Math Facts – I Think We Are Doing This Wrong

 Recently, I have been thinking a lot about what causes math anxiety for students. As a high school math teacher, it was common for students to enter my classroom on the first day and announce that they were either not good at math or they didn’t like math. Sadly, these feelings were much more common than those that exhibited great (or any) enthusiasm for math. I would respond with “You aren’t good at math yet!” or “Maybe you’ve never been taught so that the math made sense.” Eye rolls and snarky laughs were a typical response from the students.

So where does all this anxiety and hatred for such a beautiful, exciting and useful subject come from? Of course, like everything in life, the answer is complex and involves many variables. I do believe that some of this math anxiety comes from how we treat math facts. If you have spent much time around an elementary classroom or have children of your own, more than likely you have seen the infamous “Mad Minute” sheets. Regretfully, I am guilty of using these with high school students who were in my Title I classes. (Shakes head in regret.)

While I understand the intent of these timed test, I think the un-intended consequences, leave lasting negative impressions for many students. I will offer some alternative ways for students to learn facts that don’t include memorization and drill type activities. First, I would like to point out some reasons why we need to reconsider using Mad Minute type activities.

Math Anxiety

Below is a portion of an article that appeared on Edutopia titled “Tips for Tackling Timed Tests and Math Anxiety” by Youki Terada.

For many, taking math as a child was a dreadful experience. “Timed tests were the horror of my primary schooling,” Rawini Ngaamo recalled. “All I learned from them is that I was stupid and slow. I still hate maths even now because of the way it was taught.” Dozens of people agreed, recalling their own personal experiences or sharing those of their students or children.

The effects appear to be long-lasting, haunting some of the commenters for decades: “As a 57-year-old, I can still recall the anxiety of timed tests,” said Debbie Denmead Cassady, before suggesting that teachers “forget them” in the future. And the post brought back vivid, unpleasant memories of elementary school for Adina Thuransky: “In second grade, our teacher timed us on math facts (times tables)... it freaked me out so much I can still remember how much anxiety I felt!”

It is clear from these examples that timed test have the potential to cause long lasting, negative attitudes toward mathematics.

Emphasizing Understanding

Perhaps the most important expectations in the Arizona Math Standards, is to teach with a balance of conceptual understanding, application and procedural skill and fluency. Traditionally, there has been a disproportionate emphasis on procedural skill with students reproducing memorized procedures with little on no attention paid to why they work or when they might be useful. If reproducing the procedure is the goal, arguably the “drill and kill” method can prove useful. However, this leads students to believe that math is a disjointed subject where procedures have little or no connection to one another. Memorizing multiplication facts does little to help students understand how the numbers are related to one another, the patterns that exists within multiplication tables, the relationship of multiplication to other operations or even the relationship to area. These connections are what make math interesting and useful.

Effective Teaching Strategies

Teaching of mathematics often goes something like this. The teacher puts a problem on the board, then the teacher solves the problem while the students observe. Next, the students solve a problem while the teacher solves the same problem on the board using the same strategy that was used to solve the first problem. Then the students practice on another problem similar to the first two individually. While there are variations of this process, the steps are basically the same and are often referred to as “I do, you do, we do.”

John Hattie has done extensive work evaluating teaching and learning. “Through the Visible Learning research, John Hattie has identified more than 250 factors that influence student achievement. He then set about calculating a score or “effect size” for each, according to its bearing on student achievement. The average effect size of these 250 factors was 0.4, a marker that can be shown to represents an (average) year’s growth per year of schooling for a student. Any factor that has an effect size above 0.4 has an even greater positive effect on student learning.”” (“About MetaX.” Corwin Visible Learning Plus, www.visiblelearningmetax.com/research_methodology.)

Hattie identifies worked examples (I do, we do, you do) as having an effect size of .37. Thus, this is a very average strategy to use in terms of students learning over the course of a year. An alternative to using worked examples might be the use of Math Talks where students engage in sharing of ideas through discussion. Hattie rates classroom discussion with an effect size of .82. From a teacher perspective, it is often times much quicker for students to get the “right” answer when we show them how to complete a problem rather than discuss, share ideas, make mistakes, discuss, rethink and reflect. However, research is clear. Learning involves confusion, mistakes, discussion and time.

