smaller one/maybe drawn) along with a model drawing of base ten blocks to represent their work. Finally, I would build the student’s efficacy back towards abstract. All of the steps listed above are tiered strategies.

They are the foundations that should have been essentially introduced to the student in the previous grade levels to build their ability to fluently do the algorithm in their current grade.

So what tiered strategies did I list in this example?

• Base ten blocks

• Place value charts

• Model drawings

• Regrouping with models

• Steps based on place value understanding

All of these tiered strategies point back to number sense development. 

So let’s talk about “hairy money”. The process of students placing hair on top of each coin (in increments of 5) atop the nickel, dime and quarter. So a nickel would receive one hair (so students count once by 5); a dime would receive two hairs (so students skip count twice by 5) and five hairs atop the quarter (to skip count five times by 5). 

When all the coins are lined up (in order from least to greatest value...which coincidentally would be a step), students then place the appropriate hairs above each respective coin. Their final step would be to skip count by fives until all quarters, dimes and nickels are added up, and then count on by ones for every remaining penny.

The way Texas standards (TEKS) indicate to count money values, as a number sense strategy, is to have students skip count. Instruction includes tiered number sense-based strategies that include pulling out a hundreds chart (graphic organizer) and play money/coins. Then laying coins on the hundreds chart to help students connect number patterns to skip counting!

So the question is, “Is hairy money” a number sense (tiered) strategy? Or is it a trick? 

My thoughts are, it’s a scaffolded strategy. It does encompass number sense (therefore making it a strategy) but it restricts the students from fully developing the ability to skip count by twos, fives and tens which is a more efficient way to find a value. This skip counting ability transcends other concepts beyond 1st and 2nd grade! So I might reserve it for a back-up rather than introduce it as a Tier 1 approach! Ideally, as early as 1st grade, students should learn to count a set of items/objects efficiently. Grouping items and then skip counting those groups. Their mastery is evident when they can flex between skip counting (ie: skip count by 10s, then by 2s; or by 10s then 5s and 2s or 1s).

From there, students are expected to efficiently count base ten blocks where again, they skip count between 100s, 10s and 1s (or by 2s). So I would contend that students CAN be taught to skip count by 25s, then by 10s, 5s, and 1s to demonstrate mastery of efficient counting and understanding of number patterns. But those are my thoughts.

Let me know what you think! 

Free Virtual Resources for Counting Money:

Math Learning Center - Money App

Toy Theater - Money Strips

Toy Theater - Coin Bank

Illuminations - Coin Box

*Math Playground (Money Activity) - Puzzle Pic Money

Math Playground - Interactive 100s Chart

Toy Theater - 100s Chart

ABCya - 100s Chart

*PS: If you pay for Brainingcamp, they have a hundreds chart component and you can change the settings from “basic” to “money”. With this resource you can place coins on the hundreds chart to show the culminating value! It’s incredible!

Review Season Is Upon Us

I'm neck deep in studying for my SPED certification (to broaden my impact as an educator).

As I embark upon this journey, I spend some of my weekends engaging in reading, studying, and practice testing. *Can we say "bo-oooring"? But I'm doing my best to "give it the old college try", literally, LOL!

Ironically, I stumbled across a section of my reading that was dedicated to laying out the skills for studying. As it talked about the nuances of organizing, processing information and remembering what is learned, I reminisced back to my transition between high school to college.

I vividly remembering coasting through my high school courses and graduating with honors. Then getting to college and taking my freshman/sophomore courses and thinking "I must be dumb" because nobody told me how hard it is in these college streets! I had ZERO (count them) studying skills. Obviously whatever sharp wit I was using to fly through high school was powerless to get me through college. I had to quickly develop a plan if I was going to crank out scores sufficient for the amount I was paying (or going to eventually pay back) for these semester hours!

Long story, short, I made it to the fun courses of my junior and senior years (barely...on Bs) and finally found my niche (go figure). I realized and happened upon one of the most powerful tools of review or studying. It wasn't flash cards, re-reading, highlighting, taking notes and re-reading those. Nor was it taping my professors and listening to their lectures again. The key that unlocked my ability to understand and remember (regurgitate on the test day) was my the study tribe I had connected with. Others within my major who were (supposedly) headed into the same career fields as I. Gathered with this small group, engaged in discourse about our separate understandings, proved to be the lone factor that under-girded all of the other study tactics. Learning from this group of women and listening to their vantage points caused me to reflect more on my understanding and recall our conversations on that fateful day when I had to sit face to face with a "summative" test.

I say all of this to make one very important point. As the classroom teacher, your students spend hours listening to YOUR voice. During your mini-lessons, small group table instruction, whenever you're giving instructions or feedback...they hear YOU! This means they are constantly being given information. But if peer to peer discourse isn't evident in your class with student opportunities to "turn and talk", students eventually suffer from informational constipation, (backed up) if you will.
If frequent peer discourse isn't evident in your everyday Tier 1 practice, I want to challenge you to pass the conversation stick to your students.

