“Fractions…ugh”, said one teacher. “I’m getting into dividing fractions. I don’t know how to explain it to my kids any better than KEEP-CHANGE-FLIP”, she confessed.

“I don’t teach that algorithm, because it never makes sense to my students”, chimed a second teacher. “Instead”, she continued, “I use the common denominator method.”

A collective “Huh?” spread through the room. She went to the whiteboard and wrote this…

After doing a few examples, the teachers were pretty excited about the prospects of dropping the KEEP-CHANGE-FLIP mystery in exchange for this new method. That’s when I played the role of buzz-kill and asked “Why does this work?”

Uh…deer in the headlights.

**Here’s the deal**

In our search for alternative algorithms, we cannot merely exchange one meaningless algorithm with an alternative meaningless algorithm.

In our search for alternative algorithms, we cannot merely exchange one meaningless algorithm with an alternative meaningless algorithm. The purpose of alternative algorithms is to find ways to unveil the MEANING of mathematics. Often, our traditional algorithms have become the preferred algorithm because they are so efficient. But it is exactly that efficiency that obfuscates the underlying conceptual understanding of the math. If we are to use an alternative algorithm (heck…if we use ANY algorithm), we must allow students to see WHY that algorithm makes mathematical sense.

Inspired by the conversation of these awesome 6th grade teachers, I made this video explaining how common denominators can be used to divide fractions.

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I recently visited a 3rd grade class to share with the teacher ideas for teaching multiplication facts. She was particularly frustrated that her students were struggling with the larger facts – the sixes, sevens, eights, and nines. It was clear that while students had experience connecting multiplication with the idea of equal groups, students did not have many strategies for deriving the product of an unfamiliar number fact. Essentially, students only skip counted every time.

It seemed like I needed to share some additional strategies for deriving the product.

I began with the distributive property and the idea of breaking a “big” unknown multiplication fact into smaller known facts.

We started by remembering that multiplication means equal groups.

This meaning of multiplication leads to repeated addition, which leads us to the distributive property…

Here is the same image with color coding…

Eventually, students were able to imagine the repeated addition in their head and go straight to writing the fact with the distributive property.

A SECOND STRATEGY

We then moved to the second strategy: area model.

While the model is different, the conversation ended up being very similar. Students saw the distributive property lurking inside the rectangle.

FRAYER MODEL

As an ongoing routine, I shared with the teacher the Frayer model. Students folded a piece of paper into the “diamond paper”.

Then we placed a multiplication fact in the middle. Let’s say it is 7 x 6. Each quadrant serves a purpose…

Students had some SERIOUS trouble writing a story problem for 7 x 6. For 7×6, we would expect something along the lines of “There are seven girls in line. Each girl has 6 dollars. How much money do they have in all?”. Instead, students were writing addition questions like “I have 7 toys…and then Joe shows up with 6 more toys. How many toys do we have in all?” This really hit home that students spent their year trying to memorize multiplication facts rather than understanding what multiplication MEANS.

I suggest 3rd grade teachers begin Day 1 of school with the Frayer model to represent multiplication facts. Start with something small like 2 x 3. As various representations of multiplication are taught, the teacher should expect to see those representations show up in the Frayer models students are creating. Over time, students will REMEMBER all their facts…not memorize them.

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Dutifully, we’d set aside instructional minutes to give those benchmark assessments, but the results never changed. The persistent achievement gaps between various sub-groups remained. Why?

In this article, Robert Slavin shares the surprising research that benchmark assessments do not make any difference in achievement and why this is the case.

]]>Doing my best of embody the famous quotation of G.H. Hardy,* “A mathematician, like a painter or a poet, is a maker of patterns.”*, I created a table of some powers of 5. Then we filled it out gradually.

Finally, we looked for patterns in the table. Looking at the middle column, my colleague said, “See? The fives are gradually disappearing. There will be no fives for 5 to the power of 0, so the answer is zero.”

“That seems to make sense, but let’s look at the third column. What is happening as we move down the column?”, I asked.

“It divides by 5”, replied. I filled in the table only to hear her gasp in amazement. “Oh my gosh! It’s ONE!”, she almost screamed.

“But what about the middle column?”, she wondered out loud to herself.

“You were half right”, I said. “There are no fives…but the answer is one, not zero.”

“Because the pattern says so?” she asked. And before I could answer, she said “Because the pattern says so!”