Messaging – Math is About Answer Getting

When we ask students to answer 100 math fact questions in 1 minute, the message we are sending is that math is about getting the answer as quickly as possible, often from memory. This is the wrong message. Math is about using what we know and applying it to learn more about an idea or a problem. Teaching students to use what they know and understand to approach a new problem seems a much better life lesson to teach than the alternative. When facts are taught in a “memory dependent, speed is important” way we paralyze students. With this approach if you do not know the answer there is not an alternative. This is not how we solve problems in our lives. We might do research about the problem, talk to a friend or mentor, try an approach that has worked before or a combination of all these strategies.

So what instead?

I encourage all teachers that are using timed activities to “teach” facts to reconsider how to incorporate different approaches and strategies.

I will start with a simple change to the timed test approach that I incorporated. Students seemed to find the strategy more engaging and useful as well. Instead of timing students on the Mad Minute, I asked them to find 10 facts that they could answer easily and write their answers in red. I would then ask them to find 5 more problems that they could answer by using a strategy that makes sense for them. For multiplication this might be skip counting, related facts, drawing a visual model etc. Finally, I would ask them to pick 3 that were the most difficult for them and find a partner who could help them understand how to find the answer. I had moderate success with this. I never felt it was perfect, but it was better than just doing the regular 1-minute timed test.

Another approach that I highly recommend is Number Talks. You can find lots of information regarding Number Talks with a simple search. Basically, students are presented with an expression that they are asked to independently find the answer for. The work is done mentally without paper and pencil. The teacher solicits all answers and then facilitates a discussion where students consider and share strategies. The routine takes approximately 10 minutes once students and teacher are familiar with it. It is a great way to build understanding of numbers and relationships and empowers students to think about the math that makes sense to them.

How are you addressing math facts in your classroom? What do you agree with? What do you disagree with? Share your ideas so that others can consider, and we can all get better together.

 

Resources

YouCubed – Jo Boaler shares ways to help students improve their number sense and the more about the damage done by using timed test.

Number Talks – The Dana Center has a library of videos that show Number Talks being used in classroom with students. There are a variety of topics and grade levels.

Visible Learning MetaX – An interactive database that organizes and explains the work of John Hattie. Learn which influences are most effective in impacting students learning.

Classroom Culture - Elementary School vs. Jr./Sr. High School

A while back, teachers were discussing classroom management. Specifically they were discussing the role of positive and appropriate teacher/student relationships regarding student behavior. One of the teachers boldly claimed that junior and senior high students were much different than elementary students because they don't care about having a relationship with their teachers. They just want to be left alone to do their work.

I pointed out that as a former junior and senior high school teacher, I strongly disagreed with this statement and that I worked very hard at building relationships with students (especially those that might have challenging behaviors) in hopes that all of my students would learn as much as possible.

My first couple of years of teaching were challenging. It seemed that no matter what rules I put in place, there were always a handful of students that made it difficult for others to learn. My behavior often escalated situations which would lead to referrals and out of class time for students that often needed learning time the most.

I recognized that I needed to change or I would face this every year. Along with simplifying and clarifying my classroom rules, I turned my focus to building positive relationships with the students that might be the most challenging during the year. Here are some of the actions that I took that greatly improved the classroom environment.

Greet students

I can't say that I did this every day at the beginning of each class period, but I consistently met students at the door between periods. I would say hello to students as the entered. Most of the time I would say their name in a short greeting. This gave me a feel for the mood of students as they entered. A recent article on Edutopia touches on the importance of greeting students at the door.

Focus on a few

At the beginning of each school year, I would identify 5-8 students that I thought might be challenging for me to connect with. Sometimes they were the quiet ones. Sometimes there were the ones that were already demonstrating attention seeking behaviors. Other times they were ones that seemed to really dislike school and/or math. After I identified them, I would strike up conversations with them. Not school related stuff but conversations about their interest. I would make it a point to talk to them as often as I could about non-school topics. This might mean mentioning that I looked into a video game they talked about, watched a show they mentioned or listened to music they like. Whatever seemed to be relevant to that student. I wanted them to know that I was interested in them as a person. If I had to choose one action from this list as most effective, this is the one. I would often hear other teachers mention that they were having a difficult time with one of these students. Because I had built a relationship with these students, I seldom had issues and when there was a problem it was normally quite minor.

Involve students

Students want to be involved and believe that they have some control over the class and routines. This is why we would collaboratively set class guidelines and norms at the beginning of the year. I would also ask for input on how students preferred content to be delivered and how to organize and decorate the classroom. It created ownership and pride in the classroom. Students were also allowed to apply for classroom jobs. Things like taking attendance, cleaning boards, creating anchor charts etc. It kept me from having to do a lot of the daily tasks and got more students involved. 

