Monday

Math Fun - With Dates!


Last week I started throwing a little extra math into my classes, homeroom and last period (homeroom students again) - by using the date! I went to a conference years ago and this was one of the ideas in the book we were given.

I haven't used the idea in a while, so I brought it back in two ways:

1) Use all the digits in the date to create an equation. The digits should stay in the same order they are in the date, and any operation signs can be added in between any digits. The equal sign can also be placed between any digits.
Digits can be used as exponents, as in the example shown, and you could add square roots signs if you can find a way to use them.

2) The other way I used the dates was to write the date so that students have to solve an expression for each number in the date.

It's been fun to see some students writing these in the corner of their notebooks during class! Others have asked to write their equations on the board during the last period of the day.

I'm looking at the date-writing as another way to introduce notation my students haven't seen before, like the cube root.... I loved today's date!

 How would you write some of these dates?













Sunday

What's Math Got To Do With It? Chapter 5

Chapter 5: Stuck in the Slow Lane: How American Grouping Systems Perpetuate Low Achievement

Wow, what a chapter. This one has made me rethink some of my beliefs. Most of my 24 years of teaching math has included grouping by ability. It was such a big push in my early years of teaching – our smaller elementary school didn’t group, but one of the bigger elementary schools did, and the parents pushed for (and got) “equity” among the schools. So we all started grouping. Ability-grouping has become ingrained, so reading (in this chapter) that ability grouping is illegal in some countries in the world, including Finland (at the top in international achievement tests), really surprised me. Grouping is banned because when students are put in lower ability classes, they receive lower-level work, which Boaler says is damaging and suppresses achievement. Non-ability grouping provides more students with the opportunity to learn – which is needed in order to achieve. Makes sense.
Boaler states that in the US, students are typically grouped around 7th/8th grade. (In past years, our district has been grouping as early as 2nd grade....)
This chapter shares research results that support the idea that grouping in math is not the best approach for student achievement.
 
As Boaler states at the start of this chapter – parents want ability grouping because they want their “high-achieving, motivated children to be working with similar children.” Boaler shares that international studies reveal high-performing countries like Japan and Finland reject ability grouping. The countries that use ability grouping, like the US, are least successful in math.

Results collected about eight-graders from 38 different countries showed that the US had the most tracking (US was in 19th place), while Korea (highest-achieving country) had the least tracking and most equal grouping. US had the strongest link between achievement and socioeconomic success, which has also been attributed to tracking. Researchers concluded that countries that leave grouping to the latest possible moment or use the least amount of ability grouping are the highest achieving.

According to research, when students are not grouped by ability, it helps not just the students who would be on the lower tracks, but also the students who would be on the higher tracks. In mixed-ability classes, there are more occurrences of students helping other students (rather than just teacher helping students). This helps the higher students, because explaining concepts helps them learn the concepts more deeply; it also helps them find their own weaknesses - if they can’t explain, they know where they need to work. Results from one of Boaler’s studies showed that after 3 years of ability-grouped classes, students achieved at significantly lower levels than students who had been in mixed-ability groups. “High” students felt unable to say that they didn’t understand the work, which moved “too fast.”

In one California high school, all students started with algebra, taking 90 minute classes for ½ year. This gave students the opportunity to take 8 math classes through high school. By the end of high school, 47% of students took calculus and precalculus, compared to 28% of students in a typical tracked high school. It was noted the "highest students" at the "detracked" high school achieved at higher levels than high students in high tracks at other high schools.

Boaler shares the belief that tracking leads to a fixed mindset – the belief that “ability” is fixed. For “high” students, being put in a high track led them to believe they were smart. According to the growth/fixed mindset philosophy, this belief sets them up to become fearful of making mistakes and avoid more challenging work. Boaler states that this has especially devastating consequences for high-achieving girls.

Students in tracked classes are often treated as if they all have the same needs and work at same pace, which is not true. When students are not grouped, there is a greater perceived need for the teacher to differentiate both instruction and work. This results in a better match of pace and level.

