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/

]]>If you have any questions, comments, or suggestions, please Tweet us at @dhabecker and @pelelover1

Show notes at http://theothermath.com

Maggie found this wonderful article…

How teachers stay current: http://teachingcommons.cdl.edu/cdip/facultyresearch/Stayingcurrentinthefield.html (Ideas for the first 4 statements)

**Becoming an effective consumer of research (Staying current on the research)**- Awareness of the latest findings, trends, understandings, insights and conversations in the field of education places an individual’s own practice into a larger context.
- For new teachers one of the most challenging aspects of staying current is time.

**Reading key publications**- Look to professional organizations (NCTM, ASCD, CMC, CISC, CTA, NEA)
- Ask colleagues in the field about what they read
- Browse your Office of Education library/website
- Follow blogs, journals, and podcasts that publish articles of interest

**Attending key gatherings**- Professional conferences and meetings offers another venue for staying current in education.
- Many are annual events sponsored by professional organizations.
- By choosing the gatherings carefully, teachers will begin to make connections to others doing similar work, related work, or complementary work, and may form relationships that last throughout a career.
- Presentations may push traditional thinking.

**Developing a network of colleagues**- The academic life can be isolating unless teachers actively reach out to those whose work inspires, challenges, and interests them.
- Over years of such engagement, accomplished members develop an entire network of scholars that pushes their research, teaching and service in new directions.

Excerpt from – http://www.teachhub.com/professional-development-tips-teachers

The essence of a teacher is to help others. This is why it may so hard for educators to look at themselves to see what they can improve upon. Here are a few tips to help you improve your performance as a teacher.

**Read**– Join a professional scholarly journal, go online and read educational blogs, or read some literature on education. Knowledge is power.**Participate**– Go to educational conferences or workshops, or attend online seminars and webinars. Participation in these types of event will make you a more effective teacher.**Join a Group**– There are many groups you that you can join, online and off. All of these groups are a great source of information as well as inspiration. You can learn a lot from other professionals who have years of experience.**Observe Your Peers**– An effective teacher takes the time to observe other teachers. These teachers can be a great source of knowledge for you. You can find a new strategy to teach or behavior management plan to implement.**Share –**Once you have improved your performance, then you should share your knowledge with others. Contribute to your profession, and others will be thankful.

Excerpt from – https://www.teachthought.com/pedagogy/8-strategies-to-change-how-you-teach/

**Reflect, Reflect, Reflect –**Reflect on what you learned, reflect after further reading, reflect after discussing it with students or colleagues, then reflect after giving it a try. Consider using “How did it go, and how do you know?” to help frame that reflection, which forces you to both confront how you think things went, and then consider the “data” or evidence of that assessment (whether formal or informal).**Listen to Students –**They’ll let you know how you’re doing, and how any changes to your teaching are “going.” You just have to be willing to listen with an open mind.

Maggie’s Motto: Always look for at least one thing that you can learn.

A hodge-podge of ideas for growing:

**Twitter**

- Curate good people
- Follow useful hashtags
- #MTBos – https://twitter.com/search?f=tweets&vertical=default&q=%23mtbos&src=typd
- #iTeachMath – https://twitter.com/search?f=tweets&q=%23iteachmath&src=typd
- #GeoGebra – https://twitter.com/search?f=tweets&vertical=default&q=%40geogebra&src=typd
- #Desmos – https://twitter.com/search?f=tweets&q=%23Desmos&src=typd

**Regional Math Events**

- In California
- CMC-South: Palm Springs in November
- CMC-North: Monterey in December
- CMC-Central: central California in March

- Look for the math organization in your state

**National Math Events**

- NCSM Annual Conference
- NCTM Annual Conference
- NCTM regional conferences – https://www.nctm.org/Conferences-and-Professional-Development/Regional-Conferences-and-Expositions/

**Webinars**

- NCTM – https://www.nctm.org/Conferences-and-Professional-Development/Webinars-and-Webcasts/
- ASCD (Association of Supervisors of Curriculum Development) – http://www.ascd.org/professional-development/webinars/ascd-webinar-archive.aspx
- Christina Tondevold – https://buildmathminds.com/

**MOOC**

- Stanford Online – https://lagunita.stanford.edu/
- Jo Boaler’s Course – http://scpd.stanford.edu/ppc/how-learn-math-teachers

Did we leave off a favorite resource of yours? Leave it in the comments below!