But more than that, I want to challenge you to consider ways that your students can prepare for upcoming standardized forms of testing by providing them with prompts to discuss in groups of 2, 3 or 4.

Create meaningful, engaging reviews that allow students to process, justify, explain their thinking and do so with each other. Let their peers be the ones that give feedback (while we walk around clarifying and verifying...of course). Let your students learn from each other, for a change! Review season is upon us; resist the urge to control the floor! Lead4ward's Instructional Playlist is a great place to start!

Why Kids are Doomed

https://youtu.be/ejqH1ofSx8U

While scrolling Facebook, I caught a glimpse of this video (briefly) and the caption, "This is why the kids are doomed." As a lover of math, I couldn't help the burning desire to draft up a piece in efforts to stand up for and defend mathematics!

While I understand why parents and even educators (who learned the traditional algorithm) feel as if students are "doomed" for having to learn this "new math" (which actually isn't new math at all); there are so many misconceptions that have formed these thought patterns.

Now if you're like me and you've sat in front of hundreds of students who have been taught the traditional algorithm, you would concur that several students are guilty of making one or more of the following misconceptions and mistakes in this learning process:

Mistake #1 - forgetting to place the "place holder" zero on the second partial product (2nd row).

Mistake #2 - recording an incorrect answer; either because of mistake #1 or a computational error (multiplication or addition).

Mistake #3 - not regrouping appropriately or at all.

Mistake #4 - not realizing their product is incorrect (as a result of mistakes #1, 2 or 3).

As a teacher, however, the natural inclination is to get frustrated with the student and resort to fussing and/or giving MORE homework problems so they can "practice makes perfect" their way into the correct process.

However, that is a reactive approach based on a lack of understanding of appropriate progression. It's like trying to teach a kid to swim, who has a fear of water. Or trying to teach a kid to walk who has not yet developed the motor skills for crawling. Both can be done with practice. But both are not within a developmental cognitive progression. A student who learns how to embrace water in their face and floating is cognitively & thus developmentally ready to embrace the steps and strokes of swimming. A toddler that has worked through the natural motor skills needed to crawl, roll and pull up is cognitively and thus developmentally ready to conquer placing one foot before the other. And furthermore has the foundational skills to "know how" to recover when they have fallen from their attempts at walking.

So why can't we (as educators) provide the same deep foundations for students by being proactive (rather than reactive) through conceptual instruction (the why) coupled with abstract (the how) instruction?

• Why can't students know the reasons behind why the Constitution is so meaningful to our country?
• Why can't students understand and learn how and why storm clouds form?
• Why can't students learn when to appropriately use the homophones: to, too and two?
• Why can't students learn how to mentally build answers for accuracy?

I would agree that students ARE doomed...but not because they learn models such as the video on the left. Students are doomed because processes like the one on the right are forced down their brains without a rhyme or reason. They are taught to JUST DO IT (because I taught it to you) and are left with a system of steps (that can easily be forgotten, or simply incorrect) without an ability to justify whether their answer makes sense. I would argue that this is the same reason why students (as young adults) stand behind cash registers and cannot tell you whether the amount of change they are giving you is correct or incorrect (they just hope and pray they typed the right numbers into the machine). 

NO NUMBER SENSE!

NO ABILITY TO REASON!

Here's what students learn with each approach!

                                         Conceptual Approach          Abstract Approach

                                       - Place Value Concepts          - Memorization of steps

                                       - Connection to why              - Rote processes

                                       - Partial Products                   - How to get an answer

                                       - Why the answer is correct   - Lower level of thinking

                                       - Reasonableness

                                       - Deeper level of thinking

I take great pride in being a part of the paradigm shift occurring in mathematical education. I love that the conceptual components of math (that were left out when I was a student) are now evident in the TEKS. The instruction that we as educators are now able to provide students can better equip them to be problem solvers, reasoners, thinkers and developmentally appropriate mathematicians (or just productive citizens in our society, for that matter). 

Facts...or FIB?

A majority of my teaching career consisted of working with students who functioned at a Tier 2 and Tier 3 level. It's not that they could NOT learn, they were extremely bright students. It's that they either:

• Entered the US late (lacked years of schooling)
• Had very little home support
• Missed several hours or days of school (therefore Tier 1 instruction)
• Had classes in previous years where their teacher was new or left in the middle of the year (or had a sub for a great deal of the year).

Sometimes I had students who were labeled as Tier 3 and yet only really had behavioral problems; but had been labeled because in previous years they weren't being challenged to their ability, so they acted out. Their behavior (due to lack of classroom management and engagement) caused them to be sent outside the classroom so much, that by the time they got to me, they had missed a significant amount of instruction. Their learning "label" had NOTHING to do with their ability to rise to the challenge.