In research in US classes, it was found that tracked students tend to develop ideas about their own potential, but also about others – as dumb or smart, quick or slow. Students in mixed-ability classes were more respectful and learned to respect different ways of thinking.
According to Boaler, there are two critical components for mixed-ability grouping –
1) Students must be given open work that can be accessed at different levels and taken to different levels. Teachers need to provide problems that will be challenging in different ways, not problems that target small, specific pieces of content. Students will learn different things at different levels with these types of problems.
2) Students must be taught to work respectfully with each other.

At the end of the chapter, Boaler shares that she interviewed students that had been part of studies eight years earlier. Now, as young adults, those who had been in mixed-ability groups were in more professional jobs that those who had been tracked.

Very interesting chapter! So much to think about.
At this point, my grade level does not group by ability, but students were grouped for years before coming to me...

Does your school/district use ability-grouping in math?




Monday

Problem of the Week #10

In honor of Winter Storm Jonas, this week's problem solving uses some snowfall data. This is a quick one - students use the data to find mean, median and mode. It's nice that the numbers are decimals - provides some decimal adding and dividing practice!




Click to download!

To access all of the Problem of the Weeks, click here!




Sunday

What's Math Got To Do With It? - Chapter 4

Chapter 4:  Taming the Monster

I definitely enjoyed chapter 4. The chapter discusses one of our favorite topics - testing.

A few statements that Boaler makes in the chapter are:
* American children are tested more than ever and more than students in the rest of the world.
* The tests used in the U.S. are rejected by most other countries.
* Tests are damaging to schools, teachers, and students - to their health, hearts, and minds.
* It's hard to find any multiple choice questions used in Europe or in any national assessment, in any subject, at any level. (I've always taken multiple choice tests - I had no idea they are not used everywhere!  Anyone from a different country reading this - what are your tests like??) (Anyone in U.S. - are your tests mostly multiple choice?)

Boaler identifies reasons not to use multiple choice tests:
1) Multiple choice testing is known to be biased.
2) Timed multiple choice tests cause anxiety.
3) The best thing a multiple choice test shows is the ability to complete a multiple choice test.
4) Using questions that are not multiple choice allows teachers to better assess student understanding, which includes the thinking that the student does and expresses in words, numbers, and symbols. If students are simply choosing a correct answer, this information won't be available.

According to Boaler, tests in most states are extremely narrow - they don't assess thinking, or problem solving. Instead, they assess the use of procedures. She states that in the U.S., math teachers must focus on teaching what will be tested, rather than on what students need to know for work or for life. What about you? Do you agree with this statement in regards to your school/class?

Boaler states that with the Common Core era, free-response items will replace some multiple-choice items (on the state tests my students take, we've had multiple choice with several free-response for quite a long time now-before Common Core).

The results of standardized tests label students as low achievers, or "below average," and help to create low-achieving students; the label destroys their confidence and gives them the identity of a low-achiever. Research reveals that confidence in one's ability to succeed in math in an intrinsic part of success and motivation.

Click to see on Amazon
Boaler identifies "assessment for learning" as an alternative to the traditional "assessment of learning." This is a form of assessment that gives useful information to teachers and parents, but also allows students to take charge of their own learning. Assessment for learning is based on the idea that students should have:
1) A clear sense of exactly what they are learning (the concept - not the page number or chapter title):
Students have math goals to work toward - details about important concepts and how they are linked. These should be clear statements that express what students should be understanding. The example Boaler gives is, "I have understood the difference between mean and median and know when each should be used." Boaler cites an interesting study that showed great gains in low-achieving students whose class used this approach; it concluded that the students had previously been unsuccessful not because of a lack of ability, but because they hadn't known what they were supposed to be focused on.
2) Where they are in the achievement of mastery. Students can show their self-assessment by putting red, yellow, or green stickers on their work; teachers can use red, yellow, or green cups for students to display. Boaler does note that students in research studies were hesitant to show red to begin with. However, after "green cup" students were asked to explain their understanding, students who didn't understand became more willing to show red.
3) What they need to do to be successful. Teachers need to give constructive feedback about student work, rather than just giving a score. Boaler shares research that indicates that giving a grade can actually reduce achievement because the focus becomes the grade rather than what needs to happen for the student to improve. (Grades are fine at the end of the semester or term, she says). She relates this to coaching athletes - athletes aren't given grades; they are given advice and coaching to become better.