I am reminded of this story from close to 30 years ago by this **wonderful article** on the impact of teacher effectiveness on student achievement. If my colleague was unintentionally teaching her students misinformation with exponents, where else might she have been misinforming or half-informing her students. What impact might this have on her students?

Now…I’m NOT bagging on my colleague. It is not her fault that no one taught her this simple “proof” of the Zero Exponent Rule. I’m honored to have taught her a little something. Indeed, she became a great teacher, only the tiniest part because of me. For that moment, however, I performed the role of mathematics instructional coach, helping her with both mathematics AND the instructional strategies that might be used to get the mathematics across to the students. As a result, she took a huge step towards being a high performing teacher.

Why does this matter? In **this research progress report**, we learn that there is a huge difference in student academic achievement depending on whether the students experiences three consecutive years of high-performing teachers versus three consecutive years of average-performing or low-performing teachers.

In comparing 5th grade math achievement after experiencing three consecutive years with high-performing teachers versus three consecutive years with low-performing teachers, the report states, “With an even start, the difference in these two extreme sequences resulted in a range of mean student percentiles in grade five of 52 to 54 points!!”

**In a nutshell…GOOD TEACHING MATTERS!**

Unfortunately, the effect of student achievement after experiencing a low-performing for a single year can still be measured even after multiple years of having high-performing teachers. It is essential that no student is subjected to the instructional strategies of low-performing teachers. Or at least, site administrators should insure that students who experience a year with a low-performing teacher should then experience multiple years of high-performing teachers.

**GOOD TEACHING MATTERS!**

Since it is so obvious that the quality of the math instruction is of paramount importance, we teacher leaders and math leaders need to be vociferous advocates for meaningful mathematics coaching. Occasional sit-and-get professional development is not good enough. We need the kind of ongoing training and coaching that will bring about real change in the classroom.

Our students deserve it.

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]]>Now imagine a math teacher who doesn’t do math.

Unfortunately, this is often the case. Now I’m not talking about the math in the textbook. Of course, math teachers do that all the time. I’m talking about recreational math. Math just for the fun of it.

Our students will not learn to have fun with mathematics if we ourselves never have fun with mathematics.

**I propose we math teachers need to engage in more recreational mathematics…just for the fun of it. Simply because we love math and love doing math.**

Now…let me introduce you to Math Teachers’ Circle.

Math Teachers’ Circles are professional communities of K-12 mathematics teachers and mathematicians. Groups meet regularly to work on rich mathematics problems, allowing teachers to enrich their knowledge and experience of math, while building meaningful partnerships with other teachers and mathematicians.

Go to their website to find a Math Teachers’ Circle near you: https://www.mathteacherscircle.org/

In my hometown of Merced, we have a local chapter: **Merced Math Teachers’ Circle**.

MMTC connects K-12 mathematics teachers, college and university mathematics professors, and all mathematics educators and enthusiasts in the Central Valley area of California centered around Merced. They meet regularly to

- engage in fun, creative, and meaningful problem solving activities;
- share and (re)-experience the excitement of doing mathematics so that we may bring that enthusiasm into our own classrooms;
- design and discuss interesting problems to run Math Circles for our own students;
- strengthen and form connections among all mathematics educators and enthusiasts.

While you are looking for a local Math Teachers’ Circle, try some of these fun problems from @whiterosemaths. These problems are intended to be solved using some sort of a bar model (tape diagram). Do you best to avoid fancy, schmancy algebra.

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By the end of the school year, the models that were intended to make learning mathematics easier began to get jumbled in the minds of my students. No good.

Then came the empty number line, which has become a unifying model that can work in a variety of different math concepts.

The empty number line is simply a line without regular intervals and without the zero being indicated unless needed. Students can use the number line to record their thoughts for solving the problem. Below are small examples of how the empty number line can be used in various math concepts:

**Using the empty number line to add whole numbers:**

Starting on the empty number line at 347, we make a series of hops that have a sum of 285. Where we end up on the number line is the solution to 347 + 285.

**Using the empty number line to subtract whole numbers:**

To model subtraction, you begin by placing 386 and 512 on the empty number line. Then using any series of hops, find the distance from 386 to 512. In the example above, 4 is added to get to 390. Then 10 is added to get to 400. Then 100 is added to get to 500. Finally, 12 is added to get to 512.

Summing the hops from 386 to 512 gives us the difference of 512 and 386.

4 + 10 + 100 + 12 = 126

The REAL power of using the empty number line is that it builds number sense in the students. It causes the focus to shift from the students trying to memorize an algorithm to students developing number sense while making up their own series of hops along the number line.