Develop open and honest relationships

Many teachers ask students to be open with them. Some teachers actually take into account what students have to say. I would suggest that few actually change to meet the needs of their students. Although it is scary to ask students what they really think of you as a teacher, their feedback can be especially meaningful. It was tough for me to hear from students that my class was boring. However, once I took the feedback to heart and asked them for ideas on how to make it more interesting, classes became more engaging. They also had some really good ideas. Not all of their ideas were practical and certainly not all of them made it into action but the feedback was helpful in getting student more excited about learning math. I also shared my feedback with them openly. When I felt that a student or class was not focused or working as hard as we expected I let them know. Not in an accusing manner, but in a supportive "I think you can do better" way. I let them know when I thought my teaching was bad or that I failed to prepare them for a test question. I found they were willing to take the blame and change as long as I was too. Too often we blame it on the kids and forget to listen to what they are really trying to tell us. High school students are often under a lot of pressure from other teachers, parents, and life as a young adult. I think we sometime forget that.

Dealing with conflict

Of course it wasn't always smooth sailing. When a student crossed the line or broke one of our classroom norms it had to be dealt with. The biggest lesson I learned is to not take it personally. I developed a process that seemed to work...most of the time.

• Start by explaining the action that broke the norms. I tried to keep it focused on the actions. Something like, "I saw you look over John's shoulder and write the same answer that he had on his paper during the test." The "I saw" or "I heard" phrase is something I learned from a friend who is a teacher, referee and coach. He used this line when a coach would disagree with a call that he made while refereeing. He explained that if you say "I called ________ because I saw _______," it makes it really difficult to continue an argument. It's similar in a classroom situation. Although the student might disagree with the consequence and might even have an explanation of why they did something, if you apply a consequence because of an action that broke a norm, there really isn't any further explanation needed.
• If the action was severe enough that the student was asked to leave the classroom, I always held a quick conference with them when they came back. We did this away from earshot of others. I looked at it as damage control and it was an opportunity for them to give their side of the story. After all, there are always two sides. Using our scenario above, maybe the student did not have a chance to study for the test because they had to take care of a sick brother or sister the night before. It allows them to share their situation with me and be heard. I could then emphasize, "I'm sorry. I didn't know that. I'm sure it was difficult and I appreciate you letting me know. I know you don't usually cheat. However, you know that cheating is against one of our classroom norms and there is always a consequence for it." This would quickly be followed by a question from me. "What can I do to help you from doing this again?" Most of the time students would take responsibility for their actions and tell me "nothing" and it was their choice. Occasionally a student would let me know that something I did or said had upset them or missed something that was done or said by another student first. Finally I would ask them, "What can you do instead of cheating (or other behavior) next time?" In this case, if the student let me know prior to the beginning of the test, I might have let them schedule an alternate time, let them retest etc. The point of this whole conversation is not to rehash the situation and tell them what they did wrong again. It is an opportunity for the two of us to work together to problem solve so we don't end up dealing with the same situation again. In the case of a less severe incident in which a student remained in the classroom, an abbreviated conference would be held.
• It was also important for me to remember not to hold on to the conflict. I tried very hard to treat every day as a new opportunity. Holding on to grudges and emotions from previous days is not an effective way to build relationships. It can be challenging, but our attitude can change how students think about themselves. We can believe in them even if when they are having a difficult time doing so.

My efforts to change and focus on building relationships is one of my biggest accomplishments. I view it as one of my greatest areas of growth. I hope that all of my former students know that I cared about them. Not just as math students but as people.

What change can you make tomorrow to build relationships with students? Even though they may not say it, they want to be cared about by all of us.     

Concept or Context?

I was recently had the opportunity to work with a group of teachers on creating a vertical articulation for grade K-6 on Measurement, Data, Probability & Statistics. I had given them packets that included the introductory paragraphs from the original CCSS document, the critical focus areas from the AZ Mathematics Standards and the Arizona Math standards.

While circulating I listened in on a conversation between two teachers that were studying the information for grades K-3. They were sharing their concern that more emphasis was not placed on time and money in the early grades. One of the teachers is an ESS teacher. She shared that in her self-contained classroom they dedicate a significant portion of time learning about money and time. She described these ideas as "major life skills." The third grade teacher sitting at the table agreed that it is vital for all students that they have a strong understanding of time and money in order to be successful as adults.

I observed as they looked back through the standards for evidence that more time should be dedicated to these two topics. I stood back as they searched the standards and the paragraphs.