"Assessment for learning transforms students from passive receivers of knowledge to active learners who regulate their own progress and knowledge and propel themselves to higher levels of understanding."




Problem of the Week #9

It's 2016. That blows my mind. So many new ideas to think about, so many new things to try, so many hopes for the year! I hope the year will be good for you all!

This week's problem requires students to search for how many different combinations are possible for a password....can be challenging, depending on your students' background!

I hope you'll give it a try - if you do, please let me know how it goes! We have spent many days this school year giving our students the opportunity to "struggle" a bit with problems like these, and this had lead to great discussion, among the students AND the teachers.

Have a great Monday!
Click to download!
https://drive.google.com/file/d/0B121a_SUYfUBOXpEbXZhUURPSnM/view?usp=sharing


To access all of the Problem of the Weeks, click here!


Monday

What's Math Got to Do with It? - Chapter 3

Chapter 3: A Vision for a Better Future, Effective Classroom Approaches

In Chapter 3, Boaler describes two successful approaches that offered students experiences with real math work. These approaches were used in studies that Boaler conducted. The study details that she shares are very interesting (I love reading about research), and I've included the highlights here.

Boaler calls the first approach the Communicative Approach. She completed a four-year study, following about 700 students in three different high schools, to determine that this is a successful approach. Students at one particular school were detracked, algebra became the first course that all students took when entering high school, and the teachers met over several summers to design/alter their courses. In this approach the focus is on "multiple representations," like words, diagrams, tables, symbols, objects, and graphs. The students at this school explained their work to each other, and moved between different representations and communicative forms. Interestingly, these students defined math as a form of communication, or a language.

The students taught with this approach worked in groups and were taught that they are all smart, but have different strengths in different areas; everyone had something important to offer. The teachers involved in this approach reinforced the idea that being good at math involves asking questions, drawing pictures and graphs, rephrasing problems, justifying methods, and representing ideas, in addition to calculating. They also followed an instructional design (called complex instruction) that made group work more effective and promoted equity among the students. Students at this school learned to appreciate the differences in one another.

In comparison to the students in the other, more suburban high schools in the study (using traditional teacher-lecture methods), the students at this urban school ended up outperforming the others on algebra and geometry tests by the end of the second year of high school. By their senior year, 41% of the students in the urban district were taking precalculus and calculus, compared to 23% at the other schools.

The other approach Boaler describes is the Project-Based Approach. Students in two schools were followed for three years in this study, which included the observation of hundreds of hours of lessons, interviews with and surveys of students, as well as various assessments. As the name indicates, students in this group worked on projects that addressed math as a "whole" rather than as separate areas of math. In many cases, students were taught certain methods when they needed to use them in the course of a project, rather than being taught the concepts beforehand. For example, in a particular area-related project, some students ended up needing to use trig ratios, so the teacher taught them about trig ratios.
The projects were open enough that students could go in different mathematical directions - directions that interested them. Students could choose who to work with, so some worked alone, some in pairs, and some in groups.

The students at this school viewed mathematical methods as "flexible problem-solving tools," and ended up scoring higher than the national average on their exams, taken at age 16. For more details, you may want to read the book Boaler has published about this study - Experiencing School Mathematics.

As a result of her studies, Boaler concludes that students need to be actively involved in their learning and they need to be engaged in a broad form of math.

Do any readers use a project-based approach to teach math? If so, how does it work for you?


Sunday

Problem of the Week, #8

This week's problem can be solved by guessing and checking or by setting up an equation and solving algebraically, so no matter where your students "are," they can give it a shot:-)

Click to download!

 To access all of the Problem of the Weeks, click here!





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