The awesome power of empty number lines becomes very apparent with subtracting fractions.

**Using the empty number line to subtract a mixed number from a whole:**

*Example 1*

To subtract, we need to think of finding the distance between the two numbers. In the above example, we place \(5\frac{2}{9}\) and 18 on the number line and then use any series of hops to find the distance from one to the other. The drawn number line shows a hop of \(\frac{7}{9}\) and a hop of 12. This means the difference is\(12\frac{7}{9}\).

*Example 2*

**Using the empty number line to subtract mixed numbers with the same denominator:**

*Example 1*

After placing \(5\frac{8}{9}\) and\(13\frac{7}{9}\), we can use three hops to find that the distance is\(7\frac{8}{9}\).

*Example 2*

**Using the empty number line to subtract mixed numbers with different denominators:**

*Example 1*

In this example we can use three hops to find the distance from\(2\frac{2}{3}\) to\(6\frac{1}{4}\). However, we need to get a common denominator to find that the difference is\(3\frac{7}{12}\).

If you have any questions or comments on using the empty number line, please speak up!

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]]>There have been some changes…Maggie has gone back into the classroom. She is a 4th grade teacher again!

This means there will be some changes to our Infinite Insights podcast. We won’t be able to dig so deeply into the research like we did last year. Instead, we’ll be focusing on the real day-to-day mathematical issues Maggie is likely to experience in her classroom.

Look forward to Infinite Insights showing up in your podcast feed soon! Or you can listen to it here…

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]]>Why are disks better than the 100 blocks, ten sticks, and ones cubes we have been using?

It is clear the teacher is a user of Base 10 blocks. It is reasonable to ask why, if Base 10 blocks work, then why should we also learn about number disks.

Here was my reply:

**Hi there XXXXX!**

**Great question. At Grade K and Grade 1, we prefer using the Base 10 blocks you mentioned. During 2nd grade, however, we want to transition to using Number Disks. Here is why….**

**In 2nd grade students begin adding/subtracting within 1000. What would 1000 look like? To model a thousand, students would have to draw a cube, which is rather tricky. It gets worse in 3rd grade when students need to model beyond 1000, and then students in 4th grade will have to model up to 1,000,000! Within the Base 10 blocks, there is no useful geometric shape students can use to represent 10,000 or anything larger. With Number Disks, however, representing very large numbers becomes trivial.**

**It is essentially the same story with decimals. Only now, number disks become the preferred visual for very small numbers.**

**I hope this gives you food for thought!**

That was my brief email. I hope it was okay. (Please give me additional food for thought!)

I’ll go into great detail for each of the operations…

In my email response (above), I provided an example for 14,386+7,597, but I forgot to show the algorithm worked out. Here is what it would look like…

It amazes me how closely using the number disks mirrors the traditional algorithm. Try it yourself with 268 + 147.

Once you’ve given it a try, check out my video to check your work.

Here is another example that demonstrates how closely using number disks mirrors the traditional algorithm. In this case we are solving 1353 – 786.

Now give 326 – 159 a try.

Watch my video to check your work.

Using number disks for multiplication is easiest when one of the factors is small…less than 15 or so. Problems like 1326 x 9 are perfect with number disks.

Now give 4 x 267 a try.

Watch my video to check your work.

This is the one that number disks really show their power. Here is an example of the division problem 539 ➗4.

Now give 98 ➗4 a try.

Watch my video to check your work.

Try number disks with your students. Let me know how it goes in the comment section below.

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We’re a math-traumatized people, Jo Boaler says (although she uses the British locution “maths-traumatized”). It’s a belief she sees confirmed in everything from students crying over long division to MRIs that reveal young brains reacting to numbers as if they were snakes or spiders.

from Pocket

]]>Whoa. I just read an article on the Hechinger Report that got me thinking.

In the article, Deborah Loewenberg Ball shares a 1 minute 28 second long interaction between two African American female students, in which Ball identifies 20 micro-decisions the teacher present needed to make. Each decision had the power to increase or decrease the students’ identity as mathematicians and thereby perpetuate race-based and gender-based stereotypes or not.

20 decisions in 88 seconds!

The interaction between the two girls is not unlike thousands upon thousands of interactions I presided over during my teaching tenure. I wonder how many of my micro-decisions supported my students rather than tear them down. I’m humbled right now.

Please take a moment to read the article here…

http://hechingerreport.org/20-judgments-a-teacher-makes-in-1-minute-and-28-seconds/

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