This made me quite passionate about Tier 2 strategies and caused me to work diligently at strong, effective, engaging Tier 1! I wanted to not only keep students in my class (behavior problem or not), but engage them so something intrinsically triggered them to feel successful. At my small group table, I worked with different strategies to support closing their gaps.

One resource I found quite helpful in remediation with students, was SCAFFOLDING! Breaking my Tier 1 supports down into steps that were feasible for students was a gold mine!

Especially when I encountered students who didn't know their facts! I knew drill & kill was NOT the answer because if they could learn it that way, they wouldn't be struggling with their facts they way they were. 

So, a few things I found effective included: 

1. Distributive property (with representations of course)

For example, if a student struggled with a tougher fact such as their 7s...helping them break that fact down into simpler, more doable steps/fact.s

2. Using what we know (Counting Back)

So if a student didn't know their 9s, we could use their 10s (which they DID know) and help them work backwards. This helped students see a more efficient way to get an accurate answer than skip counting by 3s nine times (and possibly miss counting) or trying to remember their 9s. It also helped them see how facts are related.

3 x 10 = 30

so...3 x 9 = 30 (count back 3)

John Van De Walle also provides some GREAT do's and dont's when it comes to FACT REMEDIATION!

Check'em Out here! Enjoy!

(Concrete) Measurement Conversions in 5th grade

Having taught 5th grade and middle school math, I know that not only do students often struggle with measurement conversions but teachers struggle with how to help students GET IT! It's quite an abstract concept. So since a majority of my students benefited from remediation with concrete objects, I found myself searching for (holds my head in shame) tricks to help them understand it.

(We would teach the student to set up a ratio, multiply the two numbers that were across from each other and circle the number that didn't have a partner across from it. Then divide the product of the "bat" by the "ball")

What I wish someone would have told me (or I wish NIX THE TRICKS (click on the link for the digital book!) was published at the time) was that tricks AREN'T for kids. Tricks only put a band-aid on the open wounds of math gaps. I'll admit, SOME of my students got it, but I was still only showing students a temporary process that made very little connection to THE MATH and adding to the arsenal of math steps they had collected throughout their elementary career). While it might have helped develop their background of ratios, it didn't explain why some inches were a fractional piece of a foot.

So, here I am, some 16 years later, hoping to help YOU make sense of that math so you can "pay it forward" to your students. Believe it or not, students begin to hate math around middle school and it's primarily our fault. We have crammed tricks and steps and songs- all meaningless- into their heads. At some point, they think math is some huge mystery that can never be conquered; rather than a cool puzzle wrapped in a series of inter-related patterns with various paths to the same end.

Watch this video!!! 

How to use concrete materials, such as a ruler and color tiles, to help students visually see how the inches in between each foot serve as a fraction of a foot! I'll be trying it with some students I work with; if you try it this Spring...Let me know how it goes!

Routines, Anchor charts & Stations, oh my!

As an Elementary Math Program Coordinator, I get the privilege of visiting campuses twice a week in hopes to keep in touch with teachers (gather constructive feedback for curriculum decisions) and engage with students (to better understand how they’re learning). Upon one particular visit, I ran into a teacher who asked if she could pull me aside and into her classroom to ask about the different visuals used to teach dividing unit fractions by a whole number and dividing whole number by unit fractions.

I got goosebumps by the sheer question and personal desire this teacher had to move students from a conceptual understanding to an abstract knowledge of the algorithm associated with this concept. She urged me to watch her draw each model, encouraging me to correct her whenever I saw her deviate from the appropriate strategic approach. After she flawlessly modeled each strategy, we discussed how the process connects to contextual situations and how important it was for students to interpret the difference between each model. *for clarity, watch this video!


It was through this discussion that she began to explain to me the daily learning routine she takes her students through, that allows them to excel despite their daily hurdles of (shortened instructional time, highly disruptive behavior problems and students with tremendous gaps). 

• First, she does a quick mini-lesson by which she introduces the concept, and engages the students in the skill through manipulative exploration and/or drawings. 

• Next, she involves the students in building an anchor chart together based on their new learning.

• From there, she gives students time to complete a short, open ended formative assessment with the support of their notes and class-created anchor chart. 

• While reviewing the formative assessment data, she has her students in stations and pull students to her small group table to work with them based on their formative feedback. 

My second case of goosebumps came over my arms as I sat in awe of this teachers’ fabulous Tier 1 routines. I also couldn’t help but get distracted by the images of her class-created anchor charts hanging on the wall like a museum of art (see above), the evidence of learning from historically low performing students who felt confident enough to complete their formative without the offered support of their teacher.

I can’t explain the way a math heart leaps with excitement when it witnesses non-numeric evidence of student learning. When historically low performing students, who CAN (although their data often says otherwise), show off their true abilities.

These elements, when evident in a class can tell a story about the teacher's classroom culture, the efficacy built into his/her students and exemplifies their instructional practices.

Needless to say, my day was made! Kudos to this teacher, for her students are truly blessed!