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]]>Sometimes when I do a training, I’ll ask participants to Tweet a thought and/or image to #Duane314. This is where those tweets will go…

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You can download the report here…

If you have any questions, comments, or suggestions, please Tweet us at @dhabecker and @pelelover1

Show notes at http://theothermath.com

**Question #5:** Can I help my students learn how to learn mathematics?

A teacher’s role is to recommend or encourage the use of specific learning strategies that are the most beneficial for individual learners or the problem at hand. While no one learning strategy is perfect for all learners and all situations, PISA results indicate that students benefit throughout their schooling when they control their learning. Students who approach mathematics learning strategically are shown to have higher success rates on all types of mathematics problems, regardless of their difficulty.

WHAT ARE CONTROL STRATEGIES IN MATHEMATICS?

Control strategies are part of the METACOGNITION family of general approaches to learning. The two other strategy members of the metacognition family are memorization and elaboration. We talked about memorization strategies in Episode 12 and will talk about elaboration next episode.

Control strategies are learning strategies that allow students to set their own goals and track their own learning progress. In other words, control strategies are methods that help learners control their own learning.

This approach includes activities such as:

- organizing material
- creating a study plan and reflecting on the learning strategies used
- activities related to concepts such as efficiency, strategic learning, self-regulation and metacognition.

PISA asked students questions that measured their use of control strategies in mathematics and found that students around the world tend to use control strategies to learn mathematics more than memorization or elaboration strategies (which is looked at in detail in Question 6).

The questions to measure a student’s tendency to use control strategies were:

- When I study for a mathematics test, I try to work out what are the most important parts to learn.
- When I study mathematics, I make myself check to see if I remember the work I have already done.
- When I study mathematics, I try to figure out which concepts I still have not understood properly.
- When I cannot understand something in mathematics, I always search for more information to clarify the problem.
- When I study mathematics, I start by working out exactly what I need to learn.

Students in the United States reported using control strategies less than the OECD average…although it was just a little less than average. Countries at the top of the list (high use of control strategies) were China, Japan, Hong Kong…mathematical powerhouses. Countries at the bottom of the list were Jordan, Tunisia, Qatar. I’m not sure what to make of this because it is more complicated than just a matter of “good math teaching utilizes lots of control strategies”, because Korea, another math powerhouse, reported using control strategies far less than average.

WHAT IS THE BENEFIT OF STRATEGIC LEARNING IN MATHEMATICS?

Students who use control strategies more frequently score higher in mathematics than students who use other learning strategies such as memorization. What’s more, control strategies work equally well for nearly all mathematics problems, EXCEPT the most difficult ones.

I don’t understand this from the report:

*Control strategies might not be as effective for solving the most complex mathematics problems because too much control and strategic learning might hinder students from tapping their creativity and engaging in the deep thinking needed to solve them. *

I thought control strategies allow the student to be in control of his own learning and goals. So…I’m not too sure what this means.

What is taught in mathematics class and how learning is assessed might also limit the effectiveness of these strategies. Research suggests that the success of these strategies depends on what is being asked of students by their teachers, schools and education systems. For example, when students are only being assessed on surface-level knowledge of concepts, they won’t venture into deeper learning of mathematics on their own. Control strategies need to be practiced by the student on complex problems as well as easy ones.

IF CONTROL STRATEGIES ARE SO SUCCESSFUL, WHY SHOULDN’T I ENCOURAGE STUDENTS TO USE ONLY THESE LEARNING STRATEGIES AND NOTHING ELSE?

One size does not fit all!

Just as one teaching strategy doesn’t work for every student or every mathematical concept, control strategies aren’t appropriate for every student or every problem all the time.

Also, the use of control strategies is somewhat associated with great anxiety towards mathematics and less self-confidence.

Despite the negative characteristic associated with control strategies, students who report the highest use of control strategies score the highest on the PISA.

WHAT CAN TEACHERS DO?

- Make sure that your own teaching doesn’t prevent students from adopting control strategies.
- When teachers adopt certain teaching practices, they may be inadvertently reinforcing the use of certain learning strategies. For example, by giving homework that includes mathematics drilling exercises, you might be encouraging students to use memorization over control strategies.

- Familiarise yourself with the specific activities in the category of “control strategies”.
- Once you understand what constitutes a control strategy in mathematics, you can work to incorporate related activities into your teaching and encourage your students to use similar strategies themselves. For example, you might have your students work in groups to create a study plan for an upcoming exam and monitor their own progress.

- Encourage students to reflect on how they learn.
- Provide students with opportunities to discuss their problem-solving procedures with you and with their peers. Helping students develop a language with which to express their mathematical thinking can also help you better target any support you provide to your students.

But don’t take our word for it. Please read the report in its entirety.

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