The review confirmed what they had initially thought, time and money are only mentioned in two short standards in first, second and third grade. They also noted that these clusters were all labeled as "Supporting Clusters" in the AZ math standards.

As their dissatisfaction began to lean toward outright disgust, I stepped in and asked them if time and money are mathematical concepts or if they are contexts?

After some discussion they were able to come to the realization that time and money are a context.

Next we looked through the standards and identified where the context of time and money could be used to teach and assess the content of the standards. As we did so, the big aha moment came. They excitedly reached the conclusion that just because time and money was not mentioned in more standards, it did not prohibit them from using time and money contexts to help students understand the concepts and thus increase their understanding and skill with time and money.

I think that it is important for us to consider what contexts we should be including in order to teach the content of our grade level. What contexts can we use to help our students be more prepared for life outside of school?

Establish Mathematics Goals to Focus Learning

If you have read NCTM's "Principles to Actions," then you are familiar with the Mathematics Teaching Practices. These practices, "provide a framework for strengthening the teaching and learning of mathematics. (NCTM, 2014, p.9) If you are not familiar with "Principles to Actions" and the Math Teaching Practices, I suggest you look into them. They really do have the power to transform your teaching.

Although all of the Math Teaching Practices are important, I find the first practice, "Establish mathematics goals to focus learning," to be the most important for me to be mindful of when planning a lesson. While important, it is also the most challenging portion of the planning process. Here are the steps I take.

Step 1: Read the Standard
I mean, REALLY READ THE STANDARD! The standards are very dense and can be difficult to make sense of. I read them closely and look for parts. Lets look at a 6th grade standard.

Understand the concept of a ratio as comparing two quantities multiplicatively or joining/composing the two quantities in a way that preserves a multiplicative relationship. Use ratio language to describe a ratio relationship between two quantities. For example, "There were 2/3 as many men as women at the concert.”

I see two parts. 1) Understand what a ratio is and that it can be extended. 2) Describe ratios using appropriate ratio language (this includes written formats). At first glance, this seems pretty straight forward. Caution is needed here as understanding ratios and seeing the multiplicative relationships within them can be difficult. So although it is only 1 sentence in the standard there is a lot of intricacy in what students need to understand. However, this blog is about writing mathematical goals, so I am not going to dig deeper into that but does illustrate my point about really reading and thinking deeply about the mathematics that is the focus of the standard.

Step 2: Consult the AZ Performance Level Descriptors & Item Specifications
You can access these documents here for grades 3-5, here for grades 6-8 and here for End of Course standards (Alg. I, II, and Geometry). These two documents have become "go to" resources for me. They clearly define what students should be able to do and give some insight into what is basic understanding and/or skill and what is full understanding and/or skill. They provide information about number sets that should be included as well.

From the Performance Level Descriptors we see the following information.



As I read across the columns, I can begin to get a sense of what the standard is expecting students to not only do, but what the understanding expectations are. Generally, there are multiple ideas that students are to understand but I can begin to determine where my focus will lie. What sticks out to me is that students are to "connect between representations for ratio situations."

Here is the item specification information for the standard.



This document contains a lot of information that is helpful in thinking about the mathematical goal. There is a lot of emphasis on using language and describing. I also notice what representations students will be expected to use in the "Sample Task Demands" section. For this standard, tape diagrams and double number line graphics are specifically mentioned. This can also be helpful as you think about the mathematical goal. The "Recommended Math Practices" can also be considered when writing math goals.

Step 3: Consult other resources if needed
At this point, I might refer Acheive the Core's Coherence Map which can help when building on previous understanding or preparing for future understanding and/or skills. I will continue to consult resources if I am unclear on the standard.

Step 4: Write the mathematical goal
I will then take a stab at writing a mathematical goal. For this standard, there are a lot of mathematical goals that are appropriate, depending on the experience that students have with ratios. If students have little experience with ratios I might start with "students will understand that a ratio can be a part to part relationship or a part to whole relationship." Or, "students understand that a ratio is a multiplicative relationship and that this relationship can be extended to larger or smaller quantities." If I decide to focus on representations, the math goal might be "Students will understand that double number lines and tape diagrams can be used to extend ratios."

Once I have my math goal, I can begin to build an effective task based lesson. The goal keeps me on track and ensures that I assess for the correct understanding during the lesson. I choose better task, ask better questions and am ultimately a better teacher when I use this process.

I'd love to hear what your thoughts are and what process you use to write math goals in the comments.


Principles to Actions: Ensuring Mathematical Success for All. NCTM, National Council of Teachers of Mathematics, 2014.